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 - <p style="text-align: justify;">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.</p>