利用CESE方法模擬一維模型波滋曼方程式

Abstract

本文模擬重點是利用時空守恆元解元(CESE)對一維波滋曼方程式求解，CESE方法的好處有: (1) 在數值不連續的地方有精確的數值解， (2) 邊界幾乎不會反彈， (3) 和蒙地卡羅(DSMC)方法相比，CESE用在計算的成本相對要小很多，在本文裡，如何利用CESE對ㄧ維的尤拉方程式和波滋曼方程式的離散有詳細的介紹，我們模擬了五個Riemann problems， Riemann problem用來理解雙曲線偏微分方程式是最好用的，像歐拉方程，因為所有Shock和Rarefaction waves的現象都會出現. 模擬結果發現，CESE Method 不但可以減少大量的計算時間，在流體不連續的地方也可以的到很好的數值解，在這五個模擬的case裡，Rarefaction Wave和Shock跟Riemann problem的exact solution 很相似，但在contact 的部份差異就比較大，最後，我們還利用Test 1 的初始條件，模擬了不同的Knudsen number，從1到1E-20，結果顯示，當Knudsen number 越小，結果也就越接近Riemann problem的 exact solution，未來，我們希望可以利用CESE，並且結合Model Boltzmann equation 和 Navier-Stork equation 來模擬Shock wave的問題.

This thesis aims to develop Model Boltzmann equation solver using the conservation element/solution element (CESE) method. Advantages of the CESE method in discretizing conservation laws include(1) high accuracy in capturing discontinuity, (2) good performance in non-reflecting boundary condition treatment, and (3) Compared with DSMC(direct simulation Monte Carlo), CESE spent on the calculation of the relative costs much smaller. In this thesis, discretizations for both 1D Euler equation and model Boltzmann equation (MBE) using the CESE method are described in detail. We simulated five Riemann problems. The Riemann problem is very useful for the understanding of hyperbolic partial differential equation like the Euler equations because all properties like Shocks, Rarefaction waves appear as characteristics in the solution. The results showed that, CESE Method can reduce a lot of calculus coast, in discontinue can also be a very good simulation answer , in the five simulation cases, rarefaction and shock is very similar Riemann problem with the exact solution. But in the contact part of the relatively large difference in the end, we also use Test 1 of the initial conditions, simulated a different Knudsen number, from 1 to 1 E-20, showed that when the Knudsen number close to zero, results will be close Riemann problem of the exact solution. The future, we hope that we can use CESE Method, and the combination of Model Boltzmann equation and the Navier-Stork equation to simulate Shock wave problems.