Connection coefficients on an interval and wavelet solutions of Burgers equation. (English) Zbl 0990.65096

Summary: A definition of connection coefficients is introduced and techniques of computation are presented. We use semi-implicit time difference scheme to solve Burgers equation by applying the evaluations of connection coefficients in calculating the integrals of the variational form. Comparisons of accuracy and robustness of numerical solutions are mentioned in the examples.


65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65T60 Numerical methods for wavelets
35Q53 KdV equations (Korteweg-de Vries equations)
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[1] Bacry, E.; Mallat, S.; Papanicolau, G., A wavelet based space-time adaptive numerical method for partial differential equations, Math. modeling numer. anal., 26, 7, 793-834, (1992) · Zbl 0768.65062
[2] Basdevant, C.; Deville, M.; Haldenwang, P.; Lacroix, J.M.; Ouazzani, J.; Peyret, R.; Orlandi, P.; Patera, A.T., Spectral and finite difference solutions of the Burgers equation, Comput. fluids, 14, 23, (1986) · Zbl 0612.76031
[3] Burgers, J.M., A mathematical model illustrating the theory of turbulence, Adv. appl. mech., 1, 171-199, (1948)
[4] Daubechies, I., Orthonormal bases of compactly supported wavelets, Comm. pure appl. math., 41, 909-996, (1988) · Zbl 0644.42026
[5] Daubechies, I., Ten lectures on wavelets, (1992), SIAM Philadelphia, PA · Zbl 0776.42018
[6] Fletcher, C.A.J., Computational Galerkin methods, (1984), Springer Berlin · Zbl 0533.65069
[7] Glowinski, R., Numerical methods for nonlinear variational problems, (1984), Springer Berlin · Zbl 0575.65123
[8] R. Glowinski, W. Lawton, M. Ravachol, E. Tenenbaum, Wavelet solutions of linear and nonlinear elliptic, parabolic and hyperbolic problems in one space dimension. Proceedings of International Conference on Computing Method in Applied Science and Engineering, SIAM, Philadelphia, 1990, pp. 55-120. · Zbl 0799.65109
[9] Latto, A.; Resnikoff, H.L.; Tenenbaum, E., The evaluation of connection coefficients of compactly supported wavelets, ()
[10] Latto, A.; Tenenbaum, E., LES ondelletse a support compact et la solution numerique de l’equation de Burgers, C. R. acad. sci. France, 311, 903, (1990)
[11] J. Liandrat, Ph. Tchamitchian, Resolution of the 1D regularized Burgers equation using a spatial wavelet approximation, NASA Report, ICASE Report, 90-83, 1990.
[12] Lin, E.B.; Xiao, Z., Multi-scaling function interpolation and approximation, AMS contemp. math., 216, 129-148, (1998)
[13] Lin, E.B.; Zhou, X., Coiflet interpolation and approximate solutions of elliptic partial differential equations, Numer. method partial differential equations, 13, 303-320, (1997) · Zbl 0881.65097
[14] Mallat, S., Multiresolution approximations and wavelet orthonormal bases of L2 (R), Trans. amer. math. soc., 315, 69-87, (1989) · Zbl 0686.42018
[15] Vasilyev, O.V.; Paolucci, S., A dynamically adaptive multilevel collocation method for solving partial differential equations in a finite domain, J. comput. phys., 125, 498-512, (1996) · Zbl 0847.65073
[16] Wells, R.O.; Zhou, X., Wavelet solutions for the Dirichlet problem, Numer. math., 70, 379-396, (1995) · Zbl 0824.65108
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