Energy conservation issues in the numerical solution of the semilinear wave equation

Abstract

In this paper we discuss energy conservation issues related to the numerical solution of the nonlinear wave equation. As is well known, this problem can be cast as a Hamiltonian system that may be autonomous or not, depending on the specific boundary conditions at hand. We relate the conservation properties of the original problem to those of its semi-discrete version obtained by the method of lines. Subsequently, we show that the very same properties can be transferred to the solutions of the fully discretized problem, obtained by using energy-conserving methods in the HBVMs (Hamiltonian Boundary Value Methods) class. Similar arguments hold true for different types of Hamiltonian Partial Differential Equations, e.g., the nonlinear Schrödinger equation.

arXiv comment41 pages, 11 figure

References in corpus (13)

• Multisymplectic geometry, variational integrators, and nonlinear PDEs
• Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field" method
• Hamiltonian Boundary Value Methods (Energy Conserving Discrete Line Integral Methods)
• A unifying framework for the derivation and analysis of effective classes of one-step methods for ODEs
• On the Existence of Energy-Preserving Symplectic Integrators Based upon Gauss Collocation Formulae
• A note on the efficient implementation of Hamiltonian BVMs
• Analisys of Hamiltonian Boundary Value Methods (HBVMs): a class of energy-preserving Runge-Kutta methods for the numerical solution of polynomial Hamiltonian systems
• Efficient implementation of Gauss collocation and Hamiltonian Boundary Value Methods
• Multiple invariants conserving Runge-Kutta type methods for Hamiltonian problems
• The Lack of Continuity and the Role of Infinite and Infinitesimal in Numerical Methods for ODEs: the Case of Symplecticity
• A Two Step, Fourth Order, Nearly-Linear Method with Energy Preserving Properties
• Energy-conserving methods for Hamiltonian Boundary Value Problems and applications in astrodynamics
• Efficient implementation of geometric integrators for separable Hamiltonian problems

Cited by in corpus (14)

• Spectrally accurate space-time solution of Hamiltonian PDEs
• A linearly implicit energy-preserving exponential integrator for the nonlinear Klein-Gordon equation
• On the effectiveness of spectral methods for the numerical solution of multi-frequency highly-oscillatory Hamiltonian problems
• High-order energy-conserving Line Integral Methods for charged particle dynamics
• Analysis of Energy and QUadratic Invariant Preserving (EQUIP) methods
• Line Integral solution of Hamiltonian PDEs
• A note on the continuous-stage Runge-Kutta-(Nyström) formulation of Hamiltonian Boundary Value Methods (HBVMs)
• Line Integral Solution of Hamiltonian Systems with Holonomic Constraints
• Spectrally accurate space-time solution of Manakov systems
• Efficient energy-preserving numerical approximations for the sine-Gordon equation with Neumann boundary conditions
• Bespoke finite difference methods that preserve two local conservation laws of the modified KdV equation
• A linearly implicit and local energy-preserving scheme for the sine-Gordon equation based on the invariant energy quadratization approach
• Long-time momentum and actions behaviour of energy-preserving methods for semilinear wave equations via spatial spectral semi-discretizations
• Unconditionally optimal convergence of an energy-conserving and linearly implicit scheme for nonlinear wave equations