Volume 10, Issue 2
Error Estimate and Superconvergence of a High-Accuracy Difference Scheme for Solving Parabolic Equations with an Integral Two-Space-Variables Condition

Liping Zhou, Shi Shu & Haiyuan Yu

Adv. Appl. Math. Mech., 10 (2018), pp. 362-389.

Published online: 2018-10

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  • Abstract

The initial-boundary value problems for parabolic equations with nonlocal conditions have been widely applied in various fields. In this work, we firstly build an implicit Euler scheme for an initial-boundary value problem of one dimensional parabolic equations with an integral two-space-variables condition. Then we prove that this scheme can reach the asymptotic optimal error estimate in the maximum norm. Next, the formulas used to approximate the solution derivatives with respect to time and spatial variables are presented, and it is proved for the first time that they have superconvergence. In the end, numerical experiments demonstrate the theoretical results.

  • Keywords

Parabolic equation, integral condition, finite difference scheme, asymptotic optimal error estimate, superconvergence.

  • AMS Subject Headings

65M06, 65M12, 65T50

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-10-362, author = {}, title = {Error Estimate and Superconvergence of a High-Accuracy Difference Scheme for Solving Parabolic Equations with an Integral Two-Space-Variables Condition}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {10}, number = {2}, pages = {362--389}, abstract = {The initial-boundary value problems for parabolic equations with nonlocal conditions have been widely applied in various fields. In this work, we firstly build an implicit Euler scheme for an initial-boundary value problem of one dimensional parabolic equations with an integral two-space-variables condition. Then we prove that this scheme can reach the asymptotic optimal error estimate in the maximum norm. Next, the formulas used to approximate the solution derivatives with respect to time and spatial variables are presented, and it is proved for the first time that they have superconvergence. In the end, numerical experiments demonstrate the theoretical results.}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2017-0067}, url = {http://global-sci.org/intro/article_detail/aamm/12216.html} }
TY - JOUR T1 - Error Estimate and Superconvergence of a High-Accuracy Difference Scheme for Solving Parabolic Equations with an Integral Two-Space-Variables Condition JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 362 EP - 389 PY - 2018 DA - 2018/10 SN - 10 DO - http://dor.org/10.4208/aamm.OA-2017-0067 UR - https://global-sci.org/intro/aamm/12216.html KW - Parabolic equation, integral condition, finite difference scheme, asymptotic optimal error estimate, superconvergence. AB - The initial-boundary value problems for parabolic equations with nonlocal conditions have been widely applied in various fields. In this work, we firstly build an implicit Euler scheme for an initial-boundary value problem of one dimensional parabolic equations with an integral two-space-variables condition. Then we prove that this scheme can reach the asymptotic optimal error estimate in the maximum norm. Next, the formulas used to approximate the solution derivatives with respect to time and spatial variables are presented, and it is proved for the first time that they have superconvergence. In the end, numerical experiments demonstrate the theoretical results.
Liping Zhou, Shi Shu & Haiyuan Yu. (2020). Error Estimate and Superconvergence of a High-Accuracy Difference Scheme for Solving Parabolic Equations with an Integral Two-Space-Variables Condition. Advances in Applied Mathematics and Mechanics. 10 (2). 362-389. doi:10.4208/aamm.OA-2017-0067
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