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Zhilin Li : The Immersed Interface Method:A Numerical Approach for Interface Problems

Many physical problems involve interfaces. Examples include phase transition problems where the interface separates the solid and liquid regions, bubble simulation, Hele-Shaw flow, composite materials, and many other important physical phenomena. Mathematically, interface problems usually lead to differential equations whose input data and solutions have discontinuities or non-smoothness across interfaces. As a result, many standard numerical schemes do not work or work poorly for interface problems. This is an introductory talk about the interface problems and our immersed interface method. Through some simple examples, I will try to explain the problems of our interest and related background information. Then I will present our method for some typical model problems in two dimensions. Our method can handle both discontinuous coefficients and singular sources. The main idea is to incorporate the known jumps in the solution and its derivatives into the finite difference scheme, obtaining a modified scheme on the uniform grid for quite arbitrary interfaces. The second part of the talk will focus on applications of the methods combined with the the level set method for moving interface problems: including the Stokes flow with different surface tension, the simulation of Hele-Shaw flow, and computation of crystal growth.

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