Robust and Accurate Kinetic Schemes based on Flexible Velocities for Compressible Flows

Abstract

Kinetic theory based schemes offer a unique approach to numerically solving the hyperbolic nonlinear Euler equations. These methods start with the Boltzmann equation as the governing equation at the kinetic level and then use the moment method strategy to recover the macroscopic equations. One distinct advantage that this approach offers is the independence of the resulting scheme from the underlying eigen- structure. Traditionally, however, kinetic schemes often suffer from being numerically diffusive, which compromises their accuracy in resolving discontinuities and steep gradients. This thesis seeks to address this limitation by developing robust and accurate finite volume schemes for compressible flows by employing flexible velocities in the vector-kinetic framework.

The first part of this thesis introduces a new kinetic formulation based on flexible velocities, based on compactly supported distributions. The equilibrium distribution in this work comprises ranges of velocities centered around certain average velocities. While the average velocities are utilized to enforce Rankine-Hugoniot jump conditions in the discretization, thereby enabling exact capture of grid-aligned steady discontinuities, the variable velocity ranges are employed to provide additional numerical diffusion in expansions and smoothly varying flow regions. For this purpose, a novel discrete velocity version of relative entropy is introduced. Along with an additional criterion, this kinetic relative entropy is used to identify expansions and smooth flow regions. Further, new flow tangency and far-field boundary conditions are formulated for the proposed kinetic model and are utilized in the numerical simulations. The kinetic model is also extended to viscous flows. Some benchmark 1D and 2D compressible flow test cases are solved to demonstrate the robustness and accuracy of the solver.

The numerical scheme developed in the first part of this thesis is capable of solving a wide range of compressible flow problems. However, it remains susceptible to solution failure if negative pressures occur during computation—particularly for flows at high speeds or with large gradients—even for first order accuracy. This limitation motivates the second part of the thesis, which introduces a new kinetic model featuring flexible velocities designed to satisfy positivity preservation conditions for the Euler equations. The proposed 1D kinetic model utilizes two velocities and incorporates both asymmetrical and symmetrical formulations. Switching between these models is governed by the previously introduced kinetic relative entropy-based criteria, ensuring an accurate, entropic, and robust scheme. In 2D, a novel three velocity kinetic model is proposed, with the velocities aligned at the interface to ensure a locally 1D formulation for the resulting macroscopic interface normal flux. For first order accuracy, a limit on the time step is obtained which ensures both positivity preservation and numerical stability. The resulting numerical scheme captures grid-aligned steady shocks exactly. Further, some inviscid boundary conditions are also derived for the 2D kinetic model. The scheme is then extended to viscous flows as well. Several benchmark compressible flow test cases are solved in 1D and 2D to demonstrate the efficacy of the proposed solver.

The third part of this thesis focuses on developing a robust kinetic scheme for the multi-component Euler equations. Since a discontinuity in gas composition corresponds to a contact discontinuity in these equations, this work also aims to achieve exact resolution of steady multi-material contact discontinuities. To this end, the symmetrical kinetic model introduced in the second part is extended to multi-component gas mixtures. The velocity magnitudes are defined to satisfy conditions for preservation of positivity of density of each component as well as of overall pressure, for first order accuracy under a CFL-like time step restriction. Additionally, at a stationary contact discontinuity, the velocity definition is modified to achieve exact capture. Benchmark 1D and 2D test cases, including shock-bubble interaction problems, are solved to demonstrate the efficacy of the solver in accurately capturing the relevant flow features.