Mathematical Sciences: High Order Accurate Numerical Methods for Interface Problems

Osher 9626703许多物理问题涉及两种不同材料相互接触的界面，或者在浸泡在同一流体中的界面上存在奇异源或偶极子。在数学上，界面问题通常导致微分方程，其输入数据和解在界面上具有不连续性或非光滑性。许多为光滑解而设计的数值方法对于界面问题并不有效。研究人员的同事李志林将浸没界面法和水平集方法相结合，发展了求解界面问题的高精度、高效率的数值方法。浸没界面法是一种求解包含界面的微分方程的二阶方法，而水平集方法是一种捕捉运动锋面的有效方法。该项目发展了收敛和稳定性理论，为所提出的方法提供了理论依据。深入研究了几个具体的界面问题，包括具有固定或移动界面的椭圆型和抛物型方程，Hele-Shaw流，以及其他应用。许多重要的实际问题导致了2维或3维空间区域中的微分方程，这些区域在几何上是复杂的，并且包含解的性质改变的界面。这些方程很难精确求解，而且需要大规模的计算才能在多维区域上获得良好的解。这项工作的目标是开发有效的计算方法来近似求解这类问题。方法是将两种不同的方法结合起来，优势互补。为了验证新方法的有效性，将其应用于几个具有代表性的实际应用问题。

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.