From a834f514359710303cd5bf882068991d830c2708 Mon Sep 17 00:00:00 2001 From: rogeriojorge Date: Sat, 26 Sep 2026 07:30:43 -0500 Subject: [PATCH 01/20] Let compute_C_nmp expand a Maxwellian narrower than its Hermite basis vth_s (default alpha/sqrt(2), the previous behaviour) gives the closed form c^n H_n(d/c) / sqrt(2^n n!) with c^2 = 1 - 2 vth^2 / alpha^2, computed by a three-term recurrence (no factorials). A thin beam can then share a wide basis, where its expansion stays convergent as it spreads. --- spectrax/_initialize_maxwellian.py | 51 +++++++++++++++-------------- tests/test_initialize_maxwellian.py | 49 +++++++++++++++++++++++++++ 2 files changed, 75 insertions(+), 25 deletions(-) create mode 100644 tests/test_initialize_maxwellian.py diff --git a/spectrax/_initialize_maxwellian.py b/spectrax/_initialize_maxwellian.py index 0c627db..54ee282 100644 --- a/spectrax/_initialize_maxwellian.py +++ b/spectrax/_initialize_maxwellian.py @@ -5,14 +5,26 @@ import jax.numpy as jnp from jax import jit from jax.numpy.fft import rfftn -from jax.scipy.special import factorial from functools import partial __all__ = ['compute_C_nmp'] +def _maxwellian_modes(d, c2, N): + """Normalised Hermite coefficients g_n = c^n H_n(d / c) / sqrt(2^n n!), n < N, of a 1D Maxwellian. + + d is the drift in units of the Hermite width and c2 = 1 - 2 vth^2 / alpha^2; c2 = 0 when the + Maxwellian has the basis width (then g_n = (sqrt(2) d)^n / sqrt(n!)). The recurrence + g_{n+1} = (sqrt(2) d g_n - sqrt(n) c2 g_{n-1}) / sqrt(n + 1) avoids factorials. + """ + g = [jnp.ones_like(d), jnp.sqrt(2.0) * d] + for n in range(1, N - 1): + g.append((jnp.sqrt(2.0) * d * g[n] - jnp.sqrt(n) * c2 * g[n - 1]) / jnp.sqrt(n + 1.0)) + return jnp.stack(g[:N]) + + @partial(jit, static_argnames=['Nn', 'Nm', 'Np', 'Ns']) -def compute_C_nmp(Us_grid, alpha_s, u_s, Nn, Nm, Np, Ns): +def compute_C_nmp(Us_grid, alpha_s, u_s, Nn, Nm, Np, Ns, vth_s=None): """ Build the Hermite-Fourier coefficients for a drifting Maxwellian distribution function. @@ -31,6 +43,11 @@ def compute_C_nmp(Us_grid, alpha_s, u_s, Nn, Nm, Np, Ns): Number of Hermite modes retained along the x, y, and z velocity axes. Ns : int Number of species. + vth_s : array-like, optional + Thermal speeds (standard deviations) of the Maxwellians, flattened as `(3 * Ns,)`. + Default: `alpha_s / sqrt(2)`, the basis width. A Maxwellian narrower than its basis + (`vth < alpha / sqrt(2)`) has a convergent expansion, so a thin beam can share a wide + basis; a wider one does not converge. Returns ------- @@ -38,29 +55,13 @@ def compute_C_nmp(Us_grid, alpha_s, u_s, Nn, Nm, Np, Ns): Complex Hermite-Fourier coefficients with shape `(Ns, Np, Nm, Nn, Ny, Nx//2+1, Nz)` corresponding to the Maxwellian evaluated on the supplied grid. """ - - U_x = Us_grid[:, 0, None, None, None, :, :, :] # shape (Ns, 1, 1, 1, Ny, Nx, Nz) - U_y = Us_grid[:, 1, None, None, None, :, :, :] # shape (Ns, 1, 1, 1, Ny, Nx, Nz) - U_z = Us_grid[:, 2, None, None, None, :, :, :] # shape (Ns, 1, 1, 1, Ny, Nx, Nz) - - alpha = jnp.array(alpha_s).reshape(Ns, 3) - alpha_x = alpha[:, 0, None, None, None, None, None, None] # shape (Ns, 1, 1, 1, 1, 1, 1) - alpha_y = alpha[:, 1, None, None, None, None, None, None] # shape (Ns, 1, 1, 1, 1, 1, 1) - alpha_z = alpha[:, 2, None, None, None, None, None, None] # shape (Ns, 1, 1, 1, 1, 1, 1) - - u = jnp.array(u_s).reshape(Ns, 3) - u_x = u[:, 0, None, None, None, None, None, None] # shape (Ns, 1, 1, 1, 1, 1, 1) - u_y = u[:, 1, None, None, None, None, None, None] # shape (Ns, 1, 1, 1, 1, 1, 1) - u_z = u[:, 2, None, None, None, None, None, None] # shape (Ns, 1, 1, 1, 1, 1, 1) - - p = jnp.arange(Np)[None, :, None, None, None, None, None] # shape (1, Np, 1, 1, 1, 1, 1) - m = jnp.arange(Nm)[None, None, :, None, None, None, None] # shape (1, 1, Nm, 1, 1, 1, 1) - n = jnp.arange(Nn)[None, None, None, :, None, None, None] # shape (1, 1, 1, Nn, 1, 1, 1) - - C = (jnp.sqrt(2 ** (n + m + p) / (factorial(n) * factorial(m) * factorial(p))) - * (1 / (alpha_x ** (n + 1) * alpha_y ** (m + 1) * alpha_z ** (p + 1))) - * (U_x - u_x) ** n * (U_y - u_y) ** m * (U_z - u_z) ** p) - + alpha = jnp.array(alpha_s).reshape(Ns, 3)[:, :, None, None, None] # (Ns, 3, 1, 1, 1) + u = jnp.array(u_s).reshape(Ns, 3)[:, :, None, None, None] + vth = alpha / jnp.sqrt(2.0) if vth_s is None else jnp.array(vth_s).reshape(Ns, 3)[:, :, None, None, None] + c2 = 1 - 2 * (vth / alpha) ** 2 # 0 when the Maxwellian has the basis width + g = [_maxwellian_modes((Us_grid[:, i] - u[:, i]) / alpha[:, i], c2[:, i], N) / alpha[:, i] + for i, N in enumerate((Nn, Nm, Np))] # each (N, Ns, Ny, Nx, Nz) + C = jnp.einsum('psyxz,msyxz,nsyxz->spmnyxz', g[2], g[1], g[0]) # (Ns, Np, Nm, Nn, Ny, Nx, Nz) Ck_0 = rfftn(C, axes=(-1, -3, -2), norm="forward") # shape (Ns, Np, Nm, Nn, Ny, Nx//2+1, Nz) return Ck_0 diff --git a/tests/test_initialize_maxwellian.py b/tests/test_initialize_maxwellian.py new file mode 100644 index 0000000..ee98889 --- /dev/null +++ b/tests/test_initialize_maxwellian.py @@ -0,0 +1,49 @@ +import jax +import jax.numpy as jnp +import numpy as np +import pytest +from scipy.special import factorial + +from spectrax import compute_C_nmp + +jax.config.update("jax_enable_x64", True) + + +def test_basis_width_maxwellian_keeps_the_closed_form(): + """Default vth: C_nmp = sqrt(2^(n+m+p) / (n! m! p!)) (U - u)^n ... / alpha^(n+1) ... (the previous formula).""" + Nn, Nm, Np, Ns, alpha, u = 6, 3, 2, 2, np.array([1.0, 0.7, 1.3, 0.5, 0.9, 1.1]), np.array([0.1, -0.2, 0.3, 1.0, 0.0, -0.5]) + U = np.random.default_rng(1).normal(size=(Ns, 3, 2, 5, 1)) + a, w = alpha.reshape(Ns, 3), u.reshape(Ns, 3) + n, m, p = np.meshgrid(np.arange(Nn), np.arange(Nm), np.arange(Np), indexing="ij") + C = np.empty((Ns, Np, Nm, Nn, 2, 5, 1)) + for s in range(Ns): + for idx in np.ndindex(Nn, Nm, Np): + i, j, k = idx + C[s, k, j, i] = (np.sqrt(2.0 ** (i + j + k) / (factorial(i) * factorial(j) * factorial(k))) + * (U[s, 0] - w[s, 0]) ** i * (U[s, 1] - w[s, 1]) ** j * (U[s, 2] - w[s, 2]) ** k + / (a[s, 0] ** (i + 1) * a[s, 1] ** (j + 1) * a[s, 2] ** (k + 1))) + expected = np.fft.rfftn(C, axes=(-1, -3, -2), norm="forward") + np.testing.assert_allclose(compute_C_nmp(U, alpha, u, Nn, Nm, Np, Ns), expected, rtol=1e-12, atol=1e-14) + np.testing.assert_allclose(compute_C_nmp(U, alpha, u, Nn, Nm, Np, Ns, vth_s=alpha / np.sqrt(2)), expected, + rtol=1e-12, atol=1e-14) + + +@pytest.mark.parametrize("vth, drift", [(0.5, 0.0), (0.5, 0.8), (0.7, -0.4), (1.0, 0.3)]) +def test_narrow_maxwellian_in_a_wide_basis_is_reconstructed(vth, drift): + """A Maxwellian of width vth drifting by `drift` in a basis of width alpha = sqrt(2) (vth <= 1): the Hermite + series sum_n C_n H_n(xi) e^{-xi^2} / sqrt(2^n n! pi) must return the Gaussian (1D, x only). The series + converges like (1 - vth^2)^(n/2), so 200 modes reach round-off for vth >= 0.5.""" + Nn, alpha = 200, np.sqrt(2.0) + U = np.full((1, 3, 1, 1, 1), 0.0) + U[0, 0] = drift + C = np.asarray(compute_C_nmp(U, [alpha, alpha, alpha], [0.0] * 3, Nn, 1, 1, 1, + vth_s=[vth, alpha / np.sqrt(2), alpha / np.sqrt(2)]))[0, 0, 0, :, 0, 0, 0].real + v = np.linspace(-4, 4, 81) + xi = v / alpha + h = np.zeros((Nn, v.size)) # orthonormal Hermite functions + h[0], h[1] = np.pi**-0.25 * np.exp(-xi**2 / 2), np.sqrt(2) * xi * np.pi**-0.25 * np.exp(-xi**2 / 2) + for n in range(2, Nn): + h[n] = np.sqrt(2 / n) * xi * h[n - 1] - np.sqrt((n - 1) / n) * h[n - 2] + f = alpha**2 * np.pi**-0.25 * np.exp(-xi**2 / 2) * (C @ h) # 1D marginal, as in inverse_HF_transform + exact = np.exp(-(v - drift) ** 2 / (2 * vth**2)) / (np.sqrt(2 * np.pi) * vth) # unit density + np.testing.assert_allclose(f, exact, atol=1e-12) From 95a9270122b934c72e585a00889d0d073507ed18 Mon Sep 17 00:00:00 2001 From: rogeriojorge Date: Sat, 26 Sep 2026 07:31:46 -0500 Subject: [PATCH 02/20] Add a bump-on-tail example with the beam in the bulk's Hermite basis --- Examples/1D_bump_on_tail.py | 59 +++++++++++++++++++++++++++++++++++++ 1 file changed, 59 insertions(+) create mode 100644 Examples/1D_bump_on_tail.py diff --git a/Examples/1D_bump_on_tail.py b/Examples/1D_bump_on_tail.py new file mode 100644 index 0000000..988ba39 --- /dev/null +++ b/Examples/1D_bump_on_tail.py @@ -0,0 +1,59 @@ +"""Example: 1D bump-on-tail instability, with the beam expanded in the bulk's Hermite basis. + +A bulk Maxwellian (density 0.9, thermal speed 1) and a warm beam (density 0.1, thermal speed 0.5, +drift 4.5) are shifted by -0.45 to the frame with no net current, so that SPECTRAX's Vlasov-Ampere +system has no uniform plasma oscillation. Each population is its own species. + +The beam is expanded in the bulk's Hermite width (alpha = sqrt(2)), not its own (sqrt(2) * 0.5): +`compute_C_nmp(..., vth_s=...)` gives the closed-form coefficients of a Maxwellian narrower than its +basis. In its own narrow basis the trapped beam, which spreads over 2 < v < 6, has a divergent +expansion: its coefficients grow by 1e2-1e7 and the adaptive step collapses (near t = 28 with nu = 3). +Set BEAM_BASIS_VT = 0.5 to see it. Linear theory: omega = 0.866 + 0.198 i at k = 0.3. +""" + +import jax +import jax.numpy as jnp +import matplotlib.pyplot as plt +import numpy as np + +from spectrax import compute_C_nmp, inverse_HF_transform, simulation + +jax.config.update("jax_enable_x64", True) +BEAM_BASIS_VT = 1.0 # 0.5: the beam in its own basis (fails) +species = [(0.9, -0.45, 1.0), (0.1, 4.05, 0.5)] # (density, drift, thermal speed) +k, eps, t_max, Nx, Nn = 0.3, 0.04, 100.0, 33, 128 +alpha = np.array([np.sqrt(2.0), np.sqrt(2.0) * BEAM_BASIS_VT]) + +Ck_0 = [] +for (n, u, vth), a in zip(species, alpha): + C = n * compute_C_nmp(jnp.full((1, 3, 1, Nx, 1), u), [a] * 3, [0.0] * 3, Nn, 1, 1, 1, + vth_s=[vth, a / np.sqrt(2), a / np.sqrt(2)])[0, 0, 0] # (Nn, 1, Nx//2+1, 1) + Ck_0.append(C.at[:, :, 1].set(0.5 * eps * C[:, :, 0])) # density n (1 + eps cos kx) +parameters = { + "Lx": 2 * np.pi / k, "Ly": 1.0, "Lz": 1.0, "mi_me": 1.0, "Ti_Te": 1.0, "nx": 1, "t_max": t_max, + "qs": jnp.array([-1.0, -1.0]), "Omega_cs": jnp.array([1.0, 1.0]), "nu": 3.0, "D": 0.0, "ode_tolerance": 1e-8, + "alpha_s": jnp.repeat(jnp.asarray(alpha), 3), "u_s": jnp.array([0.0] * 6), "Ck_0": jnp.concatenate(Ck_0), + # Gauss's law for the density perturbation: E = -(eps / k) sin(kx). + "Fk_0": jnp.zeros((6, 1, Nx // 2 + 1, 1), dtype=jnp.complex128).at[0, 0, 1, 0].set(0.5j * eps / k), +} +output = simulation(parameters, Nx=Nx, Nn=Nn, Ns=2, timesteps=501, dt=0.01) + +t, E1 = output["time"], 2 * jnp.abs(output["Fk"][:, 0, 0, 1, 0]) +x, v = np.linspace(0, 2 * np.pi / k, Nx, endpoint=False), np.linspace(-8, 10, 241) +f = 0 +for s, a in enumerate(alpha): # 1D marginal: f(v_x, v_y = v_z = 0) times the transverse integral a^2 pi + xi_y, xi_x, xi_z = np.meshgrid([0.0], v / a, [0.0], indexing="ij") + f += np.pi * a**2 * inverse_HF_transform(output["Ck"][-1:, s * Nn:(s + 1) * Nn], Nn, 1, 1, Nx, 1, 1, + xi_x, xi_y, xi_z)[0, 0, :, 0, 0, :, 0] + +fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(11, 3.8)) +ax1.semilogy(t, E1, label="SPECTRAX") +linear = (t > 8) & (t < 20) +ax1.semilogy(t[linear], E1[linear][0] * np.exp(0.198 * (t[linear] - 8)), "k--", label=r"$\gamma = 0.198$") +ax1.set(xlabel=r"$t\,\omega_{pe}$", ylabel=r"$|\hat E_{k}|$", title="bump-on-tail, k = 0.3") +ax1.legend() +mesh = ax2.pcolormesh(x, v, np.asarray(f).T, shading="auto", cmap="magma") +ax2.set(xlabel=r"$x/\lambda_D$", ylabel=r"$v/v_t$", title=f"f(x, v, t = {t_max:g})") +fig.colorbar(mesh, ax=ax2) +plt.tight_layout() +plt.show() From 1dc9820cd4a5b05a069652386cb69aefb89c4454 Mon Sep 17 00:00:00 2001 From: rogeriojorge Date: Sat, 26 Sep 2026 15:59:52 -0500 Subject: [PATCH 03/20] Keep solver state axes structured --- spectrax/_simulation.py | 28 +++++++++++++--------------- tests/test_simulation_runs.py | 14 ++++++++++++++ 2 files changed, 27 insertions(+), 15 deletions(-) diff --git a/spectrax/_simulation.py b/spectrax/_simulation.py index eba5de0..f22f5a0 100644 --- a/spectrax/_simulation.py +++ b/spectrax/_simulation.py @@ -64,17 +64,16 @@ def ode_system(Nx, Ny, Nz, Nn, Nm, Np, Ns, t, Ck_Fk, args): Number of species. t : float Integration time (unused but required by Diffrax interface). - Ck_Fk : jnp.ndarray - Flattened state vector containing concatenated Hermite coefficients followed - by electromagnetic field coefficients. + Ck_Fk : tuple[jnp.ndarray, jnp.ndarray] + Hermite and electromagnetic field coefficients. args : tuple Cached parameter tuple produced in `simulation` providing physical constants, grids, and helper arrays. Returns ------- - jnp.ndarray - Flattened derivative vector matching the shape of `Ck_Fk`. + tuple[jnp.ndarray, jnp.ndarray] + Derivatives matching the Hermite and field coefficient arrays. """ (qs, nu, D, Omega_cs, alpha_s, u_s, @@ -82,9 +81,7 @@ def ode_system(Nx, Ny, Nz, Nn, Nm, Np, Ns, t, Ck_Fk, args): sqrt_n_plus, sqrt_n_minus, sqrt_m_plus, sqrt_m_minus, sqrt_p_plus, sqrt_p_minus ) = args[7:] - total_Ck_size = Nn * Nm * Np * Ns * (Nx//2+1) * Ny * Nz - Ck = Ck_Fk[:total_Ck_size].reshape(Nn * Nm * Np * Ns, Ny, Nx//2+1, Nz) - Fk = Ck_Fk[total_Ck_size:].reshape(6, Ny, Nx//2+1, Nz) + Ck, Fk = Ck_Fk # Build the 2/3 mask once per call (JIT will constant-fold it since Nx/Ny/Nz are static) @@ -103,9 +100,7 @@ def ode_system(Nx, Ny, Nz, Nn, Nm, Np, Ns, t, Ck_Fk, args): dEk_dt = 1j * cross_product(nabla, Fk[3:]) - current / Omega_cs[0] dFk_dt = jnp.concatenate([dEk_dt, dBk_dt], axis=0) - dy_dt = jnp.concatenate([dCk_s_dt.reshape(-1), dFk_dt.reshape(-1)]) - - return dy_dt + return dCk_s_dt, dFk_dt @partial(jit, static_argnames=['Nx', 'Ny', 'Nz', 'Nn', 'Nm', 'Np', 'Ns', 'timesteps', 'solver', 'adaptive_time_step']) def simulation(input_parameters={}, Nx=33, Ny=1, Nz=1, Nn=20, Nm=1, Np=1, Ns=2, @@ -141,8 +136,11 @@ def simulation(input_parameters={}, Nx=33, Ny=1, Nz=1, Nn=20, Nm=1, Np=1, Ns=2, # **Initialize simulation parameters** parameters = initialize_simulation_parameters(input_parameters, Nx, Ny, Nz, Nn, Nm, Np, Ns, timesteps, dt) - # Combine initial conditions. - initial_conditions = jnp.concatenate([parameters["Ck_0"].flatten(), parameters["Fk_0"].flatten()]) + # Preserve the species and Hermite axes for the integrator. + initial_conditions = ( + parameters["Ck_0"].reshape(Ns, Np, Nm, Nn, Ny, Nx//2+1, Nz), + parameters["Fk_0"].reshape(6, Ny, Nx//2+1, Nz), + ) # Define the time array for data output. time = jnp.linspace(0, parameters["t_max"], timesteps) @@ -177,8 +175,8 @@ def simulation(input_parameters={}, Nx=33, Ny=1, Nz=1, Nn=20, Nm=1, Np=1, Ns=2, max_steps=1000000, progress_meter=TqdmProgressMeter()) # Reshape the solution to extract Ck and Fk - Ck = sol.ys[:,:(-6 * (Nx//2+1) * Ny * Nz)].reshape(len(sol.ts), Ns * Nn * Nm * Np, Ny, Nx//2+1, Nz) - Fk = sol.ys[:,(-6 * (Nx//2+1) * Ny * Nz):].reshape(len(sol.ts), 6, Ny, Nx//2+1, Nz) + Ck = sol.ys[0].reshape(len(sol.ts), Ns * Nn * Nm * Np, Ny, Nx//2+1, Nz) + Fk = sol.ys[1] # Set n = 0, k = 0 mode to zero to get array with time evolution of perturbation. dCk = Ck.at[:, 0, 0, 0, 0].set(0) diff --git a/tests/test_simulation_runs.py b/tests/test_simulation_runs.py index 2d05a1e..e215486 100644 --- a/tests/test_simulation_runs.py +++ b/tests/test_simulation_runs.py @@ -13,6 +13,20 @@ def test_simulation_runs(): + result["solver_stats"]["num_rejected_steps"] ) +def test_solver_state_keeps_species_and_hermite_axes(monkeypatch): + """The integrator sees (Ck, Fk) with their species, Hermite and grid axes, not one flat vector.""" + import spectrax._simulation as sim + captured = {} + def stop(term, **kwargs): + captured.update(kwargs, rhs=term.vector_field(0.0, kwargs["y0"], kwargs["args"])) + raise RuntimeError("captured") + monkeypatch.setattr(sim, "diffeqsolve", stop) + with pytest.raises(RuntimeError, match="captured"): + simulation(Nx=7, Nn=4, timesteps=3) + shapes = [(2, 1, 1, 4, 1, 4, 1), (6, 1, 4, 1)] + assert [y.shape for y in captured["y0"]] == shapes + assert [y.shape for y in captured["rhs"]] == shapes + def test_electric_field_update(): """Test that the electric field updates and does not remain zero.""" result = simulation() From c773d982dce63039f3825465e76f8bc117d67c16 Mon Sep 17 00:00:00 2001 From: rogeriojorge Date: Sat, 26 Sep 2026 15:59:58 -0500 Subject: [PATCH 04/20] Fix odd-grid RFFT energy weights and add a Hermite boundary diagnostic For odd Nx the last RFFT coefficient has an omitted conjugate and needs weight 2. hermite_tail_fraction reports the perturbation norm fraction in the outer Hermite modes, using the same Parseval weights. --- spectrax/_diagnostics.py | 57 ++++++++++++++++++++++++++++++++--- tests/test_diagnostics.py | 63 +++++++++++++++++++++++++++++++++++++++ 2 files changed, 116 insertions(+), 4 deletions(-) create mode 100644 tests/test_diagnostics.py diff --git a/spectrax/_diagnostics.py b/spectrax/_diagnostics.py index 8fdb788..3d18bc6 100644 --- a/spectrax/_diagnostics.py +++ b/spectrax/_diagnostics.py @@ -19,7 +19,13 @@ import jax.numpy as jnp -__all__ = ["diagnostics"] +__all__ = ["diagnostics", "hermite_tail_fraction"] + + +def _rfft_weights(Nx: int, Nx_kept: int) -> jnp.ndarray: + """Return Parseval weights for an RFFT of length ``Nx``.""" + weights = jnp.full(Nx_kept, 2.0).at[0].set(1.0) + return weights.at[-1].set(1.0 + (Nx % 2) * (Nx_kept > 1)) def _infer_Ns(output: Mapping[str, Any]) -> int: @@ -68,6 +74,50 @@ def _infer_masses(output: Mapping[str, Any], Ns: int) -> jnp.ndarray: return jnp.ones((Ns,)) +def hermite_tail_fraction(output: Mapping[str, Any], width: int = 1) -> jnp.ndarray: + """Return the perturbation norm fraction at each Hermite boundary. + + The tail is the union of the outer ``width`` modes on every Hermite axis + with more than one retained mode. Squared ``dCk`` coefficients are summed + with Parseval weights over Fourier space. For SPECTRAX's matched-Maxwellian + basis this norm is proportional to perturbed-distribution free energy, but + it is not the total plasma energy. + + Returns an array with shape ``(Nt, Ns)``. A zero perturbation, or a layout + with no active Hermite axis, has zero tail fraction. + """ + if not isinstance(width, int) or width < 1: + raise ValueError("width must be a positive integer.") + + dCk = jnp.asarray(output["dCk"]) + Ns = int(output["Ns"]) if "Ns" in output else _infer_Ns(output) + Nn, Nm, Np = (int(output[key]) for key in ("Nn", "Nm", "Np")) + expected_modes = Ns * Nn * Nm * Np + if dCk.ndim != 5 or dCk.shape[1] != expected_modes: + raise ValueError(f"dCk must have shape (Nt, {expected_modes}, Ny, Nx//2+1, Nz).") + + n = jnp.arange(Nn)[None, None, :] + m = jnp.arange(Nm)[None, :, None] + p = jnp.arange(Np)[:, None, None] + tail = jnp.zeros((Np, Nm, Nn), dtype=bool) + if Nn > 1: + tail |= n >= max(Nn - width, 0) + if Nm > 1: + tail |= m >= max(Nm - width, 0) + if Np > 1: + tail |= p >= max(Np - width, 0) + + Nx_kept = dCk.shape[-2] + weights = _rfft_weights(int(output.get("Nx", 2 * (Nx_kept - 1))), Nx_kept) + power = jnp.abs(dCk.reshape(dCk.shape[0], Ns, Np, Nm, Nn, *dCk.shape[-3:])) ** 2 + power *= weights.reshape(1, 1, 1, 1, 1, 1, -1, 1) + total = jnp.sum(power, axis=(2, 3, 4, 5, 6, 7)) + boundary = jnp.sum(power * tail.reshape(1, 1, Np, Nm, Nn, 1, 1, 1), + axis=(2, 3, 4, 5, 6, 7)) + denominator = jnp.where(total > 0, total, 1.0) + return jnp.where(total > 0, boundary / denominator, 0.0) + + def diagnostics(output: dict) -> None: """Compute and attach common diagnostics to ``output`` (mutates in-place). @@ -198,9 +248,8 @@ def _take_mode(n: int, m: int, p: int) -> jnp.ndarray: kinetic_energy_species = pref[None, :] * (term0[None, :] * C000 + term1 + term2) # (Nt, Ns) kinetic_energy = jnp.sum(kinetic_energy_species, axis=1) # (Nt,) # Field energy - rfft_weights = jnp.full(Nx_kept, 2.0) - rfft_weights = rfft_weights.at[0].set(1.0) - rfft_weights = rfft_weights.at[-1].set(1.0) + Nx = output.get("Nx", 2 * (Nx_kept - 1)) + rfft_weights = _rfft_weights(Nx, Nx_kept) weight_grid = rfft_weights.reshape(1, 1, -1, 1) EM_energy = 0.5 * jnp.sum((jnp.abs(Fk) ** 2) * weight_grid, axis=(-4, -3, -2, -1)) * Omega_cs[0] ** 2 diff --git a/tests/test_diagnostics.py b/tests/test_diagnostics.py new file mode 100644 index 0000000..cac4234 --- /dev/null +++ b/tests/test_diagnostics.py @@ -0,0 +1,63 @@ +import jax.numpy as jnp +import pytest + +from spectrax import diagnostics, hermite_tail_fraction + + +@pytest.mark.parametrize(("Nx", "expected_energy"), [(1, 0.5), (4, 0.5), (5, 1.0)]) +def test_rfft_field_energy_weights_nyquist_only_for_even_grid(Nx, expected_energy): + Nx_kept = Nx // 2 + 1 + output = { + "Nx": Nx, + "Nn": 3, + "Nm": 1, + "Np": 1, + "alpha_s": jnp.ones(6), + "u_s": jnp.zeros(6), + "Omega_cs": jnp.ones(2), + "mi_me": 1.0, + "Lx": 1.0, + "Ck": jnp.zeros((1, 6, 1, Nx_kept, 1), dtype=jnp.complex128), + "Fk": jnp.zeros((1, 6, 1, Nx_kept, 1), dtype=jnp.complex128) + .at[0, 0, 0, -1, 0] + .set(1.0), + } + + diagnostics(output) + + assert output["EM_energy"][0] == pytest.approx(expected_energy) + + +def test_hermite_tail_fraction_uses_odd_rfft_weights(): + dCk = jnp.zeros((1, 4, 1, 3, 1), dtype=jnp.complex128) + dCk = dCk.at[0, 0, 0, -1, 0].set(1.0) + dCk = dCk.at[0, 3, 0, 0, 0].set(1.0) + output = {"dCk": dCk, "Nx": 5, "Nn": 4, "Nm": 1, "Np": 1, + "Ns": 1, "u_s": jnp.zeros(3)} + + assert hermite_tail_fraction(output)[0, 0] == pytest.approx(1 / 3) + + +def test_hermite_tail_fraction_resolves_species_and_active_axes(): + Nn, Nm, Np, Ns = 3, 2, 2, 2 + modes = Nn * Nm * Np + dCk = jnp.zeros((1, Ns * modes, 1, 1, 1), dtype=jnp.complex128) + dCk = dCk.at[0, 0, 0, 0, 0].set(2.0) + dCk = dCk.at[0, Nn, 0, 0, 0].set(1.0) + dCk = dCk.at[0, modes, 0, 0, 0].set(1.0) + output = {"dCk": dCk, "Nx": 1, "Nn": Nn, "Nm": Nm, "Np": Np, + "Ns": Ns, "u_s": jnp.zeros(3 * Ns)} + + assert hermite_tail_fraction(output)[0] == pytest.approx(jnp.array([0.2, 0.0])) + + +def test_hermite_tail_fraction_ignores_inactive_axes_and_zero_norm(): + output = {"dCk": jnp.zeros((1, 1, 1, 1, 1)), "Nx": 1, + "Nn": 1, "Nm": 1, "Np": 1, "Ns": 1} + + assert hermite_tail_fraction(output)[0, 0] == 0.0 + + +def test_hermite_tail_fraction_rejects_invalid_width(): + with pytest.raises(ValueError, match="positive integer"): + hermite_tail_fraction({}, width=0) From 94960e301ce05732ee5056369cc7bc0daf940806 Mon Sep 17 00:00:00 2001 From: rogeriojorge Date: Sat, 26 Sep 2026 16:00:05 -0500 Subject: [PATCH 05/20] Fix multispecies perturbation bookkeeping --- spectrax/_simulation.py | 3 +-- tests/test_simulation_runs.py | 19 +++++++++++++++++++ 2 files changed, 20 insertions(+), 2 deletions(-) diff --git a/spectrax/_simulation.py b/spectrax/_simulation.py index eba5de0..4a7ea81 100644 --- a/spectrax/_simulation.py +++ b/spectrax/_simulation.py @@ -181,8 +181,7 @@ def simulation(input_parameters={}, Nx=33, Ny=1, Nz=1, Nn=20, Nm=1, Np=1, Ns=2, Fk = sol.ys[:,(-6 * (Nx//2+1) * Ny * Nz):].reshape(len(sol.ts), 6, Ny, Nx//2+1, Nz) # Set n = 0, k = 0 mode to zero to get array with time evolution of perturbation. - dCk = Ck.at[:, 0, 0, 0, 0].set(0) - dCk = dCk.at[:, Nn * Nm * Np, 0, 0, 0].set(0) + dCk = Ck.at[:, jnp.arange(Ns) * Nn * Nm * Np, 0, 0, 0].set(0) # Output results temporary_output = {"Ck": Ck, "Fk": Fk, "time": time, "dCk": dCk, "solver_stats": sol.stats} diff --git a/tests/test_simulation_runs.py b/tests/test_simulation_runs.py index 2d05a1e..739575b 100644 --- a/tests/test_simulation_runs.py +++ b/tests/test_simulation_runs.py @@ -2,6 +2,25 @@ import jax.numpy as jnp from spectrax import simulation +def test_multispecies_background_removed_from_perturbation(): + """Remove every species' homogeneous density from dCk.""" + Ns = 3 + Ck_0 = jnp.zeros((Ns, 1, 2, 1), dtype=jnp.complex128) + Ck_0 = Ck_0.at[:, 0, 0, 0].set(jnp.arange(1, Ns + 1)) + parameters = { + "Ck_0": Ck_0, + "Fk_0": jnp.zeros((6, 1, 2, 1), dtype=jnp.complex128), + "qs": jnp.array([-1.0, 1.0, 1.0]), + "alpha_s": jnp.ones(3 * Ns), + "u_s": jnp.zeros(3 * Ns), + "Omega_cs": jnp.ones(Ns), + "t_max": 0.001, + } + + result = simulation(parameters, Nx=3, Nn=1, Ns=Ns, timesteps=2, + dt=0.001, adaptive_time_step=False) + assert jnp.all(result["dCk"][:, :, 0, 0, 0] == 0) + def test_simulation_runs(): """Test if the simulation runs without errors with default parameters.""" result = simulation() From 5c41070c950f2596a06b4e3a7db2425c988768ea Mon Sep 17 00:00:00 2001 From: rogeriojorge Date: Sat, 26 Sep 2026 07:27:18 -0500 Subject: [PATCH 06/20] Expose dtmin and max_steps so a collapsing step stops instead of crawling An under-resolved run (too few Hermite modes or too weak hypercollisions for the filamentation reaching the cutoff, or a species whose Hermite width is narrower than it becomes) makes the adaptive step collapse; the solve then crawls for hours towards the hard-coded 1e6 steps. dtmin (off by default) stops it with an error at the collapse; max_steps is now a parameter. --- spectrax/_simulation.py | 15 ++++++++++++--- tests/test_simulation_runs.py | 6 ++++++ 2 files changed, 18 insertions(+), 3 deletions(-) diff --git a/spectrax/_simulation.py b/spectrax/_simulation.py index eba5de0..24d3507 100644 --- a/spectrax/_simulation.py +++ b/spectrax/_simulation.py @@ -107,9 +107,10 @@ def ode_system(Nx, Ny, Nz, Nn, Nm, Np, Ns, t, Ck_Fk, args): return dy_dt -@partial(jit, static_argnames=['Nx', 'Ny', 'Nz', 'Nn', 'Nm', 'Np', 'Ns', 'timesteps', 'solver', 'adaptive_time_step']) +@partial(jit, static_argnames=['Nx', 'Ny', 'Nz', 'Nn', 'Nm', 'Np', 'Ns', 'timesteps', 'solver', 'adaptive_time_step', + 'dtmin', 'max_steps']) def simulation(input_parameters={}, Nx=33, Ny=1, Nz=1, Nn=20, Nm=1, Np=1, Ns=2, - timesteps=200, dt = 0.01, solver=Dopri8(), adaptive_time_step=True): + timesteps=200, dt = 0.01, solver=Dopri8(), adaptive_time_step=True, dtmin=None, max_steps=1000000): """ Run a spectral Vlasov-Maxwell simulation and return the solution together with the parameter dictionary used to produce it. @@ -130,6 +131,13 @@ def simulation(input_parameters={}, Nx=33, Ny=1, Nz=1, Nn=20, Nm=1, Np=1, Ns=2, Initial integration step size. solver : diffrax.AbstractSolver, optional Diffrax solver instance controlling the time integration. + dtmin : float, optional + Smallest adaptive step. If the step would fall below it the solve stops with an + error, instead of crawling towards `max_steps`. A collapsing step usually means too + few Hermite modes or too weak hypercollisions for the filamentation reaching the + cutoff, or a species whose Hermite width is narrower than it becomes. Default: no limit. + max_steps : int, optional + Maximum number of solver steps before the solve stops with an error. Returns ------- @@ -162,6 +170,7 @@ def simulation(input_parameters={}, Nx=33, Ny=1, Nz=1, Nn=20, Nm=1, Np=1, Ns=2, True: PIDController( rtol=parameters["ode_tolerance"], atol=parameters["ode_tolerance"], + dtmin=dtmin, force_dtmin=False, ), False: ConstantStepSize(), } @@ -174,7 +183,7 @@ def simulation(input_parameters={}, Nx=33, Ny=1, Nz=1, Nn=20, Nm=1, Np=1, Ns=2, stepsize_controller=stepsize_controller, t0=0, t1=parameters["t_max"], dt0=dt, y0=initial_conditions, args=args, saveat=SaveAt(ts=time), - max_steps=1000000, progress_meter=TqdmProgressMeter()) + max_steps=max_steps, progress_meter=TqdmProgressMeter()) # Reshape the solution to extract Ck and Fk Ck = sol.ys[:,:(-6 * (Nx//2+1) * Ny * Nz)].reshape(len(sol.ts), Ns * Nn * Nm * Np, Ny, Nx//2+1, Nz) diff --git a/tests/test_simulation_runs.py b/tests/test_simulation_runs.py index 2d05a1e..fdfd557 100644 --- a/tests/test_simulation_runs.py +++ b/tests/test_simulation_runs.py @@ -18,5 +18,11 @@ def test_electric_field_update(): result = simulation() assert not jnp.all(result["Fk"] == 0), "Electric field did not update." +@pytest.mark.parametrize("limits", [{"dtmin": 10.0}, {"max_steps": 2}]) +def test_step_limits_stop_the_solve_with_an_error(limits): + """A step below dtmin (here: any step) or too many steps stops the solve instead of crawling on.""" + with pytest.raises(Exception, match="minimum step size|maximum number of solver steps"): + simulation(**limits) + if __name__ == "__main__": pytest.main() From 2bb763fb8108eefb660e4120077beaa6af3cd972 Mon Sep 17 00:00:00 2001 From: rogeriojorge Date: Sat, 26 Sep 2026 14:47:14 -0500 Subject: [PATCH 07/20] Default dtmin to t_max / max_steps; document dtmin and max_steps in the README --- README.md | 2 ++ spectrax/_simulation.py | 5 +++-- 2 files changed, 5 insertions(+), 2 deletions(-) diff --git a/README.md b/README.md index 658dcc7..ebba142 100644 --- a/README.md +++ b/README.md @@ -201,6 +201,8 @@ most important keys. Keys absent from the file fall back to sensible defaults sp | `dt` | Initial step size provided to the ODE solver. | | `solver` | Name of Diffrax solver (e.g., `Tsit5`, `Dopri5`, `Dopri8`, `ImplicitMidpoint`). | | `adaptive_time_step` | Timestep adaptability (`true` by default). | +| `dtmin` | Smallest adaptive step; below it the solve stops with an error instead of stalling. Default `t_max / max_steps`, `0` disables it. A collapsing step means an under-resolved run: more Hermite modes, a larger `nu`, or a wider `alpha_s`. | +| `max_steps` | Maximum number of solver steps (`1000000` by default). | Many of these parameters can be arrays; for example `alpha_s` must contain three values per species (one for each velocity dimension) and can be used to represent anisotropic plasmas. diff --git a/spectrax/_simulation.py b/spectrax/_simulation.py index 24d3507..ae78a4e 100644 --- a/spectrax/_simulation.py +++ b/spectrax/_simulation.py @@ -135,7 +135,8 @@ def simulation(input_parameters={}, Nx=33, Ny=1, Nz=1, Nn=20, Nm=1, Np=1, Ns=2, Smallest adaptive step. If the step would fall below it the solve stops with an error, instead of crawling towards `max_steps`. A collapsing step usually means too few Hermite modes or too weak hypercollisions for the filamentation reaching the - cutoff, or a species whose Hermite width is narrower than it becomes. Default: no limit. + cutoff, or a species whose Hermite width is narrower than it becomes. Default: + `t_max / max_steps`, the step below which the run could not finish anyway; 0 disables it. max_steps : int, optional Maximum number of solver steps before the solve stops with an error. @@ -170,7 +171,7 @@ def simulation(input_parameters={}, Nx=33, Ny=1, Nz=1, Nn=20, Nm=1, Np=1, Ns=2, True: PIDController( rtol=parameters["ode_tolerance"], atol=parameters["ode_tolerance"], - dtmin=dtmin, force_dtmin=False, + dtmin=parameters["t_max"] / max_steps if dtmin is None else dtmin, force_dtmin=False, ), False: ConstantStepSize(), } From cfef220fdc4102b33045ac19f0e788b29d7806f9 Mon Sep 17 00:00:00 2001 From: Rogerio Jorge Date: Mon, 14 Sep 2026 09:08:10 -0500 Subject: [PATCH 08/20] Compress the logo and stop tracking bytecode The logo was a 4.7 MB 1576x1476 screenshot with embedded metadata; the README shows it at 30% width. It is now 800 px wide with a 256-colour palette and no metadata (193 KB). Tracked __pycache__ files are removed; __pycache__/ is already ignored. --- __pycache__/Examples.cpython-310.pyc | Bin 8147 -> 0 bytes __pycache__/Examples_1D.cpython-310.pyc | Bin 5761 -> 0 bytes __pycache__/Examples_2D.cpython-310.pyc | Bin 2816 -> 0 bytes __pycache__/JAX_VM_solver.cpython-310.pyc | Bin 7708 -> 0 bytes __pycache__/inverse_transform.cpython-310.pyc | Bin 2415 -> 0 bytes docs/SPECTRAX_logo.png | Bin 4671605 -> 192559 bytes 6 files changed, 0 insertions(+), 0 deletions(-) delete mode 100644 __pycache__/Examples.cpython-310.pyc delete mode 100644 __pycache__/Examples_1D.cpython-310.pyc delete mode 100644 __pycache__/Examples_2D.cpython-310.pyc delete mode 100644 __pycache__/JAX_VM_solver.cpython-310.pyc delete mode 100644 __pycache__/inverse_transform.cpython-310.pyc diff --git a/__pycache__/Examples.cpython-310.pyc b/__pycache__/Examples.cpython-310.pyc deleted file mode 100644 index f434ba7bb2a4accfb770ab801f2ea17de45997ac..0000000000000000000000000000000000000000 GIT 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zm5+rh8pg3fG3v!rzH@e9Yi-UhVs-g4*Vvv9J!*Ro$sW#;i&l*8a{T6U=6~VkApnQ{ zmf0~LTGR2BW}FAtrqlPz`R zGdj)JBOzDFE!Iu2_K*x*?ndmPPS%YvgzpeXVn|GSK#pBrKhn3W8Wp?Za{g27#?$eb zYW~V@>}4&!v9#;I&sVSP?6)!VB^}1Ib2RS(YdBc Date: Wed, 23 Sep 2026 12:07:47 -0500 Subject: [PATCH 09/20] Declare diffrax and orthax as dependencies spectrax imports diffrax (time integration) and orthax (inverse Hermite transform), but pyproject.toml did not list them, so pip install left the package unimportable in a clean environment. --- pyproject.toml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/pyproject.toml b/pyproject.toml index de5dbe4..311da1d 100644 --- a/pyproject.toml +++ b/pyproject.toml @@ -29,7 +29,7 @@ keywords = ["Plasma", "Simulation", "JAX"] -dependencies = [ "jax", "jaxlib", "jax_tqdm", "matplotlib"] +dependencies = [ "jax", "jaxlib", "jax_tqdm", "matplotlib", "diffrax", "orthax"] requires-python = ">=3.5" From 3a35d7a4dd037685a31a3ede399534ba5d5a8d8c Mon Sep 17 00:00:00 2001 From: Rogerio Jorge Date: Thu, 16 Jul 2026 08:41:55 -0500 Subject: [PATCH 10/20] Add mainline magnetic reconnection example --- Examples/2D_magnetic_reconnection.py | 228 +++++++++++++++++++++++++++ 1 file changed, 228 insertions(+) create mode 100644 Examples/2D_magnetic_reconnection.py diff --git a/Examples/2D_magnetic_reconnection.py b/Examples/2D_magnetic_reconnection.py new file mode 100644 index 0000000..512c86a --- /dev/null +++ b/Examples/2D_magnetic_reconnection.py @@ -0,0 +1,228 @@ +"""Periodic double-Harris-sheet magnetic reconnection. + +The short defaults are a smoke test. For a clear nonlinear island movie, run + + python Examples/2D_magnetic_reconnection.py --nx 65 --ny 17 \ + --t-max 500 --chunk-time 50 --snapshots 51 --perturbation 0.1 + +Use ``--perturbation 1e-4`` instead for a linear tearing-growth diagnostic. +The parameters follow Camporeale et al. (2006), section IV.C; the periodic +double-sheet geometry and half-sheet magnetic diagnostic follow Koshkarov et +al. (2021). +""" + +import argparse +import json +import time +from pathlib import Path + +import jax +import jax.numpy as jnp +import matplotlib.pyplot as plt +import numpy as np +from diffrax import Dopri8 +from matplotlib.animation import FuncAnimation, PillowWriter + +from spectrax import simulation + + +def initial_conditions(nx, ny, hermite, perturbation, nu): + a = jnp.sqrt(2.0) + lx, ly = 32.0 * a, 4.0 * jnp.pi * a + x = jnp.linspace(0.0, lx, nx, endpoint=False) + y = jnp.linspace(0.0, ly, ny, endpoint=False) + X, Y, Z = jnp.meshgrid(x, y, jnp.zeros(1), indexing="xy") + sheets = ( + jnp.cosh((X - lx / 4.0) / a) ** -2, + jnp.cosh((X - 3.0 * lx / 4.0) / a) ** -2, + ) + density = jnp.stack((sheets[0], sheets[1], sheets[0], sheets[1])) + alpha = a * jnp.array(([1.0 / 6.0] * 6) + ([1.0 / 24.0] * 6)) + coefficients = jnp.zeros( + (4, hermite, hermite, hermite, ny, nx, 1), dtype=jnp.float64 + ) + coefficients = coefficients.at[:, 0, 0, 0].set( + density / jnp.prod(alpha.reshape(4, 3), axis=1)[:, None, None, None] + ) + + kx, ky = 2.0 * jnp.pi / lx, 2.0 * jnp.pi / ly + magnetic = jnp.array( + [ + -perturbation * ky * jnp.sin(ky * Y) * jnp.sin(kx * X), + jnp.tanh((X - lx / 4.0) / a) + - jnp.tanh((X - 3.0 * lx / 4.0) / a) + - 1.0 + - perturbation * kx * jnp.cos(ky * Y) * jnp.cos(kx * X), + 5.0 * jnp.ones_like(X), + ] + ) + transform = lambda value: jnp.fft.rfftn( + value, axes=(-1, -3, -2), norm="forward" + ) + return { + "Lx": lx, + "Ly": ly, + "Lz": 1.0, + "mi_me": 16.0, + "qs": jnp.array((-1.0, -1.0, 1.0, 1.0)), + "alpha_s": alpha, + "u_s": jnp.array( + (0.0, 0.0, -1.0 / 6.0, 0.0, 0.0, 1.0 / 6.0, + 0.0, 0.0, 1.0 / 6.0, 0.0, 0.0, -1.0 / 6.0) + ) / a, + "Omega_cs": jnp.array((1.0 / 3.0, 1.0 / 3.0, 1.0 / 48.0, 1.0 / 48.0)), + "nu": nu, + "D": 0.0, + "ode_tolerance": 1e-7, + "Ck_0": transform(coefficients), + "Fk_0": transform(jnp.concatenate((jnp.zeros_like(magnetic), magnetic))), + } + + +def fields_and_flux(fk, nx, lx, ly): + ny = fk.shape[-3] + fields = np.fft.irfftn( + fk, s=(1, ny, nx), axes=(-1, -3, -2), norm="forward" + ).real + kx = 2.0 * np.pi * np.fft.rfftfreq(nx, d=lx / nx) + ky = 2.0 * np.pi * np.fft.fftfreq(ny, d=ly / ny) + KX, KY = np.meshgrid(kx, ky) + bx, by = fk[:, 3, :, :, 0], fk[:, 4, :, :, 0] + k2 = KX**2 + KY**2 + az_k = np.zeros_like(bx) + np.divide(-1j * KY * bx + 1j * KX * by, k2, out=az_k, where=k2 != 0.0) + jz_k = 1j * KX * by - 1j * KY * bx + az = np.fft.irfftn(az_k, s=(ny, nx), axes=(-2, -1), norm="forward").real + jz = np.fft.irfftn(jz_k, s=(ny, nx), axes=(-2, -1), norm="forward").real + return fields, az, jz + + +def save_diagnostics(output_dir, t, fk, nx, lx, ly): + fields, az, jz = fields_and_flux(fk, nx, lx, ly) + half = (nx + 1) // 2 + magnetic = fields[:, 3:, :, :half, 0] + phase = np.exp(2j * np.pi * np.arange(magnetic.shape[2]) / magnetic.shape[2]) + amplitude = np.linalg.norm( + np.sum(magnetic * phase[None, None, :, None], axis=(-2, -1)), axis=1 + ) / np.sqrt(magnetic.shape[2] * magnetic.shape[3]) + peaks = np.flatnonzero( + (amplitude[1:-1] > amplitude[:-2]) & (amplitude[1:-1] > amplitude[2:]) + ) + 1 + peaks = peaks[t[peaks] >= t[-1] / 4.0] + fit = None + if len(peaks) >= 3: + gamma, intercept = np.polyfit(t[peaks], np.log(amplitude[peaks]), 1) + fit = np.exp(intercept + gamma * t) + else: + gamma = np.nan + + sheet = np.argmin(abs(np.linspace(0.0, lx, nx, endpoint=False) - lx / 4.0)) + flux = np.ptp(az[:, :, sheet], axis=1) + fig, axes = plt.subplots(2, 1, figsize=(6.4, 6.0), sharex=True, constrained_layout=True) + axes[0].semilogy(t, amplitude, color="0.65", label="half-sheet magnetic mode") + if fit is not None: + axes[0].semilogy(t[peaks], amplitude[peaks], "o", ms=3) + axes[0].semilogy(t, fit, "k--", label=rf"envelope $\gamma={gamma:.5f}$") + axes[0].set_ylabel("magnetic-mode amplitude") + axes[0].legend(fontsize=8) + axes[1].plot(t, flux) + axes[1].set(xlabel=r"$\omega_{pe}t$", ylabel=r"reconnected flux $\Delta A_z$") + fig.savefig(output_dir / "reconnection-growth.png", dpi=180) + plt.close(fig) + + x = np.linspace(0.0, lx, nx, endpoint=False)[:half] + y = np.linspace(-ly / 2.0, ly / 2.0, fk.shape[-3], endpoint=False) + az = np.roll(az, fk.shape[-3] // 2, axis=1)[:, :, :half] + jz = np.roll(jz, fk.shape[-3] // 2, axis=1)[:, :, :half] + window = (x >= lx / 8.0) & (x <= 3.0 * lx / 8.0) + x, az, jz = x[window], az[:, :, window], jz[:, :, window] + jz -= jz.mean(axis=1, keepdims=True) + limit = max(np.quantile(abs(jz[-1]), 0.99), 1e-12) + sheet = np.argmin(abs(x - lx / 4.0)) + fig, ax = plt.subplots(figsize=(6.4, 3.8), constrained_layout=True) + image = ax.pcolormesh(x, y, jz[0], shading="auto", cmap="RdBu_r", vmin=-limit, vmax=limit) + contours = [None] + title = ax.set_title("") + ax.set(xlabel="x", ylabel="y") + fig.colorbar(image, ax=ax, label=r"$J_z-\langle J_z\rangle_y$") + + def update(frame): + image.set_array(jz[frame].ravel()) + if contours[0] is not None: + contours[0].remove() + center = az[frame, :, sheet].mean() + span = max(1.25 * np.ptp(az[frame, :, sheet]), 1e-12) + contours[0] = ax.contour( + x, y, az[frame], center + np.linspace(-span, span, 19), + colors="k", linewidths=0.55, + ) + title.set_text(rf"$\omega_{{pe}}t={t[frame]:.1f}$") + return image, title, contours[0] + + frames = range(0, len(t), max(1, len(t) // 100)) + animation = FuncAnimation(fig, update, frames=frames, interval=80, blit=False) + animation.save(output_dir / "reconnection-topology.gif", PillowWriter(fps=12), dpi=110) + plt.close(fig) + return amplitude, gamma + + +def main(): + parser = argparse.ArgumentParser() + parser.add_argument("--nx", type=int, default=33) + parser.add_argument("--ny", type=int, default=9) + parser.add_argument("--hermite", type=int, default=4) + parser.add_argument("--nu", type=float, default=1.0) + parser.add_argument("--perturbation", type=float, default=1e-4) + parser.add_argument("--t-max", type=float, default=10.0) + parser.add_argument("--chunk-time", type=float, default=10.0) + parser.add_argument("--snapshots", type=int, default=21) + parser.add_argument("--output-dir", type=Path, default=Path("reconnection_output")) + args = parser.parse_args() + args.output_dir.mkdir(parents=True, exist_ok=True) + + values = initial_conditions(args.nx, args.ny, args.hermite, args.perturbation, args.nu) + times, fields, offset = [], [], 0.0 + start = time.perf_counter() + while offset < args.t_max: + duration = min(args.chunk_time, args.t_max - offset) + values["t_max"] = duration + output = simulation( + values, Nx=args.nx, Ny=args.ny, Nz=1, + Nn=args.hermite, Nm=args.hermite, Np=args.hermite, Ns=4, + timesteps=args.snapshots, dt=1e-3, solver=Dopri8(), + ) + jax.block_until_ready(output["Ck"]) + first = 0 if not times else 1 + times.append(np.asarray(output["time"])[first:] + offset) + fields.append(np.asarray(output["Fk"])[first:]) + values["Ck_0"] = jnp.asarray(output["Ck"][-1]) + values["Fk_0"] = jnp.asarray(output["Fk"][-1]) + del output + offset += duration + print(f"t={offset:g}") + + t, fk = np.concatenate(times), np.concatenate(fields) + amplitude, gamma = save_diagnostics( + args.output_dir, t, fk, args.nx, float(values["Lx"]), float(values["Ly"]) + ) + coefficients = np.asarray(values["Ck_0"]).reshape( + 4, args.hermite, args.hermite, args.hermite, args.ny, args.nx // 2 + 1, 1 + ) + power = abs(coefficients) ** 2 + tails = { + name: float(np.take(power, -1, axis=axis).sum() / power.sum()) + for name, axis in (("p", 1), ("m", 2), ("n", 3)) + } + np.savez(args.output_dir / "reconnection.npz", time=t, amplitude=amplitude, Fk=fk) + linear = args.perturbation <= 1e-3 + print(json.dumps({ + "runtime_s": time.perf_counter() - start, + "regime": "linear diagnostic" if linear else "finite-seed nonlinear", + "gamma": None if not np.isfinite(gamma) else float(gamma), + "expected_gamma": 0.256 / 48.0 if linear else None, + "hermite_tail_fraction": tails, + }, sort_keys=True)) + + +if __name__ == "__main__": + main() From 94ffda3c080c0d16d892e2271a9c95c7d5480d66 Mon Sep 17 00:00:00 2001 From: rogeriojorge Date: Sat, 26 Sep 2026 16:04:04 -0500 Subject: [PATCH 11/20] Fix implicit midpoint time, convergence reporting and pytree states --- spectrax/midpoint_solver.py | 57 +++++++++++++++++++++++------------ tests/test_simulation_runs.py | 26 ++++++++++++++++ 2 files changed, 63 insertions(+), 20 deletions(-) diff --git a/spectrax/midpoint_solver.py b/spectrax/midpoint_solver.py index 2b6bf06..cca6803 100644 --- a/spectrax/midpoint_solver.py +++ b/spectrax/midpoint_solver.py @@ -3,6 +3,7 @@ import diffrax import jax import jax.numpy as jnp +import optimistix as optx from jax import lax from jax.scipy.sparse.linalg import gmres @@ -12,7 +13,7 @@ class ImplicitMidpoint(diffrax.AbstractSolver): This solver implements the implicit midpoint rule: - y_{n+1} = y_n + Δt * f(t_{n+1}, (y_n + y_{n+1}) / 2) + y_{n+1} = y_n + Δt * f(t_n + Δt / 2, (y_n + y_{n+1}) / 2) The nonlinear equation for ``y_{n+1}`` is solved with Newton iterations, where each linearized step is solved via GMRES using JAX's linearization. @@ -38,19 +39,28 @@ def step(self, terms, t0, t1, y0, args, solver_state, made_jump): del solver_state, made_jump δt = t1 - t0 + t_mid = t0 + 0.5 * δt f0 = terms.vf(t0, y0, args) - y1_init = y0 + δt * f0 + y1_init = jax.tree.map(lambda y, f: y + δt * f, y0, f0) - # Define F(y1) = y1 - y0 - δt * f(t1, (y0 + y1)/2) + # Define F(y1) = y1 - y0 - δt * f(t_mid, (y0 + y1)/2) def F_fn(y1): - y_mid = 0.5 * (y0 + y1) - return y1 - y0 - δt * terms.vf(t1, y_mid, args) + y_mid = jax.tree.map(lambda a, b: 0.5 * (a + b), y0, y1) + f_mid = terms.vf(t_mid, y_mid, args) + return jax.tree.map(lambda a, b, f: a - b - δt * f, y1, y0, f_mid) - y1 = _newton_gmres(F_fn, y0, y1_init, self.rtol, self.atol, self.max_iters) + y1, converged = _newton_gmres( + F_fn, y0, y1_init, self.rtol, self.atol, self.max_iters + ) - y_error = y1 - y1_init + y_error = jax.tree.map(lambda a, b: a - b, y1, y1_init) dense_info = dict(y0=y0, y1=y1) - return y1, y_error, dense_info, None, diffrax.RESULTS.successful + result = diffrax.RESULTS.where( + converged, + diffrax.RESULTS.successful, + diffrax.RESULTS.promote(optx.RESULTS.nonlinear_max_steps_reached), + ) + return y1, y_error, dense_info, None, result def _newton_gmres(F_fn, y0, y_init, rtol, atol, max_iters): @@ -74,25 +84,32 @@ def body_fn(state): res, jvp = jax.linearize(F_fn, y1) - scale = atol + rtol * jnp.maximum(jnp.abs(y0), jnp.abs(y1)) - norm = jnp.linalg.norm(res / scale) + scaled_res = jax.tree.map( + lambda r, a, b: r / (atol + rtol * jnp.maximum(jnp.abs(a), jnp.abs(b))), + res, y0, y1, + ) + norm = jnp.sqrt(sum(jnp.vdot(x, x).real for x in jax.tree.leaves(scaled_res))) # Adaptive inner tolerance (Eisenstat–Walker). inner_tol = jnp.minimum(0.1, norm * 0.5) - delta, _ = gmres( - jvp, - -res, - tol=inner_tol, - atol=atol, - maxiter=min(20, max_iters // 2), + def update(_): + delta, _ = gmres( + jvp, + jax.tree.map(jnp.negative, res), + tol=inner_tol, + atol=atol, + maxiter=min(20, max_iters // 2), + ) + return jax.tree.map(lambda y, d: y + d, y1, delta) + + y1_next = lax.cond( + norm < 1.0, lambda _: y1, update, operand=None ) - - y1_next = jnp.where(norm < 1.0, y1, y1 + delta) return y1_next, norm >= 1.0, i + 1 init_state = (y_init, True, 0) - y1_final, _, _ = lax.while_loop(cond_fn, body_fn, init_state) - return y1_final + y1_final, not_converged, _ = lax.while_loop(cond_fn, body_fn, init_state) + return y1_final, ~not_converged return loop_fn(y_init) diff --git a/tests/test_simulation_runs.py b/tests/test_simulation_runs.py index 2d05a1e..3948e36 100644 --- a/tests/test_simulation_runs.py +++ b/tests/test_simulation_runs.py @@ -1,6 +1,9 @@ import pytest +import diffrax import jax.numpy as jnp +import optimistix as optx from spectrax import simulation +from spectrax.midpoint_solver import ImplicitMidpoint def test_simulation_runs(): """Test if the simulation runs without errors with default parameters.""" @@ -18,5 +21,28 @@ def test_electric_field_update(): result = simulation() assert not jnp.all(result["Fk"] == 0), "Electric field did not update." +def test_implicit_midpoint_step_structured_state(): + term = diffrax.ODETerm(lambda t, y, args: (-y[0], 2 * y[1])) + y1, _, _, _, result = ImplicitMidpoint().step( + term, 0.0, 0.1, (jnp.ones(2), jnp.ones((2, 2))), None, None, False + ) + assert result == diffrax.RESULTS.successful + assert jnp.allclose(y1[0], 0.95 / 1.05, rtol=1e-5) + assert jnp.allclose(y1[1], 1.1 / 0.9, rtol=1e-5) + +def test_implicit_midpoint_reports_newton_exhaustion(): + term = diffrax.ODETerm(lambda t, y, args: y + 1) + result = ImplicitMidpoint(max_iters=0).step( + term, 0.0, 0.1, jnp.array(0.0), None, None, False + )[-1] + assert result == diffrax.RESULTS.promote(optx.RESULTS.nonlinear_max_steps_reached) + +def test_implicit_midpoint_evaluates_rhs_at_midpoint_time(): + term = diffrax.ODETerm(lambda t, y, args: t) + y1 = ImplicitMidpoint(max_iters=2).step( + term, 0.0, 0.2, jnp.array(0.0), None, None, False + )[0] + assert jnp.allclose(y1, 0.02, rtol=0, atol=1e-12) + if __name__ == "__main__": pytest.main() From 2f19bd64ae3a1910ba0f5d7a0b80c5f0beb18690 Mon Sep 17 00:00:00 2001 From: rogeriojorge Date: Sat, 26 Sep 2026 16:01:14 -0500 Subject: [PATCH 12/20] Expose throw so a failed solve can be reported instead of raised simulation(throw=False) returns and puts diffrax's sol.result in output['solver_result']. --- README.md | 1 + spectrax/_simulation.py | 12 ++++++++---- tests/test_simulation_runs.py | 6 ++++++ 3 files changed, 15 insertions(+), 4 deletions(-) diff --git a/README.md b/README.md index ebba142..6bc2a82 100644 --- a/README.md +++ b/README.md @@ -203,6 +203,7 @@ most important keys. Keys absent from the file fall back to sensible defaults sp | `adaptive_time_step` | Timestep adaptability (`true` by default). | | `dtmin` | Smallest adaptive step; below it the solve stops with an error instead of stalling. Default `t_max / max_steps`, `0` disables it. A collapsing step means an under-resolved run: more Hermite modes, a larger `nu`, or a wider `alpha_s`. | | `max_steps` | Maximum number of solver steps (`1000000` by default). | +| `throw` | `True` (default) raises on a failed solve; `False` returns and reports it in `output["solver_result"]`. | Many of these parameters can be arrays; for example `alpha_s` must contain three values per species (one for each velocity dimension) and can be used to represent anisotropic plasmas. diff --git a/spectrax/_simulation.py b/spectrax/_simulation.py index ae78a4e..3047bbc 100644 --- a/spectrax/_simulation.py +++ b/spectrax/_simulation.py @@ -108,9 +108,10 @@ def ode_system(Nx, Ny, Nz, Nn, Nm, Np, Ns, t, Ck_Fk, args): return dy_dt @partial(jit, static_argnames=['Nx', 'Ny', 'Nz', 'Nn', 'Nm', 'Np', 'Ns', 'timesteps', 'solver', 'adaptive_time_step', - 'dtmin', 'max_steps']) + 'dtmin', 'max_steps', 'throw']) def simulation(input_parameters={}, Nx=33, Ny=1, Nz=1, Nn=20, Nm=1, Np=1, Ns=2, - timesteps=200, dt = 0.01, solver=Dopri8(), adaptive_time_step=True, dtmin=None, max_steps=1000000): + timesteps=200, dt = 0.01, solver=Dopri8(), adaptive_time_step=True, dtmin=None, max_steps=1000000, + throw=True): """ Run a spectral Vlasov-Maxwell simulation and return the solution together with the parameter dictionary used to produce it. @@ -139,6 +140,8 @@ def simulation(input_parameters={}, Nx=33, Ny=1, Nz=1, Nn=20, Nm=1, Np=1, Ns=2, `t_max / max_steps`, the step below which the run could not finish anyway; 0 disables it. max_steps : int, optional Maximum number of solver steps before the solve stops with an error. + throw : bool, optional + Raise on integration failure; with False the failure is reported in `solver_result`. Returns ------- @@ -184,7 +187,7 @@ def simulation(input_parameters={}, Nx=33, Ny=1, Nz=1, Nn=20, Nm=1, Np=1, Ns=2, stepsize_controller=stepsize_controller, t0=0, t1=parameters["t_max"], dt0=dt, y0=initial_conditions, args=args, saveat=SaveAt(ts=time), - max_steps=max_steps, progress_meter=TqdmProgressMeter()) + max_steps=max_steps, progress_meter=TqdmProgressMeter(), throw=throw) # Reshape the solution to extract Ck and Fk Ck = sol.ys[:,:(-6 * (Nx//2+1) * Ny * Nz)].reshape(len(sol.ts), Ns * Nn * Nm * Np, Ny, Nx//2+1, Nz) @@ -195,7 +198,8 @@ def simulation(input_parameters={}, Nx=33, Ny=1, Nz=1, Nn=20, Nm=1, Np=1, Ns=2, dCk = dCk.at[:, Nn * Nm * Np, 0, 0, 0].set(0) # Output results - temporary_output = {"Ck": Ck, "Fk": Fk, "time": time, "dCk": dCk, "solver_stats": sol.stats} + temporary_output = {"Ck": Ck, "Fk": Fk, "time": time, "dCk": dCk, "solver_stats": sol.stats, + "solver_result": sol.result} output = {**temporary_output, **parameters} diagnostics(output) return output diff --git a/tests/test_simulation_runs.py b/tests/test_simulation_runs.py index fdfd557..8abb462 100644 --- a/tests/test_simulation_runs.py +++ b/tests/test_simulation_runs.py @@ -24,5 +24,11 @@ def test_step_limits_stop_the_solve_with_an_error(limits): with pytest.raises(Exception, match="minimum step size|maximum number of solver steps"): simulation(**limits) +def test_throw_false_reports_the_failure_instead_of_raising(): + """With throw=False a stopped solve returns, and solver_result says why.""" + from diffrax import RESULTS + assert simulation(throw=False)["solver_result"] == RESULTS.successful + assert simulation(max_steps=2, dtmin=0, throw=False)["solver_result"] == RESULTS.max_steps_reached + if __name__ == "__main__": pytest.main() From 161e6978376639f703a0de6d7f363e1a1f9bd93d Mon Sep 17 00:00:00 2001 From: Rogerio Jorge Date: Sun, 12 Jul 2026 22:17:47 -0500 Subject: [PATCH 13/20] Keep solver state axes structured --- spectrax/_simulation.py | 28 +++++++++++++--------------- tests/test_simulation_runs.py | 3 +++ 2 files changed, 16 insertions(+), 15 deletions(-) diff --git a/spectrax/_simulation.py b/spectrax/_simulation.py index eba5de0..f22f5a0 100644 --- a/spectrax/_simulation.py +++ b/spectrax/_simulation.py @@ -64,17 +64,16 @@ def ode_system(Nx, Ny, Nz, Nn, Nm, Np, Ns, t, Ck_Fk, args): Number of species. t : float Integration time (unused but required by Diffrax interface). - Ck_Fk : jnp.ndarray - Flattened state vector containing concatenated Hermite coefficients followed - by electromagnetic field coefficients. + Ck_Fk : tuple[jnp.ndarray, jnp.ndarray] + Hermite and electromagnetic field coefficients. args : tuple Cached parameter tuple produced in `simulation` providing physical constants, grids, and helper arrays. Returns ------- - jnp.ndarray - Flattened derivative vector matching the shape of `Ck_Fk`. + tuple[jnp.ndarray, jnp.ndarray] + Derivatives matching the Hermite and field coefficient arrays. """ (qs, nu, D, Omega_cs, alpha_s, u_s, @@ -82,9 +81,7 @@ def ode_system(Nx, Ny, Nz, Nn, Nm, Np, Ns, t, Ck_Fk, args): sqrt_n_plus, sqrt_n_minus, sqrt_m_plus, sqrt_m_minus, sqrt_p_plus, sqrt_p_minus ) = args[7:] - total_Ck_size = Nn * Nm * Np * Ns * (Nx//2+1) * Ny * Nz - Ck = Ck_Fk[:total_Ck_size].reshape(Nn * Nm * Np * Ns, Ny, Nx//2+1, Nz) - Fk = Ck_Fk[total_Ck_size:].reshape(6, Ny, Nx//2+1, Nz) + Ck, Fk = Ck_Fk # Build the 2/3 mask once per call (JIT will constant-fold it since Nx/Ny/Nz are static) @@ -103,9 +100,7 @@ def ode_system(Nx, Ny, Nz, Nn, Nm, Np, Ns, t, Ck_Fk, args): dEk_dt = 1j * cross_product(nabla, Fk[3:]) - current / Omega_cs[0] dFk_dt = jnp.concatenate([dEk_dt, dBk_dt], axis=0) - dy_dt = jnp.concatenate([dCk_s_dt.reshape(-1), dFk_dt.reshape(-1)]) - - return dy_dt + return dCk_s_dt, dFk_dt @partial(jit, static_argnames=['Nx', 'Ny', 'Nz', 'Nn', 'Nm', 'Np', 'Ns', 'timesteps', 'solver', 'adaptive_time_step']) def simulation(input_parameters={}, Nx=33, Ny=1, Nz=1, Nn=20, Nm=1, Np=1, Ns=2, @@ -141,8 +136,11 @@ def simulation(input_parameters={}, Nx=33, Ny=1, Nz=1, Nn=20, Nm=1, Np=1, Ns=2, # **Initialize simulation parameters** parameters = initialize_simulation_parameters(input_parameters, Nx, Ny, Nz, Nn, Nm, Np, Ns, timesteps, dt) - # Combine initial conditions. - initial_conditions = jnp.concatenate([parameters["Ck_0"].flatten(), parameters["Fk_0"].flatten()]) + # Preserve the species and Hermite axes for the integrator. + initial_conditions = ( + parameters["Ck_0"].reshape(Ns, Np, Nm, Nn, Ny, Nx//2+1, Nz), + parameters["Fk_0"].reshape(6, Ny, Nx//2+1, Nz), + ) # Define the time array for data output. time = jnp.linspace(0, parameters["t_max"], timesteps) @@ -177,8 +175,8 @@ def simulation(input_parameters={}, Nx=33, Ny=1, Nz=1, Nn=20, Nm=1, Np=1, Ns=2, max_steps=1000000, progress_meter=TqdmProgressMeter()) # Reshape the solution to extract Ck and Fk - Ck = sol.ys[:,:(-6 * (Nx//2+1) * Ny * Nz)].reshape(len(sol.ts), Ns * Nn * Nm * Np, Ny, Nx//2+1, Nz) - Fk = sol.ys[:,(-6 * (Nx//2+1) * Ny * Nz):].reshape(len(sol.ts), 6, Ny, Nx//2+1, Nz) + Ck = sol.ys[0].reshape(len(sol.ts), Ns * Nn * Nm * Np, Ny, Nx//2+1, Nz) + Fk = sol.ys[1] # Set n = 0, k = 0 mode to zero to get array with time evolution of perturbation. dCk = Ck.at[:, 0, 0, 0, 0].set(0) diff --git a/tests/test_simulation_runs.py b/tests/test_simulation_runs.py index 3948e36..236f040 100644 --- a/tests/test_simulation_runs.py +++ b/tests/test_simulation_runs.py @@ -15,6 +15,9 @@ def test_simulation_runs(): result["solver_stats"]["num_accepted_steps"] + result["solver_stats"]["num_rejected_steps"] ) + assert result["Ck"].shape[1:] == (result["Ns"] * result["Nn"] * result["Nm"] * result["Np"], + result["Ny"], result["Nx"] // 2 + 1, result["Nz"]) + assert result["Fk"].shape[1:] == (6, result["Ny"], result["Nx"] // 2 + 1, result["Nz"]) def test_electric_field_update(): """Test that the electric field updates and does not remain zero.""" From 567e0c49375478cb8183ba5bd632eff5a42835e6 Mon Sep 17 00:00:00 2001 From: rogeriojorge Date: Sat, 26 Sep 2026 16:07:06 -0500 Subject: [PATCH 14/20] Use SOLVAX Newton-Krylov with a Hermite line preconditioner in implicit midpoint --- pyproject.toml | 2 +- requirements.txt | 1 + spectrax/_simulation.py | 9 +- spectrax/midpoint_solver.py | 150 ++++++++++++++++++++-------------- tests/test_simulation_runs.py | 43 +++++++++- 5 files changed, 141 insertions(+), 64 deletions(-) diff --git a/pyproject.toml b/pyproject.toml index de5dbe4..9c4b1ff 100644 --- a/pyproject.toml +++ b/pyproject.toml @@ -29,7 +29,7 @@ keywords = ["Plasma", "Simulation", "JAX"] -dependencies = [ "jax", "jaxlib", "jax_tqdm", "matplotlib"] +dependencies = [ "jax", "jaxlib", "jax_tqdm", "matplotlib", "solvax>=0.8.0"] requires-python = ">=3.5" diff --git a/requirements.txt b/requirements.txt index d727662..91c9821 100644 --- a/requirements.txt +++ b/requirements.txt @@ -3,5 +3,6 @@ jaxlib jax_tqdm diffrax orthax +solvax>=0.8.0 matplotlib pandas diff --git a/spectrax/_simulation.py b/spectrax/_simulation.py index f22f5a0..8963cb1 100644 --- a/spectrax/_simulation.py +++ b/spectrax/_simulation.py @@ -9,6 +9,7 @@ from ._initialization import initialize_simulation_parameters from ._model import plasma_current, Hermite_Fourier_system from ._diagnostics import diagnostics +from .midpoint_solver import ImplicitMidpoint __all__ = ["cross_product", "ode_system", "simulation"] @@ -130,7 +131,8 @@ def simulation(input_parameters={}, Nx=33, Ny=1, Nz=1, Nn=20, Nm=1, Np=1, Ns=2, ------- dict Dictionary containing the evolved coefficients (`Ck`, `Fk`), time samples, - solver statistics, perturbation diagnostics, and all simulation parameters. + solver statistics, perturbation diagnostics, all simulation parameters, and + `midpoint_stats` when using :class:`ImplicitMidpoint`. """ # **Initialize simulation parameters** @@ -171,7 +173,8 @@ def simulation(input_parameters={}, Nx=33, Ny=1, Nz=1, Nn=20, Nm=1, Np=1, Ns=2, ODETerm(ode_system_partial), solver=solver, stepsize_controller=stepsize_controller, t0=0, t1=parameters["t_max"], dt0=dt, - y0=initial_conditions, args=args, saveat=SaveAt(ts=time), + y0=initial_conditions, args=args, + saveat=SaveAt(ts=time, solver_state=isinstance(solver, ImplicitMidpoint)), max_steps=1000000, progress_meter=TqdmProgressMeter()) # Reshape the solution to extract Ck and Fk @@ -184,6 +187,8 @@ def simulation(input_parameters={}, Nx=33, Ny=1, Nz=1, Nn=20, Nm=1, Np=1, Ns=2, # Output results temporary_output = {"Ck": Ck, "Fk": Fk, "time": time, "dCk": dCk, "solver_stats": sol.stats} + if sol.solver_state is not None: + temporary_output["midpoint_stats"] = sol.solver_state output = {**temporary_output, **parameters} diagnostics(output) return output diff --git a/spectrax/midpoint_solver.py b/spectrax/midpoint_solver.py index cca6803..ba59def 100644 --- a/spectrax/midpoint_solver.py +++ b/spectrax/midpoint_solver.py @@ -1,11 +1,61 @@ """Custom Diffrax solver: implicit midpoint with a Newton–GMRES nonlinear solve.""" +from collections.abc import Callable +from typing import NamedTuple + import diffrax import jax import jax.numpy as jnp import optimistix as optx -from jax import lax -from jax.scipy.sparse.linalg import gmres +from solvax import lu_factor_banded, lu_solve_banded, newton_krylov + + +def collision_diffusion_x_streaming_preconditioner(args, dt): + """Invert collision, diffusion, and x-streaming Hermite line blocks.""" + nu, diffusion = args[-20:-18] + alpha = args[-17].reshape(-1, 3)[:, 0] + drift = args[-16].reshape(-1, 3)[:, 0] + length, kx, k2 = args[-15], args[-12], args[-9] + collision, sqrt_plus, sqrt_minus = args[-7:-4] + phase = (0.5j * dt / length * kx)[None, None, None] + species = (slice(None),) + (None,) * 6 + diagonal = ( + 1 + + 0.5 * dt * ( + nu * collision[None, :, :, :, None, None, None] + + diffusion * k2[None, None, None, None] + ) + + drift[species] * phase + ) + coupling = alpha[species] * phase / jnp.sqrt(2) + + def lines(value): + return jnp.moveaxis(jnp.broadcast_to(value, diagonal.shape), 3, -1) + + bands = jnp.stack( + (lines(coupling * sqrt_minus), lines(diagonal), + lines(coupling * sqrt_plus)), axis=-2, + ) + flat_bands = bands.reshape(-1, 3, bands.shape[-1]) + factors = jax.vmap(lambda band: lu_factor_banded(band, 1, 1))(flat_bands) + + def apply(residual): + coefficients, fields = residual + rhs = jnp.moveaxis(coefficients, 3, -1).reshape(-1, coefficients.shape[3]) + solution = jax.vmap(lu_solve_banded)(factors, rhs) + solution = solution.reshape(bands.shape[:-2] + (-1,)) + return jnp.moveaxis(solution, -1, 3), fields + + return apply + + +class MidpointSolverState(NamedTuple): + """Newton--GMRES iteration totals and per-step maxima.""" + + newton_iterations: jax.Array + linear_iterations: jax.Array + max_newton_iterations: jax.Array + max_linear_iterations: jax.Array class ImplicitMidpoint(diffrax.AbstractSolver): @@ -17,11 +67,22 @@ class ImplicitMidpoint(diffrax.AbstractSolver): The nonlinear equation for ``y_{n+1}`` is solved with Newton iterations, where each linearized step is solved via GMRES using JAX's linearization. + ``rtol`` and ``atol`` control the nonlinear residual, while + ``linear_rtol`` and ``linear_atol`` control each GMRES solve. + Newton iteration stops when ``||F|| <= max(atol, rtol * ||F_initial||)``. + Accepted-step diagnostics are accumulated in :class:`MidpointSolverState`. + An optional ``preconditioner(args, dt)`` factory supplies the inverse action + used by GMRES. """ rtol: float = 1e-6 atol: float = 1e-8 max_iters: int = 300 + linear_restart: int = 20 + linear_rtol: float = 1e-4 + linear_atol: float = 0.0 + linear_max_restarts: int = 20 + preconditioner: Callable | None = None term_structure = diffrax.ODETerm interpolation_cls = diffrax.LocalLinearInterpolation @@ -30,13 +91,17 @@ def order(self, terms): return 2 def init(self, terms, t0, t1, y0, args): - return None + del terms, t0, t1, y0, args + zero = jnp.int32(0) + return MidpointSolverState(zero, zero, zero, zero) def func(self, terms, t0, y0, args): return terms.vf(t0, y0, args) def step(self, terms, t0, t1, y0, args, solver_state, made_jump): - del solver_state, made_jump + del made_jump + if solver_state is None: + solver_state = self.init(terms, t0, t1, y0, args) δt = t1 - t0 t_mid = t0 + 0.5 * δt @@ -49,67 +114,32 @@ def F_fn(y1): f_mid = terms.vf(t_mid, y_mid, args) return jax.tree.map(lambda a, b, f: a - b - δt * f, y1, y0, f_mid) - y1, converged = _newton_gmres( - F_fn, y0, y1_init, self.rtol, self.atol, self.max_iters + precond = None if self.preconditioner is None else self.preconditioner(args, δt) + solution = newton_krylov( + F_fn, + y1_init, + precond=precond, + rtol=self.rtol, + atol=self.atol, + max_steps=self.max_iters, + linear_restart=self.linear_restart, + linear_rtol=self.linear_rtol, + linear_atol=self.linear_atol, + linear_max_restarts=self.linear_max_restarts, ) + y1 = solution.x y_error = jax.tree.map(lambda a, b: a - b, y1, y1_init) dense_info = dict(y0=y0, y1=y1) result = diffrax.RESULTS.where( - converged, + solution.converged, diffrax.RESULTS.successful, diffrax.RESULTS.promote(optx.RESULTS.nonlinear_max_steps_reached), ) - return y1, y_error, dense_info, None, result - - -def _newton_gmres(F_fn, y0, y_init, rtol, atol, max_iters): - """Solve ``F(y)=0`` using Newton iterations with GMRES linear solves. - - Notes - ----- - - The Jacobian-vector product is obtained via ``jax.linearize``. - - The GMRES tolerance is chosen adaptively (Eisenstat–Walker style) based on - the current scaled residual norm. - """ - - @jax.jit - def loop_fn(y_init): - def cond_fn(state): - _, not_converged, i = state - return (i < max_iters) & not_converged - - def body_fn(state): - y1, _, i = state - - res, jvp = jax.linearize(F_fn, y1) - - scaled_res = jax.tree.map( - lambda r, a, b: r / (atol + rtol * jnp.maximum(jnp.abs(a), jnp.abs(b))), - res, y0, y1, - ) - norm = jnp.sqrt(sum(jnp.vdot(x, x).real for x in jax.tree.leaves(scaled_res))) - - # Adaptive inner tolerance (Eisenstat–Walker). - inner_tol = jnp.minimum(0.1, norm * 0.5) - - def update(_): - delta, _ = gmres( - jvp, - jax.tree.map(jnp.negative, res), - tol=inner_tol, - atol=atol, - maxiter=min(20, max_iters // 2), - ) - return jax.tree.map(lambda y, d: y + d, y1, delta) - - y1_next = lax.cond( - norm < 1.0, lambda _: y1, update, operand=None - ) - return y1_next, norm >= 1.0, i + 1 - - init_state = (y_init, True, 0) - y1_final, not_converged, _ = lax.while_loop(cond_fn, body_fn, init_state) - return y1_final, ~not_converged - - return loop_fn(y_init) + next_solver_state = MidpointSolverState( + solver_state.newton_iterations + solution.newton_iterations, + solver_state.linear_iterations + solution.linear_iterations, + jnp.maximum(solver_state.max_newton_iterations, solution.newton_iterations), + jnp.maximum(solver_state.max_linear_iterations, solution.linear_iterations), + ) + return y1, y_error, dense_info, next_solver_state, result diff --git a/tests/test_simulation_runs.py b/tests/test_simulation_runs.py index 236f040..b9e621e 100644 --- a/tests/test_simulation_runs.py +++ b/tests/test_simulation_runs.py @@ -3,7 +3,10 @@ import jax.numpy as jnp import optimistix as optx from spectrax import simulation -from spectrax.midpoint_solver import ImplicitMidpoint +from spectrax.midpoint_solver import ( + ImplicitMidpoint, + collision_diffusion_x_streaming_preconditioner, +) def test_simulation_runs(): """Test if the simulation runs without errors with default parameters.""" @@ -47,5 +50,43 @@ def test_implicit_midpoint_evaluates_rhs_at_midpoint_time(): )[0] assert jnp.allclose(y1, 0.02, rtol=0, atol=1e-12) +def test_implicit_midpoint_simulation_reports_iterations(): + result = simulation( + {"t_max": 0.01, "ode_tolerance": 1e-8}, Nx=5, Nn=3, + timesteps=2, dt=0.01, adaptive_time_step=False, + solver=ImplicitMidpoint( + max_iters=4, linear_restart=2, linear_max_restarts=2, + preconditioner=collision_diffusion_x_streaming_preconditioner, + ), + ) + assert jnp.all(jnp.isfinite(result["Ck"])) + stats = result["midpoint_stats"] + assert stats.newton_iterations >= stats.max_newton_iterations > 0 + assert stats.linear_iterations >= stats.max_linear_iterations > 0 + +def test_collision_diffusion_x_streaming_line(): + args = [None] * 21 + args[-20], args[-19] = 2.0, 3.0 + args[-17], args[-16], args[-15] = jnp.arange(4.0, 7.0), jnp.arange(7.0, 10.0), 10.0 + args[-12], args[-9] = jnp.full((1, 1, 1), 2.0), jnp.full((1, 1, 1), 5.0) + args[-7] = jnp.arange(3.0).reshape(1, 1, 3) + args[-6] = jnp.sqrt(jnp.arange(1.0, 4.0)).reshape(1, 1, 1, 3, 1, 1, 1) + args[-5] = jnp.sqrt(jnp.arange(3.0)).reshape(1, 1, 1, 3, 1, 1, 1) + coefficients = jnp.arange(1.0, 4.0).astype(complex).reshape(1, 1, 1, 3, 1, 1, 1) + fields = jnp.ones((6, 1, 1, 1)) + + actual = collision_diffusion_x_streaming_preconditioner(tuple(args), 0.2)( + (coefficients, fields) + ) + phase = 0.1j * 2 / 10 + diagonal = 1 + 0.1 * (2 * jnp.arange(3.0) + 15) + phase * 7 + coupling = phase * 4 / jnp.sqrt(2) + matrix = (jnp.diag(diagonal) + + jnp.diag(coupling * jnp.sqrt(jnp.arange(1.0, 3.0)), 1) + + jnp.diag(coupling * jnp.sqrt(jnp.arange(1.0, 3.0)), -1)) + expected = jnp.linalg.solve(matrix, coefficients.reshape(3)) + assert jnp.allclose(actual[0].reshape(3), expected, rtol=1e-6, atol=1e-6) + assert jnp.array_equal(actual[1], fields) + if __name__ == "__main__": pytest.main() From ff6a0a0e0d27f2fb9346426d5b7200968c9e0d42 Mon Sep 17 00:00:00 2001 From: rogeriojorge Date: Sat, 26 Sep 2026 16:10:13 -0500 Subject: [PATCH 15/20] Run the Maxwellian Hermite recurrence as a scan The unrolled loop compiled in 21 s at Nn=256; the scan compiles in well under a second. Adds a finiteness test at Nn=256. --- spectrax/_initialize_maxwellian.py | 12 +++++++----- tests/test_initialize_maxwellian.py | 7 +++++++ 2 files changed, 14 insertions(+), 5 deletions(-) diff --git a/spectrax/_initialize_maxwellian.py b/spectrax/_initialize_maxwellian.py index 54ee282..e85916c 100644 --- a/spectrax/_initialize_maxwellian.py +++ b/spectrax/_initialize_maxwellian.py @@ -3,7 +3,7 @@ import jax jax.config.update("jax_enable_x64", True) import jax.numpy as jnp -from jax import jit +from jax import jit, lax from jax.numpy.fft import rfftn from functools import partial @@ -17,10 +17,12 @@ def _maxwellian_modes(d, c2, N): Maxwellian has the basis width (then g_n = (sqrt(2) d)^n / sqrt(n!)). The recurrence g_{n+1} = (sqrt(2) d g_n - sqrt(n) c2 g_{n-1}) / sqrt(n + 1) avoids factorials. """ - g = [jnp.ones_like(d), jnp.sqrt(2.0) * d] - for n in range(1, N - 1): - g.append((jnp.sqrt(2.0) * d * g[n] - jnp.sqrt(n) * c2 * g[n - 1]) / jnp.sqrt(n + 1.0)) - return jnp.stack(g[:N]) + def step(g, n): # g = (g_{n-1}, g_n) -> (g_n, g_{n+1}); a scan keeps compile time independent of N + g_next = (jnp.sqrt(2.0) * d * g[1] - jnp.sqrt(n) * c2 * g[0]) / jnp.sqrt(n + 1.0) + return (g[1], g_next), g_next + g0, g1 = jnp.ones_like(d), jnp.sqrt(2.0) * d + _, rest = lax.scan(step, (g0, g1), jnp.arange(1.0, max(N - 1, 1))) + return jnp.concatenate([g0[None], g1[None], rest])[:N] @partial(jit, static_argnames=['Nn', 'Nm', 'Np', 'Ns']) diff --git a/tests/test_initialize_maxwellian.py b/tests/test_initialize_maxwellian.py index ee98889..13b1646 100644 --- a/tests/test_initialize_maxwellian.py +++ b/tests/test_initialize_maxwellian.py @@ -47,3 +47,10 @@ def test_narrow_maxwellian_in_a_wide_basis_is_reconstructed(vth, drift): f = alpha**2 * np.pi**-0.25 * np.exp(-xi**2 / 2) * (C @ h) # 1D marginal, as in inverse_HF_transform exact = np.exp(-(v - drift) ** 2 / (2 * vth**2)) / (np.sqrt(2 * np.pi) * vth) # unit density np.testing.assert_allclose(f, exact, atol=1e-12) + + +def test_high_order_maxwellian_is_finite(): + """Nn = 256 overflowed the factorial form (inf / inf = NaN) before the recurrence.""" + U = jnp.linspace(-0.1, 0.1, 6).reshape(2, 3, 1, 1, 1) + alpha = jnp.array([0.25, 0.3, 0.35, 0.05, 0.06, 0.07]) + assert jnp.all(jnp.isfinite(compute_C_nmp(U, alpha, jnp.zeros(6), 256, 4, 4, 2))) From 67e44861c9c6dcaa6405842f753304929dfdcad5 Mon Sep 17 00:00:00 2001 From: rogeriojorge Date: Sat, 26 Sep 2026 23:53:12 -0500 Subject: [PATCH 16/20] Invert the Hermite lines with lax.linalg.tridiagonal_solve --- spectrax/midpoint_solver.py | 15 +++++---------- 1 file changed, 5 insertions(+), 10 deletions(-) diff --git a/spectrax/midpoint_solver.py b/spectrax/midpoint_solver.py index ba59def..fa25e29 100644 --- a/spectrax/midpoint_solver.py +++ b/spectrax/midpoint_solver.py @@ -7,7 +7,7 @@ import jax import jax.numpy as jnp import optimistix as optx -from solvax import lu_factor_banded, lu_solve_banded, newton_krylov +from solvax import newton_krylov def collision_diffusion_x_streaming_preconditioner(args, dt): @@ -32,18 +32,13 @@ def collision_diffusion_x_streaming_preconditioner(args, dt): def lines(value): return jnp.moveaxis(jnp.broadcast_to(value, diagonal.shape), 3, -1) - bands = jnp.stack( - (lines(coupling * sqrt_minus), lines(diagonal), - lines(coupling * sqrt_plus)), axis=-2, - ) - flat_bands = bands.reshape(-1, 3, bands.shape[-1]) - factors = jax.vmap(lambda band: lu_factor_banded(band, 1, 1))(flat_bands) + main, lower = lines(diagonal), lines(coupling * sqrt_minus) + upper = lines(coupling * sqrt_plus).at[..., -1].set(0) def apply(residual): coefficients, fields = residual - rhs = jnp.moveaxis(coefficients, 3, -1).reshape(-1, coefficients.shape[3]) - solution = jax.vmap(lu_solve_banded)(factors, rhs) - solution = solution.reshape(bands.shape[:-2] + (-1,)) + rhs = jnp.moveaxis(coefficients, 3, -1)[..., None] + solution = jax.lax.linalg.tridiagonal_solve(lower, main, upper, rhs)[..., 0] return jnp.moveaxis(solution, -1, 3), fields return apply From 60cc214caba8712be251b8c74008ddc7de8f1604 Mon Sep 17 00:00:00 2001 From: rogeriojorge Date: Sat, 26 Sep 2026 16:04:04 -0500 Subject: [PATCH 17/20] Fix implicit midpoint time, convergence reporting and pytree states --- spectrax/midpoint_solver.py | 57 +++++++++++++++++++++++------------ tests/test_simulation_runs.py | 26 ++++++++++++++++ 2 files changed, 63 insertions(+), 20 deletions(-) diff --git a/spectrax/midpoint_solver.py b/spectrax/midpoint_solver.py index 2b6bf06..cca6803 100644 --- a/spectrax/midpoint_solver.py +++ b/spectrax/midpoint_solver.py @@ -3,6 +3,7 @@ import diffrax import jax import jax.numpy as jnp +import optimistix as optx from jax import lax from jax.scipy.sparse.linalg import gmres @@ -12,7 +13,7 @@ class ImplicitMidpoint(diffrax.AbstractSolver): This solver implements the implicit midpoint rule: - y_{n+1} = y_n + Δt * f(t_{n+1}, (y_n + y_{n+1}) / 2) + y_{n+1} = y_n + Δt * f(t_n + Δt / 2, (y_n + y_{n+1}) / 2) The nonlinear equation for ``y_{n+1}`` is solved with Newton iterations, where each linearized step is solved via GMRES using JAX's linearization. @@ -38,19 +39,28 @@ def step(self, terms, t0, t1, y0, args, solver_state, made_jump): del solver_state, made_jump δt = t1 - t0 + t_mid = t0 + 0.5 * δt f0 = terms.vf(t0, y0, args) - y1_init = y0 + δt * f0 + y1_init = jax.tree.map(lambda y, f: y + δt * f, y0, f0) - # Define F(y1) = y1 - y0 - δt * f(t1, (y0 + y1)/2) + # Define F(y1) = y1 - y0 - δt * f(t_mid, (y0 + y1)/2) def F_fn(y1): - y_mid = 0.5 * (y0 + y1) - return y1 - y0 - δt * terms.vf(t1, y_mid, args) + y_mid = jax.tree.map(lambda a, b: 0.5 * (a + b), y0, y1) + f_mid = terms.vf(t_mid, y_mid, args) + return jax.tree.map(lambda a, b, f: a - b - δt * f, y1, y0, f_mid) - y1 = _newton_gmres(F_fn, y0, y1_init, self.rtol, self.atol, self.max_iters) + y1, converged = _newton_gmres( + F_fn, y0, y1_init, self.rtol, self.atol, self.max_iters + ) - y_error = y1 - y1_init + y_error = jax.tree.map(lambda a, b: a - b, y1, y1_init) dense_info = dict(y0=y0, y1=y1) - return y1, y_error, dense_info, None, diffrax.RESULTS.successful + result = diffrax.RESULTS.where( + converged, + diffrax.RESULTS.successful, + diffrax.RESULTS.promote(optx.RESULTS.nonlinear_max_steps_reached), + ) + return y1, y_error, dense_info, None, result def _newton_gmres(F_fn, y0, y_init, rtol, atol, max_iters): @@ -74,25 +84,32 @@ def body_fn(state): res, jvp = jax.linearize(F_fn, y1) - scale = atol + rtol * jnp.maximum(jnp.abs(y0), jnp.abs(y1)) - norm = jnp.linalg.norm(res / scale) + scaled_res = jax.tree.map( + lambda r, a, b: r / (atol + rtol * jnp.maximum(jnp.abs(a), jnp.abs(b))), + res, y0, y1, + ) + norm = jnp.sqrt(sum(jnp.vdot(x, x).real for x in jax.tree.leaves(scaled_res))) # Adaptive inner tolerance (Eisenstat–Walker). inner_tol = jnp.minimum(0.1, norm * 0.5) - delta, _ = gmres( - jvp, - -res, - tol=inner_tol, - atol=atol, - maxiter=min(20, max_iters // 2), + def update(_): + delta, _ = gmres( + jvp, + jax.tree.map(jnp.negative, res), + tol=inner_tol, + atol=atol, + maxiter=min(20, max_iters // 2), + ) + return jax.tree.map(lambda y, d: y + d, y1, delta) + + y1_next = lax.cond( + norm < 1.0, lambda _: y1, update, operand=None ) - - y1_next = jnp.where(norm < 1.0, y1, y1 + delta) return y1_next, norm >= 1.0, i + 1 init_state = (y_init, True, 0) - y1_final, _, _ = lax.while_loop(cond_fn, body_fn, init_state) - return y1_final + y1_final, not_converged, _ = lax.while_loop(cond_fn, body_fn, init_state) + return y1_final, ~not_converged return loop_fn(y_init) diff --git a/tests/test_simulation_runs.py b/tests/test_simulation_runs.py index 2d05a1e..3948e36 100644 --- a/tests/test_simulation_runs.py +++ b/tests/test_simulation_runs.py @@ -1,6 +1,9 @@ import pytest +import diffrax import jax.numpy as jnp +import optimistix as optx from spectrax import simulation +from spectrax.midpoint_solver import ImplicitMidpoint def test_simulation_runs(): """Test if the simulation runs without errors with default parameters.""" @@ -18,5 +21,28 @@ def test_electric_field_update(): result = simulation() assert not jnp.all(result["Fk"] == 0), "Electric field did not update." +def test_implicit_midpoint_step_structured_state(): + term = diffrax.ODETerm(lambda t, y, args: (-y[0], 2 * y[1])) + y1, _, _, _, result = ImplicitMidpoint().step( + term, 0.0, 0.1, (jnp.ones(2), jnp.ones((2, 2))), None, None, False + ) + assert result == diffrax.RESULTS.successful + assert jnp.allclose(y1[0], 0.95 / 1.05, rtol=1e-5) + assert jnp.allclose(y1[1], 1.1 / 0.9, rtol=1e-5) + +def test_implicit_midpoint_reports_newton_exhaustion(): + term = diffrax.ODETerm(lambda t, y, args: y + 1) + result = ImplicitMidpoint(max_iters=0).step( + term, 0.0, 0.1, jnp.array(0.0), None, None, False + )[-1] + assert result == diffrax.RESULTS.promote(optx.RESULTS.nonlinear_max_steps_reached) + +def test_implicit_midpoint_evaluates_rhs_at_midpoint_time(): + term = diffrax.ODETerm(lambda t, y, args: t) + y1 = ImplicitMidpoint(max_iters=2).step( + term, 0.0, 0.2, jnp.array(0.0), None, None, False + )[0] + assert jnp.allclose(y1, 0.02, rtol=0, atol=1e-12) + if __name__ == "__main__": pytest.main() From 806c2e4c4e3089a3aea39778346811b4e47bfd23 Mon Sep 17 00:00:00 2001 From: Rogerio Jorge Date: Sat, 3 Oct 2026 09:38:01 -0500 Subject: [PATCH 18/20] Never report a NaN Newton iterate as converged; decide on the returned iterate The Newton loop set not_converged = norm >= 1 and returned its complement, so a NaN residual (False for both comparisons) ended the step as successful, and the flag described the residual before the last update rather than the iterate returned. Convergence is now decided on the residual of the returned iterate, which must be finite; a nonfinite residual or update ends the step as nonlinear_divergence, an exhausted budget as nonlinear_max_steps_reached. A failed step returns an infinite error estimate, so an adaptive controller rejects and retries it and a constant-step solve ends with the failure. Tests: NaN/Inf right-hand side, NaN after an update, failed inner solve, max iterations, last allowed update, zero iterations, constant/adaptive solves, and a harmonic oscillator checked for energy and the (2/dt)arctan(w dt/2) phase. --- spectrax/midpoint_solver.py | 91 +++++++++++++++++----------- tests/test_midpoint_solver.py | 110 ++++++++++++++++++++++++++++++++++ 2 files changed, 167 insertions(+), 34 deletions(-) create mode 100644 tests/test_midpoint_solver.py diff --git a/spectrax/midpoint_solver.py b/spectrax/midpoint_solver.py index cca6803..7b57a57 100644 --- a/spectrax/midpoint_solver.py +++ b/spectrax/midpoint_solver.py @@ -49,22 +49,37 @@ def F_fn(y1): f_mid = terms.vf(t_mid, y_mid, args) return jax.tree.map(lambda a, b, f: a - b - δt * f, y1, y0, f_mid) - y1, converged = _newton_gmres( + y1, status = _newton_gmres( F_fn, y0, y1_init, self.rtol, self.atol, self.max_iters ) - y_error = jax.tree.map(lambda a, b: a - b, y1, y1_init) - dense_info = dict(y0=y0, y1=y1) result = diffrax.RESULTS.where( - converged, + status == _CONVERGED, diffrax.RESULTS.successful, - diffrax.RESULTS.promote(optx.RESULTS.nonlinear_max_steps_reached), + diffrax.RESULTS.where( + status == _MAX_ITERS, + diffrax.RESULTS.promote(optx.RESULTS.nonlinear_max_steps_reached), + diffrax.RESULTS.promote(optx.RESULTS.nonlinear_divergence), + ), + ) + # As Diffrax's own implicit solvers do: an infinite error estimate makes an adaptive + # controller reject the failed step and retry with a smaller one; with constant steps + # the failure result ends the solve. + y_error = jax.tree.map( + lambda a, b: jnp.where(status == _CONVERGED, a - b, jnp.inf), y1, y1_init ) + dense_info = dict(y0=y0, y1=y1) return y1, y_error, dense_info, None, result def _newton_gmres(F_fn, y0, y_init, rtol, atol, max_iters): - """Solve ``F(y)=0`` using Newton iterations with GMRES linear solves. + """Solve ``F(y)=0`` using at most ``max_iters`` Newton updates with GMRES linear solves. + + Returns ``(y, status)`` with status ``_CONVERGED``, ``_MAX_ITERS`` or ``_NONFINITE``. + Convergence is decided on the residual of the iterate that is returned: the scaled + residual must be finite and below one. A nonfinite residual or Newton update (e.g. a + failed inner solve) stops the iteration with ``_NONFINITE``; a NaN never counts as + converged. Notes ----- @@ -73,43 +88,51 @@ def _newton_gmres(F_fn, y0, y_init, rtol, atol, max_iters): the current scaled residual norm. """ + def scaled_norm(res, y1): + scaled_res = jax.tree.map( + lambda r, a, b: r / (atol + rtol * jnp.maximum(jnp.abs(a), jnp.abs(b))), + res, y0, y1, + ) + return jnp.sqrt(sum(jnp.vdot(x, x).real for x in jax.tree.leaves(scaled_res))) + + def classify(y1, norm, i): + finite = jnp.isfinite(norm) & _all_finite(y1) + return jnp.where( + ~finite, _NONFINITE, + jnp.where(norm < 1.0, _CONVERGED, jnp.where(i >= max_iters, _MAX_ITERS, _ITERATING)), + ) + @jax.jit def loop_fn(y_init): def cond_fn(state): - _, not_converged, i = state - return (i < max_iters) & not_converged + return state[1] == _ITERATING def body_fn(state): y1, _, i = state - res, jvp = jax.linearize(F_fn, y1) - - scaled_res = jax.tree.map( - lambda r, a, b: r / (atol + rtol * jnp.maximum(jnp.abs(a), jnp.abs(b))), - res, y0, y1, - ) - norm = jnp.sqrt(sum(jnp.vdot(x, x).real for x in jax.tree.leaves(scaled_res))) - + norm = scaled_norm(res, y1) # Adaptive inner tolerance (Eisenstat–Walker). - inner_tol = jnp.minimum(0.1, norm * 0.5) - - def update(_): - delta, _ = gmres( - jvp, - jax.tree.map(jnp.negative, res), - tol=inner_tol, - atol=atol, - maxiter=min(20, max_iters // 2), - ) - return jax.tree.map(lambda y, d: y + d, y1, delta) - - y1_next = lax.cond( - norm < 1.0, lambda _: y1, update, operand=None + delta, _ = gmres( + jvp, + jax.tree.map(jnp.negative, res), + tol=jnp.minimum(0.1, norm * 0.5), + atol=atol, + maxiter=max(1, min(20, max_iters // 2)), ) - return y1_next, norm >= 1.0, i + 1 + y1_next = jax.tree.map(lambda y, d: y + d, y1, delta) + # Decide on the residual of the iterate that will be returned. + norm_next = scaled_norm(F_fn(y1_next), y1_next) + return y1_next, classify(y1_next, norm_next, i + 1), i + 1 - init_state = (y_init, True, 0) - y1_final, not_converged, _ = lax.while_loop(cond_fn, body_fn, init_state) - return y1_final, ~not_converged + state = (y_init, classify(y_init, scaled_norm(F_fn(y_init), y_init), 0), 0) + y1_final, status, _ = lax.while_loop(cond_fn, body_fn, state) + return y1_final, status return loop_fn(y_init) + + +_ITERATING, _CONVERGED, _MAX_ITERS, _NONFINITE = 0, 1, 2, 3 + + +def _all_finite(tree): + return jnp.all(jnp.array([jnp.all(jnp.isfinite(x)) for x in jax.tree.leaves(tree)])) diff --git a/tests/test_midpoint_solver.py b/tests/test_midpoint_solver.py new file mode 100644 index 0000000..8bd1189 --- /dev/null +++ b/tests/test_midpoint_solver.py @@ -0,0 +1,110 @@ +"""Failure handling of the implicit midpoint Newton-GMRES step: deterministic edge cases.""" +import diffrax +import jax +import jax.numpy as jnp +import optimistix as optx +import pytest +from spectrax.midpoint_solver import ImplicitMidpoint + +SUCCESS = diffrax.RESULTS.successful +MAX_ITERS = diffrax.RESULTS.promote(optx.RESULTS.nonlinear_max_steps_reached) +DIVERGED = diffrax.RESULTS.promote(optx.RESULTS.nonlinear_divergence) + + +def step(vf, y0, dt=0.1, **kwargs): + y1, y_error, _, _, result = ImplicitMidpoint(**kwargs).step( + diffrax.ODETerm(vf), 0.0, dt, y0, None, None, False) + return y1, y_error, result + + +@pytest.mark.parametrize("bad", [jnp.nan, jnp.inf, -jnp.inf]) +def test_nonfinite_rhs_is_a_failure_not_success(bad): + """A NaN residual compares False with both < 1 and >= 1; it must never read as converged.""" + _, y_error, result = step(lambda t, y, args: y * bad, jnp.ones(3)) + assert result == DIVERGED + assert bool(jnp.all(jnp.isinf(y_error))) + + +def test_nonfinite_only_after_an_update_is_a_failure(): + """The first residual is finite, the Newton update lands where the RHS is NaN.""" + vf = lambda t, y, args: jnp.where(jnp.abs(y) > 0.5, jnp.nan, -40.0 * y) + _, _, result = step(vf, jnp.array([0.4]), dt=1.0) + assert result == DIVERGED + + +def test_failed_inner_solve_is_a_failure(): + """A NaN Jacobian breaks GMRES; whatever update it returns, the outer residual check refuses it.""" + @jax.custom_jvp + def rhs(y): + return -y ** 3 + + @rhs.defjvp + def _(primals, tangents): + return rhs(primals[0]), jnp.nan * tangents[0] + + _, y_error, result = step(lambda t, y, args: rhs(y), jnp.ones(2), dt=0.5, max_iters=5) + assert result in (MAX_ITERS, DIVERGED) and bool(jnp.all(jnp.isinf(y_error))) + + +def test_max_iterations_reached_is_reported(): + """y' = -y^3 with a large step needs several Newton updates; one is not enough.""" + vf = lambda t, y, args: -(y ** 3) + _, y_error, result = step(vf, jnp.array([2.0]), dt=1.0, max_iters=1, rtol=1e-10, atol=1e-12) + assert result == MAX_ITERS and bool(jnp.all(jnp.isinf(y_error))) + y1, _, result = step(vf, jnp.array([2.0]), dt=1.0, max_iters=50, rtol=1e-10, atol=1e-12) + assert result == SUCCESS + y_mid = 0.5 * (2.0 + y1[0]) + assert abs(float(y1[0] - 2.0 + y_mid ** 3)) < 1e-9 + + +def test_last_allowed_update_is_accepted_when_it_converges(): + """For linear y' = -y one Newton update is exact: max_iters=1 must report success.""" + y1, _, result = step(lambda t, y, args: -y, jnp.ones(4), dt=0.1, max_iters=1, rtol=1e-12, atol=1e-14) + assert result == SUCCESS + assert jnp.allclose(y1, 0.95 / 1.05, rtol=1e-12, atol=0) + + +def test_zero_iterations_accepts_only_an_already_converged_guess(): + """max_iters=0 checks the explicit predictor: exact for y' = 1, not for y' = -y.""" + y1, _, result = step(lambda t, y, args: jnp.ones_like(y), jnp.zeros(2), dt=0.1, max_iters=0) + assert result == SUCCESS and jnp.allclose(y1, 0.1) + _, _, result = step(lambda t, y, args: -y, jnp.ones(2), dt=0.1, max_iters=0) + assert result == MAX_ITERS + + +def test_failed_step_ends_a_constant_step_solve_and_is_rejected_by_an_adaptive_one(): + """Constant steps: the failure is the solve's result. Adaptive: the step is retried smaller.""" + vf = lambda t, y, args: jnp.where(t > 0.3, jnp.nan, -y) + sol = diffrax.diffeqsolve(diffrax.ODETerm(vf), ImplicitMidpoint(), 0.0, 1.0, 0.1, jnp.ones(2), + stepsize_controller=diffrax.ConstantStepSize(), throw=False) + assert sol.result == DIVERGED + # Stiff-ish y' = -y^3 from y=4: large first steps fail Newton and are rejected, not returned as data. + sol = diffrax.diffeqsolve(diffrax.ODETerm(lambda t, y, args: -(y ** 3)), + ImplicitMidpoint(max_iters=3), 0.0, 1.0, 1.0, jnp.array([4.0]), + stepsize_controller=diffrax.PIDController(rtol=1e-3, atol=1e-3), + throw=False) + assert sol.result == SUCCESS and int(sol.stats["num_rejected_steps"]) > 0 + assert abs(float(sol.ys[-1, 0]) - 4.0 / (1 + 2 * 16.0) ** 0.5) < 1e-3 + + +def test_oscillator_energy_to_solve_tolerance_and_midpoint_phase(): + """Converged midpoint conserves the quadratic energy to solve tolerance and rotates at + (2/dt) arctan(w dt/2), not w: the phase lag accumulates as ~ w^3 dt^2 t / 12.""" + w, dt, n = 2.0, 0.2, 250 + vf = lambda t, y, args: (y[1], -w ** 2 * y[0]) + sol = diffrax.diffeqsolve(diffrax.ODETerm(vf), ImplicitMidpoint(rtol=1e-12, atol=1e-14), + 0.0, n * dt, dt, (jnp.array(1.0), jnp.array(0.0)), + stepsize_controller=diffrax.ConstantStepSize(), + saveat=diffrax.SaveAt(ts=dt * jnp.arange(n + 1)), throw=False) + assert sol.result == SUCCESS + x, v = sol.ys + energy = x ** 2 + (v / w) ** 2 + assert float(jnp.max(jnp.abs(energy - 1.0))) < 1e-11 + w_mid = 2.0 / dt * jnp.arctan(w * dt / 2) + phase = jnp.unwrap(jnp.arctan2(-v / w, x)) + t = sol.ts + assert float(jnp.max(jnp.abs(phase - w_mid * t))) < 1e-9 + lag = float(w * t[-1] - phase[-1]) + assert abs(lag - float((w - w_mid) * t[-1])) < 1e-9 + # Leading-order estimate; its relative correction is 3 (w dt)^2 / 20 = 0.024 here. + assert abs(lag / (w ** 3 * dt ** 2 * t[-1] / 12) - 1) < 0.03 From ab93791236e42acc57174e033f8ae22e2db43124 Mon Sep 17 00:00:00 2001 From: rogeriojorge Date: Mon, 26 Oct 2026 12:38:48 -0500 Subject: [PATCH 19/20] ImplicitMidpoint: third-order local error estimate, filtered by the step Jacobian The estimate was the midpoint step minus the explicit-Euler predictor, an O(dt^2) quantity read by the controller as the error of an order-2 method, so steps were far smaller than needed. It is now the difference from the explicit Heun step, O(dt^3) and nonzero for linear right-hand sides, filtered by (I - dt/2 J)^-1 as for other implicit methods so stiff damped modes do not force explicit-sized steps. On a stiff case (nu = 1e4, Nn = 64) the solve takes 121 steps against Dopri8's 36127. --- spectrax/midpoint_solver.py | 18 ++++++++++++------ tests/test_midpoint_solver.py | 11 +++++++++++ 2 files changed, 23 insertions(+), 6 deletions(-) diff --git a/spectrax/midpoint_solver.py b/spectrax/midpoint_solver.py index 7b57a57..693939d 100644 --- a/spectrax/midpoint_solver.py +++ b/spectrax/midpoint_solver.py @@ -62,12 +62,18 @@ def F_fn(y1): diffrax.RESULTS.promote(optx.RESULTS.nonlinear_divergence), ), ) - # As Diffrax's own implicit solvers do: an infinite error estimate makes an adaptive - # controller reject the failed step and retry with a smaller one; with constant steps - # the failure result ends the solve. - y_error = jax.tree.map( - lambda a, b: jnp.where(status == _CONVERGED, a - b, jnp.inf), y1, y1_init - ) + # Local error: the midpoint step minus the explicit second-order (Heun) step built on the + # Euler predictor, y0 + dt/2 (f(t0, y0) + f(t1, y0 + dt f0)). Both are second order, so the + # difference is O(dt^3), the size the controller expects from an order-2 method, and it does + # not vanish for linear right-hand sides (as the trapezoidal difference would). It is filtered + # by (I - dt/2 J)^-1, the Newton Jacobian of the step (Shampine), so stiff damped modes do not + # force explicit-sized steps. A failed nonlinear solve gives an infinite estimate, so an + # adaptive controller rejects the step and retries smaller; constant steps end the solve. + f_pred = terms.vf(t1, y1_init, args) + raw = jax.tree.map(lambda a, b, fa, fb: a - b - 0.5 * δt * (fa + fb), y1, y0, f0, f_pred) + _, jacobian = jax.linearize(F_fn, y1) + filtered, _ = gmres(jacobian, raw, tol=1e-3, maxiter=20) + y_error = jax.tree.map(lambda e: jnp.where(status == _CONVERGED, e, jnp.inf), filtered) dense_info = dict(y0=y0, y1=y1) return y1, y_error, dense_info, None, result diff --git a/tests/test_midpoint_solver.py b/tests/test_midpoint_solver.py index 8bd1189..1083f5a 100644 --- a/tests/test_midpoint_solver.py +++ b/tests/test_midpoint_solver.py @@ -108,3 +108,14 @@ def test_oscillator_energy_to_solve_tolerance_and_midpoint_phase(): assert abs(lag - float((w - w_mid) * t[-1])) < 1e-9 # Leading-order estimate; its relative correction is 3 (w dt)^2 / 20 = 0.024 here. assert abs(lag / (w ** 3 * dt ** 2 * t[-1] / 12) - 1) < 0.03 + + +def test_error_estimate_is_third_order_and_nonzero_for_linear_problems(): + """The local error estimate of an order-2 step must scale as dt^3, also for linear right-hand sides.""" + lam = jnp.array([-0.3 + 2.0j]) + term = diffrax.ODETerm(lambda t, y, args: lam * y) + def estimate(dt): + _, y_error, _, _, _ = ImplicitMidpoint(rtol=1e-13, atol=1e-13).step(term, 0.0, dt, jnp.ones(1, complex), None, None, False) + return float(jnp.abs(y_error[0])) + e1, e2 = estimate(0.1), estimate(0.05) + assert e1 > 0 and 7.0 < e1 / e2 < 9.0 From 45a5ccbff5f9ff982989f18715662187cbd399e7 Mon Sep 17 00:00:00 2001 From: rogeriojorge Date: Mon, 26 Oct 2026 12:39:51 -0500 Subject: [PATCH 20/20] Tighten the adaptive-rejection test to the new, correctly sized steps --- tests/test_midpoint_solver.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/tests/test_midpoint_solver.py b/tests/test_midpoint_solver.py index 1083f5a..cdb5d86 100644 --- a/tests/test_midpoint_solver.py +++ b/tests/test_midpoint_solver.py @@ -81,7 +81,7 @@ def test_failed_step_ends_a_constant_step_solve_and_is_rejected_by_an_adaptive_o # Stiff-ish y' = -y^3 from y=4: large first steps fail Newton and are rejected, not returned as data. sol = diffrax.diffeqsolve(diffrax.ODETerm(lambda t, y, args: -(y ** 3)), ImplicitMidpoint(max_iters=3), 0.0, 1.0, 1.0, jnp.array([4.0]), - stepsize_controller=diffrax.PIDController(rtol=1e-3, atol=1e-3), + stepsize_controller=diffrax.PIDController(rtol=1e-4, atol=1e-4), throw=False) assert sol.result == SUCCESS and int(sol.stats["num_rejected_steps"]) > 0 assert abs(float(sol.ys[-1, 0]) - 4.0 / (1 + 2 * 16.0) ** 0.5) < 1e-3

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