利用漸近保守高解析算則解波茲曼模型方程式

Abstract

本文使用相空間之直接解法，模擬利用鬆弛時間近似之古典波茲曼模型方程式及半古典橢球波茲曼模型方程式，並同時處理三種統計之粒子。在時間離散中，因為非線性波茲曼模型方程式之碰撞項(源項)非常複雜，故引入漸近保守算則，使得碰撞項與鬆弛時間能做拆解。引入漸近保守算則後，使用四階準確之Runge-Kutta法將時間離散。在空間離散中，本文採用五階準確性之加權基本不震盪算則；在速度離散中，使用分立坐標法使得波茲曼模型方程式與速度空間相互獨立。本文利用波茲曼模型方程式求解二維黎曼(Riemann)問題，且調整鬆弛時間至極小，使得波茲曼模型方程式達到尤拉極限(Euler Limit)，並與文獻中所模擬之結果做比較。基於氣體分子動力學理論，本文成功解析波茲曼模型方程式在不同初始條件之下交互作用之複雜震波、Slip Line、膨脹波等現象，且利用較少網格得到較佳之結果。且在半古典波茲曼模型方程式中，成功模擬三種不同統計粒子之量子傳輸現象。

An accurate and direct algorithm for solving the classical Boltzmann equation and the semiclassical Boltzmann equation with relaxation time approximation in phase space is presented for parallel treatment of rarefied gas flows of particles of three statistics. In time domain, we use asymptotic-preserving method for solving two-dimensional Riemann problem by the classical Boltzmann equation and the semiclassical Boltzmann equation with very small relaxation time. After using asymptotic-preserving, we use fourth-order Runge-Kutta method to discrete time domain. In space domain, we use fifth-order weighted essentially non-oscillatory scheme to evolve the flux term. The discrete ordinate method is applied to remove the microscopic velocity dependency of the distribution function that renders the Boltzmann BGK equation in phase space to a set of hyperbolic conservation laws with source terms in physical space. Computational examples of two-dimensional Riemann problems for rarefied gas flows at very small relaxation time are presented. By using WENO scheme, the results show good resolution in capturing the main flow features while using grids with few good points.