Proceedings of the 2010 ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis
The Fast Generalized Gauss Transform
SIAM Journal on Scientific Computing
Fast integral equation methods for Rothe's method applied to the isotropic heat equation
Computers & Mathematics with Applications
Artificial Boundary Conditions for the Simulation of the Heat Equation in an Infinite Domain
SIAM Journal on Scientific Computing
A fast Galerkin method for parabolic space-time boundary integral equations
Journal of Computational Physics
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We discuss the numerical evaluation of single and double layer heat potentials in two dimensions on stationary and moving boundaries. One of the principal difficulties in designing high order methods concerns the local behavior of the heat kernel, which is both weakly singular in time and rapidly decaying in space. We show that standard quadrature schemes suffer from a poorly recognized form of inaccuracy, which we refer to as “geometrically induced stiffness,” but that rules based on product integration of the full heat kernel in time are robust. When combined with previously developed fast algorithms for the evolution of the “history part” of layer potentials, diffusion processes in complex, moving geometries can be computed accurately and in nearly optimal time.