A new discretization technique for initial value problems based on a variational principle

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 functional with doubled degrees of freedom, which is discretized using a single regularized summation-by-parts (SBP) operator. The variational principle provides a straight forward recipe to formulate the corresponding optimization functional for a large class of differential equations. The novel regularization we develop here 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.