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：本文研究了非线性双曲初值问题的逐步时间近似方法。所采用的技术是最小化移动方法的一种推广，利用了两个时间尺度：一个用于速度，另一个（可能远大于前者）用于加速度。主要应用来自弹性动力学，即所谓的广义固体。演化过程遵循一个潜在变分结构，该结构通过逐步最小化加以利用。我们针对一大类（弹性）能量证明了所提出格式的稳定性，允许最高阶非线性。若最高阶可假设为线性，我们证明极限解是正则的，并且最小化移动方法以最优线性速率收敛。因此，这项工作将数值时间步长最小化方法扩展到了双曲问题的领域。