Error analysis of Runge--Kutta discontinuous Galerkin methods for linear time-dependent partial differential equations

TL;DR

This paper provides comprehensive error estimates for fully discrete Runge--Kutta discontinuous Galerkin methods applied to various linear time-dependent PDEs, covering multiple spatial discretizations and demonstrating optimal error bounds.

Contribution

It offers a unified error analysis framework for Runge--Kutta DG schemes across different spatial discretizations and PDE types, including an alternative proof for high-order derivative equations.

Findings

Error estimates valid for explicit Runge--Kutta schemes of any order

Applicable to various DG spatial discretizations like upwind-biased, central, local, and ultra-weak

Provides optimal error bounds without energy inequalities for high-order derivatives

Abstract

In this paper, we present error estimates of fully discrete Runge--Kutta discontinuous Galerkin (DG) schemes for linear time-dependent partial differential equations. The analysis applies to explicit Runge--Kutta time discretizations of any order. For spatial discretization, a general discrete operator is considered, which covers various DG methods, such as the upwind-biased DG method, the central DG method, the local DG method and the ultra-weak DG method. We obtain error estimates for stable and consistent fully discrete schemes, if the solution is sufficiently smooth and a spatial operator with certain properties exists. Applications to schemes for hyperbolic conservation laws, the heat equation, the dispersive equation and the wave equation are discussed. In particular, we provide an alternative proof of optimal error estimates of local DG methods for equations with high order derivatives in one dimension, which does not rely on energy inequalities of auxiliary unknowns.