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Numerical approximation of the generalized regularized long wave equation using Petrov–Galerkin finite element method
Author(s) -
Bhowmik Samir Kumar,
Karakoc Seydi B. G.
Publication year - 2019
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22410
Subject(s) - mathematics , petrov–galerkin method , mathematical analysis , finite element method , galerkin method , quadratic equation , nonlinear system , rate of convergence , numerical analysis , physics , geometry , channel (broadcasting) , quantum mechanics , electrical engineering , thermodynamics , engineering
The generalized regularized long wave (GRLW) equation has been developed to model a variety of physical phenomena such as ion‐acoustic and magnetohydrodynamic waves in plasma, nonlinear transverse waves in shallow water and phonon packets in nonlinear crystals. This paper aims to develop and analyze a powerful numerical scheme for the nonlinear GRLW equation by Petrov–Galerkin method in which the element shape functions are cubic and weight functions are quadratic B‐splines. The proposed method is implemented to three reference problems involving propagation of the single solitary wave, interaction of two solitary waves and evolution of solitons with the Maxwellian initial condition. The variational formulation and semi‐discrete Galerkin scheme of the equation are firstly constituted. We estimate rate of convergence of such an approximation. Using Fourier stability analysis of the linearized scheme we show that the scheme is unconditionally stable. To verify practicality and robustness of the new scheme error norms L 2 , L ∞ and three invariants I 1 , I 2 , and I 3 are calculated. The computed numerical results are compared with other published results and confirmed to be precise and effective.