Implicit-explicit finite-difference lattice boltzmann method for compressible flows

We propose an implicit-explicit finite-difference lattice Boltzmann method for compressible flows in this work. The implicit-explicit Runge Kutta scheme, which solves the relaxation term of the discrete velocity Boltzmann equation implicitly and other terms explicitly, is adopted for the time discretization. Owing to the characteristic of the collision invariants in the lattice Boltzmann method, the implicitness can be completely eliminated, and thus no iteration is needed in practice. In this fashion, problems (no matter stiff or not) can be integrated quickly with large Courant Friedriche-Lewy numbers. As a result, with our implicit-explicit finite-difference scheme the computational convergence rate can be significantly improved compared with previous finite-difference and standard lattice Boltzmann methods. Numerical simulations of the Riemann problem, Taylor vortex flow, Couette flow, and oscillatory compressible flows with shock waves show that our implicit-explicit finite-difference lattice Boltzmann method is accurate and efficient. In addition, it is demonstrated that with the proposed scheme non-uniform meshes can also be implemented with ease.