A robust common-weights WENO scheme based on the flux vector splitting for Euler equations

Abstract

This paper proposes a common-weights weighted essentially non-oscillatory (Co-WENO) scheme for solving the Euler equations of gas dynamics. Different from the usual component-wise weighting methods, common-weights means that, on one global stencil, a set of weights is commonly shared by the split flux vector of Euler equations in one spatial dimension. The common-weights WENO scheme has two significant advantages. First, since only one set of weights is calculated and used for the split flux vector, the method has an improved computational efficiency. Second, for a stencil (or each cell on the stencil), the Co-WENO scheme keeps the same contribution on each component numerical flux in a hyperbolic system of equations. How to calculate the weights is one of the vital issues in developing this kind of Co-WENO schemes. In this paper, based on the flux vector split method, the product of density, pressure, and the split flux of energy equation(Γ±=ρpfE±) is proposed to calculate the common weights. This is based on the following considerations: (1) the density jumps at shocks and contact discontinuities; (2) the split energy flux contains the term of the third power of the velocity (for example, u3) and makes the resulting scheme has upwind characteristic; (3) the pressure always jumps at shocks, and it can help improve the stability in high speed flows, in which the kinetic energy is much larger than the internal energy. Numerical experiments also show that the proposed common-weights WENO scheme has good robustness and low numerical dissipation, and it can help suppress phase errors.