在有不規則曲面的定義域上的微分方程式的數值計算方法

Abstract

這是一個二年期的計劃。在計算微分方程近似解的過程中，若是要計算的微分方程式的定義域是不規則的情形，常常在不規則的邊界附近，必須做一些特別的處理。然而，在不規則的邊界附近，做微分方程式的離散化並不容易，即使完成了離散化，離散化之後的代數式子的解的計算，若代數式子沒有好的矩陣結構，也需要較長的時間的計算才能得到好的近似結果。在我們平常接觸到的例子中，有不少的例子是以上的情形。譬如，要模擬海岸邊洋流的變化，水流通過彎曲通道的問題，血管中血液的流動變化，自由邊界的問題，模擬飛機機翼附近氣流的變化，彈性力學的問題等。 在此計劃中我們希望能發展出一套容易離散化微分方程式的技巧，也希望能發展出一套程式撰寫能夠簡單化、能夠得到準確近似值的方法。此計劃若能完成，對很多的例子的數值模擬有很大的幫助。

This is a two-year project. To discretize a differential equation in an irregular domain, special treatment is needed around the irregular boundary of the domain in order to keep the accuracy and consistency of the approximation. It depends on the way of the discretization, in many cases, the resulting algebraic equations could be complicated and have no good matrix structure. In this case, numerical solvers to solve the algebraic equations may not be efficient. Many practical examples share this feature. For example, simulation of the ocean current around coast, simulation of blood motion in vessels, some free boundary value problems, numerical simulation of flow over airfoils, some elasticity problems, and so on. In this project, we plan to develop a technique to discretize a differential equation around the irregular boundary. We expect that after the discretization, the approximation has good accuracy and the resulting algebraic equations have good matrix structure so that they can be solved easily by existing solvers.