[Paper Review] Strong convergence rates for a full discretization of stochastic wave equation with nonlinear damping

Meng Cai, David Cohen+1 more|arXiv (Cornell University)|Jul 5, 2023

Stochastic processes and financial applicationsEconomics, Econometrics and Finance46 references3 citations

TL;DR

This paper establishes strong convergence rates for a full discretization of the stochastic wave equation with nonlinear damping in one and two spatial dimensions. Using a spectral Galerkin method in space and a modified implicit exponential Euler scheme in time, the authors achieve a convergence rate of order $\frac{1}{2}$ in space and order $1$ in time for dimension one, with a reduced rate in dimension two due to an infinitesimal factor, without requiring moment bounds on the full approximations.

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 $ frac12$ 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.

Motivation & Objective

• To establish strong convergence rates for a full discretization of the stochastic wave equation with nonlinear damping in one and two spatial dimensions.
• To address the challenges posed by super-linearly growing damping terms in the error analysis.
• To develop a numerical scheme that achieves optimal convergence rates without requiring moment bounds on the full approximations.
• To validate the theoretical findings through numerical experiments in both 1D and 2D settings.

Proposed method

• A spectral Galerkin method is applied for spatial discretization, projecting the SPDE onto a finite-dimensional subspace spanned by eigenfunctions of the Laplacian.
• A modified implicit exponential Euler scheme is used for temporal discretization, designed to handle the nonlinear damping and preserve stability.
• The error analysis is conducted via upper mean-square error bounds, leveraging Itô's formula and stochastic integral estimates.
• Key estimates involve bounding the residual terms using Sobolev embedding inequalities and moment bounds on the solution and its increments.
• The analysis avoids requiring a priori moment bounds on the full numerical approximations, enhancing the robustness of the convergence results.
• Theoretical convergence rates are derived by carefully estimating the conditional expectation of the residual term in the discrete scheme.

Experimental results

Research questions

• RQ1 What is the strong convergence rate of a full discretization scheme for the stochastic wave equation with nonlinear damping in one spatial dimension?
• RQ2 How does the presence of super-linearly growing damping affect the error analysis and convergence rates in the numerical scheme?
• RQ3 Can optimal convergence rates be achieved in two spatial dimensions without imposing moment bounds on the full numerical approximations?
• RQ4 What is the impact of spatial dimension on the convergence rate, particularly in the context of Sobolev embeddings and infinitesimal factors?
• RQ5 How do the theoretical convergence rates compare with empirical results from numerical experiments?

Key findings

• In one spatial dimension, the proposed full discretization achieves a strong convergence rate of order $\frac{1}{2}$ in space and order $1$ in time.
• In two spatial dimensions, the convergence rate is reduced due to an infinitesimal factor arising from Sobolev embedding inequalities, resulting in a rate of order $\tau^{4 - \frac{4\epsilon}{2 + \epsilon}}$ for small $\epsilon > 0$.
• The error analysis does not require a priori moment bounds on the full numerical approximations, which strengthens the theoretical framework.
• Numerical experiments confirm the theoretical convergence rates: a temporal convergence order of $1$ in both 1D and 2D, and a spatial convergence order of $\frac{1}{2}$ in 1D.
• The log-log plots of the mean-square errors from numerical simulations align closely with the predicted theoretical rates, validating the analysis.
• The reference solution computed with a fine time step ($\tau = 2^{-10}$) ensures reliable error estimation, with Monte–Carlo errors found to be negligible.