Explicit stabilized implementation of singly diagonally implicit Runge-Kutta methods

Jul 16, 2026

Implicit Runge–Kutta methods provide an effective way to integrate stiff differential equations by avoiding restrictive time-step limits of standard explicit schemes. Explicit stabilized integrators offer an alternative, especially for high-dimensional advection–diffusion–reaction problems with diffusive terms. A new explicit stabilized implementation for a class of diagonally implicit Runge–Kutta methods is presented, enabling high-order singly-diagonally-implicit schemes for PDEs with a computational cost analogous to classical explicit stabilized methods. The method rewrites the implicit update as a steady state of a modified auxiliary system, solved via a partitioned Runge–Kutta–Chebyshev procedure inspired by optimization.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages25

Reading Time1 h 3 min

Introduction

Stiff ODEs from PDE discretization

Implicit Runge-Kutta and diagonally implicit structure

Explicit stabilized methods and Runge-Kutta-Chebyshev families

arXiv :2607 .07497v1 [math .NA] 8 Jul 2026 Explicit stabilized implementation of singly diagonally implicit Runge-Kutta methods Ibrahim Almuslimani∗, Gilles Vilmart†, and Konstantinos Zygalakis‡ July 9, 2026 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 highdimensional 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. Keywords: diagonally implicit Runge-Kutta methods, explicit stabilised methods, partitioned Runge-Kutta methods, stiff problems, advection-diffusion-reaction problems. AMS subject classification (2010): 65L20, 65M12, 65M20 1 Introduction In this paper, we consider systems of ordinary differential equations (ODEs) representing space discretisations of partial differential equations (PDEs) of the form y˙ = F (y), (1a) F (y) := FD (y) + FA (y) + FR (y) . (1b) where FD (y), FA (y), FR (y) 2 Rd represent diffusion, advection and reaction terms respectively. Typically (1) arises from the spatial discretisation of an advection-diffusion-reaction problem in N spatial dimensions, and assuming a spatial grid of size ∆x, the dimension of the ODE grows like d = O((∆x)−N ) . The numerical time integration of (1) is very challenging since this is a stiff ODE. Following [13], we call a differential equation stiff when a standard explicit numerical integrator such asa standard explicit Runge-Kutta method faces a severe time-step restriction. In particular, in the case of (1) the eigenvalues of the Jacobian of FD are typically distributed along the negative ∗ École Polytechnique Fédérale de Lausanne (EPFL), Swiss Plasma Center (SPC), CH-1015 Lausanne, Switzer[land. Ibrahim.Almuslimani@epfl.ch](land. Ibrahim.Almuslimani@epfl.ch) †Section de Mathématiques, Université de Genève, CP 64, 1211 Genève 4, [Switzerland. Gilles.Vilmart@unige.ch](Switzerland. Gilles.Vilmart@unige.ch) ‡School of Mathematics and the Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, EH9 3FD, [UK. K.Zygalakis@ed.ac.uk](UK. K.Zygalakis@ed.ac.uk) real axis in an interval that grows as [􀀀O(∆x−2) , 0] (for a symmetric diffusion operator), which implies that the time step ∆t that can be used by the explicit Euler method suffers from the CFL ∆t 􀀔 C∆x2 . Furthermore, the eigenvalues of the Jacobian of FA are typically distributed along the imaginary axis in an interval that grows as [􀀀iO(∆x−1) , iO(∆x−1)], while the eigenvalues of the Jacobian of the reaction term FR even though are usually not related to ∆x can have a ratio maxj jRλj j/ minj jRλj j that can be very large or vary over several orders of magnitude when modeling multiscale reaction terms. There are two different approaches to dealing with the issue of stiffness within the class of Runge-Kutta methods. The first relates to the design of implicit methods, with a prime example being the implicit Euler method. These methods have the advantage of achieving stability unconditionally with respect to the stepsize, which comes with the cost of solving a system of nonlinear equations. When the dimension of the corresponding differential equation is low, the nonlinear system can be solved eﬀiciently using Newton’s method [17] . As the dimension of the dif

FAQs

What problem motivates explicit stabilized methods in this paper?

The paper targets stiff ODE systems arising from spatial discretizations of advection–diffusion–reaction PDEs, where standard explicit integrators face severe time-step restrictions.

How does the proposed approach relate implicit updates to an auxiliary steady-state system?

It recasts the implicit Runge–Kutta update as the steady state of a modified auxiliary system, which is then computed numerically.

What numerical technique is used to compute the auxiliary system’s steady state?

A partitioned Runge–Kutta–Chebyshev method is employed, inspired by optimization techniques, to obtain an explicit stabilized implementation with provable cost.

Jul 16, 2026

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