Motivated by the fact that both the classical and quantum description of nature rest on causality and a variational principle, we develop a novel and highly versatile discretization prescription for classical initial value problems (IVPs). It is based on an optimization (action) functional with doubled degrees of freedom, which is discretized using a single regularized summation-by-parts (SBP) operator. Formulated as optimization task it allows us to obtain classical trajectories without the need to derive an equation of motion. The novel regularization we develop in this context is inspired by the weak imposition of initial data, often deployed in the modern treatment of IVPs and is implemented using affine coordinates. We demonstrate numerically the stability, accuracy and convergence properties of our approach in systems with classical equations of motion featuring both first and second order derivatives in time. onvergence properties of our approach in systems with classical equations of motion featuring both first and second order derivatives in time.
翻译:自然的古典和量子描述都以因果关系和变异原则为基础,我们为古典初始价值问题(IVPs)制定了一套新颖和高度多用途的分解处方,其基础是优化(行动)功能,自由度翻倍,使用单一的按部和部分类的常规汇总操作员进行分解。它作为优化设计,使我们能够获得古典轨迹,而不必得出运动的方程式。我们在这方面发展的新式正规化的灵感来自对初步数据的薄弱强制实施,这些数据往往部署在现代的IVPs处理中,并使用直角坐标加以执行。我们用数字展示了我们方法在系统上的稳定、准确性和趋同特性,其典型的动作方程式具有第一和第二级衍生物的特性。我们采用典型的动作方程式,其典型方程式同时具有第一和第二级衍生物的特性。