A second-order efficientL-stable numerical method for space fractional reaction–diffusion equations

2018 ◽  
Vol 95 (6-7) ◽  
pp. 1408-1422 ◽  
Author(s):  
M. Yousuf
Keyword(s):  
Numerical Method ◽  
Reaction Diffusion ◽  
Second Order ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Stable Numerical Method ◽  
Fractional Reaction

scholarly journals Splitting spectral element method for fractional reaction-diffusion equations

2020 ◽  
Vol 14 ◽  
pp. 174830262096670
Author(s):  
Qi Li ◽  
Fangying Song
Keyword(s):  
Spectral Element Method ◽  
Operator Splitting ◽  
Reaction Diffusion ◽  
Spectral Element ◽  
Second Order ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Nonlinear Solver ◽  
Fractional Reaction ◽  
Element Method

In this paper, we propose a second-order operator splitting spectral element method for solving fractional reaction-diffusion equations. In order to achieve a fast second-order scheme in time, we decompose the original equation into linear and nonlinear sub-equations, and combine a quarter-time nonlinear solver and a half-time linear solver followed by final quarter-time nonlinear solver. The spatial discretization is eigen-decomposition based on spectral element method. Since this method gives a full diagonal representation of the fractional operator and gets an exponential convergence in space. We have an accurate and efficient approach for solving spacial fractional reaction-diffusion equations. Some numerical experiments are carried out to demonstrate the accuracy and efficiency of this method. Finally, we apply the proposed method to investigate the effect of the fractional order in the fractional reaction-diffusion equations.

scholarly journals On the Long Time Simulation of Reaction-Diffusion Equations with Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Dongfang Li ◽  
Chengjian Zhang
Keyword(s):  
Numerical Method ◽  
Numerical Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Open Problems ◽  
Time Simulation ◽  
Long Time ◽  
Continuous Problem ◽  
Equations With Delay

For a consistent numerical method to be practically useful, it is widely accepted that it must preserve the asymptotic stability of the original continuous problem. However, in this study, we show that it may lead to unreliable numerical solutions in long time simulation even if a classical numerical method gives a larger stability region than that of the original continuous problem. Some numerical experiments on the reaction-diffusion equations with delay are presented to confirm our findings. Finally, some open problems on the subject are proposed.

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