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Nonlinear stability and convergence of finite-difference methods for the ”good” Boussinesq equation. (English) Zbl 0749.65082
Summary: The “good” Boussinesq equation \(u_{tt}=-u_{xxxx}+u_{xx}+(u^ 2)_{xx}\) has recently been found to possess an interesting soliton- interaction mechanism. In this paper we study the nonlinear stability and the convergence of some simple finite-difference schemes for the numerical solution of problems involving the “good” Boussinesq equation. Numerical experiments are also reported.

MSC:
65Z05 Applications to the sciences
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
35Q51 Soliton equations
35L75 Higher-order nonlinear hyperbolic equations
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[1] de Frutos, J., Sanz-Serna, J.M.:h-dependent stability thresholds avoid the need for a priori bounds in nonlinear convergence proofs. In: Fatunla, S.O. (ed.) Computational Mathematics III, Proceedings of the Third International Conference held in Benin City, Nigeria, January 1988. Dublin: Boole Press (to appear)
[2] de Frutos, J., Sanz-Serna, J.M.: Split-step spectral schemes for nonlinear Dirac systems. J. Comput. Phys.83, 407-423 (1989) · Zbl 0675.65131 · doi:10.1016/0021-9991(89)90127-7
[3] L?pez-Marcos, J.C.: Estabilidad de Discretizaciones no Lineales. Tesis Doctoral, Universidad de Valladolid. Valladolid 1985
[4] L?pez-Marcos, J.C., Sanz-Serna, J.M.: A definition of stability for nonlinear problems. In: Strehmel, K. (ed.) Numerical treatment of differential equations,Proceedings of the fourth seminar ?NUMDIFF-4? held in Halle, 1987, pp. 216-226. Leipzig. Teubner-Texte zur Mathematik 1988
[5] L?pez-Marcos, J.C., Sanz-Serna, J.M.: Stability and convergence in numerical analysis III: Linear investigation of nonlinear stability. IMA J. Numer. Anal.7, 71-84 (1988) · Zbl 0695.65042 · doi:10.1093/imanum/8.1.71
[6] Manoranjan, V.S., Mitchell, A.R., Morris J.LL.: Numerical solution of the ?good? Boussinesq equation. SIAM J. Sci. Stat. Comput.5, 946-957 (1984) · Zbl 0555.65080 · doi:10.1137/0905065
[7] Manoranjan, V.S., Ortega, T., Sanz-Serna, J.M.: Soluton and anti-soliton interactions in the ?good? Boussinesq equation. J. Math. Phys.29, 1964-1968 (1988) · Zbl 0673.35089 · doi:10.1063/1.527850
[8] Ortega, T.: Soluci?n Num?rica de la Ecuaci?n ?Buena? de Boussinesq. Tesis Doctoral, Universidad de Valladolid. Valladolid 1988
[9] Sanz-Serna, J.M.: Stability and convergence in numerical analysis I: Linear problems, a simple, comprehensive account. In: Halle, J.K., Martinez-Amores, P. (eds.) Nonlinear differential equations and applications, pp. 64-113. Boston. Pitman 1985
[10] Sanz-Serna, J.M., Palencia, C.: A general equivalence theorem in the theory of discretization methods. Math. Comput.45, 143-152 (1985) · Zbl 0599.65034 · doi:10.1090/S0025-5718-1985-0790648-7
[11] Stetter, H.J.: Analysis of discretization methods for ordinary differential equations. Berlin: Springer 1973 · Zbl 0276.65001
[12] S?li, E.: Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equation. Numer. Math.53, 459-484 (1988) · Zbl 0637.76024 · doi:10.1007/BF01396329
[13] Taha, T.R., Ablowitz, M.J.: Analytical and numerical aspects of certain nonlinear evolution equations. II: Numerical nonlinear Schroedinger equation. J. Comput. Phys.55, 203-230 (1984) · Zbl 0541.65082 · doi:10.1016/0021-9991(84)90003-2
[14] Weideman, J.A.C., Herbst, B.M.: Split-step methods for the nonlinear Schroedinger equation. SIAM J. Numer. Anal.23, 485-507 (1986) · Zbl 0597.76012 · doi:10.1137/0723033
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