New conservative difference schemes with fourth‐order accuracy for some model equation for nonlinear dispersive waves

Abstract

In this article, some high‐order accurate difference schemes of dispersive shallow water waves with Rosenau‐KdV‐RLW‐equation are presented. The corresponding conservative quantities are discussed. Existence of the numerical solution has been shown. A priori estimates, convergence, uniqueness, and stability of the difference schemes are proved. The convergence order is O ( h 4 + k 2 ) in the uniform norm without any restrictions on the mesh sizes. At last numerical results are given to support the theoretical analysis.

Numerical solution of the Rosenau–KdV–RLW equation by operator splitting techniques based on B‐spline collocation method

“…Firstly, Equation (7) is solved using the initial condition (2). Then, Equation (8) is solved by using this newly obtained solution as the initial condition of the equation.…”

“…The numerical solutions at the increasing time levels are similarly obtained as in the first time step. As it can be seen in the Lie-Trotter technique, the solution of the Rosenau-KdV-RLW equation (1) subject to initial and boundary conditions (2) and (3) is obtained by solving two initial boundary value problems (7) and (8) combined with the initial conditions. The approximate solutions of Equations (7) and (8) can be found by any numerical solution methods.…”

“…Moreover, a is the drifting coefficient of the waves; b is the nonlinearity coefficient; c and e are the third and fifth order spatio-temporal dispersion coefficients, respectively; d is the third order spatial dispersion coefficient and finally, p is the power law nonlinearity parameter. It is necessary to note that p ≥ 2 for the solitons to exist [1,2]. Equation (1) models the dynamics of dispersive shallow water waves along lake shores and beaches.…”

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…”

A conservative numerical scheme for Rosenau-RLW equation based on multiple integral finite volume method

“…The theoretical studies, on the existence, uniqueness and regularity for the solution of (1.3), have been performed by Park [5]. Various numerical techniques have been used to solve the Rosenau equation [6][7][8][9][10][11][12], particularly including the discontinuous Galerkin method, the C1-conforming finite element method [13], the finite difference method and the orthogonal cubic spline collocation method. More detailed solving processes can be obtained in Refs.…”