High Order Accurate Hermite Schemes on Curvilinear Grids with Compatibility Boundary Conditions

TL;DR

This paper introduces high order Hermite schemes for wave equations on curvilinear grids, employing compatibility boundary conditions to enhance accuracy and stability, with demonstrated effectiveness through numerical examples.

Contribution

It develops new Hermite schemes with compatibility boundary conditions for wave equations on curvilinear domains, improving accuracy and stability over traditional methods.

Findings

Achieves high order accuracy in space and time.

Demonstrates stability and convergence in 2D numerical tests.

Provides a framework for boundary treatment using compatibility conditions.

Abstract

High order accurate Hermite methods for the wave equation on curvilinear domains are presented. Boundaries are treated using centered compatibility conditions rather than more standard one-sided approximations. Both first-order-in-time (FOT) and second-order-in-time (SOT) Hermite schemes are developed. Hermite methods use the solution and multiple derivatives as unknowns and achieve space-time orders of accuracy $2m-1$ (FOT) and $2m$ (SOT) for methods using $(m+1)^d$ degree of freedom per node in $d$ dimensions. The compatibility boundary conditions (CBCs) are based on taking time derivatives of the boundary conditions and using the governing equations to replace the time derivatives with spatial derivatives. These resulting constraint equations augment the Hermite scheme on the boundary. The solvability of the equations resulting from the compatibility conditions are analyzed. Numerical examples demonstrate the accuracy and stability of the new schemes in two dimensions.