Localized implicit time stepping for the wave equation

TL;DR

It is proved that the superposition of localized solutions is appropriately close to the solution of the (global) implicit scheme, so it is justified that the localized computation on multiple overlapping subdomains is reasonable.

Abstract

This work proposes a discretization of the acoustic wave equation with possibly oscillatory coefficients based on a superposition of discrete solutions to spatially localized subproblems computed with an implicit time discretization. Based on exponentially decaying entries of the global system matrices and an appropriate partition of unity, it is proved that the superposition of localized solutions is appropriately close to the solution of the (global) implicit scheme. It is thereby justified that the localized (and especially parallel) computation on multiple overlapping subdomains is reasonable. Moreover, a re-start is introduced after a certain amount of time steps to maintain a moderate overlap of the subdomains. Overall, the approach may be understood as a domain decomposition strategy in space on successive short time intervals that completely avoids inner iterations. Numerical examples are presented.

Figures (5)

• Figure 3.1: Values of the inverse system matrix of the Crank--Nicolson scheme in two dimensions on a uniform and lexicographically ordered mesh with logarithmic color coding; mesh size (and time step) $h = \tau = 2^{-4}$ (left) and $h= \tau = 2^{-6}$ (right).
• Figure 4.1: Errors between the local superposition method and the global Crank--Nicolson method for $A \equiv 1$; left: with respect to $H$ for $\ell = 2H/h$ and variable $h$; right: with respect to $\ell$ for $h = 2^{-8}$ and $H = 2^{-4}$.
• Figure 5.1: Errors between the local superposition method and the global Crank--Nicolson method for $A \equiv 1$; left: with respect to $H$ for $\ell = 2H/h$ and $h = 2^{-9}$, and variable $\tau/h$; right: with respect to $\ell$ for $h = 2^{-8}$, $H = 2^{-4}$, and variable $\tau/h$.
• Figure 5.2: Errors between the local superposition method and the global Crank--Nicolson method for $h = 2^{-8}$, $H = 2^{-4}$ with respect to $\ell$; left: $A \equiv 1$ and variable choices of $\tau$ and $T$; right: oscillatory $A$, $\tau = h$, and $T = H/\beta$.
• Figure 5.3: Errors between the local superposition method and the exact solution in $2d$ for $h = 2^{-11}$, $\tau = 2^{-8}$, and $H = T = 2^{-4}$ (left) and in $1d$ for $h = 2^{-16}$, $\tau = 2^{-8}$, and $H = T = 2^{-4}$ (right) with respect to $\ell$.

Theorems & Definitions (10)

• Theorem 2.1: Energy conservation of the Crank--Nicolson method
• proof
• Lemma 3.1: Decaying discrete solution
• proof
• Theorem 3.2: Localization error
• proof
• Remark 3.3: Choice of $\ell$
• Theorem 4.1: Error of the local superposition method
• proof : Proof of Theorem \ref{['thm:superpositionError']}
• Remark 4.2: Generalizations