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': 0.003213143000010632, 'tracker': 5.136999999422187e-05, 'compilation': 6.116227676000008}, 'solver_start': '2026-10-16 13:23:33.862641+00:00', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.003267', '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': 0.0005955889999995634, 'tracker': 5.643999999449534e-05, 'compilation': 7.2662837550000035}, 'solver_start': '2026-10-16 13:23:41.138438+00:00', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.000644', 't_final': 1.0}, 'package_version': 'unknown', 'solver': {'class': 'RungeKuttaSolver', 'pde_class': 'DiffusionPDE', 'dt': 0.20236408647918536, 'steps': 12, 'dt_adaptive': True, 'stochastic': False, 'backend': {'name': 'numba', 'implementation': 'numba'}, 'dt_statistics': {'min': 0.001, 'max': 0.1719755546555488, 'mean': 0.08333333333333331, 'std': 0.05265612935068183, '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': 0.014164979999989669, 'tracker': 5.1492000011421624e-05, 'compilation': 4.260625953000002}, 'solver_start': '2026-10-16 13:23:45.409722+00:00', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.014210', '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': 0.019666666000006217, 'tracker': 5.168799999921703e-05, 'compilation': 6.88918108499999}, 'solver_start': '2026-10-16 13:23:52.321538+00:00', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.019714', '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': 3.7730858939999905, 'tracker': 6.069100000161143e-05, 'compilation': 1.9152614150000034}, 'solver_start': '2026-10-16 13:23:54.268055+00:00', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:03.773625', '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': 0.6230274810000083, 'tracker': 4.9849999982143345e-05, 'compilation': 0.0016289309999990564}, 'solver_start': '2026-10-16 13:23:58.046260+00:00', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.623165', '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))