A flux split based finite-difference two-stage boundary variation diminishing scheme with application to the Euler equations

Abstract

In this paper, we present a new finite difference (FD) two stage boundary variation diminishing (BVD) P4T2 (fourth-degree polynomial (P4) and Tangent of Hyperbola for Interface Capturing (THINC) function with a two-level steepness (T2)) scheme for the compressible Euler equations. First, after splitting the flux vector into positive and negative flux vectors using an appropriate flux vector splitting methods, we use the P4 and THINC functions with a two-level steepness to obtain the upwind numerical fluxes at cell boundaries for both the positive and negative flux vectors. Then, we adopt the corresponding downwind version of the P4 and THINC functions with a two-level steepness to obtain the downwind numerical fluxes at cell boundaries for both the positive and negative flux vectors. The final upwind numerical flux is determined for the positive/negative flux respectively through the two-stage BVD principle, which minimizes the jumps of the reconstructed values at the cell boundaries. The final scheme, called FD-P4T2-BVD, exhibits a better resolution, compared with WENO5 scheme, for small-scale structures, particularly the contact discontinuities in various one-dimensional and two-dimensional cases.