Mathematical Sciences: High Order Accurate Numerical Methods for Interface Problems

数学科学：接口问题的高阶精确数值方法

基本信息

批准号：
9626703
负责人：
Stanley Osher
金额：
$ 6万
依托单位：
University of California-Los Angeles
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
1996
资助国家：
美国
起止时间：
1996-08-15 至 2000-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626703&HistoricalAwards=false
关键词：
Mathematical Sciences Order Accurate Numerical

项目摘要

Osher 9626703 Many physical problems involve interfaces where two different materials contact with each other, or singular sources or dipoles are present along the interfaces immersed in the same fluid. Mathematically, interface problems usually lead to differential equations whose input data and solutions have discontinuities or non-smoothness across interfaces. Many numerical methods designed for smooth solutions do not work efficiently for interface problems. The investigator's colleague Zhilin Li combines the immersed interface method, a second order method for solving differential equations involving interfaces, with the level set approach, an efficient method for capturing moving fronts, to develop high order accurate and efficient numerical methods for interface problems. The project develops convergence and stability theory to provide theoretical justification for the proposed methods. Several specific interface problems, including elliptic and parabolic equations with fixed or moving interfaces, Hele-Shaw flow, and other applications, are studied in depth. Many important practical problems lead to differential equations in regions of 2- or 3-dimensional space that are geometrically complicated, and that contain interfaces across which the nature of the solution changes. These equations can rarely be solved exactly, and large-scale computation is required to obtain well-resolved solutions over multi-dimensional regions. The goal of this work is to develop efficient computational methods to approximate solutions of such problems. The approach is to combine two different methods with complementary strengths. To test the new method, it is applied to several problems representative of those with arising in practical applications.

Osher 9626703许多物理问题涉及界面，其中两种不同的材料彼此接触，或者沿浸入同一流体中的界面存在奇异源或偶极子。从数学上讲，界面问题通常会导致微分方程，其输入数据和解决方案在跨接口方面具有不连续性或不平滑度。许多用于平滑解决方案的数值方法在接口问题上无法有效地工作。研究者的同事Zhilin li结合了浸入式接口方法，这是一种用于求解涉及接口的微分方程的二阶方法，其水平设置方法是一种捕获移动前部的有效方法，以开发高阶准确，有效的数值方法来解决界面问题。该项目开发了融合和稳定理论，以为所提出的方法提供理论上的理由。深入研究了几个特定的界面问题，包括具有固定或移动接口的椭圆形和抛物线方程，Hele-shaw流量和其他应用。许多重要的实际问题导致在几何复杂的2-或3维空间区域的微分方程中，并且包含溶液性质在变化的接口。这些方程式很少可以准确地解决，并且需要进行大规模计算以在多维区域获得良好的解决方案。这项工作的目的是开发有效的计算方法来近似此类问题的解决方案。该方法是将两种不同的方法与互补强度相结合。为了测试新方法，它应用于代表实际应用中出现的几个问题。