Splitting schemes for FitzHugh–Nagumo stochastic partial differential equations

Charles-Edouard Bréhier; David Cohen; Giuseppe Giordano

doi:10.3934/dcdsb.2023094

Article Contents

2024, Volume 29, Issue 1: 214-244. Doi: 10.3934/dcdsb.2023094

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Splitting schemes for FitzHugh–Nagumo stochastic partial differential equations

1.
Université de Pau et des Pays de l'Adour, E2S UPPA, CNRS, LMAP, Pau, France
2.
Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-41296 Gothenburg, Sweden
3.
Department of Mathematics, University of Salerno, I-84084 Fisciano, Italy

^*Corresponding author: Charles-Edouard Bréhier
^*Corresponding author: Charles-Edouard Bréhier

Received: January 2023

Early access: May 2023

Published: January 2024

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Abstract

We design and study splitting integrators for the temporal discretization of the stochastic FitzHugh–Nagumo system. This system is a model for signal propagation in nerve cells where the voltage variable is the solution of a one-dimensional parabolic PDE with a cubic nonlinearity driven by additive space-time white noise. We first show that the numerical solutions have finite moments. We then prove that the splitting schemes have, at least, the strong rate of convergence $ 1/4 $. Finally, numerical experiments illustrating the performance of the splitting schemes are provided.

Keywords:

FitzHugh–Nagumo equation,
stochastic partial differential equations,
splitting schemes,
strong error estimates.

Mathematics Subject Classification: Primary: 60H35, 65C30, 60H15.

Citation:

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Figure 1. Space-time evolution plots of $ u $ and $ v $ using the Lie–Trotter splitting schemes LTexact, LTexpo, and LTimp

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Figure 2. Mean-square errors as a function of the time step: Lie–Trotter splitting schemes, $ \phi_\tau = \phi_\tau^{\rm L}\circ\phi_\tau^{\rm NL} $ (top) and $ \phi_\tau = \phi_\tau^{\rm NL}\circ\phi_\tau^{\rm L} $ (bottom), ($ \diamond $ for LTexact, $ \square $ for LTexpo, ☆ for LTimp). The dotted lines have slopes $ 1/2 $ and $ 1/4 $

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