Stability analysis of the numerical Method of characteristics applied to a class of energy-preserving systems. Part I: Periodic boundary conditions

TL;DR

This paper investigates the numerical stability of the Method of Characteristics (MoC) with different solvers when applied to energy-preserving hyperbolic PDEs with periodic boundary conditions, revealing unexpected instability in the Leap-frog method.

Contribution

It provides a detailed stability analysis of Euler, modified Euler, and Leap-frog solvers within the MoC framework for non-dissipative PDEs, highlighting their stability properties.

Findings

Euler and modified Euler show mild, unconditional instability.

Leap-frog exhibits the strongest instability among tested solvers.

Runge-Kutta's stability within MoC is also discussed.

Abstract

We study numerical (in)stability of the Method of characteristics (MoC) applied to a system of non-dissipative hyperbolic partial differential equations (PDEs) with periodic boundary conditions. We consider three different solvers along the characteristics: simple Euler (SE), modified Euler (ME), and Leap-frog (LF). The two former solvers are well known to exhibit a mild, but unconditional, numerical instability for non-dissipative ordinary differential equations (ODEs). They are found to have a similar (or stronger, for the MoC-ME) instability when applied to non-dissipative PDEs. On the other hand, the LF solver is known to be stable when applied to non-dissipative ODEs. However, when applied to non-dissipative PDEs within the MoC framework, it was found to have by far the strongest instability among all three solvers. We also comment on the use of the fourth-order Runge--Kutta solver within the MoC framework.