Abstract
A predictor–corrector (P-C) scheme is applied successfully to a nonlinear method arising from the use of rational approximants to the matrix-exponential term in a three-time level recurrence relation. The resulting nonlinear finite-difference scheme, which is analyzed for local truncation error and stability, is solved using a P-C scheme, in which the predictor and the corrector are explicit schemes of order 2. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behaviour of the P-C/MPC schemes is tested numerically on the Boussinesq equation already known from the bibliography free of boundary conditions. The numerical results are derived for both the bad and the good Boussinesq equation and conclusions from the relevant known results are derived.
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References
Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform, SIAM Studies in Applied Mathematics 4. Society for Industrial and Applied Mathematics, Philadelphia (1981)
Ablowitz, M.J., Clarkson, P.A.: Solitons, nonlinear evolution equations and inverse scattering. London Math. Soc. Lect. Note Ser. 149 (1991)
Airy, G.B.: Tides and waves. Encycl. Metrop. Lond. Art 192, 241–396 (1845)
Attili, B.S.: The Adomian decomposition method for solving the Boussinesq equation arising in water wave propagation. Numer. Methods Partial Differ. Equ. 22, 1337–1347 (2006)
Bogolubsky, I.L.: Some examples of inelastic soliton interaction. Comp. Phys. Commun. 13, 149–155 (1977)
Boussinesq, M.J.: Théorie de l’intumescence appelée onde solitaire ou de translation se propageant dans un canal rectangulaire. C. R. Acad. Sci. Paris 72, 755–759 (1871)
Boussinesq, M.J.: Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblemant perielles de la surface au fond. J. Math. Pures Appl. 17(2), 55–108 (1872)
Bratsos, A.G.: The solution of the Boussinesq equation using the method of lines. Comput. Methods Appl. Mech. Eng. 157, 33–44 (1998)
Bratsos, A.G.: A parametric scheme for the numerical solution of the Boussinesq equation. Korean J. Comput. Appl. Math. 8(1), 45–57 (2001)
Bratsos, A.G., Tsitouras, Ch., Natsis, D.G.: Linearized numerical schemes for the Boussinesq equation. Appl. Num. Anal. Comp. Math. 2(1), 34–53 (2005)
Bratsos, A.G.: A modified predictor–corrector scheme for the two-dimensional sine-Gordon equation. Numer. Algorithms 43(2), 1–14 (2007)
Bratsos, A.G.: A third order numerical scheme for the two-dimensional sine-Gordon equation. Math. Comput. Simul. (in press)
Bratsos, A.G., Famelis, I.Th., Prospathopoulos, A.M.: A parametric finite-difference method for shallow sea waves. Int. J. Numer. Methods Fluids 53, 129–147 (2007)
Daripa, P., Hua, W.: A numerical study of an ill-posed Boussinesq equation arising in water waves and nonlinear lattices: filtering and regularization techniques. Appl. Math. Comput. 101(2–3), 159–207 (1999)
Defrutos, J., Ortega, T., Sanz-Sema, J.M.: A Hamiltonian explicit algorithm with spectral accuracy for the “good” Boussinesq equation. Comput. Methods Appl. Mech. Eng. 80, 417–423 (1990)
Defrutos, J., Ortega, T., Sanz-Sema, J.M.: Pseudospectral method for the “good” Boussinesq equation. Math. Comput. 57, 109–122 (1991)
El-Zoheiry, H.: Numerical investigation for the solitary waves interaction of the “good” Boussinesq equation. Appl. Numer. Math. 45, 161–173 (2003)
Feng, Z.S.: Traveling solitary solutions to the generalized Boussinesq equation. Wave Motion 37(1), 17–23 (2003)
Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge (1998)
Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett. 19, 1095–1097 (1967)
Goursat, E.: Le problème de Bäcklund. Mem. Sci. Math. Fasc. 6, Gauthier-Villars, Paris (1925)
Hirota, R.: Exact envelope-soliton solutions of a nonlinear wave. J. Math. Phys. 14, 805–809 (1973)
Hirota, R.: Exact N-soliton solutions of the wave of long waves in shallow-water and in nonlinear lattices. J. Math. Phys. 14, 810–814 (1973)
Ismail, M.S., Bratsos, A.G.: A predictor–corrector scheme for the numerical solution of the Boussinesq equation. J. Appl. Math. Comput. 13(1–2), 11–27 (2003)
Korteweg, D.J., de Vries, G.G.: On the change of form of long-waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 39(5), 422–443 (1895)
Lamb, G.L.: Elements of Soliton Theory. Wiley, New York (1980)
McKean H.P.: Boussinesq’s equation on a circle. Commun. Pure Appl. Math. 34, 599–691 (1981)
Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60(7), 650–654 (1992)
Manoranjan, V.S., Mitchell, A.R., Morris, J.LI.: Numerical solutions of the Good Boussinesq equation. SIAM J. Sci. Statist. Comput. 5, 946–957 (1984)
Matsuo, T.: New conservative schemes with discrete variational derivatives for nonlinear wave equations. J. Comput. Appl. Math. 203, 32–56 (2007)
Miura, R.M.: Bäcklund transformations. In: Dold, A., Eckman, B. (eds.) Lecture Notes in Mathematics, vol. 515. Springer, Berlin (1976)
Nimmo, J.J.C., Freeman, N.C.: A method of obtaining the N-soliton solutions of the Boussinesq equation in terms of a Wronskian. Phys. Lett. 95A, 4–6 (1983)
Nimmo, J.J.C., Freeman, N.C.: The use of Bäcklund transformation in obtaining the N-soliton solutions in Wronskian form. J. Phys. A 17, 1415–1424 (1984)
Olver, P.J.: Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, vol. 107. Springer, Berlin (1986)
Pani, A.K., Saranga, H.: Finite element Galerkin method for the “Good” Boussinesq equation. Nonlinear Anal. 29(8), 937–956 (1997)
Rayleigh, J.W.S.: On waves. Philos. Mag. Ser. 1(5), 257–279 (1876)
Scott, R.J.: Report of the Committee of Waves, Report of the 7th Meeting of the British Association for the Advancement of Science, pp. 417–496. Murray, Liverpool (1838)
Scott, R.J.: Report on Waves, Report of the 14th Meeting of the British Association for the Advancement of Science, pp. 311–390. Murray, London (1844)
Stokes, G.: On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455 (1847)
Twizell, E.H.: Computational Methods for Partial Differential Equations. Ellis Horwood Limited, England (1984)
Wahlquist, H., Estabrook, F.B.: Bäcklund transformation for solutions of the Korteweg-de Vries equation. Phys. Rev. Lett. 31, 1386–1390 (1973)
Wazwaz, A.M.: Constructions of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method. Chaos, Solitons Fractals 12, 1549–1556 (2001)
Wazwaz, A. M.: A sine-cosine method for handling nonlinear wave equations. Math. Comput. Model. 40, 499–508 (2004)
Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)
Zabusky, N.J., Kruskal, M.D.: Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965)
Zakharov, V. E.: Randomization problem for 1-dimensional chains of nonlinear operators. Z. Eksp. Teor. Fiz. 65, 219–228 (1973)
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Bratsos, A.G. A second order numerical scheme for the solution of the one-dimensional Boussinesq equation. Numer Algor 46, 45–58 (2007). https://doi.org/10.1007/s11075-007-9126-y
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DOI: https://doi.org/10.1007/s11075-007-9126-y