An Efficient Hybrid Method for Solving Euler Equations

In this paper, a hybrid method suitable for solving the Euler equations using high order methods has been proposed. The method was implemented and validated with a seventh order WENO scheme in OpenFOAM^®. The hybrid method combines a simple MUSCL-type flux approach and a characteristic flux approach. In the MUSCL-type flux approach, the inviscid fluxes are computed using approximate Riemann solvers HLL and HLLC schemes based on the WENO-reconstructed state variables. Hence, this is dubbed as the VF (variable-based flux) approach. In critical regions where VF may produce spurious oscillations, a novel, low-dissipation HLL-based CF (characteristic flux) approach is applied. Critical regions were identified using a modified Bhagatwala–Lele shock sensor. The VF/CF hybrid method has been shown to produce high-resolution, essentially non-oscillatory results for a number of 1D and 2D problems at a fraction of the cost of a pure CF approach. Moreover, a 2D advection problem was designed to investigate the choice of state variables and flux schemes. The results have shed more light on the relation between Kelvin–Helmholtz roll-ups and numerical instabilities along slip lines.