BGK Schemes Based On BGK Boltzmann Equation

According to the disadvantage that the simulation of open channel flows cannot meet the entropy condition based on the St. Venant equation, this paper tried to apply BGK schemes to Boltzmann Equation to simulate the open channel flows, based on the gas-kinetic schemes for the compressible Euler Equation. The governing equations for open channel flows are derived from the BGK Boltzmann equation by using basic relationships between the particle velocity distribution and macroscopic variables. It is shown that the St. Venant Equations are the special cases of the BGK model under the equilibrium state. After that, a BGK numerical model for open channel flows which satisfies the entropy condition is established by the finite volume method. The proposal model is applied to solve typical open channel flow problem such as the dam-break problem. Through the comparison between the numerical results and analytical solution (Ritter solution and Stoker solution), we can get the accuracy of the model. It is shown that the BGK model has its advantages in the computational accuracy and stability and it dose not meet any entropy fixes. This model can accurately simulate discontinuous flows and is free of unphysical oscillations.