Discontinuous Galerkin spectral element method with hybrid sub-element order reduction for shock-capturing with Navier–Stokes equations

The current study presents a high-order methodology for the simulation of three-dimensional compressible viscous flows with shocks in complex geometries. The method is developed based on the framework of a split-form discontinuous Galerkin spectral element method (DGSEM) with summation-by-parts (SBP) operators. The Bassi and Rebay (Bassi and Rebay, 1997) (BR1) scheme is employed for the discretization of the viscous terms. For shock capturing, a hybrid sub-element order reduction methodology is developed which is based on a mixed functional space that blends high-order polynomial basis functions with piecewise-constant functions supported on sub-element volumes. An amount of blending is determined based on a modified Ducros indicator which has excellent shock detecting capabilities in viscous turbulent flows. The performance of the methodology is demonstrated on the example of eight test cases, featuring 1D, 2D and 3D inviscid and viscous flows with and without shocks.