Computing Stochastic Traveling Waves

• Published in: SIAM journal on scientific computing Vol. 34; no. 1; pp. B24 - B43
• Main Authors: Lord, G. J., Thümmler, V.
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2012
• Subjects: Algebra; Noise; Numerical analysis
• ISSN: 1064-8275; 1095-7197
• DOI: 10.1137/100784734

The aim of this paper is to investigate new numerical methods for computing traveling wave solutions and new ways for estimating characteristic properties such as wave speed for stochastically forced partial differential equations. As a particular example we consider the Nagumo equation with multiplicative noise which we mainly consider in the Stratonovich sense. A standard approach for determining the position and hence speed of a wave is to compute the evolution of a level set. We compare this approach against an alternative where the wave position is found by minimizing the $L^2$ norm against a fixed reference profile. This approach can be used to freeze (or stop) the wave and obtain a stochastic partial differential algebraic equation that we then discretize and solve. Although attractive because it leads to a smaller domain size, it can be numerically unstable due to large convection terms. We compare numerically the different approaches for estimating the wave speed. We use these techniques to investigate the effect of both Itô and Stratonovich noise on the Nagumo equation as correlation length and noise intensity increase.