Leapfrog time-stepping for Hermite methods

TL;DR

This paper introduces Hermite-leapfrog methods combining leapfrog time-stepping with Hermite spatial discretization for wave systems, achieving high-order accuracy, stability, and efficient GPU implementation.

Contribution

The paper presents a novel Hermite-leapfrog scheme that staggers variables in time and space, extending Hermite methods to wave equations with proven stability and high-order convergence.

Findings

Method conserves variables in 1D

Achieves high-order accuracy

Demonstrates efficient GPU implementation

Abstract

We introduce Hermite-leapfrog methods for first order wave systems. The new Hermite-leapfrog methods pair leapfrog time-stepping with the Hermite methods of Goodrich and co-authors. The new schemes stagger field variables in both time and space and are high-order accurate. We provide a detailed description of the method and demonstrate that the method conserves variable quantities in one-space dimension. Higher dimensional versions of the method are constructed via a tensor product construction. Numerical evidence and rigorous analysis in one space dimension establish stability and high-order convergence. Experiments demonstrating efficient implementations on a graphics processing unit are also presented.