Explicit unconditionally stable methods for the heat equation via potential theory

Abstract

We study the stability properties of explicit marching schemes for second-kind Volterra integral equations that arise when solving boundary value problems for the heat equation by means of potential theory. It is well known that explicit finite-difference or finite-element schemes for the heat equation are stable only if the time step $\mathrm{\Delta }t$ is of the order $\mathcal{O}\left(\mathrm{\Delta }{x}^2\right)$, where $\mathrm{\Delta }x$ is the finest spatial grid spacing. In contrast, for the Dirichlet and Neumann problems on the unit ball in all dimensions $d\ge 1$, we show that the simplest Volterra marching scheme, i.e., the forward Euler scheme, is unconditionally stable. Our proof is based on an explicit spectral radius bound of the marching matrix, leading to an estimate that an $L^2$-norm of the solution to the integral equation is bounded by $c_d{T}^{d∕2}$ times the norm of the right-hand side. For the Robin problem on the half-space in any dimension, with constant Robin (heat transfer) coefficient $\kappa$, we exhibit a constant $C$ such that the forward Euler scheme is stable if $\mathrm{\Delta }t<C∕{\kappa }^2$, independent of any spatial discretization. This relies on new lower bounds on the spectrum of real symmetric Toeplitz matrices defined by convex sequences. Finally, we show that the forward Euler scheme is unconditionally stable for the Dirichlet problem on any smooth convex domain in any dimension, in the $L^{\infty }$-norm.