A new variational discretization technique for initial value problems bypassing governing equations

TL;DR

This paper introduces a novel variational discretization method for initial value problems that bypasses deriving equations of motion, using an optimization functional with doubled degrees of freedom and a regularized SBP operator.

Contribution

It presents a new discretization technique based on an optimization principle with regularization, enabling trajectory computation without explicit equations of motion.

Findings

Demonstrates stability and accuracy in classical systems

Shows convergence properties for first and second order derivatives

Applicable to both classical and quantum-inspired systems

Abstract

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.