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This paper modifies the peculiar velocity upwind (PVU) Boltzmann scheme to accurately resolve contact discontinuities in gas dynamics by redesigning the numerical discretization of the peculiar velocity term. Analysis revealed that the original scheme's peculiar velocity component introduced non-vanishing numerical diffusion at low Mach numbers, hindering contact discontinuity capture. The proposed variants achieve low numerical diffusion and exact capture of stationary contact discontinuities in 1D and slip surfaces in 2D, while retaining the robustness of the original PVU scheme.
Contact discontinuities are now exactly captured in 1D and slip surfaces in 2D by a robust Boltzmann scheme, eliminating a key limitation of previous peculiar velocity methods.
Starting from the robust foundation of the peculiar velocity based upwind (PVU) Boltzmann scheme, which fails to recognize contact discontinuities, this paper introduces new variants for the numerical solutions of equations of gas dynamics. A modified partial differential equation (MPDE) analysis of the original scheme reveals that the numerical diffusion coefficient corresponding to the peculiar velocity part does not vanish as the Mach number goes to zero. The numerical discretization of the peculiar velocity term is redesigned so that the numerical diffusion depends only on the bulk fluid velocity across the stationary contact discontinuities, enabling its exact capture in 1D and exact capture of slip surfaces in 2D. The resulting variants achieve low numerical diffusion while preserving the robustness of the original scheme, as demonstrated through several 1D and 2D benchmark compressible flow test cases.