A new conservative fourth‐order accurate difference scheme for solving a model of nonlinear dispersive equations

Abstract

This article is devoted to the study of a nonlinear conservative fourth‐order difference scheme for a model of nonlinear dispersive equations that is governed by the RLW‐KdV equation. The existence of the approximate solution and the convergence of the difference scheme are proved, by using the energy method. In addition, the convergent order in maximum norm is 2 in temporal direction and 4 in spatial direction. The unconditional stability as well as uniqueness of the difference scheme is also derived. An application on the RLW and MRLW equations is discussed numerically in details. Furthermore, interaction of solitary waves with different amplitudes are shown. The 3 invariants of the motion are evaluated to determine the conservation proprieties of the system. The temporal evaluation of a Maxwellian initial pulse is then studied. Some numerical examples are given to validate the theoretical results.

An efficient numerical simulation of nonlinear‐dispersive model of shallow water wave

“…Because our scheme is linear, iterative numerical calculation is not required. All the results show that present linearized conservative difference scheme is the most effective in terms of accuracy and time consumption and gives a summary of the advantages of our method over the existing methods [3–5, 9, 28, 36–39]. Future work is planned to study the compact linearized difference scheme for RLW‐KdV equations in two dimension.…”

“…From these Tables, we have noticed that our scheme conserves the invariants to at least four decimal places. Calculated under the same parameters, our results were compared with those enlisted in [28], [39], and [9]. It is seen that our scheme provides quite stable results in both Tables.…”

Numerical approximation of the generalized regularized long wave equation using Petrov–Galerkin finite element method

“…MRLW equation was solved numerically by various methods . We would refer for an application for such models of nonlinear dispersive equations.…”

Optimal H2‐error estimates of conservative compact difference scheme for the Zakharov equation in two‐space dimension

“…[15][16][17] Among the existing numerical methods for the Zakharov equation, most of the methods and error estimates are accomplished in only one space dimension. In fact, their proofs 10,14,18,19 for difference scheme rely strongly on not only the conservation laws of the method but also the discrete version of the Sobolev inequality in one space dimension…”

“…Among the existing numerical methods for the Zakharov equation, most of the methods and error estimates are accomplished in only one space dimension. In fact, their proofs() for difference scheme rely strongly on not only the conservation laws of the method but also the discrete version of the Sobolev inequality in one space dimension However, the above Sobolev inequality is no longer valid in high space dimensions. So few L ∞ ‐error estimates is available in the literatures of numerical methods for the Zakharov equation in high space dimensions.…”