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Compact structure-preserving approach to solitary wave in shallow water modeled by the Rosenau-RLW equation. (English) Zbl 1428.76142

Summary: A mass-preserving scheme, a nonlinear algorithm based on modification of a finite difference method to the Rosenau-RLW equation, is proposed subject to homogeneous boundary conditions. The key feature of the method for improving the accuracy of approximate solutions is to develop a compact higher-order scheme together with an iterative algorithm for solving the nonlinear implicit scheme. The derivatives for space discretization are approximated by using the algorithm dealing with a five-point stencil. In addition, a three-level average difference technique is used to perform time discretization. The conservation of mass and both the existence and uniqueness of the numerical solution are proved. The stability and convergence of the numerical solution with order \(O(\tau^4 + \tau^2 h^2 + h^4)\) are also confirmed. For efficiency analysis, numerical results show that the computational efficiency of the compact scheme is much higher than that of non-compact schemes. Moreover, long-time behavior is also used to validate the capability of the present method.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
76B25 Solitary waves for incompressible inviscid fluids
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