Taking insight from the theory of general relativity, where space and time are treated on the same footing, we develop a novel geometric variational discretization for second order initial value problems (IVPs). By discretizing the dynamics along a world-line parameter, instead of physical time directly, we retain manifest translation symmetry and conservation of the associated continuum Noether charge. A non-equidistant time discretization emerges dynamically, realizing a form of automatic adaptive mesh refinement (AMR), guided by the system symmetries. Using appropriately regularized summation by parts finite difference operators, the continuum Noether charge, defined via the Killing vector associated with translation symmetry, is shown to be exactly preserved in the interior of the simulated time interval. The convergence properties of the approach are demonstrated with two explicit examples.

翻译：受广义相对论中时空统一处理思想的启发，我们针对二阶初始值问题(IVPs)提出了一种新型几何变分离散化方法。通过沿世界线参数对动力学过程进行离散化（而非直接采用物理时间），我们保持了显式平移对称性及相应连续诺特荷的守恒性。动力学过程自然产生非等距时间离散化，在系统对称性引导下实现了自适应网格加密(AMR)的自动形式。采用适当正则化的分部求和有限差分算子后，由平移对称性对应的基灵矢量定义的连续诺特荷，在模拟时间区间内部被证明能够精确保持。通过两个显式算例验证了该方法的收敛特性。