Variable Time-Step θ-Scheme for Nonlinear Second Order Evolution Inclusion
Krzysztof Bartosz
Int. J. Numer. Anal. Mod., 14 (2017), pp. 842-868
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.
