[Paper Review] Numerical solutions of some hyperbolic stochastic partial differential equations with mixed derivatives including sine-Gordon equation

Henry C. Tuckwell|arXiv (Cornell University)|Aug 2, 2015

Stochastic processes and financial applications28 references3 citations

TL;DR

This paper presents an explicit numerical scheme for solving hyperbolic stochastic partial differential equations (SPDEs) with mixed derivatives, including the sine-Gordon equation, driven by space-time Gaussian white noise. The method accurately captures wave-like structures in both deterministic and stochastic regimes, with solutions showing robust convergence and emergence of coherent dynamics even under zero initial conditions when driven purely by noise, particularly for small noise intensities (σ ≤ 0.25).

ABSTRACT

We consider linear and nonlinear hyperbolic SPDEs with mixed derivatives with additive space-time Gaussian white noise of the form $Y_{xt}=F(Y) + σW_{xt}.$ Such equations, which transform to linear and nonlinear wave equations, including Klein-Gordon, Liouville's and the sine-Gordon equation, are related to what Zimmerman (1972) called a diffusion equation. An explicit numerical scheme is employed in both deterministic and stochastic examples. The scheme is checked for accuracy against known exact analytical solutions for deterministic equations. In the stochastic case with $F=0$, solutions yield sample paths for the Brownian sheet whose statistics match well exact values. Generally the boundary conditions are chosen to be initial values $Y(x,0)$ and boundary values $Y(0,t)$ on the quarter-plane or subsets thereof, which have been shown to lead to existence and uniqueness of solutions. For the linear case solutions are compared at various grid sizes and wave-like solutions were found, with and without noise, for non-zero initial and boundary conditions. Surprisingly, wave-like structures seemed to emerge with zero initial and boundary conditions and purely noise source terms with no signal. Equations considered with nonlinear $F$ included quadratic and cubic together with the sine-Gordon equation. For the latter, wave-like structures were apparent with $σ\le 0.25$ but they tended to be shattered at larger values of $σ$. Previous work on stochastic sine-Gordon equations is briefly reviewed.

Motivation & Objective

• To develop and validate a numerical scheme for solving hyperbolic SPDEs with mixed partial derivatives and additive space-time white noise.
• To investigate the emergence of wave-like behavior in stochastic SPDEs, particularly when initial and boundary conditions are zero but noise is present.
• To assess the accuracy and convergence of the numerical method by comparing with known analytical solutions for deterministic cases.
• To explore the behavior of nonlinear SPDEs, including the sine-Gordon equation, under varying noise intensities (σ).
• To examine the relationship between solutions of the mixed-derivative SPDE form and classical wave equations via coordinate transformation.

Proposed method

• An explicit finite difference scheme is applied to solve the SPDE of the form $ Y_{xt} = F(Y) + \sigma W_{xt} $, where $ W_{xt} $ is space-time white noise.
• The scheme is validated against exact analytical solutions for deterministic linear and nonlinear cases, including the wave equation and Liouville’s equation.
• For the stochastic case, the method simulates sample paths of the Brownian sheet, with statistical properties compared to theoretical expectations.
• Numerical experiments are conducted on the quarter-plane domain $[0, T] \times [0, X]$ with constant initial and boundary conditions $ Y(x,0) $ and $ Y(0,t) $, ensuring solution uniqueness.
• Grid refinement studies are performed to assess convergence, with comparisons between solutions at different grid sizes (e.g., 400×400 and 800×800) and varying numbers of trials.
• The method is applied to nonlinear cases including quadratic, cubic, and sine-Gordon $ F(Y) $, with noise intensity $ \sigma $ systematically varied.

Experimental results

Research questions

• RQ1 Can an explicit numerical scheme accurately simulate hyperbolic SPDEs with mixed derivatives and additive space-time white noise?
• RQ2 Do wave-like structures emerge in the solution when both initial and boundary conditions are zero but the system is driven solely by noise?
• RQ3 How does the noise intensity $ \sigma $ affect the stability and coherence of wave-like solutions in the stochastic sine-Gordon equation?
• RQ4 To what extent does the numerical solution converge to the deterministic solution as $ \sigma \to 0 $, and how does grid resolution influence this?
• RQ5 What is the relationship between solutions of the mixed-derivative SPDE $ Y_{xt} = F(Y) + \sigma W_{xt} $ and classical wave equations $ \phi_{tt} - \phi_{xx} = G(\phi) $?

Key findings

• The explicit numerical scheme accurately reproduces known analytical solutions for deterministic linear and nonlinear SPDEs, confirming its reliability.
• For the linear SPDE with $ F = 0 $, the simulated sample paths of the Brownian sheet matched theoretical statistical properties, validating the noise integration.
• Wave-like structures emerged even with zero initial and boundary conditions when driven by noise, particularly for small $ \sigma $, indicating intrinsic noise-driven pattern formation.
• For the sine-Gordon equation, coherent wave-like structures were stable for $ \sigma \leq 0.25 $, but became fragmented at higher noise intensities.
• Grid refinement (e.g., 800×800 vs. 400×400) improved the definition of wave structures and reduced standard deviation, indicating convergence with finer resolution.
• Good agreement between the mean of stochastic simulations and the deterministic solution was observed for $ \sigma = 0.1 $, especially with 800×800 grids and 50 trials, suggesting numerical stability for small noise.