2.4.9 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': 0.004881362000006106, 'tracker': 2.6800000000548607e-05, 'compilation': 4.007911159999992}, 'jit_count': {'make_stepper': 10, 'simulation': 0}, 'solver_start': '2025-09-30 15:55:20.062583', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.004904', 't_final': 1.0}, 'package_version': 'unknown', 'solver': {'class': 'ExplicitSolver', 'pde_class': 'DiffusionPDE', 'scheme': 'euler', 'backend': 'numba', 'dt': 0.001, 'dt_adaptive': False, 'steps': 1000, 'post_step_data': None, 'stochastic': False}}

Diagnostic information for explicit, adaptive Runge-Kutta solver:
{'controller': {'mpi_run': False, 't_start': 0, 't_end': 1.0, 'profiler': {'solver': 0.0009397510000042075, 'tracker': 2.7209999998945023e-05, 'compilation': 0.6125792349999983}, 'jit_count': {'make_stepper': 1, 'simulation': 0}, 'solver_start': '2025-09-30 15:55:20.680429', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.000962', 't_final': np.float64(1.0)}, 'package_version': 'unknown', 'solver': {'class': 'ExplicitSolver', 'pde_class': 'DiffusionPDE', 'dt_statistics': {'min': 0.001, 'max': 0.17316814567512445, 'mean': 0.08333333333333334, 'std': 0.052599296030970966, 'count': 12.0}, 'scheme': 'runge-kutta', 'backend': 'numba', 'dt': 0.001, 'dt_adaptive': True, 'steps': 12, 'stochastic': False, 'post_step_data': None}}

Diagnostic information for implicit solver:
{'controller': {'mpi_run': False, 't_start': 0, 't_end': 1.0, 'profiler': {'solver': 0.858154380000002, 'tracker': 4.090999999561973e-05, 'compilation': 0.6119605050000132}, 'jit_count': {'make_stepper': 1, 'simulation': 0}, 'solver_start': '2025-09-30 15:55:21.293789', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.858297', 't_final': 1.0}, 'package_version': 'unknown', 'solver': {'class': 'ImplicitSolver', 'pde_class': 'DiffusionPDE', 'function_evaluations': 0, 'scheme': 'implicit-euler', 'stochastic': False, 'backend': 'numba', 'dt': 0.001, 'dt_adaptive': False, 'steps': 1000, 'post_step_data': None}}

Diagnostic information for Crank-Nicolson solver:
{'controller': {'mpi_run': False, 't_start': 0, 't_end': 1.0, 'profiler': {'solver': 0.8247405780000037, 'tracker': 4.199999999343618e-05, 'compilation': 0.6147173760000015}, 'jit_count': {'make_stepper': 1, 'simulation': 0}, 'solver_start': '2025-09-30 15:55:22.767310', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.824914', 't_final': 1.0}, 'package_version': 'unknown', 'solver': {'class': 'CrankNicolsonSolver', 'pde_class': 'DiffusionPDE', 'function_evaluations': 0, 'scheme': 'implicit-euler', 'stochastic': False, 'backend': 'numba', 'dt': 0.001, 'dt_adaptive': False, 'steps': 1000, 'post_step_data': None}}

Diagnostic information for Adam-Bashforth solver:
{'controller': {'mpi_run': False, 't_start': 0, 't_end': 1.0, 'profiler': {'solver': 0.01031574400001034, 'tracker': 2.8849999992530684e-05, 'compilation': 0.614233286000001}, 'jit_count': {'make_stepper': 1, 'simulation': 0}, 'solver_start': '2025-09-30 15:55:24.207001', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.010339', 't_final': 1.0}, 'package_version': 'unknown', 'solver': {'class': 'AdamsBashforthSolver', 'pde_class': 'DiffusionPDE', 'backend': 'numba', 'dt': 0.001, 'dt_adaptive': False, 'steps': 1000, 'post_step_data': None, 'stochastic': False}}

Diagnostic information for scipy solver:
{'controller': {'mpi_run': False, 't_start': 0, 't_end': 1.0, 'profiler': {'solver': 0.0013335009999906333, 'tracker': 2.8720000003090718e-05, 'compilation': 0.6122638950000123}, 'jit_count': {'make_stepper': 1, 'simulation': 0}, 'solver_start': '2025-09-30 15:55:24.829887', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.001358', 't_final': np.float64(1.0)}, 'package_version': 'unknown', 'solver': {'class': 'ScipySolver', 'pde_class': 'DiffusionPDE', 'dt': 0.001, 'steps': 55, 'stochastic': False, 'backend': '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.ExplicitSolver(eq), "explicit Euler"),
    run_solver(
        pde.ExplicitSolver(eq, scheme="runge-kutta", 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))