A conservative, skew-symmetric Finite Difference Scheme for the compressible Navier--Stokes Equations

TL;DR

This paper introduces a fully conservative, skew-symmetric finite difference scheme for the compressible Navier--Stokes equations that preserves kinetic energy and enhances stability without complex averaging, using point-wise operations.

Contribution

The scheme is fully conservative, skew-symmetric, and adaptable, avoiding special averaging and enabling high-order derivatives for improved stability and damping control.

Findings

Preserves kinetic energy by design.

Avoids central instability and numerical damping.

Compatible with high-order derivatives.

Abstract

We present a fully conservative, skew-symmetric finite difference scheme on transformed grids. The skew-symmetry preserves the kinetic energy by first principles, simultaneously avoiding a central instability mechanism and numerical damping. In contrast to other skew-symmetric schemes no special averaging procedures are needed. Instead, the scheme builds purely on point-wise operations and derivatives. Any explicit and central derivative can be used, permitting high order and great freedom to optimize the scheme otherwise. This also allows the simple adaption of existing finite difference schemes to improve their stability and damping properties.