反應擴散方程在移動曲面上的數值模擬

Abstract

這篇論文的主要目的是使用快速傅立葉轉換的優點來數值上地解反應擴散方程組，並且解以移動的曲面為定義域的方程組。在未來的工作上希望能夠找出近似心臟的舒張與收縮的移動曲面方程式以結合反應擴散方程式來做出更好的心臟電流傳導的數值模擬。首先，在球面座標與橢球面座標下利用譜方法和一些二階精確的數值方法分別做球面上及橢球面上的熱方程快速解法。接著，使用這兩種快速解法並且結合時間分解的方式產生反應擴散方程的快速解法。再來，使用在曲線座標下的曲面拉普拉斯算子來數值上地解熱方程，利用這個數值解可以計算定義在移動曲面上的熱方程，進一步用它來計算傳導擴散方程。在以上的各種數值解下，我們皆舉了四個可以涵蓋其他各種情形的例子去觀察質量的變化。最後，對這些數值解和質量的變化我們做出了結論及它們的應用性。

The central objective of this thesis is to use the fast fourier transform to numerically solve the reaction-diffusion systems and solve another systems whose domain moves with time. Hope that we could produce the equations on the moving domain which approximates the diastole and systole in human hearts. First, use the spectral method and some second-order numerical methods to produce the fast heat solvers on the spherical and ellipsoid surface domain in spherical coordinates and ellipsoid coordinates, respectively. Then, couple these two fast solvers with the time splitting method to produce the fast solver of the reaction-diffusion equation. Next, we use the surface Laplacian operator in Curvilinear coordinates to numerically compute the heat equation. Therefore, this heat solver could compute the heat equation on the moving surface domain. Furthermore, use it to compute the convection-diffusion equation. In each of above numerical solvers, we give four examples which could cover other situations to observe the changes in the mass. Finally, we summarize the applications and results for these numerical solvers and changes in the mass.