TY - JOUR T1 - Variable Time-Step θ-Scheme for Nonlinear Second Order Evolution Inclusion 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.