Numerical methods for differential equations
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Abstract
Abstract
Implicit methods are a natural approach for the integration of stiff differential equations, to avoid time-step restrictions faced by standard explicit integrators. Explicit stabilised integrators are an alternative to implicit methods, which can be particularly efficient in high-dimensional applications with diffusive terms. Towards the best of both worlds, we introduce a new explicit stabilised implementation of a class of diagonally implicit Runge-Kutta methods. This allows us to implement high-order singly-diagonally-implicit Runge-Kutta methods for advection-diffusion-reaction PDEs with a provable computational cost analogous to that of standard explicit stabilised methods. The main ingredient is to recast the implicit Runge-Kutta update as the steady state of a modified auxiliary system, which is then computed using a partitioned Runge-Kutta-Chebyshev method inspired by optimisation techniques.
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@article{Almuslimani2026Explicit,
title = {Explicit stabilized implementation of singly diagonally implicit Runge-Kutta methods},
author = {Ibrahim Almuslimani and Gilles Vilmart and Konstantinos Zygalakis},
journal = {arXiv (Cornell University)},
year = {2026},
url = {https://arxiv.org/abs/2607.07497}
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