Surrogate Lagrangians for Variational Integrators: High Order Convergence with Low Order Schemes

TL;DR

This paper introduces surrogate Lagrangians in variational integrators to achieve fourth-order convergence with low-order schemes, enhancing accuracy without increasing computational complexity.

Contribution

The paper proposes a novel method using surrogate Lagrangians to improve variational integrators' order of accuracy while maintaining low-order computational schemes.

Findings

Achieves fourth-order convergence with second-order schemes.

Applicable to Hamiltonian systems with constraints and external forces.

Demonstrates effectiveness through numerical experiments.

Abstract

Variational integrators are momentum-preserving and symplectic numerical methods used to propagate the evolution of Hamiltonian systems. In this paper, we introduce a new class of variational integrators that achieve fourth-order convergence despite having the same integration scheme as traditional second-order variational integrators. The new class of integrators are created by replacing a dynamical system's Lagrangian in the variational integration algorithm with its surrogate Lagrangian. By incorporating the surrogate Lagrangian the propagation errors induced by variational integrators, up to a given order, are eliminated. Furthermore, no assumption on the Lagrangian's structure is made and, therefore, the proposed approach is applicable to a large range of dynamical systems. In addition, surrogate variational integrators are also constructed for Hamiltonian systems subjected to holonomic constraints and external forces. Finally, the methodology is extended to derive higher-order surrogate variational integrators that achieve an arbitrary order of accuracy but retain second-order complexity in the integration scheme. Several numerical experiments are presented to demonstrate the efficacy of our approach.