Explicit unconditionally stable methods for the heat equation via potential theory
Explicit unconditionally stable methods for the heat equation via potential theory
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DOI：
10.2140/paa.2019.1.709
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发表时间：
2019-02
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影响因子：
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通讯作者：
A. Barnett;C. Epstein;L. Greengard;Shidong Jiang;Jun Wang
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文献类型：
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作者：
A. Barnett;C. Epstein;L. Greengard;Shidong Jiang;Jun Wang
本文研究了用势理论求解热方程边值问题时出现的第二类Volterra积分方程的显式推进格式的稳定性。众所周知，只有当时间步长$\Delta t$为$O(\Delta x^2)$阶时，热方程的显式有限差分格式或有限元格式才稳定，其中$\Delta x$为最细的空间网格间距。相反，对于单位球在所有维度$d\ge 1$上的Dirichlet和Neumann问题，我们证明了最简单的Volterra行进格式，即正演欧拉格式是无条件稳定的。我们的证明基于行进矩阵的显式谱半径界，从而得出积分方程解的$L^2$ -范数以$c_dT^{d/2}$乘以右侧范数为界的估计。对于任意维度半空间上的Robin问题，具有恒定的Robin（传热）系数$\kappa$，我们展示了一个恒定的$C$，使得正向欧拉格式在$\Delta t < C/\kappa^2$时是稳定的，独立于任何空间离散化。这依赖于由凸序列定义的实对称Toeplitz矩阵谱上新的下界。最后，我们证明了在$L^\infty$ -范数范围内，对于任意维的光滑凸域上的Dirichlet问题，正演Euler格式是无条件稳定的。
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 $\Delta t$ is of the order $O(\Delta x^2)$, where $\Delta x$ 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 $c_dT^{d/2}$ 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 $\Delta t < C/\kappa^2$, 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 $L^\infty$-norm.