Computing Stochastic Traveling Waves

Gabriel James Lord, Vera Thuemmler

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

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.
Original languageEnglish
Pages (from-to)B24-B43
Number of pages20
JournalSIAM Journal on Scientific Computing
Volume34
Issue number1
DOIs
Publication statusPublished - 2012

Keywords

  • stochastic traveling waves
  • Stochastic partial differential equations
  • stochastic partial differential algebraic equations
  • Nagumo equation
  • wave speed

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