Spectral Approximation of the Free-Space Heat Kernel

Many problems in applied mathematics, physics, and engineering require the solution of the heat equation in unbounded domains. Integral equation methods are particularly appropriate in this setting for several reasons: they are unconditionally stable, they are insensitive to the complexity of the geometry, and they do not require the artificial truncation of the computational domain as do finite difference and finite element techniques. Methods of this type, however, have not become widespread due to the high cost of evaluating heat potentials. When mpoints are used in the discretization of the initial data, Mpoints are used in the discretization of the boundary, and Ntime steps are computed, an amount of work of the order O(N²M²+NMm) has traditionally been required. In this paper, we present an algorithm which requires an amount of work of the order O(NMlog M+mlog m) and which is based on the evolution of the continuousspectrum of the solution. The method generalizes an earlier technique developed by Greengard and Strain (1990, Comm. Pure Appl. Math.43, 949) for evaluating layer potentials in bounded domains.