1,588 research outputs found
Error Estimate for a Fully Discrete Spectral Scheme for Korteweg-de Vries-Kawahara Equation
We are concerned with the convergence of spectral method for the numerical
solution of the initial-boundary value problem associated to the Korteweg-de
Vries-Kawahara equation (in short Kawahara equation), which is a transport
equation perturbed by dispersive terms of 3rd and 5th order. This equation
appears in several fluid dynamics problems. It describes the evolution of small
but finite amplitude long waves in various problems in fluid dynamics. These
equations are discretized in space by the standard Fourier- Galerkin spectral
method and in time by the explicit leap-frog scheme. For the resulting fully
discrete, conditionally stable scheme we prove an L2-error bound of spectral
accuracy in space and of second-order accuracy in time.Comment: 15 page
Numerical integration of coupled Korteweg-de Vries System
We introduce a numerical method for general coupled Korteweg-de Vries
systems. The scheme is valid for solving Cauchy problems for arbitrary number
of equations with arbitrary constant coefficients. The numerical scheme takes
its legality by proving its stability and convergence which gives the
conditions and the appropriate choice of the grid sizes. The method is applied
to Hirota-Satsuma (HS) system and compared with its known explicit solution
investigating the influence of initial conditions and grid sizes on accuracy.
We also illustrate the method to show the effects of constants with a
transition to non-integrable cases.Comment: 11 pages, 13 figure
The design of conservative finite element discretisations for the vectorial modified KdV equation
We design a consistent Galerkin scheme for the approximation of the vectorial
modified Korteweg-de Vries equation. We demonstrate that the scheme conserves
energy up to machine precision. In this sense the method is consistent with the
energy balance of the continuous system. This energy balance ensures there is
no numerical dissipation allowing for extremely accurate long time simulations
free from numerical artifacts. Various numerical experiments are shown
demonstrating the asymptotic convergence of the method with respect to the
discretisation parameters. Some simulations are also presented that correctly
capture the unusual interactions between solitons in the vectorial setting
Optimally convergent hybridizable discontinuous Galerkin method for fifth-order Korteweg-de Vries type equations
We develop and analyze the first hybridizable discontinuous Galerkin (HDG)
method for solving fifth-order Korteweg-de Vries (KdV) type equations. We show
that the semi-discrete scheme is stable with proper choices of the
stabilization functions in the numerical traces. For the linearized fifth-order
equations, we prove that the approximations to the exact solution and its four
spatial derivatives as well as its time derivative all have optimal convergence
rates. The numerical experiments, demonstrating optimal convergence rates for
both the linear and nonlinear equations, validate our theoretical findings
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