6 research outputs found
A Fast Adaptive Method for the Evaluation of Heat Potentials in One Dimension
We present a fast adaptive method for the evaluation of heat potentials,
which plays a key role in the integral equation approach for the solution of
the heat equation, especially in a non-stationary domain. The algorithm
utilizes a sum-of-exponential based fast Gauss transform that evaluates the
convolution of a Gaussian with either discrete or continuous volume
distributions. The latest implementation of the algorithm allows for both
periodic and free space boundary conditions. The history dependence is overcome
by splitting the heat potentials into a smooth history part and a singular
local part. We discuss the resolution of the history part on an adaptive volume
grid in detail, providing sharp estimates that allow for the construction of an
optimal grid, justifying the efficiency of the bootstrapping scheme. While the
discussion in this paper is restricted to one spatial dimension, the
generalization to two and three dimensions is straightforward. The performance
of the algorithm is illustrated via several numerical examples.Comment: 14 pages, 3 table