| The development of gas-kinetic methods for compressible flow simulations has attracted much attention in the last decade. Gas-kinetic schemes are based on the Boltzmann equation, which represents a better physical description of the behavior of the flows than either the Euler equations or the Navier-Stokes equations. Due to the complications of the collision integral term of the Boltzmann equation, most gas-kinetic schemes solve the collisionless Boltzmann equation instead by neglecting the collision integral term. The gas-kinetic Bhatnagar-Gross-Krook (BGK) method offers a simplified model by linearizing the collision integral term of the Boltzmann equation. Since the BGK model includes the collision term, which simulates flow transition from a non-equilibrium state to an equilibrium one, it is an excellent shock-capturing method, especially for multidimensional compressible flows and strong shock tubes, with only moderate extra effort on a computer.; The results of his dissertation have shown that the gas-kinetic BGK scheme is robust and accurate for shock applications, especially for three-dimensional transonic and supersonic flows. Some of the computational results are shown to compare well with the exact results, numerical results of other schemes, or experimental results. In this dissertation, the positivity properties of the first-order BGK method are analyzed in detail for the linear advection equation and the Euler equations. It is proved here that the gas-kinetic BGK scheme is a positivity-preserving scheme. The positivity conditions, which are dependent on the collision time, are also given for these two governing equations. In conclusion, the gas-kinetic BGK scheme is a very good shock-capturing scheme. It is positivity preserving, and it satisfies the entropy condition automatically. Since it is based on the kinetic theory, it also has a potential to extend to rarefied gas flows. |
