Vol. 1, No. 4, 2019

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Explicit unconditionally stable methods for the heat equation via potential theory

Alex Barnett, Charles L. Epstein, Leslie Greengard, Shidong Jiang and Jun Wang

Vol. 1 (2019), No. 4, 709–742
DOI: 10.2140/paa.2019.1.709
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 Δt is of the order O(Δx2), where Δx is the finest spatial grid spacing. In contrast, for the Dirichlet and Neumann problems on the unit ball in all dimensions d 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 L2-norm of the solution to the integral equation is bounded by cdTd2 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 κ, we exhibit a constant C such that the forward Euler scheme is stable if Δt < Cκ2, 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-norm.

Keywords
heat equation, Abel equation, forward Euler scheme, Volterra integral equation, stability analysis, Toeplitz matrix, convex sequence, modified Bessel function of the first kind
Mathematical Subject Classification 2010
Primary: 00A05
Milestones
Received: 1 March 2019
Revised: 20 April 2019
Accepted: 10 June 2019
Published: 12 October 2019
Authors
Alex Barnett
Flatiron Institute
Simons Foundation
New York, NY
United States
Charles L. Epstein
Department of Mathematics
University of Pennsylvania
Philadelphia, PA
United States
Leslie Greengard
Courant Institute of Mathematical Sciences
New York University
New York, NY
United States
Flatiron Institute
Simons Foundation
New York, NY
United States
Shidong Jiang
Department of Mathematical Sciences
New Jersey Institute of Technology
Newark, NJ
United States
Jun Wang
Flatiron Institute
Simons Foundation
New York, NY
United States