A New Technique for Preserving Conservation Laws

Abstract

This paper introduces a new symbolic-numeric strategy for finding semidiscretizations of a given PDE that preserve multiple local conservation laws. We prove that for one spatial dimension, various one-step time integrators from the literature preserve fully discrete local conservation laws whose densities are either quadratic or a Hamiltonian. The approach generalizes to time integrators with more steps and conservation laws of other kinds; higher-dimensional PDEs can be treated by iterating the new strategy. We use the Boussinesq equation as a benchmark and introduce new families of schemes of order two and four that preserve three conservation laws. We show that the new technique is practicable for PDEs with three dependent variables, introducing as an example new families of second-order schemes for the potential Kadomtsev–Petviashvili equation.

Exponentially fitted methods that preserve conservation laws

“…Typically, at the end of the procedure above one obtains a family of methods that depend on some remaining parameters that can be arbitrarily chosen [29][30][31][32]. In general, optimal values of these parameters are not available a priori.…”

“…represent the conservation laws of charge and momentum, respectively. Although in Section 2 we have only considered a single PDE, the strategy can be straightforwardly extended to deal with systems of PDEs (see [31]). A wide range of numerical methods for the NLS equation that preserve both the charge and momentum conservation laws has been derived in [32].…”

“…Finite difference methods that preserve conservation laws have been introduced for a range of numerical equations by using a novel technique in [29][30][31][32]. Although these particular geometric integrators perform better than standard methods, when the solution of the problem is highly oscillatory they require very small stepsizes in order to correctly reproduce the oscillations of the solution.…”

“…In the context of PDEs symplectic Runge-Kutta methods preserve all the conservation laws with either linear or quadratic density G of a suitable space discretization [31].…”

“…The paper is organized as follows. In Section 2 we show how to use the technique in [31] to define space discretizations that preserve conservation laws. In Section 3 we deal with the time discretization and prove original results on the conservation properties of EFRK methods.…”