Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field” method

Abstract

a b s t r a c tWe give a systematic method for discretizing Hamiltonian partial differential equations (PDEs) with constant symplectic structure, while preserving their energy exactly. The same method, applied to PDEs with constant dissipative structure, also preserves the correct monotonic decrease of energy. The method is illustrated by many examples. In the Hamiltonian case these include: the sine-Gordon, Korteweg-de Vries, nonlinear Schrödinger, (linear) time-dependent Schrödinger, and Maxwell equations. In the dissipative case the examples are: the Allen-Cahn, Cahn-Hilliard, Ginzburg-Landau, and heat equations.

A local discontinuous Galerkin method for the Burgers–Poisson equation

“…For j = N , we use the periodic extension to define (Q θ ω) + N +1/2 , in order to be consistent with the numerical flux defined in (7).…”

“…Let (u h , p h , φ h ) be obtained from (6) with the choice of fluxes (7). Let θ ∈ [0, 1] be such that both Q θ and Q 1−θ are uniquely defined, then the following inequality holds for all t > 0.…”

“…It is believed that numerical methods preserving more invariants are advantageous: besides the high accuracy of numerical solutions, an invariant preserving scheme can preserve good stability properties after long-time numerical integration. Much more effort has been devoted in this topic for different integrable PDEs recently [7,18,30,31].…”

Geometric-integration tools for the simulation of musical sounds

“…The piano sounds are computed by integrating a Hamiltonian partial differential equation (PDE) model describing the oscillations of the string and an ordinary differential equation (ODE) model describing the dynamics of the hammer. The procedure has its basis on the semi-discretization method by Celledoni et al [13], which semi-discretizes the PDE to a system of ODEs while preserving the Hamiltonian structure. This method allows the application of geometric integrators, which are numerical integrators with excellent quality for Hamiltonian ODEs.…”

Energy preserving integration of the strongly coupled nonlinear Schrödinger equation

“…Using the B-series, it can shown that the AVF method for canonical and non-canonical Hamiltonian systems is conjugate to symplectic or Poisson integrators [6,7]. Application of the AVF method for various nonlinear evolutionary partial differential equation was given in [8].…”