An integration factor method for stochastic and stiff reaction–diffusion systems

Catherine Ta,Dongyong Wang,Qing Nie

doi:10.1016/j.jcp.2015.04.028

Catherine Ta, Dongyong Wang + Show 1 more

Open Access

https://doi.org/10.1016/j.jcp.2015.04.028

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Abstract

Stochastic effects are often present in the biochemical systems involving reactions and diffusions. When the reactions are stiff, existing numerical methods for stochastic reaction diffusion equations require either very small time steps for any explicit schemes or solving large nonlinear systems at each time step for the implicit schemes. Here we present a class of semi-implicit integration factor methods that treat the diffusion term exactly and reaction implicitly for a system of stochastic reaction–diffusion equations. Our linear stability analysis shows the advantage of such methods for both small and large amplitudes of noise. Direct use of the method to solving several linear and nonlinear stochastic reaction–diffusion equations demonstrates good accuracy, efficiency, and stability properties. This new class of methods, which are easy to implement, will have broader applications in solving stochastic reaction–diffusion equations arising from models in biology and physical sciences.

Highlights

Complex patterns can be extensively found in nature, from the skin of zebrafish to the disposition of feather buds in chicks and hair follicles in mice
The approach is based on the semi-Implicit Integration Factor (IIF) methods [23] [24] [25] [26], which has been found to be effective for the stiff reaction-diffusion equations with better stability constraints imposed on the time steps associated with both reaction and diffusion
Through choosing different values of a, which corresponds to the size of diffusion, and different values of b, which corresponds to the strength and stiffness of reactions, we evaluate the convergence and stability of IIF-Maruyama methods

Summary

Introduction

Complex patterns can be extensively found in nature, from the skin of zebrafish to the disposition of feather buds in chicks and hair follicles in mice. The approach is based on the semi-Implicit Integration Factor (IIF) methods [23] [24] [25] [26], which has been found to be effective for the stiff reaction-diffusion equations with better stability constraints imposed on the time steps associated with both reaction and diffusion. Because of the nice decoupling properties in the IIF method, we will treat the deterministic diffusion and reaction terms in a similar fashion [23], while dealing with the stochastic term explicitly as in the Euler Maruyama method [27] We compare this approach with constructed schemes whose main difference is in the treatment of the deterministic part of the equation, which can be approximated using ETD, Crank-Nicolson, or Implicit Euler methods. We use this approach to study a nonlinear activator-substrate system of two diffusion species and lastly, make our conclusion

Implicit Integration Factor methods

Numerical Simulations

Tests on Stochastic Ordinary Differential Equations

Tests on Stochastic Partial Differential Equations

A Turing patterning system with noise

Conclusions and Discussion