Anti-diffusive, non-oscillatory central difference scheme (adNOC) suitable for highly nonlinear advection-dominated problems

TL;DR

This paper introduces an anti-diffusive correction to the Nessyahu-Tadmor central difference scheme, significantly reducing numerical dissipation in highly nonlinear advection problems without requiring eigenstructure knowledge.

Contribution

A novel correction method for non-oscillatory central difference schemes that enhances accuracy in nonlinear advection problems by reducing numerical diffusion.

Findings

The corrected scheme effectively resolves sharp discontinuities.

It maintains stability and can be TVD under certain conditions.

Performance is validated against analytical and published results.

Abstract

Explicit non-oscillatory central difference schemes become excessively diffusive when applied to highly nonlinear advection problems where small time steps are necessary to maintain stability. Here, we present a correction to reduce such numerical dissipation for this class of problems. The correction is obtained by selecting the appropriate finite difference approximations for calculating the slopes utilized to reconstruct the solution from the cell averages. The anti-diffusive central scheme does not require any knowledge of the eigenstructure and is fully central. The proposed correction is applied to the widely used Nessyahu-Tadmor scheme to demonstrate the utility of the correction. The stability of the corrected scheme is discussed and the condition for the scheme to become TVD (total variation diminishing) is presented. The corrected scheme is finally tested with a number of test cases and the results are compared with analytical solutions and published results showing the ability of the corrected scheme to effectively resolve sharp discontinuities.