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Strong convergence rates for a full discretization of stochastic wave equation with nonlinear damping

Meng Cai, David Cohen, Xiaojie Wang

TL;DR

The paper addresses strong convergence for the full discretization of a stochastic wave equation with nonlinear damping in 1D and 2D. It combines a spectral Galerkin spatial discretization with a modified implicit exponential Euler temporal scheme and introduces an auxiliary analysis to obtain mean-square error bounds without requiring high-moment bounds of the fully discrete solution. The main results show a spatial rate of $\lambda_N^{-\frac12}$ and a temporal rate of $\tau$ in 1D, and a spatial rate of $\lambda_N^{-\frac12+\varepsilon}$ with a temporal rate of $\tau^{1-\varepsilon}$ in 2D, with $\varepsilon>0$ arbitrary. Numerical experiments corroborate the theoretical rates and demonstrate the practicality of the approach for SWE with nonlinear damping.

Abstract

The paper establishes the strong convergence rates of a spatio-temporal full discretization of the stochastic wave equation with nonlinear damping in dimension one and two. We discretize the SPDE by applying a spectral Galerkin method in space and a modified implicit exponential Euler scheme in time. The presence of the super-linearly growing damping in the underlying model brings challenges into the error analysis. To address these difficulties, we first achieve upper mean-square error bounds, and then obtain mean-square convergence rates of the considered numerical solution. This is done without requiring the moment bounds of the full approximations. The main result shows that, in dimension one, the scheme admits a convergence rate of order $\tfrac12$ in space and order $1$ in time. In dimension two, the error analysis is more subtle and can be done at the expense of an order reduction due to an infinitesimal factor. Numerical experiments are performed and confirm our theoretical findings.

Strong convergence rates for a full discretization of stochastic wave equation with nonlinear damping

TL;DR

The paper addresses strong convergence for the full discretization of a stochastic wave equation with nonlinear damping in 1D and 2D. It combines a spectral Galerkin spatial discretization with a modified implicit exponential Euler temporal scheme and introduces an auxiliary analysis to obtain mean-square error bounds without requiring high-moment bounds of the fully discrete solution. The main results show a spatial rate of and a temporal rate of in 1D, and a spatial rate of with a temporal rate of in 2D, with arbitrary. Numerical experiments corroborate the theoretical rates and demonstrate the practicality of the approach for SWE with nonlinear damping.

Abstract

The paper establishes the strong convergence rates of a spatio-temporal full discretization of the stochastic wave equation with nonlinear damping in dimension one and two. We discretize the SPDE by applying a spectral Galerkin method in space and a modified implicit exponential Euler scheme in time. The presence of the super-linearly growing damping in the underlying model brings challenges into the error analysis. To address these difficulties, we first achieve upper mean-square error bounds, and then obtain mean-square convergence rates of the considered numerical solution. This is done without requiring the moment bounds of the full approximations. The main result shows that, in dimension one, the scheme admits a convergence rate of order in space and order in time. In dimension two, the error analysis is more subtle and can be done at the expense of an order reduction due to an infinitesimal factor. Numerical experiments are performed and confirm our theoretical findings.
Paper Structure (5 sections, 10 theorems, 103 equations, 2 figures)

This paper contains 5 sections, 10 theorems, 103 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The stochastic wave equation with nonlinear damping
  3. Strong convergence rates of the spatial discretization
  4. Strong convergence rates of the full discretization
  5. Numerical experiments

Key Result

lemma 1

For $t \geq s \geq 0$ and $\gamma \in [0,1]$, we have for some constant $\widehat{C}>0$. Moreover, for $X\in\mathbb{H}^\gamma$ it holds that for some constant $\widehat{C}>0$.

Figures (2)

  • Figure 1: Strong convergence rates of the temporal discretization given by the implicit exponential Euler scheme (IEE) (left: $d=1$, right: $d=2$).
  • Figure 2: Strong convergence rates of the spatial discretization given by the spectral Galerkin method.

Theorems & Definitions (18)

  • lemma 1
  • theorem 3
  • proof
  • lemma 2
  • proof
  • lemma 3
  • proof
  • lemma 4
  • proof
  • theorem 4: Strong convergence rates of the spatial semi-discretization
  • ...and 8 more