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Volume 42, Issue 2
Stability and Convergence of Stepsize-Dependent Linear Multistep Methods for Nonlinear Dissipative Evolution Equations in Banach Space

Wansheng Wang

DOI: 10.4208/jcm.2207-m2021-0064

J. Comp. Math., 42 (2024), pp. 337-354.

Published online: 2024-01

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  • Abstract

Stability and global error bounds are studied for a class of stepsize-dependent linear multistep methods for nonlinear evolution equations governed by $ω$-dissipative vector fields in Banach space. To break through the order barrier $p ≤ 1$ of unconditionally contractive linear multistep methods for dissipative systems, strongly dissipative systems are introduced. By employing the error growth function of the methods, new contractivity and convergence results of stepsize-dependent linear multistep methods on infinite integration intervals are provided for strictly dissipative systems $(ω < 0)$ and strongly dissipative systems. Some applications of the main results to several linear multistep methods, including the trapezoidal rule, are supplied. The theoretical results are also illustrated by a set of numerical experiments.

  • Keywords

Nonlinear evolution equation, Linear multistep methods, ω-dissipative operators, Stability, Convergence, Banach space.

  • AMS Subject Headings

65J15, 65M12, 65M15, 65J08

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-337, author = {Wang , Wansheng}, title = {Stability and Convergence of Stepsize-Dependent Linear Multistep Methods for Nonlinear Dissipative Evolution Equations in Banach Space }, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {42}, number = {2}, pages = {337--354}, abstract = {

Stability and global error bounds are studied for a class of stepsize-dependent linear multistep methods for nonlinear evolution equations governed by $ω$-dissipative vector fields in Banach space. To break through the order barrier $p ≤ 1$ of unconditionally contractive linear multistep methods for dissipative systems, strongly dissipative systems are introduced. By employing the error growth function of the methods, new contractivity and convergence results of stepsize-dependent linear multistep methods on infinite integration intervals are provided for strictly dissipative systems $(ω < 0)$ and strongly dissipative systems. Some applications of the main results to several linear multistep methods, including the trapezoidal rule, are supplied. The theoretical results are also illustrated by a set of numerical experiments.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2207-m2021-0064}, url = {http://global-sci.org/intro/article_detail/jcm/22883.html} }
TY - JOUR T1 - Stability and Convergence of Stepsize-Dependent Linear Multistep Methods for Nonlinear Dissipative Evolution Equations in Banach Space AU - Wang , Wansheng JO - Journal of Computational Mathematics VL - 2 SP - 337 EP - 354 PY - 2024 DA - 2024/01 SN - 42 DO - http://doi.org/10.4208/jcm.2207-m2021-0064 UR - https://global-sci.org/intro/article_detail/jcm/22883.html KW - Nonlinear evolution equation, Linear multistep methods, ω-dissipative operators, Stability, Convergence, Banach space. AB -

Stability and global error bounds are studied for a class of stepsize-dependent linear multistep methods for nonlinear evolution equations governed by $ω$-dissipative vector fields in Banach space. To break through the order barrier $p ≤ 1$ of unconditionally contractive linear multistep methods for dissipative systems, strongly dissipative systems are introduced. By employing the error growth function of the methods, new contractivity and convergence results of stepsize-dependent linear multistep methods on infinite integration intervals are provided for strictly dissipative systems $(ω < 0)$ and strongly dissipative systems. Some applications of the main results to several linear multistep methods, including the trapezoidal rule, are supplied. The theoretical results are also illustrated by a set of numerical experiments.

Wansheng Wang. (2024). Stability and Convergence of Stepsize-Dependent Linear Multistep Methods for Nonlinear Dissipative Evolution Equations in Banach Space . Journal of Computational Mathematics. 42 (2). 337-354. doi:10.4208/jcm.2207-m2021-0064
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