Variations on Hermite methods for wave propagation

TL;DR

This paper introduces three new variations of Hermite methods for wave propagation that avoid dual grid time evolution, demonstrating their stability, high order accuracy, and potential for coupling with DG methods.

Contribution

The paper presents novel Hermite method variations that simplify implementation and maintain high order accuracy without dual grid time evolution.

Findings

Methods exhibit stability and high order convergence.

Dispersion and dissipation properties are favorable.

Coupling with DG methods enhances geometric flexibility.

Abstract

Hermite methods, as introduced by Goodrich et al., combine Hermite interpolation and staggered (dual) grids to produce stable high order accurate schemes for the solution of hyperbolic PDEs. We introduce three variations of this Hermite method which do not involve time evolution on dual grids. Computational evidence is presented regarding stability, high order convergence, and dispersion/dissipation properties for each new method. Hermite methods may also be coupled to discontinuous Galerkin (DG) methods for additional geometric flexibility. An example illustrates the simplification of this coupling of this coupling for the Hermite methods.