Numerical solution of evolution equations by the Haar wavelet method. (English) Zbl 1110.65097

Summary: An efficient numerical method for the solution of nonlinear evolution equations based on the Haar wavelets approach is proposed. The method is tested in the case of Burgers and sine-Gordon equations. Numerical results, obtained by computer simulation, are compared with other available solutions. These calculations demonstrate that the accuracy of the Haar wavelet solutions is quite high even in the case of a small number of grid points.


65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
65T60 Numerical methods for wavelets
Full Text: DOI


[1] Basdevant, C.; Deville, M.; Haldenwang, P.; Lacroix, J.M.; Ouazzani, J.; Peyret, R.; Orlandi, P.; Patera, A., Spectral and finite difference solutions of the Burgers equation, Comput. fluids, 14, 23-41, (1986) · Zbl 0612.76031
[2] Comincioli, V.; Naldi, G.; Scapolla, T., A wavelet-based method for numerical solution of nonlinear evolution equations, Appl. numer. math., 33, 291-297, (2000) · Zbl 0964.65112
[3] Helal, M.A., Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics, Chaos, solitons & fractals, 13, 1917-1929, (2002) · Zbl 0997.35063
[4] Quian, S.; Weiss, J., Wavelets and the numerical solution of partial differential equations, J. comput. phys., 106, 155-175, (1993) · Zbl 0771.65072
[5] Bertoluzza, S.; Naldi, G.; Ravel, J.C., Wavelet methods for the numerical solution of boundary value problems on the interval, (), 425-428 · Zbl 0845.65040
[6] Chen, M.-Q.; Hwang, C.; Shih, Y.-P., The computation of wavelet-Galerkin approximation on a bounded interval, Int. J. numer. methods eng., 39, 2921-2944, (1996) · Zbl 0884.76058
[7] Avudainayagam, A.; Vani, C., Wavelet-Galerkin solutions of quasilinear hyperbolic conservation equations, Commun. numer. methods eng., 15, 589-601, (1999) · Zbl 0942.65114
[8] Comincioli, V.; Naldi, G.; Scapolla, T., A wavelet-based method for numerical solution of nonlinear evolution equations, Appl. numer math., 33, 291-297, (2000) · Zbl 0964.65112
[9] G. Beylkin, J.M. Kaiser, An adaptive pseudo-wavelet approach for solving nonlinear partial differential equations, in: W. Dahmen, A.J. Kurdila, P. Ostwald (Eds.), Multiscale Methods for PDEs, 1997, pp. 137-197.
[10] Walden, J., Filterbank methods for hyperbolic pdes, SIAM J. numer. anal., 36, 1183-1233, (1999) · Zbl 0934.65094
[11] Lazaar, S.; Ponenti, P.; Liandrat, J.; Tchamitchian, P., Wavelet algorithms for numerical resolution of partial differential equations, Comput. methods appl. mech. eng., 116, 309-314, (1994) · Zbl 0820.65059
[12] Restrepo, J.M.; Leaf, G.K., Inner product computations using periodized Daubechies wavelets, Int. J. numer. methods eng., 40, 3557-3578, (1997) · Zbl 0906.65138
[13] Bertoluzza, S.; Canuto, C.; Urban, K., On adaptive computation of integrals of wavelets, Appl. numer. math., 34, 13-38, (2000) · Zbl 0981.65134
[14] Jameson, L., A wavelet-optimized, very much order adaptive grid and order numerical method, SIAM: J. sci. comput., 19, 1980-2013, (1998) · Zbl 0913.65090
[15] Cattani, C., Haar wavelet spline, J. interdisciplinary math., 4, 35-47, (2001) · Zbl 1019.65107
[16] Chen, C.F.; Hsiao, C.H., Haar wavelet method for solving lumped and distributed parameter systems, IEE-proc.: control theory appl., 144, 87-94, (1997) · Zbl 0880.93014
[17] Newland, D.E., An introduction to random vibrations. spectral and wavelet analysis, (1993), Longman New York · Zbl 0588.70001
[18] Vasilyev, O.V.; Paolucci, S.; Sen, M., A multilevel wavelet collocation method for solving partial differential equations in finite domain, J. comput. phys., 120, 33-47, (1995) · Zbl 0837.65113
[19] Chiavassa, G.; Guichaoua, M.; Lindrat, J., Two adaptive wavelet algorithms for non-linear parabolic partial differential equations, Comput. fluids, 31, 467-480, (2002) · Zbl 0996.65105
[20] Ablowitz, M.J.; Herbst, B.M.; Schober, C.M., On the numerical solution of the sine-Gordon equation. II. performance of numerical schemes, J. comput. phys., 131, 354-367, (1997) · Zbl 0874.65076
[21] Lu, X.; Schmid, R., Sympletic integration of sine-Gordon type systems, Math. comput. simul., 50, 255-263, (1999)
[22] Nana, L.; Kofane, T.C.; Kaptonom, E., Subharmonic and homoclinic bifurcations in the driven and damped sine-Gordon system, Physica D, 134, 61-74, (1999) · Zbl 0939.37045
[23] Lu, X., Sympletic computation of solitary waves for general sine Gordon equations, Math. comput. simul., 55, 519-532, (2001) · Zbl 0987.65132
[24] Chawla, M.M.; Al-Zanaidi, M.A., A class of one-step time integration schemes for second-order hyperbolic differential equations, Math. comput. model., 33, 431-443, (2001) · Zbl 0973.65073
[25] González, J.A.; Bellorin, A.; Guerrero, L.E., How to excite the internal modes of sine-Gordon solitons, Chaos, solitons & fractals, 17, 907-919, (2003) · Zbl 1030.35138
[26] Forinash, K.; Willis, C.R., Nonlinear response of the sine-Gordon breather to an a.c. driver, Physica D, 149, 95-106, (2001) · Zbl 0974.35106
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.