Variable Time-Step θ-Scheme for Nonlinear Second Order Evolution Inclusion

Volume 14, Issue 6

Krzysztof Bartosz

DOI:

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 θ-scheme with θ ∈ [\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.

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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 θ-scheme with θ ∈ [\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 KW - hemivariational inequality KW - second order inclusion KW - time discretization KW - 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 θ-scheme with θ ∈ [\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. (1970). 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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