Improved Difference Scheme for a Singularly Perturbed Parabolic Reaction-Diffusion Equation with Discontinuous Initial Condition

2009 ◽  
pp. 116-127
Author(s):  
Grigory Shishkin
Keyword(s):  
Diffusion Equation ◽  
Difference Scheme ◽  
Reaction Diffusion ◽  
Singularly Perturbed ◽  
Reaction Diffusion Equation ◽  
Initial Condition

scholarly journals A new parallel difference algorithm based on improved alternating segment Crank–Nicolson scheme for time fractional reaction–diffusion equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xiaozhong Yang ◽  
Xu Dang
Keyword(s):  
Parallel Computing ◽  
Diffusion Equation ◽  
Difference Scheme ◽  
Reaction Diffusion ◽  
Difference Schemes ◽  
Reaction Diffusion Equation ◽  
Stability And Convergence ◽  
Parallel Difference ◽  
Fractional Reaction ◽  
Crank Nicolson

Abstract The fractional reaction–diffusion equation has profound physical and engineering background, and its rapid solution research is of important scientific significance and engineering application value. In this paper, we propose a parallel computing method of mixed difference scheme for time fractional reaction–diffusion equation and construct a class of improved alternating segment Crank–Nicolson (IASC–N) difference schemes. The class of parallel difference schemes constructed in this paper, based on the classical Crank–Nicolson (C–N) scheme and classical explicit and implicit schemes, combines with alternating segment techniques. We illustrate the unique existence, unconditional stability, and convergence of the parallel difference scheme solution theoretically. Numerical experiments verify the theoretical analysis, which shows that the IASC–N scheme has second order spatial accuracy and $2-\alpha $ 2 − α order temporal accuracy, and the computational efficiency is greatly improved compared with the implicit scheme and C–N scheme. The IASC–N scheme has ideal computation accuracy and obvious parallel computing properties, showing that the IASC–N parallel difference method is effective for solving time fractional reaction–diffusion equation.

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