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2019 Explicit unconditionally stable methods for the heat equation via potential theory
Alex Barnett, Charles L. Epstein, Leslie Greengard, Shidong Jiang, Jun Wang
Pure Appl. Anal. 1(4): 709-742 (2019). DOI: 10.2140/paa.2019.1.709


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 d1, 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.


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Alex Barnett. Charles L. Epstein. Leslie Greengard. Shidong Jiang. Jun Wang. "Explicit unconditionally stable methods for the heat equation via potential theory." Pure Appl. Anal. 1 (4) 709 - 742, 2019.


Received: 1 March 2019; Revised: 20 April 2019; Accepted: 10 June 2019; Published: 2019
First available in Project Euclid: 29 October 2019

zbMATH: 07142204
MathSciNet: MR4026552
Digital Object Identifier: 10.2140/paa.2019.1.709

Primary: 00A05

Keywords: Abel equation , convex sequence , forward Euler scheme , heat equation , modified Bessel function of the first kind , Stability analysis , Toeplitz matrix , Volterra integral equation

Rights: Copyright © 2019 Mathematical Sciences Publishers


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Vol.1 • No. 4 • 2019
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