Convergence of difference scheme for heat equation in unbounded domains using artificial boundary conditions (Q596576)

The one-dimensional heat equation on the half-line \([-1,\infty)\) is discretized by a finite difference scheme, after introducing an artificial boundary at \(x=0\). The support of the right-hand side is assumed to be in \([-1,0]\times [0,\infty)\ni (x,t)\), the support of the initial condition lies in \([-1,0]\). The method, which is a fractional step approximation, is shown to be of order 2 in space and of order 3/2 in time if the solution is in \(C^{4,3}_{x,t}\). The authors also prove \textit{a priori} stability estimates for the discrete solution.