A priori subcell limiting based on compact nonuniform nonlinear weighted schemes of high-order CPR method for hyperbolic conservation laws

TL;DR

This paper introduces a novel a priori subcell limiting technique for high-order CPR methods using compact nonuniform nonlinear weighted schemes, improving shock capturing and conservation in hyperbolic conservation laws.

Contribution

It develops a new shock capturing approach combining CNNW schemes with CPR, enabling adaptive limiting based on shock detection for high-order accuracy and robustness.

Findings

Enhanced shock capturing robustness demonstrated in numerical tests.

Improved conservation properties and resolution balance achieved.

Adaptive limiting effectively detects and treats shock regions.

Abstract

This paper develops a shock capturing approach for high-order correction procedure via reconstruction (CPR) method with Legendre-Gauss solution points. Shock regions are treated by novel compact nonuniform nonlinear weighted (CNNW) schemes, which have the same solution points as the CPR method. CNNW schemes are constructed by discretizing flux derivatives based on Riemann fluxes at flux points in one cell and using nonuniform nonlinear weighted (NNW) interpolations to obtain the left and right values at flux points. Then, a priori subcell p-adaptive CNNW limiting of the CPR method is proposed for hyperbolic conservation laws. Firstly, a troubled cell indicator is used to detect shock regions and to quantify solution smoothness. Secondly, according to the magnitude of the indicator, CNNW schemes with varying accuracy orders are chosen adaptively for the troubled cells. The spectral property and discrete conservation laws are mathematically analyzed. Various numerical experiments show that the CPR method with subcell CNNW limiting has superiority in satisfying discrete conservation laws and in good balance between resolution and shock capturing robustness.