Higher order discrete gradient method by the discontinuous Galerkin time-stepping method

TL;DR

This paper introduces a higher-order, structure-preserving numerical method for gradient systems by reformulating the system and applying a discontinuous Galerkin time-stepping approach, achieving improved accuracy.

Contribution

It develops a novel higher-order scheme that extends the discrete gradient method using discontinuous Galerkin techniques for gradient systems.

Findings

Order of accuracy is (k+1) generally.

Order of accuracy is (2k+1) at nodal times.

Numerical examples confirm the theoretical accuracy improvements.

Abstract

Many differential equations with physical backgrounds are described as gradient systems, which are evolution equations driven by the gradient of some functionals, and such problems have energy conservation or dissipation properties. For numerical computation of gradient systems, numerical schemes that inherit the energy structure of the equation play important roles, which are called structure-preserving. The discrete gradient method is one of the most classical framework of structure-preserving methods, which is at most second order accurate. In this paper, we develop a higher-order structure-preserving numerical method for gradient systems, which includes the discrete gradient method. We reformulate the gradient system as a coupled system and then apply the discontinuous Galerkin time-stepping method. Numerical examples suggests that the order of accuracy of our scheme is $(k+1)$ in general and $(2k+1)$ at nodal times, where $k$ is the degree of polynomials.