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Generic Numerical Analysis of Stochastic Reaction Diffusion Model with applications in excitable media

Yahya Alnashri, Hasan Alzubaidi

TL;DR

The paper addresses numerical analysis for stochastic reaction-diffusion equations with multiplicative noise driving traveling waves in excitable media. It introduces the gradient discretisation method (GDM) as a unifying framework to obtain convergence for a broad class of schemes, proving the existence of weak martingale solutions and deriving discrete energy estimates that ensure stability. Complementary finite-volume tests using the hybrid mimetic mixed (HMM) method examine how multiplicative noise affects traveling waves, revealing phenomena such as wave propagation failure and wave backfiring. The results provide a theoretically sound and practically applicable approach for simulating noisy RD systems and assessing noise-driven wave dynamics in excitable media.

Abstract

The stochastic reaction-diffusion model driven by a multiplicative noise is examined. We construct the gradient discretisation method (GDM), an abstract framework combining several numerical method families. The paper provides the discretisation and proves the convergence of the approximate schemes using a compactness argument that works under natural assumptions on data. We also investigate, using a finite volume method, known as the hybrid mixed mimetic (HMM) approach, the effects of multiplicative noise on the dynamics of the travelling waves in the excitable media displayed by the model. Particularly, we consider how sufficiently high noise can cause waves to backfire or fail to propagate.

Generic Numerical Analysis of Stochastic Reaction Diffusion Model with applications in excitable media

TL;DR

The paper addresses numerical analysis for stochastic reaction-diffusion equations with multiplicative noise driving traveling waves in excitable media. It introduces the gradient discretisation method (GDM) as a unifying framework to obtain convergence for a broad class of schemes, proving the existence of weak martingale solutions and deriving discrete energy estimates that ensure stability. Complementary finite-volume tests using the hybrid mimetic mixed (HMM) method examine how multiplicative noise affects traveling waves, revealing phenomena such as wave propagation failure and wave backfiring. The results provide a theoretically sound and practically applicable approach for simulating noisy RD systems and assessing noise-driven wave dynamics in excitable media.

Abstract

The stochastic reaction-diffusion model driven by a multiplicative noise is examined. We construct the gradient discretisation method (GDM), an abstract framework combining several numerical method families. The paper provides the discretisation and proves the convergence of the approximate schemes using a compactness argument that works under natural assumptions on data. We also investigate, using a finite volume method, known as the hybrid mixed mimetic (HMM) approach, the effects of multiplicative noise on the dynamics of the travelling waves in the excitable media displayed by the model. Particularly, we consider how sufficiently high noise can cause waves to backfire or fail to propagate.
Paper Structure (7 sections, 7 theorems, 78 equations, 3 figures, 2 tables)

This paper contains 7 sections, 7 theorems, 78 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Approximate Scheme
  4. Discrete Energy Estimates
  5. Main Results
  6. Numerical results
  7. Conclusion

Key Result

Lemma 4.1

Let $v$ be a solution to the approximate problem gs-rm-weak. Then, for some positive constant $C$, we have

Figures (3)

  • Figure 6.1: Samples of the various 2D meshes
  • Figure 6.2: Travelling wave solutions on the range $[-5,5]^2$, taking at levels of noise $\rho=0$, $\rho=1$, $\rho=5$ and $\rho=10$, with parameters chosen as $\delta t=0.001$, $T=1$, and $c=0.3$.
  • Figure 6.3: Travelling wave solutions on the range $[-5,5]^2$, taking at levels of noise $\rho=0$, $\rho=1$, $\rho=2$ and $\rho=3$, with parameters chosen as $\delta t=0.001$, $T=1$, and $c=0.3$.

Theorems & Definitions (18)

  • Definition 2.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 8 more