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

翻译：我们研究非线性双曲型初值问题的步进时间逼近方法。本文采用的技术是“最小化移动法”的推广，利用两个时间尺度：一个用于速度，另一个（可能显著更大）用于加速度。主要应用来自弹性动力学，即所谓的广义固体。演化过程遵循由步进极小化所利用的底层变分结构。我们证明，对于一大类（弹性）能量，所引入的格式是稳定的，可容许最高阶非线性。若最高阶可假设为线性，我们证明极限解是正则的，且最小化移动法以最优线性速率收敛。因此，本文将数值时间步最小化方法推广至双曲型问题领域。