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': 0.004973414000005505, 'tracker': 3.391999999280415e-05, 'compilation': 4.043484868000007}, 'jit_count': {'make_stepper': 10, 'simulation': 0}, 'solver_start': '2025-12-10 07:35:52.695938+00:00', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.005001', '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': 0.0009715500000027077, 'tracker': 3.177000000675889e-05, 'compilation': 0.6150535529999956}, 'jit_count': {'make_stepper': 1, 'simulation': 0}, 'solver_start': '2025-12-10 07:35:53.316411+00:00', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.000997', 't_final': np.float64(1.0)}, 'package_version': 'unknown', 'solver': {'class': 'ExplicitSolver', 'pde_class': 'DiffusionPDE', 'dt_statistics': {'min': 0.001, 'max': 0.16377518593819976, 'mean': 0.08333333333333333, 'std': 0.05095272967314988, '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': 0.8390226650000017, 'tracker': 4.411000000459353e-05, 'compilation': 0.6160930239999942}, 'jit_count': {'make_stepper': 1, 'simulation': 0}, 'solver_start': '2025-12-10 07:35:53.933960+00:00', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.839105', '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': 0.8213862320000089, 'tracker': 4.3739999995295875e-05, 'compilation': 0.6140412130000072}, 'jit_count': {'make_stepper': 1, 'simulation': 0}, 'solver_start': '2025-12-10 07:35:55.387662+00:00', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.821499', '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': 0.010337458000009292, 'tracker': 3.3549999983506495e-05, 'compilation': 0.6123047009999993}, 'jit_count': {'make_stepper': 1, 'simulation': 0}, 'solver_start': '2025-12-10 07:35:56.822024+00:00', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.010366', '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': 0.0014126310000079911, 'tracker': 3.3059999992701705e-05, 'compilation': 0.6131197520000029}, 'jit_count': {'make_stepper': 1, 'simulation': 0}, 'solver_start': '2025-12-10 07:35:57.445829+00:00', 'successful': True, 'stop_reason': 'Reached final time', 'solver_duration': '0:00:00.001441', 't_final': np.float64(1.0)}, 'package_version': 'unknown', 'solver': {'class': 'ScipySolver', 'pde_class': 'DiffusionPDE', 'dt': 0.001, 'steps': 61, '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))