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Volume 14, Issue 6
Variable Time-Step θ-Scheme for Nonlinear Second Order Evolution Inclusion

Krzysztof Bartosz

Int. J. Numer. Anal. Mod., 14 (2017), pp. 842-868.

Published online: 2017-10

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  • Abstract

We deal with a multivalued second order dynamical system involving a Clarke subdifferential of a locally Lipschitz functional. We apply a time discretization procedure to construct a sequence of solutions to a family of the approximate problems and show its convergence to a solution of the exact problem as the time step size vanishes. We consider a nonautonomous problem in which both the viscosity and the multivalued operators depend on time explicitly. The time discretization method we use, is the $\theta$-scheme with $\theta \in [\frac{1}{2}, 1]$, thus, in particular, the Crank-Nicolson scheme and the implicit Euler scheme are included. We apply our result to a class of dynamic hemivariational inequalities.

  • Keywords

Clarke subdifferential, hemivariational inequality, second order inclusion, time discretization, numerical methods.

  • AMS Subject Headings

35G55, 74H15, 74H20, 65J15, 35R70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-14-842, author = {Krzysztof Bartosz}, title = {Variable Time-Step θ-Scheme for Nonlinear Second Order Evolution Inclusion}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2017}, volume = {14}, number = {6}, pages = {842--868}, abstract = {

We deal with a multivalued second order dynamical system involving a Clarke subdifferential of a locally Lipschitz functional. We apply a time discretization procedure to construct a sequence of solutions to a family of the approximate problems and show its convergence to a solution of the exact problem as the time step size vanishes. We consider a nonautonomous problem in which both the viscosity and the multivalued operators depend on time explicitly. The time discretization method we use, is the $\theta$-scheme with $\theta \in [\frac{1}{2}, 1]$, thus, in particular, the Crank-Nicolson scheme and the implicit Euler scheme are included. We apply our result to a class of dynamic hemivariational inequalities.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/10483.html} }
TY - JOUR T1 - Variable Time-Step θ-Scheme for Nonlinear Second Order Evolution Inclusion AU - Krzysztof Bartosz JO - International Journal of Numerical Analysis and Modeling VL - 6 SP - 842 EP - 868 PY - 2017 DA - 2017/10 SN - 14 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/10483.html KW - Clarke subdifferential, hemivariational inequality, second order inclusion, time discretization, numerical methods. AB -

We deal with a multivalued second order dynamical system involving a Clarke subdifferential of a locally Lipschitz functional. We apply a time discretization procedure to construct a sequence of solutions to a family of the approximate problems and show its convergence to a solution of the exact problem as the time step size vanishes. We consider a nonautonomous problem in which both the viscosity and the multivalued operators depend on time explicitly. The time discretization method we use, is the $\theta$-scheme with $\theta \in [\frac{1}{2}, 1]$, thus, in particular, the Crank-Nicolson scheme and the implicit Euler scheme are included. We apply our result to a class of dynamic hemivariational inequalities.

Krzysztof Bartosz. (2017). Variable Time-Step θ-Scheme for Nonlinear Second Order Evolution Inclusion. International Journal of Numerical Analysis and Modeling. 14 (6). 842-868. doi:
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