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

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.