[Paper Review] Energy-preserving fully-discrete schemes for nonlinear stochastic wave equations with multiplicative noise

Jialin Hong, Baohui Hou+1 more|arXiv (Cornell University)|May 31, 2021

Advanced Numerical Methods in Computational Mathematics36 references16 citations

TL;DR

This paper proposes energy-preserving fully-discrete schemes for nonlinear stochastic wave equations with multiplicative noise by combining compact finite difference (CFD) or interior penalty discontinuous Galerkin (IPDG) methods in space with the discrete gradient method and Padé approximation in time. The key contribution is proving that the resulting schemes exactly preserve the discrete averaged energy evolution law, even for multiplicative noise, and numerical experiments confirm the theoretical findings with linear energy growth and first-order temporal convergence.

ABSTRACT

In this paper, we focus on constructing numerical schemes preserving the averaged energy evolution law for nonlinear stochastic wave equations driven by multiplicative noise. We first apply the compact finite difference method and the interior penalty discontinuous Galerkin finite element method to discretize space variable and present two semi-discrete schemes, respectively. Then we make use of the discrete gradient method and the Pad\'e approximation to propose efficient fully-discrete schemes. These semi-discrete and fully-discrete schemes are proved to preserve the discrete averaged energy evolution law. In particular, we also prove that the proposed fully-discrete schemes exactly inherit the averaged energy evolution law almost surely if the considered model is driven by additive noise. Numerical experiments are given to confirm theoretical findings.

Motivation & Objective

• To develop fully-discrete numerical schemes that preserve the averaged energy evolution law for nonlinear stochastic wave equations driven by multiplicative noise.
• To address the lack of existing fully-discrete schemes that exactly inherit the averaged energy evolution law under multiplicative noise.
• To combine high-order spatial discretization (CFD or IPDG) with energy-preserving time integration via discrete gradient and Padé approximation.
• To prove that the proposed schemes preserve the discrete averaged energy evolution law both theoretically and numerically.
• To validate the schemes through numerical experiments demonstrating linear energy growth and first-order convergence.

Proposed method

• Apply the compact finite difference (CFD) method and interior penalty discontinuous Galerkin (IPDG) finite element method to spatially discretize the stochastic wave equation, resulting in semi-discrete stochastic differential equations.
• Use the discrete gradient method to handle the nonlinear drift term and the Padé approximation to treat the diffusion term in time, ensuring energy preservation.
• Prove that the resulting fully-discrete schemes preserve the discrete version of the averaged energy evolution law for both multiplicative and additive noise cases.
• For additive noise, show that the schemes preserve the averaged energy evolution law almost surely.
• Construct six numerical schemes: CFD-I, CFD-II, CFD-CNM, DG-I, DG-II, DG-CNM, based on combinations of spatial and temporal discretization methods.
• Validate the schemes numerically using test problems with f(u) = sin(u), f(u) = u^3, and g(u) = 1, sin(u), u, comparing energy evolution and convergence rates.

Experimental results

Research questions

• RQ1 Can fully-discrete schemes be constructed that exactly preserve the averaged energy evolution law for nonlinear stochastic wave equations with multiplicative noise?
• RQ2 How do compact finite difference and interior penalty discontinuous Galerkin methods perform in preserving the discrete averaged energy when used for spatial discretization?
• RQ3 Does the combination of the discrete gradient method and Padé approximation in time ensure energy preservation in the fully-discrete setting?
• RQ4 Do the proposed schemes maintain linear energy growth and first-order convergence in time, as predicted by theory?
• RQ5 How do the proposed schemes compare to standard schemes like C-N and BEM in preserving the averaged energy evolution law?

Key findings

• The proposed fully-discrete schemes based on CFD or IPDG in space and discrete gradient/Padé in time exactly preserve the discrete averaged energy evolution law for nonlinear stochastic wave equations with multiplicative noise.
• For additive noise (g(u) = constant), the schemes preserve the averaged energy evolution law almost surely, confirming theoretical consistency.
• Numerical experiments show that CFD-I, CFD-II, DG-I, and DG-II schemes exhibit linear growth of averaged energy that matches the exact solution, confirming energy preservation.
• In contrast, CFD-CNM and DG-CNM schemes fail to preserve the averaged energy evolution law, showing non-linear and faster energy growth.
• The temporal convergence order of all six schemes (CFD-I, CFD-II, CFD-CNM, DG-I, DG-II, DG-CNM) is approximately one, as evidenced by log-log plots of error vs. step size.
• The averaged energy grows faster when g(u) = u than when g(u) = sin(u), which is consistent with the theoretical growth rate λ²/2 Tr(Q) for additive noise.