Explicit Unconditionally Stable Methods For the Heat Equation via Potential Theory

Alex Barnett Flatiron Institute, Simons Foundation Charles L. Epstein Flatiron Institute, Simons Foundation Leslie Greengard Courant Institute, New York University Shidong Jiang Flatiron Institute, Simons Foundation Jun Wang Yau Mathematical Sciences Center, Tsinghua University

Numerical Analysis and Scientific Computing mathscidoc:2205.25003

Pure and Applied Analysis, 1, (4), 2019.1
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 􏰃t is of the order O(􏰃x2), where 􏰃x 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 cd 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 κ, we exhibit a constant C such that the forward Euler scheme is stable if 􏰃t < C /κ 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∞-norm.
heat equation, Abel equation, forward Euler scheme, Volterra integral equation, stability analysis
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@inproceedings{alex2019explicit,
  title={Explicit Unconditionally Stable Methods For the Heat Equation via Potential Theory},
  author={Alex Barnett, Charles L. Epstein, Leslie Greengard, Shidong Jiang, and Jun Wang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220517122434195796187},
  booktitle={Pure and Applied Analysis},
  volume={1},
  number={4},
  year={2019},
}
Alex Barnett, Charles L. Epstein, Leslie Greengard, Shidong Jiang, and Jun Wang. Explicit Unconditionally Stable Methods For the Heat Equation via Potential Theory. 2019. Vol. 1. In Pure and Applied Analysis. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220517122434195796187.
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