The immersed interface method: numerical solutions of PDEs involving interfaces and irregular domains

Interface problems arise when there are two different materials, such as water and oil, or the same material at different states, such as water and ice. If partial or ordinary differential equations are used to model these applications, the parameters in the governing equations are typically discontinuous across the interface separating the two materials or states, and the source terms are often singular to reflect source/sink distributions along codimensional interfaces. Because of these irregularities, the solutions to the differential equations are typically nonsmooth or even discontinuous. As a result, many standard numerical methods based on the assumption of smoothness of solutions do not work or work poorly for interface problems.

The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains provides an introduction to the immersed interface method (IIM), a powerful numerical method for solving interface problems and problems defined on irregular domains for which analytic solutions are rarely available. This book gives a complete description of the IIM, discusses recent progress in the area, and describes numerical methods for a number of classic interface problems. It also contains many numerical examples that can be used as benchmark problems for numerical methods designed for interface problems on irregular domains.

The IIM is a sharp interface method that has been coupled with evolution schemes such as the level set and front tracking methods and has been used in both finite difference and finite element formulations to solve several moving interface and free boundary problems. In particular, the authors discuss the IIM’s applications to Stefan problems and unstable crystal growth, incompressible Stokes and Navier–Stokes flows with moving interfaces, an inverse problem identifying unknown shapes in a region, a nonlinear interface problem of magnetorheological fluids containing iron particles, and other problems. The book also contains several applications of free boundary and moving interface problems, including examples from physics, computational fluid mechanics, mathematical biology, material science, and other fields.

The IIM, which is based on uniform or adaptive Cartesian/polar/spherical grids or triangulations, is simple enough to be implemented by researchers and graduate students with a reasonable background in differential equations and numerical analysis yet powerful enough to solve complicated problems with high-order accuracy. Since interfaces or irregular boundaries are one dimension lower than solution domains, the extra costs in dealing with interfaces or irregular boundaries are generally insignificant, and many software packages based on uniform Cartesian/polar/spherical grids, such as the FFT and fast Poisson solvers, can be applied easily with the IIM. The most recent IIM computer codes and packages are available online.

Audience This book will be a useful resource for mathematicians, numerical analysts, engineers, graduate students, and anyone who uses numerical methods to solve computational problems, particularly problems with fixed and moving interfaces, free boundary problems, and problems on irregular domains.