On the Stability of Explicit Finite Difference Methods for Advection-Diffusion Equations

TL;DR

This paper analyzes the stability conditions of explicit finite difference methods for advection-diffusion equations, highlighting the impact of spatial and temporal discretization orders on stability, and extends the analysis to wave systems.

Contribution

It provides a comprehensive stability analysis for high-order spatial discretizations combined with explicit time integrators, including new results for wave systems.

Findings

Stable semi-discretizations lead to conditionally stable fully discretized methods.

High-order spatial discretization requires sufficiently high-order time integration for stability.

Numerical experiments confirm the theoretical stability predictions.

Abstract

In this paper we study the stability of explicit finite difference discretizations of linear advection-diffusion equations (ADE) with arbitrary order of accuracy in the context of method of lines. The analysis first focuses on the stability of the system of ordinary differential equations (ODE) that is obtained by discretizing the ADE in space and then extends to fully discretized methods where explicit Runge-Kutta methods are used for integrating the ODE system. In particular, it is proved that all stable semi-discretization of the ADE gives rise to a conditionally stable fully discretized method if the time-integrator is at least first-order accurate, whereas high-order spatial discretization of the advection equation cannot yield a stable method if the temporal order is too low. In the second half of this paper, we extend the analysis to a partially dissipative wave system and obtain the stability results for both semi-discretized and fully-discretized methods. Finally, the major theoretical predictions are verified numerically.