Modified Equations for Variational Integrators

Abstract

It is well-known that if a symplectic integrator is applied to a Hamiltonian system, then the modified equation, whose solutions interpolate the numerical solutions, is again Hamiltonian. We investigate this property from the variational side. We present a technique to construct a Lagrangian for the modified equation from the discrete Lagrangian of a variational integrator.

Cited by in corpus (11)

• Contact variational integrators
• Symplectic integration of learned Hamiltonian systems
• Variational Learning of Euler-Lagrange Dynamics from Data
• Continuum limits of pluri-Lagrangian systems
• Performance Assessment of Energy-preserving, Adaptive Time-step Variational Integrators
• Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery
• The inverse problem of the calculus of variations for discrete systems
• Backward error analysis for conjugate symplectic methods
• Modified equations and the Basel problem
• Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification
• Superconvergence of Galerkin variational integrators