基於 Liouville 方程式的方法計算波方程式的高頻震盪解

Computing highly oscillatory solutions for wave equations based on a Liouville-equation approach

周函融(Han-Jung Chou)

指導教授 ： 薛克民

國立臺灣大學/理學院/數學研究所/碩士(2014年)

https://doi.org/10.6342/NTU.2014.00158

摘要

在科學上，高頻震盪波方程式的初值問題有廣泛的應用。當波的頻率非常高時，傳統的數值方法會花費相當多的計算時間。因為這樣的理由，許多對這個問題的近似估計方法被發展起來。我們使用論文中所介紹的基於Liouville方程的估計方法作近似估計。在本論文中，我們以一些例子展示當波速不連續的時候，由Jin和Wen [ J. Comput. Phys. 214 (2006), no. 2, 672-697. ] 所介紹的Hamiltonian-preserving的數值法會比傳統的上風法還要好。我們也藉一些例子展示高頻震盪波方程的能量分布會弱收歛到基於Liouville方程式方法的能量分布。我們會決定當頻率多高的時候，高頻震盪的能量分布的$ L^1 $-norm會收斂到Liouville方程式方法的能量的分布的$ L^1 $-norm。但它們卻不會在$ L^1 $中收斂。最後，我們會給出在不同類型波速下的Liouville方程式的解的推導過程。

關鍵字

高頻震盪初始值問題 ； Wigner變換 ； Liouville方程式 ； Hamiltonian-preserving方法 ； 不連續波速

並列摘要

The highly oscillatory initial value problem of wave equations has wide applications in science. When the frequency of the wave is very high, traditional numerical methods take much computational time. For this reason, many approximation approaches to this problem are developed. We use an approximation method based on the Liouville equation introduced in the thesis. In this thesis, we use some numerical examples to show that the Hamiltonian-preserving method which is introduced by Jin and Wen [ J. Comput. Phys. 214 (2006), no. 2, 672-697. ] is better than traditional upwind method when the wave speed is discontinuous. We also show that the highly oscillatory energy density of the highly oscillatory initial value problem weakly converges to the Liouville energy density based on the Liouville-quation approach via some examples. In these examples, We determine how high the frequency is, the $ L^1 $-norm of highly oscillatory energy density converges to the $ L^1 $-norm of Liouville energy density. But highly oscillatory energy density does not converge to Liouville energy density in $ L^1 $. Finally, We also give the derivation of solutions of Liouville equaiton with different type of wave speeds.

並列關鍵字

highly oscillatory initial value problem ； Wigner transform ； Liouville equation ； Hamiltonian-preserving method ； discontinuous wave speed

參考文獻

[1] Lawrence C. Evans ”Partial Differential Equations.” American Mathematical Society,1998.

[2] Jin, Shi and Wen, Xin, ”Hamiltonian-preserving schemes for the Liouville equation

[4] Jin, Shi; Liu, Hailiang; Osher, Stanley; Tsai, Yen-Hsi Richard, ”Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation.” J. Comput. Phys. 205 (2005), no. 1, 222–241.

[5] S. Jin and D. Yin, ”Computational High Frequency Wave Diffraction By A Corner Via The Liouville Equation And Geometric Theory Of Diffraction.” AIMS Pages: 295–316, Volume 4, Issue 1, March 2011

[6] S. Jin and D. Yin, ”Computational high frequency waves through curved interfaces via the Liouville equation and Geometric Theory of Diffraction.” J. Comput. Phys. 227 (2008), no. 12, 6106–-6139.