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Volume 16, Issue 3
Conservative Three-Level Linearized Finite Difference Schemes for the Fisher Equation and Its Maximum Error Estimates

Guang-Hua Gao, Biao Ge & Zhi-Zhong Sun

DOI: 10.4208/nmtma.OA-2022-0123

Numer. Math. Theor. Meth. Appl., 16 (2023), pp. 634-667.

Published online: 2023-08

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  • Abstract

A three-level linearized difference scheme for solving the Fisher equation is firstly proposed in this work. It has the good property of discrete conservative energy. By the discrete energy analysis and mathematical induction method, it is proved to be uniquely solvable and unconditionally convergent with the second-order accuracy in both time and space. Then another three-level linearized compact difference scheme is derived along with its discrete energy conservation law, unique solvability and unconditional convergence of order two in time and four in space. The resultant schemes preserve the maximum bound principle. The analysis techniques for convergence used in this paper also work for the Euler scheme, the Crank-Nicolson scheme and others. Numerical experiments are carried out to verify the computational efficiency, conservative law and the maximum bound principle of the proposed difference schemes.

  • Keywords

Fisher equation, linearized difference scheme, solvability, convergence, conservation.

  • AMS Subject Headings

65M06, 65M12, 65M15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-16-634, author = {Gao , Guang-HuaGe , Biao and Sun , Zhi-Zhong}, title = {Conservative Three-Level Linearized Finite Difference Schemes for the Fisher Equation and Its Maximum Error Estimates}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2023}, volume = {16}, number = {3}, pages = {634--667}, abstract = {

A three-level linearized difference scheme for solving the Fisher equation is firstly proposed in this work. It has the good property of discrete conservative energy. By the discrete energy analysis and mathematical induction method, it is proved to be uniquely solvable and unconditionally convergent with the second-order accuracy in both time and space. Then another three-level linearized compact difference scheme is derived along with its discrete energy conservation law, unique solvability and unconditional convergence of order two in time and four in space. The resultant schemes preserve the maximum bound principle. The analysis techniques for convergence used in this paper also work for the Euler scheme, the Crank-Nicolson scheme and others. Numerical experiments are carried out to verify the computational efficiency, conservative law and the maximum bound principle of the proposed difference schemes.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2022-0123}, url = {http://global-sci.org/intro/article_detail/nmtma/21961.html} }
TY - JOUR T1 - Conservative Three-Level Linearized Finite Difference Schemes for the Fisher Equation and Its Maximum Error Estimates AU - Gao , Guang-Hua AU - Ge , Biao AU - Sun , Zhi-Zhong JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 634 EP - 667 PY - 2023 DA - 2023/08 SN - 16 DO - http://doi.org/10.4208/nmtma.OA-2022-0123 UR - https://global-sci.org/intro/article_detail/nmtma/21961.html KW - Fisher equation, linearized difference scheme, solvability, convergence, conservation. AB -

A three-level linearized difference scheme for solving the Fisher equation is firstly proposed in this work. It has the good property of discrete conservative energy. By the discrete energy analysis and mathematical induction method, it is proved to be uniquely solvable and unconditionally convergent with the second-order accuracy in both time and space. Then another three-level linearized compact difference scheme is derived along with its discrete energy conservation law, unique solvability and unconditional convergence of order two in time and four in space. The resultant schemes preserve the maximum bound principle. The analysis techniques for convergence used in this paper also work for the Euler scheme, the Crank-Nicolson scheme and others. Numerical experiments are carried out to verify the computational efficiency, conservative law and the maximum bound principle of the proposed difference schemes.

Gao , Guang-HuaGe , Biao and Sun , Zhi-Zhong. (2023). Conservative Three-Level Linearized Finite Difference Schemes for the Fisher Equation and Its Maximum Error Estimates. Numerical Mathematics: Theory, Methods and Applications. 16 (3). 634-667. doi:10.4208/nmtma.OA-2022-0123
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