Near-field infinity-simulating boundary conditions for the heat equation

The numerical simulation of various physical phenomena in infinite domains poses great difficulties. Truncation of the domain is usually necessary in such cases. To ensure stability of the resulting problem on the restricted domain, appropriate boundary conditions should be applied. Here, we develop a boundary condition for the case in which the heat equation is satisfied outside the domain of interest with no restrictions on the equation inside. The condition employs a thin layer encasing the computational domain. The resulting condition, combined with techniques similar to those proposed by Jiang and Greengard [Jiang S, Greengard L (2004) Fast evaluation of nonreflecting boundary conditions for the Schrödinger equation in one dimension. Comp Math Appl 47:955–966.] promises to be more accurate and computationally efficient than previously described techniques.