An optimized dispersion-relation-preserving combined compact difference scheme to solve advection equations

TL;DR

This paper introduces an improved sixth-order dispersion-relation-preserving combined compact difference scheme for advection equations, enhancing stability and accuracy, and couples it with a WENO scheme for better performance on benchmark problems.

Contribution

It presents a novel hybrid scheme combining an improved DRPCCD5 with WENO5, offering enhanced stability, accuracy, and computational efficiency for advection problems.

Findings

The hybrid scheme avoids oscillations near discontinuities.

It handles large gradients effectively.

It achieves similar accuracy with coarser meshes, reducing computational cost.

Abstract

In this study, we first present an improved version of the classical sixth-order combined compact difference (CCD6) scheme to enhance the convective stability of advection equations through an increased dispersion accuracy. This improved fifth-order dispersion-relation-preserving combined compact difference scheme (DRPCCD5) has been rigorously analyzed through the dispersion, phase speed anisotropy and stability analyses. We then couple the DRPCCD5 scheme with the previous fifth-order compact-reconstruction weighted essentially non-oscillatory (CRWENO5) scheme using a novel hybrid strategy based on the monotonicity-maintenance criteria. To verify the resulting "optimized" hybrid scheme (ODRPCCD5), several benchmark problems with available exact solution are investigated. The comparison to the previous fifth-order WENO (WENO5) scheme shows that the ODRPCCD5 avoids numerical oscillation around discontinuities, handles large gradients well, and is much faster at the same accuracy because a coarser mesh can be used.