Variational Formulations for Explicit Runge-Kutta Methods

TL;DR

This paper demonstrates that explicit Runge-Kutta methods can be formulated as discontinuous Petrov-Galerkin methods in space and time, enabling new adaptive algorithms for PDEs.

Contribution

It establishes a variational framework for explicit Runge-Kutta methods, which was previously only known for implicit schemes.

Findings

Explicit Runge-Kutta methods can be expressed as discontinuous Petrov-Galerkin methods.

Trial and test spaces are constructed for the linear diffusion equation.

This approach facilitates the design of explicit adaptive algorithms.

Abstract

Variational space-time formulations for Partial Differential Equations have been of great interest in the last decades. While it is known that implicit time marching schemes have variational structure, the Galerkin formulation of explicit methods in time remains elusive. In this work, we prove that the explicit Runge-Kutta methods can be expressed as discontinuous Petrov-Galerkin methods both in space and time. We build trial and test spaces for the linear diffusion equation that lead to one, two, and general stage explicit Runge-Kutta methods. This approach enables us to design explicit time-domain (goal-oriented) adaptive algorithms