A WENO-Based Upwind Rotated Lattice Boltzmann Flux Solver with Lower Numerical Dissipation for Simulating Compressible Flows with Contact Discontinuities and Strong Shock Waves

Abstract

This paper presents a WENO-based upwind rotated lattice Boltzmann flux solver (WENO-URLBFS) in the finite difference framework for simulating compressible flows with contact discontinuities and strong shock waves. In the method, the original rotating lattice Boltzmann flux solver is improved by applying the theoretical solution of the Euler equation in the tangential direction of the cell interface to reconstruct the tangential flux so that the numerical dissipation can be reduced. The fluxes at each interface are evaluated using a weighted summation of lattice Boltzmann solutions in two local perpendicular directions decomposed from the direction vector so that the stability performance can be improved. To achieve high-order accuracy, both fifth and seventh-order WENO reconstructions of the flow variables in the characteristic spaces are carried out. The order accuracy of the WENO-URLBFS is evaluated and compared with the traditional Lax–Friedrichs scheme, Roe scheme, and the LBFS by simulating the advection of the density disturbance problem. It is shown that the fifth and seventh-order accuracy can be achieved by all considered flux-evaluation schemes, and the present WENO-URLBFS has the lowest numerical dissipation. The performance of the WENO-URLBFS is further examined by simulating several 1D and 2D examples, including shock tube problems, Shu–Osher problems, blast wave problems, double Mach reflections, 2D Riemann problems, K-H instability problems, and High Mach number astrophysical jets. Good agreements with published data have been achieved quantitatively. Moreover, complex flow structures, including shock waves and contact discontinuities, are successfully captured. The present WENO-URLBFS scheme seems to present an effective numerical tool with high-order accuracy, lower numerical dissipation, and strong robustness for simulating challenging compressible flow problems.