Gardner, L. R. T.; Gardner, G. A.; Dag, I. A \(B\)-spline finite element method for the regularized long wave equation. (English) Zbl 0819.65125 Commun. Numer. Methods Eng. 11, No. 1, 59-68 (1995). For the solution of the equation \(u_ t + u_ x + auu_ x + bu_{xxt} = 0\) the authors consider a Galerkin method with quadratic \(B\)-splines, applying Crank-Nicolson for time stepping. (The solution of the nonlinearity is not mentioned.) The method is reported to be linearly stable. An exact solitary wave solution is used to test the accuracy of the approach; finally, it is applied to the simulation of an undular bore. Here too, no stability problems were experienced. Reviewer: G.Stoyan (Budapest) Cited in 62 Documents MSC: 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 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 35L75 Higher-order nonlinear hyperbolic equations Keywords:quadratic \(B\)-spline; Crank-Nicolson method; finite element method; regularized long wave equation; Galerkin method; time stepping; solitary wave solution; stability PDFBibTeX XMLCite \textit{L. R. T. Gardner} et al., Commun. Numer. Methods Eng. 11, No. 1, 59--68 (1995; Zbl 0819.65125) Full Text: DOI References: [1] Jain, Numerical solutions of regularised long wave equation, Commum. numer. methods eng. 9 pp 587– (1993) [2] Peregrine, Calculations of the development of an undular bore, J. Fluid Mech. 25 pp 321– (1966) [3] Eilbeck, Numerical study of RLW equation I: numerical methods, J. Comput. Phys. 19 pp 43– (1975) · Zbl 0325.65054 [4] Jain, Numerical solution of the RLW equation, Comput. Methods Appl. Mech. Eng. 20 pp 195– (1979) · Zbl 0409.73012 [5] Bona, Numerical schemes for a model of nonlinear dispersive waves, J. Comput. Phys. 60 pp 167– (1985) · Zbl 0578.65120 [6] Gardner, Solitary waves of the regularised long wave equation, J. Comput. Phys. 91 pp 441– (1990) · Zbl 0717.65072 [7] Gardner, Computational Mechanics pp 1555– (1991) [8] Ali, A collocation solution for Burger’s equation, Comput. Methods Appl. Mech. Eng. 100 pp 325– (1992) · Zbl 0762.65072 [9] Davies, A numerical investigation of errors arising in applying the Galerkin method to the solution of nonlinear pdes, Comput. Methods Appl. Mech. Eng. 11 pp 341– (1977) · Zbl 0364.65093 [10] Bona, Fully discrete Galerkin methods for the Korteweg-de Vries equation, Comput. Math. Appl. 12A pp 859– (1986) · Zbl 0597.65072 [11] Christie, Product approximations for nonlinear problems in the finite element method, IMA J. Numer. Anal. 1 pp 253– (1981) · Zbl 0469.65072 [12] P. M. Prenter Splines and variational methods 1975 [13] Olver, Euler operators and conservation laws of the BBM equation, Math. Proc. Camb. Phil. Soc. 85 pp 143– (1979) · Zbl 0387.35050 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.