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Volume 19, Issue 5
First-Order and Second-Order, Chaos-Free, Finite Difference Schemes for Fisher Equation

Geng Sun, Hua-Mo Wu & Li-Er Wang

J. Comp. Math., 19 (2001), pp. 519-530.

Published online: 2001-10

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  • Abstract

A new class of finite difference schemes is constructed for Fisher partial differential equation i.e. the reaction-diffusion equation with stiff source term: $au(1-u)$. These schemes have the properties that they reduce to high fidelity algorithms in the diffusion-free case namely in which the numerical solutions preserve the properties in the exact solutions for arbitrary time step-size and reaction coefficient α>0 and all nonphysical spurious solutions including bifurcations and chaos that normally appear in the standard discrete models of Fisher partial differential equation will not occur. The implicit schemes so developed obtain the numerical solutions by solving a single linear algebraic system at each step. The boundness and asymptotic behaviour of numerical solutions obtained by all these schemes are given. The approach constructing the above schemes can be extended to reaction-diffusion equations with other stiff source terms.  

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@Article{JCM-19-519, author = {Sun , GengWu , Hua-Mo and Wang , Li-Er}, title = {First-Order and Second-Order, Chaos-Free, Finite Difference Schemes for Fisher Equation}, journal = {Journal of Computational Mathematics}, year = {2001}, volume = {19}, number = {5}, pages = {519--530}, abstract = {

A new class of finite difference schemes is constructed for Fisher partial differential equation i.e. the reaction-diffusion equation with stiff source term: $au(1-u)$. These schemes have the properties that they reduce to high fidelity algorithms in the diffusion-free case namely in which the numerical solutions preserve the properties in the exact solutions for arbitrary time step-size and reaction coefficient α>0 and all nonphysical spurious solutions including bifurcations and chaos that normally appear in the standard discrete models of Fisher partial differential equation will not occur. The implicit schemes so developed obtain the numerical solutions by solving a single linear algebraic system at each step. The boundness and asymptotic behaviour of numerical solutions obtained by all these schemes are given. The approach constructing the above schemes can be extended to reaction-diffusion equations with other stiff source terms.  

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9004.html} }
TY - JOUR T1 - First-Order and Second-Order, Chaos-Free, Finite Difference Schemes for Fisher Equation AU - Sun , Geng AU - Wu , Hua-Mo AU - Wang , Li-Er JO - Journal of Computational Mathematics VL - 5 SP - 519 EP - 530 PY - 2001 DA - 2001/10 SN - 19 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9004.html KW - Reaction-diffusion equation, Fidelity algorithm. AB -

A new class of finite difference schemes is constructed for Fisher partial differential equation i.e. the reaction-diffusion equation with stiff source term: $au(1-u)$. These schemes have the properties that they reduce to high fidelity algorithms in the diffusion-free case namely in which the numerical solutions preserve the properties in the exact solutions for arbitrary time step-size and reaction coefficient α>0 and all nonphysical spurious solutions including bifurcations and chaos that normally appear in the standard discrete models of Fisher partial differential equation will not occur. The implicit schemes so developed obtain the numerical solutions by solving a single linear algebraic system at each step. The boundness and asymptotic behaviour of numerical solutions obtained by all these schemes are given. The approach constructing the above schemes can be extended to reaction-diffusion equations with other stiff source terms.  

Geng Sun, Hua-Mo Wu & Li-Er Wang. (1970). First-Order and Second-Order, Chaos-Free, Finite Difference Schemes for Fisher Equation. Journal of Computational Mathematics. 19 (5). 519-530. doi:
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