Partially implicit Runge-Kutta methods for wave-like equations

TL;DR

This paper introduces a new class of partially implicit Runge-Kutta methods tailored for wave-like hyperbolic systems, offering improved stability and accuracy over explicit methods without increased computational cost.

Contribution

The paper develops a novel class of partially implicit RK methods for wave-like equations, simplifying derivation and implementation compared to traditional IMEX methods, especially at higher orders.

Findings

Better stability allowing larger time steps

Smaller discretization errors compared to explicit methods

No operator inversion required, reducing computational cost

Abstract

In this work we present a new class of Runge-Kutta (RK) methods for solving systems of hyperbolic equations with a particular structure, generalization of a wave-equation. The new methods are {\it partially implicit} in the sense that a proper subset of the equations of the system contains some terms which are treated implicitly. These methods can be viewed as a particular case of the implicit-explicit (IMEX) RK methods for systems of equations with wave-like structure. For these systems, the optimal methods with the new structure are easier to derive than the IMEX ones, specially when aiming at higher-order (up to fourth-order in this work). The methods are constructed considering the classical strong-stability-preserving optimal explicit RK methods for the purely explicit part. The resulting partially implicit RK methods do not require any inversion of operators and hence their computational cost per iteration is similar to those of explicit RK methods. We analyse the stability and convergence properties and show their practical applicability in several numerical examples. Our results show that, compared with explicit RK methods, the new methods have better stability properties (larger steps are allowed) and in general show smaller discretization error.