Many problems in physics and engineering require the solution of the heat equation in moving geometry. Integral representations are particularly appropriate in this setting since they satisfy the governing equation automatically and, in the homogeneous case, require the discretization of the space-time boundary alone. Unlike methods based on direct discretization of the partial differential equation, they are unconditonally stable. Moreover, while a naive implementation of this approach is impractical, several efforts have been made over the past few years to reduce the overall computational cost. Of particular note are Fourier-based methods which achieve optimal complexity so long as the time step Δt is of the same order as Δx, the mesh size in the spatial variables. As the time step goes to zero, however, the cost of the Fourier-based fast algorithms grows without bound. A second difficulty with existing schemes has been the lack of efficient, high-order local-in-time quadratures for layer heat potentials.
In this dissertation, we present a new method for evaluating heat potentials that makes use of a spatially adaptive mesh instead of a Fourier series, a new version of the fast Gauss transform, and a new hybrid asymptotic/numerical method for local-in-time quadrature. The method is robust and efficient for any Δt, with essentially optimal computational complexity. We demonstrate its performance with numerical examples and discuss its implications for subsequent work in diffusion, heat flow, solidification and fluid dynamics.
|Advisor:||Greengard, Leslie F.|
|Commitee:||Cerfon, Antoine, Jiang, Shidong, O'Neil, Michael, Shelley, Michael|
|School:||New York University|
|School Location:||United States -- New York|
|Source:||DAI-B 79/02(E), Dissertation Abstracts International|
|Subjects:||Applied Mathematics, Mathematics, Computer science|
|Keywords:||Differential equation, Fast algorithm, Integral equation, Numerical analysis|
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