# Energy conservative SBP discretizations of the acoustic wave equation in covariant form on staggered curvilinear grids

## TL;DR

This paper introduces an energy conserving numerical method for the acoustic wave equation on curvilinear grids, utilizing covariant basis decomposition and high-order SBP operators to enhance accuracy and stability.

## Contribution

The paper presents a novel covariant basis decomposition approach combined with SBP operators for energy conserving discretization on curvilinear grids, improving rotational invariance and stability.

## Key findings

- Outperforms Cartesian basis on rotated grids
- Provides a conditionally stable discretization method
- Offers bounds for stability evaluation

## Abstract

We develop a numerical method for solving the acoustic wave equation in covariant form on staggered curvilinear grids in an energy conserving manner. The use of a covariant basis decomposition leads to a rotationally invariant scheme that outperforms a Cartesian basis decomposition on rotated grids. The discretization is based on high order Summation-By-Parts (SBP) operators and preserves both symmetry and positive definiteness of the contravariant metric tensor. To improve accuracy and decrease computational cost, we also derive a modified discretization of the metric tensor that leads to a conditionally stable discretization. Bounds are derived that yield a point-wise condition that can be evaluated to check for stability of the modified discretization. This condition shows that the interpolation operators should be constructed such that their norm is close to one.