Fast high-order integral equation methods for PDEs with moving interfaces.

摘要

Many problems in Science and Engineering require the solution of partial differential equations (PDEs) on moving domains. Stencil-based numerical techniques like the Finite Element Method (FEM) tend to be computationally expensive for such problems. For certain classes of PDEs, there are promising alternatives that are based on integral equations. Unlike FEM-based schemes, they do not require unstructured mesh-generation and remeshing. In this work, we construct fast, high-order solvers based on integral equations for two problems: Solving the heat equation on moving domains; Simulating the dynamics of deformable vesicles suspended in viscous fluid flows.;For the heat equation, we describe a fast high-order accurate method for its solution on domains with moving Dirichlet or Neumann boundaries and distributed forces. We assume that the motion of the boundary is prescribed. Our method extends the work of L. Greengard and J. Strain, "A fast algorithm for the evaluation of heat potentials", Comm. Pure & Applied Math. 1990. Our scheme is based on a time-space Chebyshev pseudo-spectral collocation discretization, which is combined with a recursive product quadrature rule to accurately and efficiently approximate convolutions with the Green's function for the heat equation. We present numerical results that exhibit up to sixteenth-order convergence rates. Assuming N time steps and M spatial discretization points, the evaluation of the solution of the heat equation at the same number of points in space-time requires O (N M log M) work.;Vesicle flows model numerous biophysical phenomena that involve deforming particles interacting with a Stokesian fluid. While conventional techniques can be used to simulate isolated vesicles, new approaches are needed for large number of interacting vesicles. An integral equation formulation leads to a system of nonlinear integro-differential equations whose unknowns reside on the fluid-vesicle interfaces. We have developed a novel numerical scheme for such equations. It incorporates a new time-stepping scheme that allows much larger time-steps than the existing explicit schemes. The associated linear systems are solved in optimal time using spectral preconditioners, FFTs and the Fast Multipole Method.