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Volume 39, Issue 5
Analysis on a Numerical Scheme with Second-Order Time Accuracy for Nonlinear Diffusion Equations

Xia Cui, Guangwei Yuan & Fei Zhao

DOI: 10.4208/jcm.2007-m2020-0058

J. Comp. Math., 39 (2021), pp. 777-800.

Published online: 2021-08

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  • Abstract

A nonlinear fully implicit finite difference scheme with second-order time evolution for nonlinear diffusion problem is studied. The scheme is constructed with two-layer coupled discretization (TLCD) at each time step. It does not stir numerical oscillation, while permits large time step length, and produces more accurate numerical solutions than the other two well-known second-order time evolution nonlinear schemes, the Crank-Nicolson (CN) scheme and the backward difference formula second-order (BDF2) scheme. By developing a new reasoning technique, we overcome the difficulties caused by the coupled nonlinear discrete diffusion operators at different time layers, and prove rigorously the TLCD scheme is uniquely solvable, unconditionally stable, and has second-order convergence in both space and time. Numerical tests verify the theoretical results, and illustrate its superiority over the CN and BDF2 schemes.

  • Keywords

Nonlinear diffusion problem, Nonlinear two-layer coupled discrete scheme, Second-order time accuracy, Property analysis, Unique existence, Convergence.

  • AMS Subject Headings

65M06, 65M12, 65M15

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

cui_xia@iapcm.ac.cn (Xia Cui)

yuan_guangwei@iapcm.ac.cn (Guangwei Yuan)

zhaofei17@gscaep.ac.cn (Fei Zhao)

  • BibTex
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  • TXT
@Article{JCM-39-777, author = {Cui , XiaYuan , Guangwei and Zhao , Fei}, title = {Analysis on a Numerical Scheme with Second-Order Time Accuracy for Nonlinear Diffusion Equations}, journal = {Journal of Computational Mathematics}, year = {2021}, volume = {39}, number = {5}, pages = {777--800}, abstract = {

A nonlinear fully implicit finite difference scheme with second-order time evolution for nonlinear diffusion problem is studied. The scheme is constructed with two-layer coupled discretization (TLCD) at each time step. It does not stir numerical oscillation, while permits large time step length, and produces more accurate numerical solutions than the other two well-known second-order time evolution nonlinear schemes, the Crank-Nicolson (CN) scheme and the backward difference formula second-order (BDF2) scheme. By developing a new reasoning technique, we overcome the difficulties caused by the coupled nonlinear discrete diffusion operators at different time layers, and prove rigorously the TLCD scheme is uniquely solvable, unconditionally stable, and has second-order convergence in both space and time. Numerical tests verify the theoretical results, and illustrate its superiority over the CN and BDF2 schemes.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2007-m2020-0058}, url = {http://global-sci.org/intro/article_detail/jcm/19380.html} }
TY - JOUR T1 - Analysis on a Numerical Scheme with Second-Order Time Accuracy for Nonlinear Diffusion Equations AU - Cui , Xia AU - Yuan , Guangwei AU - Zhao , Fei JO - Journal of Computational Mathematics VL - 5 SP - 777 EP - 800 PY - 2021 DA - 2021/08 SN - 39 DO - http://doi.org/10.4208/jcm.2007-m2020-0058 UR - https://global-sci.org/intro/article_detail/jcm/19380.html KW - Nonlinear diffusion problem, Nonlinear two-layer coupled discrete scheme, Second-order time accuracy, Property analysis, Unique existence, Convergence. AB -

A nonlinear fully implicit finite difference scheme with second-order time evolution for nonlinear diffusion problem is studied. The scheme is constructed with two-layer coupled discretization (TLCD) at each time step. It does not stir numerical oscillation, while permits large time step length, and produces more accurate numerical solutions than the other two well-known second-order time evolution nonlinear schemes, the Crank-Nicolson (CN) scheme and the backward difference formula second-order (BDF2) scheme. By developing a new reasoning technique, we overcome the difficulties caused by the coupled nonlinear discrete diffusion operators at different time layers, and prove rigorously the TLCD scheme is uniquely solvable, unconditionally stable, and has second-order convergence in both space and time. Numerical tests verify the theoretical results, and illustrate its superiority over the CN and BDF2 schemes.

Cui , XiaYuan , Guangwei and Zhao , Fei. (2021). Analysis on a Numerical Scheme with Second-Order Time Accuracy for Nonlinear Diffusion Equations. Journal of Computational Mathematics. 39 (5). 777-800. doi:10.4208/jcm.2007-m2020-0058
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