Implementation of a Wiener Chaos Expansion Method for the Numerical Solution of the Stochastic Generalized Kuramoto-Sivashinsky Equation driven by Brownian motion forcing

TL;DR

This paper develops a Wiener Chaos Expansion method to numerically solve the stochastic generalized Kuramoto-Sivashinsky equation driven by Brownian motion, demonstrating its accuracy and effectiveness compared to semi-analytical solutions.

Contribution

The paper introduces a novel Wiener Chaos Expansion approach for solving the stochastic generalized Kuramoto-Sivashinsky equation, with detailed error analysis and validation.

Findings

Absolute error is proportional to time with a small constant.

Relative error is approximately 1% or less.

WCE-based solutions are effective for stochastic evolution equations.

Abstract

Numerical computations based on the Wiener Chaos Expansion (WCE) are carried out to approximate the solutions of the stochastic generalized Kuramoto--Sivashinsky (SgKS) equation driven by Brownian motion forcing. In the assessment of the accuracy of the WCE based approximate numerical solutions, the WCE based solutions are contrasted with semi-analytical solutions, and the absolute and relative errors are evaluated. It is found that the absolute error is $O(\varsigma t)$, where $\varsigma$ is small constant and $t$ is the time variabe; and the relative error is order $10^{-2}$ or less. This demonstrates that numerical methods based on the WCE are powerful tools to solve the SgKS equation or other related stochastic evolution equations.