Abstract
We construct three finite difference methods to solve a linearized Korteweg–de-Vries (KdV) equation with advective and dispersive terms and specified initial and boundary conditions. Two numerical experiments are considered; case 1 is when the coefficient of advection is greater than the coefficient of dispersion, while case 2 is when the coefficient of dispersion is greater than the coefficient of advection. The three finite difference methods constructed include classical, multisymplectic and a modified explicit scheme. We obtain the stability region and study the consistency and dispersion properties of the various finite difference methods for the two cases. This is one of the rare papers that analyse dispersive properties of methods for dispersive partial differential equations. The performance of the schemes are gauged over short and long propagation times. Absolute and relative errors are computed at a given time at the spatial nodes used.
Highlights
In this paper, we solve a linearised Korteweg-de-Vries equation with specified initial and boundary conditions
This paper considers, proposes and analyzes three finite difference schemes for two linear dispersive KdV equations
One of the cases is such that the advective term dominates the dispersive term, while in the second case, the dispersive term dominates the advective term
Summary
We solve a linearised Korteweg-de-Vries equation with specified initial and boundary conditions. The performance of the four methods is compared in regard to dispersion and dissipation errors and ability to conserve mass, momentum and energy by using two numerical experiments that involve solitons. Plots of relative phase error vs w at some values of ∆x, ∆t for a few schemes discretizing 1D linear advection and 1D advection-diffusion equation were obtained in [21] and [24], respectively.
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