# Numerical anisotropy study of a class of compact schemes

## TL;DR

This paper investigates the numerical anisotropy in compact difference schemes for hyperbolic PDEs, proposing methods to reduce anisotropy error and improve stability, resulting in more efficient multidimensional schemes with preserved accuracy.

## Contribution

It introduces a prefactorization and predictor-corrector approach to reduce anisotropy and enhance stability in compact schemes, making multidimensional schemes more efficient without sacrificing accuracy.

## Key findings

- Reduced anisotropy error for large wave numbers
- Significantly improved stability restrictions
- Multidimensional schemes become more efficient

## Abstract

We study the numerical anisotropy existent in compact difference schemes as applied to hyperbolic partial differential equations, and propose an approach to reduce this error and to improve the stability restrictions based on a previous analysis applied to explicit schemes. A prefactorization of compact schemes is applied to avoid the inversion of a large matrix when calculating the derivatives at the next time level, and a predictor-corrector time marching scheme is used to update the solution in time. A reduction of the isotropy error is attained for large wave numbers and, most notably, the stability restrictions associated with MacCormack time marching schemes are considerably improved. Compared to conventional compact schemes of similar order of accuracy, the multidimensional schemes employ larger stencils which would presumably demand more processing time, but we show that the new stability restrictions render the multidimensional schemes to be in fact more efficient, while maintaining the same dispersion and dissipation characteristics of the one dimensional schemes.