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, undergoing large deformations. 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逐步时间逼近方法。本文采用的技术是极小化移动方法的推广，使用两个时间尺度：一个用于速度，另一个（可能更大）用于加速度。主要应用领域为弹性动力学，即经历大变形的所谓广义固体。演化过程遵循潜在的变分结构，该结构通过逐步极小化加以利用。我们证明，对于一大类（弹性）能量，所提出的格式是稳定的；允许最高阶非线性项的存在。若最高阶项可假设为线性，我们证明极限解是正则的，并且极小化移动方法以最优线性速率收敛。因此，本文将数值时间步极小化方法扩展至双曲问题领域。