Variational Integrators for Nonvariational Partial Differential Equations

TL;DR

This paper extends variational integrators to nonvariational PDEs by embedding them into larger Lagrangian systems using formal Lagrangians, enabling geometric numerical methods for a broader class of problems.

Contribution

It introduces a novel approach to apply variational integrators to nonvariational PDEs via formal Lagrangians, expanding their applicability to systems like advection-diffusion equations.

Findings

Successfully applied to advection and vorticity equations

Preserves discrete energy exactly in vorticity integrator

Combines Arakawa's discretisation with symplectic time stepping

Abstract

Variational integrators for Lagrangian dynamical systems provide a systematic way to derive geometric numerical methods. These methods preserve a discrete multisymplectic form as well as momenta associated to symmetries of the Lagrangian via Noether's theorem. An inevitable prerequisite for the derivation of variational integrators is the existence of a variational formulation for the considered problem. Even though for a large class of systems this requirement is fulfilled, there are many interesting examples which do not belong to this class, e.g., equations of advection-diffusion type frequently encountered in fluid dynamics or plasma physics. On the other hand, it is always possible to embed an arbitrary dynamical system into a larger Lagrangian system using the method of formal (or adjoint) Lagrangians. We investigate the application of the variational integrator method to formal Lagrangians, and thereby extend the application domain of variational integrators to include potentially all dynamical systems. The theory is supported by physically relevant examples, such as the advection equation and the vorticity equation, and numerically verified. Remarkably, the integrator for the vorticity equation combines Arakawa's discretisation of the Poisson brackets with a symplectic time stepping scheme in a fully covariant way such that the discrete energy is exactly preserved. In the presentation of the results, we try to make the geometric framework of variational integrators accessible to non specialists.