Abstract
We study the numerical error in solitary wave solutions of nonlinear dispersive wave equations. A number of existing results for discretizations of solitary wave solutions of particular equations indicate that the error grows quadratically in time for numerical methods that do not conserve energy, but grows only linearly for conservative methods. We provide numerical experiments suggesting that this result extends to a very broad class of equations and numerical methods.
Highlights
In this work we study the behavior of numerical approximations of solitary wave solutions of nonlinear wave partial differential equations (PDEs)
We demonstrate for the first time that the aforementioned favorable behavior can be observed in the numerical solution of a system that does not possess solitary wave solutions with simple translation or rotation symmetries
We observed a similar behavior of orthogonal projection methods for other nonlinear dispersive wave equations such as the BBM equation
Summary
In this work we study the behavior of numerical approximations of solitary wave solutions of nonlinear wave partial differential equations (PDEs). We investigate in a broad setting the phenomenon, already studied for certain equations, that numerical methods that conserve (up to rounding errors) this functional (or the Hamiltonian of the system) exhibit much smaller errors over long times. This behavior has been demonstrated for numerical approximations of certain solutions of Hamiltonian systems. These solutions are referred to as relative equilibria, and are those whose trajectory lies on a manifold defined by a symmetry group. Numerical methods that exactly conserve certain invariants give rise to a leading term in the global
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