A Linearly Implicit Splitting Method for Solving Time Dependent Semilinear Reaction-Diffusion Systems

2020 ◽  
pp. 129-141
Author(s):  
Carmelo Clavero ◽  
Juan Carlos Jorge
Keyword(s):  
Splitting Method ◽  
Reaction Diffusion ◽  
Time Dependent ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems

Turing and Benjamin–Feir instability mechanisms in non-autonomous systems

2020 ◽  
Vol 476 (2238) ◽  
pp. 20200003 ◽  
Author(s):  
Robert A. Van Gorder
Keyword(s):  
Autonomous Systems ◽  
Reaction Diffusion ◽  
Reaction Rates ◽  
Time Dependent ◽  
Model Parameters ◽  
Time Varying ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Dependent Model ◽  
Spatio Temporal

The Turing and Benjamin–Feir instabilities are two of the primary instability mechanisms useful for studying the transition from homogeneous states to heterogeneous spatial or spatio-temporal states in reaction–diffusion systems. We consider the case when the underlying reaction–diffusion system is non-autonomous or has a base state which varies in time, as in this case standard approaches, which rely on temporal eigenvalues, break down. We are able to establish respective criteria for the onset of each instability using comparison principles, obtaining inequalities which involve the in general time-dependent model parameters and their time derivatives. In the autonomous limit where the base state is constant in time, our results exactly recover the respective Turing and Benjamin–Feir conditions known in the literature. Our results make the Turing and Benjamin–Feir analysis amenable for a wide collection of applications, and allow one to better understand instabilities emergent due to a variety of non-autonomous mechanisms, including time-varying diffusion coefficients, time-varying reaction rates, time-dependent transitions between reaction kinetics and base states which change in time (such as heteroclinic connections between unique steady states, or limit cycles), to name a few examples.

scholarly journals Stochastic operator-splitting method for reaction-diffusion systems

2012 ◽  
Vol 137 (18) ◽  
pp. 184102 ◽  
Author(s):  
TaiJung Choi ◽  
Mano Ram Maurya ◽  
Daniel M. Tartakovsky ◽  
Shankar Subramaniam
Keyword(s):  
Operator Splitting ◽  
Splitting Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Operator Splitting Method ◽  
Stochastic Operator ◽  
Diffusion Systems

scholarly journals A novel time efficient structure-preserving splitting method for the solution of two-dimensional reaction-diffusion systems

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Nauman Ahmed ◽  
Alper Korkmaz ◽  
M. Rafiq ◽  
Dumitru Baleanu ◽  
Ali Saleh Alshomrani ◽  
...  
Keyword(s):  
Splitting Method ◽  
Reaction Diffusion ◽  
Two Dimensional ◽  
Structure Preserving ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems
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