Method overview

Built-in families

DESolve organizes time integrators by method-table modules. The main families in the repository are:

• methods_rk.py for explicit Runge-Kutta schemes
• methods_symplectic.py for explicit symplectic splitting methods
• methods_esdirk.py for singly diagonally implicit RK variants
• methods_ark.py for additive Runge-Kutta methods
• methods_glee.py and methods_glee_eimex.py for GLEE-related methods
• methods_mrk.py and methods_imex_mrk.py for multirate formulations
• methods_etrs.py and methods_ide.py for additional research-specific tables

How registration works

During DESolver.setup(), the solver calls _RegisterDefaultMethods(), which collects method dictionaries from each module and stores them in the internal method registry.

From a user perspective, that means two things:

• adding a new method is mostly a data-entry problem in the appropriate method table
• a method is not available until it is registered inside DESolver

Choosing a first method

Use this rough rule of thumb:

• start with RK4 for simple explicit baselines
• use the symplectic family for separable Hamiltonian systems when long-time phase-space structure matters
• move to ESDIRK or ARK variants when stiffness or splitting matters
• use IMEX and multirate methods when you want explicit/implicit separation or multiple time scales

Extending the library

When adding a method:

1. Put the coefficient data in the relevant desolve/methods_*.py file.
2. Keep the naming consistent with the existing method registry.
3. Register it in _RegisterDefaultMethods().
4. Add a small regression test or notebook reproduction.

Detailed IMEX-MRK notes

The IMEX multirate methods have their own dedicated documentation page because their implementation uses four coupled tableaux (AB, AF, AS, AT) and maps directly onto the 2022 Applied Mathematics Letters paper that introduced the one-implicit-stage extension.

See the IMEX-MRK implementation notes.

Symplectic notes

The new symplectic family uses the paper-style drift and kick coefficient tables a and b rather than a Runge-Kutta tableau. The dedicated page shows how to solve the same harmonic-oscillator problem with either RK4 or a symplectic composition method.

See the symplectic methods page.