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We study step-wise time approximations of non-linear hyperbolic initial valueproblems. The technique used here is a generalization of the minimizingmovements method, using two time-scales: one for velocity, the other(potentially much larger) for acceleration. The main applications are fromelastodynamics namely so-called generalized solids, undergoing largedeformations. The evolution follows an underlying variational structureexploited by step-wise minimisation.  We show for a large family of (elastic) energies that the introduced schemeis stable; allowing for non-linearities of highest order. If the highest ordercan assumed to be linear, we show that the limit solutions are regular and thatthe minimizing movements scheme converges with optimal linear rate. Thus thiswork extends numerical time-step minimization methods to the realm ofhyperbolic problems.

Stability and convergence of in time approximations of hyperbolic elastodynamics via stepwise minimization