Jacobian-free explicit multiderivative Runge-Kutta methods for hyperbolic conservation laws

TL;DR

This paper introduces Jacobian-free multiderivative Runge-Kutta methods for hyperbolic conservation laws, enhancing stability and efficiency by avoiding analytical derivatives and enabling practical higher-order time integration.

Contribution

It presents a novel Jacobian-free multiderivative Runge-Kutta scheme that improves stability and efficiency for hyperbolic conservation laws, overcoming previous analytical derivative challenges.

Findings

Higher-order schemes achieved without analytical derivatives

Stability and CFL conditions depend on the number of derivatives

Numerical tests confirm the method's effectiveness

Abstract

Based on the recent development of Jacobian-free Lax-Wendroff (LW) approaches for solving hyperbolic conservation laws [Zorio, Baeza and Mulet, Journal of Scientific Computing 71:246-273, 2017], [Carrillo and Par\'es, Journal of Scientific Computing 80:1832-1866, 2019], a novel collection of explicit Jacobian-free multistage multiderivative solvers for hyperbolic conservation laws is presented in this work. In contrast to Taylor time-integration methods, multiderivative RungeKutta (MDRK) techniques achieve higher-order of consistency not only through the excessive addition of higher temporal derivatives, but also through the addition of Runge-Kutta-type stages. This adds more flexibility to the time integration in such a way that more stable and more efficient schemes could be identified. The novel method permits the practical application of MDRK schemes. In their original form, they are difficult to utilize as higher-order flux derivatives have to be computed analytically. Here we overcome this by adopting a Jacobian-free approximation of those derivatives. In this paper, we analyze the novel method with respect to order of consistency and stability. We show that the linear CFL number varies significantly with the number of derivatives used. Results are verified numerically on several representative testcases.