Conserved quantities of some Hamiltonian wave equations after full discretization

Abstract

Hamiltonian PDEs have some invariant quantities, which would be good to conserve with the numerical integration. In this paper, we concentrate on the nonlinear wave and Schrödinger equations. Under hypotheses of regularity and periodicity, we study how a symmetric space discretization makes that the space discretized system also has some invariants or 'nearly' invariants which well approximate the continuous ones. We conjecture some facts which would explain the good numerical approximation of them after time integration when using symplectic Runge-Kutta methods or symmetric linear multistep methods for second-order systems.

Energy preserving schemes for nonlinear Hamiltonian systems of wave equations: Application to the vibrating piano string

“…It appears that results can easily be found in the case of scalar equations (the unknown function takes scalar values), of course in the context of ordinary differential equations ODEs (see for instance [16,26]) but also of some partial differential equations, particularly semi-linear wave equations (see for instance [6,11,12,23,25,31]). The case of systems has been much less investigated and it seems that most results are restricted to very particular systems : see for instance in [27], [15] or [8] in the context of systems of ODE's and [14] (for nonlinear elasticity) or [5] (for nonlinear strings) in the context of systems of PDE's.…”

Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations

“…Cano [4] also considers the nonlinear wave equation and aims at extending the classical backward error analysis to this situation. Long-time conservation properties are obtained under a list of unverified conditions formulated as conjectures.…”

Propagation of Gevrey regularity over long times for the fully discrete Lie Trotter splitting scheme applied to the linear Schrödinger equation

“…By Proposition 3.10, C 1 only depends on V , M (3) , ρ V , α, ν, γ and d. Hence, there exists a positive constant λ 0 ∈ (0, 1) depending only on these parameters such that for all λ ∈ (0, λ 0 ), we have Moreover, Proposition 3.12 ensures that…”

“…Eventually, Section 5 is a collection of technical lemmas needed in the course of the analysis. Related works and tools, that are related with the techniques used in this article and the questions we raise, may be found in [1,8,15] (where KAM tools are developed in the infinite dimensional setting), in [3,4,12,14] (where numerical methods in the Hamiltonian setting are developed and analyzed), or in [11] (where considerations on splitting schemes are developed).…”