Second-generation wavelet collocation method for the solution of partial differential equations. (English) Zbl 0984.65105

Vasilyev, Oleg V.; Bowman, Christopher

J. Comput. Phys. 165, No. 2, 660-693 (2000).

A general framework for constructing accurate and efficient numerical methods of collocation type for solving nonlinear partial differential equations of the form \({{\partial u}\over{\partial t}}=F(x,t,u,\nabla u)\) with boundary conditions (and possibly constraints) is developed. The considered methods are based on second-generation wavelets – in the paper lifted interpolating wavelets are applied.
The grid of collocation points is adapted dynamically with respect to time and reflects local changes in the solution. It is achieved by applying wavelet decompositions. Moreover, a new hierarchical finite difference scheme is described for calculating spatial derivatives of a function on an adaptive grid.
The proposed numerical method is applied to solving one-dimensional Burgers and modified Burgers equations and the one-dimensional diffusion flame problem. The numerical results indicate efficient adaptivity of the computational grid and associated wavelets to the local irregularities of the solution.

Reviewer: Teresa Regińska (Warszawa)

MSC:

65M70	Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65T60	Numerical methods for wavelets
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q53	KdV equations (Korteweg-de Vries equations)
65M50	Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65M55	Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs

Keywords:

multilevel wavelet collocation method; second-generation wavelets; interpolating wavelets; lifting scheme; grid adaptation; hierarchical finite difference scheme; Burgers equation; numerical results

References:

[1]	Daubechies, I., Ten lectures on wavelets, (1992) · Zbl 0776.42018
[2]	Louis, A.K.; Maaß, P.; Rieder, A., Wavelets: theory and applications, (1997) · Zbl 0897.42019
[3]	Y. Meyer, Wavelets and Operators, (translated by, D. H. Salinger, Cambridge Studies in Advanced Mathematics, Cambridge Univ. Press, Cambridge, UK, 1992, No. 37.
[4]	Strang, G.; Nguyen, T., Wavelets and filter banks, (1996) · Zbl 1254.94002
[5]	Liandrat, J.; Tchamitchian, P., Resolution of the 1d regularized Burgers equation using a spatial wavelet approximation, (1990)
[6]	Bacry, E.; Mallat, S.; Papanicolaou, G., Wavelet based space-time adaptive numerical method for partial differential equations, Math. model. numer. anal., 26, 793, (1992) · Zbl 0768.65062
[7]	Beylkin, G.; Keiser, J., On the adaptive numerical solution of nonlinear partial differential equations in wavelet bases, J. comput. phys., 132, 233, (1997) · Zbl 0880.65076
[8]	Holmstrom, M.; Walden, J., Adaptive wavelet methods for hyperbolic pdes, J. sci. comput., 13, 19, (1998) · Zbl 0907.65093
[9]	Harten, A., Adaptive multiresolution schemes for shock computations, J. comput. phys., 115, 319, (1994) · Zbl 0925.65151
[10]	Harten, A., Multiresolution algorithms for the numerical solution of hyperbolic conservation laws, Commun. pure appl. math., 48, 1305, (1995) · Zbl 0860.65078
[11]	Vasilyev, O.V.; Paolucci, S.; Sen, M., A multilevel wavelet collocation method for solving partial differential equations in a finite domain, J. comput. phys., 120, 33, (1995) · Zbl 0837.65113
[12]	Cai, W.; Wang, J.Z., Adaptive multiresolution collocation methods for initial boundary value problems of nonlinear pdes, SIAM J. numer. anal., 33, 937, (1996) · Zbl 0856.65115
[13]	Fröhlich, J.; Schneider, K., An adaptive wavelet – vaguelette algorithm for the solution of pdes, J. comput. phys., 130, 174, (1997) · Zbl 0868.65067
[14]	Vasilyev, O.V.; Paolucci, S., A fast adaptive wavelet collocation algorithm for multidimensional pdes, J. comput. phys., 125, 16, (1997) · Zbl 0887.65116
[15]	Jameson, L., A wavelet-optimized, very high order adaptive grid and order numerical method, SIAM J. sci. comput., 19, 1980, (1998) · Zbl 0913.65090
[16]	Holmstrom, M., Solving hyperbolic PDEs using interpolating wavelets, SIAM J. sci. comput., 21, 405, (1999) · Zbl 0959.65109
[17]	Vasilyev, O.V.; Paolucci, S., A dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain, J. comput. phys., 125, 498, (1996) · Zbl 0847.65073
[18]	Jameson, L., Wavelets and numerical methods, (1993)
[19]	Walden, J., Filter bank methods for hyperbolic pdes, SIAM J. numer. anal., 36, 1183, (1999) · Zbl 0934.65094
[20]	C. K. Chui, and, E. Quak, Wavelets on a bounded interval, in, Numerical Methods of Approximation Theory, edited by, D. Braess and L. L. Schumaker, International Series of Numerical Mathematics, Birkhäuser Verlag, Basel, 1992, Vol, 105, p, 53. · Zbl 0815.42016
[21]	Cohen, A.; Daubechies, I., Wavelets on the interval and fast wavelet transforms, Appl. comput. harmon. anal., 1, 54, (1993) · Zbl 0795.42018
[22]	L. Andersson, N. Hall, B. Jawerth, and, G. Peters, Wavelets on a closed subset of the real line, in, Recent Advances in Wavelet Analysis, edited by, L. L. Schumaker and G. Webb, Academic Press, San Diego, 1994, p, 1. · Zbl 0808.42019
[23]	Sweldens, W., The lifting scheme: A construction of second generation wavelets, SIAM J. math. anal., 29, 511, (1998) · Zbl 0911.42016
[24]	Cohen, A.; Daubechies, I.; Feauveau, J., Bi-orthogonal bases of compactly supported wavelets, Commun. pures appl. math., 45, 485, (1992) · Zbl 0776.42020
[25]	Donoho, D.L., Interpolating wavelet transforms, (1992)
[26]	Deslauriers, G.; Dubuc, S., Symmetric iterative interpolation process, Constr. approx., 5, 49, (1989) · Zbl 0659.65004
[27]	S. Bertoluzza, G. Naldi, and, J. C. Ravel, Wavelet methods for the numerical solution of boundary value problems on the interval, in, Wavelets: Theory, Algorithms, and Applications, edited by, C. K. Chui, L. Montefusco, and L. Puccio, Academic Press, San Diego, 1994, p, 425. · Zbl 0845.65040
[28]	Daubechies, I.; Sweldens, W., Factoring wavelet transforms into lifting steps, J. Fourier anal. appl., 4, 245, (1998) · Zbl 0913.42027
[29]	Saito, N.; Beylkin, G., Multiresolution representations using the autocorrelation functions of compactly supported wavelets, IEEE trans. signal process., 41, 3584, (1993) · Zbl 0841.94019
[30]	Sweldens, W., The lifting scheme: A custom-design construction of biorthogonal wavelets, Appl. comput. harmon. anal., 3, 186, (1996) · Zbl 0874.65104
[31]	G. R. Ruetsch, Personal communication, 1998.
[32]	Williams, F.A., Combustion theory, (1986) · Zbl 0603.35048
[33]	Poinsot, T.; Lele, S., Boundary conditions for direct simulations of compressible viscous flows, J. comput. phys., 101, 104, (1992) · Zbl 0766.76084