Conservation of wave action under multisymplectic discretizations

In this paper we discuss the conservation of wave action under numerical discretization by variational and multisymplectic methods. Both the general wave action conservation defined with respect to a smooth, periodic, one-parameter ensemble of flow realizations and the specific wave action based on an approximated and averaged Lagrangian are addressed in the numerical context. It is found that the discrete variational formulation gives rise in a natural way not only to the discrete wave action conservation law but to a generalization of the numerical dispersion relation to the case of variable coefficients. Indeed a fully discrete analog of the modulation equations arises. On the other hand the multisymplectic framework gives easy access to the conservation law for the general class of multisymplectic Runge-Kutta methods. A numerical experiment confirms conservation of wave action to machine precision and suggests that the solution of the discrete modulation equations approximates the numerical solution to order O(e) on intervals of O(

Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems

We are interested in high-order linear multistep schemes for time discretization of adjoint equations arising within optimal control problems. First we consider optimal control problems for ordinary differential equations and show loss of accuracy for Adams-Moulton and Adams-Bashford methods, whereas BDF methods preserve high--order accuracy. Subsequently we extend these results to semi--lagrangian discretizations of hyperbolic relaxation systems. Computational results illustrate theoretical findings

Resolving Entropy Growth from Iterative Methods

We consider entropy conservative and dissipative discretizations of nonlinear conservation laws with implicit time discretizations and investigate the influence of iterative methods used to solve the arising nonlinear equations. We show that Newton's method can turn an entropy dissipative scheme into an anti-dissipative one, even when the iteration error is smaller than the time integration error. We explore several remedies, of which the most performant is a relaxation technique, originally designed to fix entropy errors in time integration methods. Thus, relaxation works well in consort with iterative solvers, provided that the iteration errors are on the order of the time integration method. To corroborate our findings, we consider Burgers' equation and nonlinear dispersive wave equations. We find that entropy conservation results in more accurate numerical solutions than non-conservative schemes, even when the tolerance is an order of magnitude larger.Comment: 25 pages, 6 figure

Multi-Symplectic Integrators for Nonlinear Wave Equations

Symplectic (area-preserving) integrators for Hamiltonian ordinary differential equations have shown to be robust, efficient and accurate in long-term calculations. In this thesis, we show how symplectic integrators have a natural generalization to Hamiltonian PDEs by introducing the concept of multi-symplectic partial differential equations (PDEs). In particular, we show that multi-symplectic PDEs have an underlying spatio-temporal multi-symplectic structure characterized by a multi-symplectic conservation law MSCL). Then multi-symplectic integrators (MSIs) are numerical schemes that preserve exactly the MSCL. Remarkably, we demonstrate that, although not designed to do so, MSIs preserve very well other associated local conservation laws and global invariants, such as the energy and the momentum, for very long periods of time. We develop two types of MSIs, based on finite differences and Fourier spectral approximations, and illustrate their superior performance over traditional integrators by deriving new numerical schemes to the well known 1D nonlinear Schrödinger and sine-Gordon equations and the 2D Gross-Pitaevskii equation. In sensitive regimes, the spectral MSIs are not only more accurate but are better at capturing the spatial features of the solutions. In particular, for the sine-Gordon equation we show that its phase space, as measured by the nonlinear spectrum associated with it, is better preserved by spectral MSIs than by spectral non-symplectic Runge-Kutta integrators. Finally, to further understand the improved performance of MSIs, we develop a backward error analysis of the multi-symplectic centered-cell discretization for the nonlinear Schrödinger equation. We verify that the numerical solution satisfies to higher order a nearby modified multi-symplectic PDE and its modified multi-symplectic energy conservation law. This implies, that although the numerical solution is an approximation, it retains the key feature of the original PDE, namely its multi-symplectic structure

An Overview of Variational Integrators

The purpose of this paper is to survey some recent advances in variational integrators for both finite dimensional mechanical systems as well as continuum mechanics. These advances include the general development of discrete mechanics, applications to dissipative systems, collisions, spacetime integration algorithms, AVI’s (Asynchronous Variational Integrators), as well as reduction for discrete mechanical systems. To keep the article within the set limits, we will only treat each topic briefly and will not attempt to develop any particular topic in any depth. We hope, nonetheless, that this paper serves as a useful guide to the literature as well as to future directions and open problems in the subject

Geometric Numerical Integration

The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods