2.4.10 Solver comparison

This example shows how to set up solvers explicitly and how to extract diagnostic information.

Diagnostic information for explicit Euler solver:
{'controller': {'mpi_run': False, 't_start': 0, 't_end': 1.0, 'profiler': {'solver': 0.0030161580000083177, 'tracker': 4.450999998084626e-05, 'compilation': 5.728593246000003}, 'solver_start': '2026-07-10 11:05:49.221811+00:00', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.003053', 't_final': 1.0}, 'package_version': 'unknown', 'solver': {'class': 'EulerSolver', 'pde_class': 'DiffusionPDE', 'dt': 0.001, 'steps': 1000, 'dt_adaptive': False, 'stochastic': False, 'backend': {'name': 'numba', 'implementation': 'numba'}, 'post_step_data': None}}

Diagnostic information for explicit, adaptive Runge-Kutta solver:
{'controller': {'mpi_run': False, 't_start': 0, 't_end': 1.0, 'profiler': {'solver': 0.00046139999999184056, 'tracker': 3.765000001010321e-05, 'compilation': 6.788288657999999}, 'solver_start': '2026-07-10 11:05:56.018554+00:00', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.000493', 't_final': 1.0}, 'package_version': 'unknown', 'solver': {'class': 'RungeKuttaSolver', 'pde_class': 'DiffusionPDE', 'dt': 0.19682423106794425, 'steps': 12, 'dt_adaptive': True, 'stochastic': False, 'backend': {'name': 'numba', 'implementation': 'numba'}, 'dt_statistics': {'min': 0.001, 'max': 0.16456777293881025, 'mean': 0.08333333333333333, 'std': 0.05128293419195853, 'count': 12.0}, 'post_step_data': None}}

Diagnostic information for implicit solver:
{'controller': {'mpi_run': False, 't_start': 0, 't_end': 1.0, 'profiler': {'solver': 0.013426164999998491, 'tracker': 5.397000001039487e-05, 'compilation': 3.870954999999995}, 'solver_start': '2026-07-10 11:05:59.898735+00:00', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.013472', 't_final': 1.0}, 'package_version': 'unknown', 'solver': {'class': 'ImplicitSolver', 'pde_class': 'DiffusionPDE', 'dt': 0.001, 'steps': 1000, 'dt_adaptive': False, 'stochastic': False, 'backend': {'name': 'numba', 'implementation': 'numba'}, 'function_evaluations': 0, 'post_step_data': None}}

Diagnostic information for Crank-Nicolson solver:
{'controller': {'mpi_run': False, 't_start': 0, 't_end': 1.0, 'profiler': {'solver': 0.01840291300000274, 'tracker': 4.477999999608073e-05, 'compilation': 5.748427328000005}, 'solver_start': '2026-07-10 11:06:05.667401+00:00', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.018441', 't_final': 1.0}, 'package_version': 'unknown', 'solver': {'class': 'CrankNicolsonSolver', 'pde_class': 'DiffusionPDE', 'dt': 0.001, 'steps': 1000, 'dt_adaptive': False, 'stochastic': False, 'backend': {'name': 'numba', 'implementation': 'numba'}, 'function_evaluations': 0, 'post_step_data': None}}

Diagnostic information for Adam-Bashforth solver:
{'controller': {'mpi_run': False, 't_start': 0, 't_end': 1.0, 'profiler': {'solver': 3.5817983630000043, 'tracker': 4.5818999993230136e-05, 'compilation': 1.8146645910000103}, 'solver_start': '2026-07-10 11:06:07.510225+00:00', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:03.582152', 't_final': 1.0}, 'package_version': 'unknown', 'solver': {'class': 'AdamsBashforthSolver', 'pde_class': 'DiffusionPDE', 'dt': 0.001, 'steps': 1000, 'dt_adaptive': False, 'stochastic': False, 'backend': {'name': 'numba', 'implementation': 'numba'}, 'post_step_data': None}}

Diagnostic information for scipy solver:
{'controller': {'mpi_run': False, 't_start': 0, 't_end': 1.0, 'profiler': {'solver': 0.5990516800000023, 'tracker': 3.103999999609641e-05, 'compilation': 0.0009618899999992436}, 'solver_start': '2026-07-10 11:06:11.096022+00:00', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.599109', 't_final': np.float64(1.0)}, 'package_version': 'unknown', 'solver': {'class': 'ScipySolver', 'pde_class': 'DiffusionPDE', 'dt': 0.001, 'steps': 61, 'stochastic': False, 'backend': {'name': 'numba', 'implementation': 'numba'}}}

import pde

# initialize the grid, an initial condition, and the PDE
grid = pde.UnitGrid([32, 32])
field = pde.ScalarField.random_uniform(grid, -1, 1)
eq = pde.DiffusionPDE()


def run_solver(solver, label):
    """Helper function testing the solver."""
    controller = pde.Controller(solver, t_range=1, tracker=None)
    sol = controller.run(field, dt=1e-3)
    sol.label = label + " solver"
    print(f"Diagnostic information for {sol.label}:")
    print(controller.diagnostics)
    print()
    return sol


# try different solvers
solutions = [
    run_solver(pde.EulerSolver(eq), "explicit Euler"),
    run_solver(
        pde.RungeKuttaSolver(eq, adaptive=True), "explicit, adaptive Runge-Kutta"
    ),
    run_solver(pde.ImplicitSolver(eq), "implicit"),
    run_solver(pde.CrankNicolsonSolver(eq), "Crank-Nicolson"),
    run_solver(pde.AdamsBashforthSolver(eq), "Adam-Bashforth"),
    run_solver(pde.ScipySolver(eq), "scipy"),
]

# plot both fields and give the deviation as the title
deviations = [(solutions[0] - sol).fluctuations for sol in solutions]
title = "Deviation: " + ", ".join(f"{deviation:.2g}" for deviation in deviations[1:])
pde.FieldCollection(solutions).plot(title=title, arrangement=(2, 3))