Simulating compressible flows with shock waves using the finite-difference Lattice Boltzmann Method

The finite-difference method is combined with the Lattice Boltzmann Method (LBM) to simulate compressible flows with shock waves. The LBM model used here is based on a polynomial kernel function in the phase space, and can be recovered to the compressible Navier Stokes equations with a flexible specific-heat ratio and Prandtl number. The third-order implicit-explicit Runge Kutta scheme and the fifth-order weighted essentially non-oscillatory scheme are adopted for time and space discretisation, respectively. Three problems, including the Sod and the Lax shock tubes, and the strong shock wave problem, are simulated. Numerical solutions agree well with the exact solutions.