A Fast Adaptive Method for the Heat Equation with Moving or Free Boundaries in One Dimension

Abstract: 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.