An adaptive wavelet-vaguelette algorithm for the solution of PDEs

Journal Article · Wed Jan 15 04:00:00 EST 1997 · Journal of Computational Physics

DOI:https://doi.org/10.1006/jcph.1996.5573· OSTI ID:530788

Froehlich, J ^[1]; Schneider, K ^[2]

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1. Konrad-Zuse-Zentrum Berlin (Germany)
2. Universitaet Kaiserslautern (Germany)

The paper first describes a fast algorithm for the discrete orthonormal wavelet transform and its inverse without using the scaling function. This approach permits to compute the decomposition of a function into a lacunary wavelet basis, i.e., a basis constituted of a subset of all basis functions up to a certain scale, without modification. The construction is then extended to operator-adapted biorthogonal wavelets. This is relevant for the solution of certain nonlinear evolutionary PDEs where a priori information about the significant coefficients is available. We pursue the approach which is based on the explicit computation of the scalewise contributions of the approximated function to the values at points of hierarchical grids. Here, we present an improved construction employing the cardinal function of the multiresolution. The new method is applied to the Helmholtz equation and illustrated by comparative numerical results. It is then extended for the solution of a nonlinear parabolic PDE with semi-implicit discretization in time and self-adaptive wavelet discretization in space. Results with full adaptivity of the spatial wavelet discretization are presented for a one-dimensional flame front as well as for a two-dimensional problem. 50 refs., 4 figs., 3 tabs.

• Sponsoring Organization: USDOE
• OSTI ID: 530788
• Journal Information: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: 2 Vol. 130; ISSN 0021-9991; ISSN JCTPAH
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
ALGORITHMS
FOURIER ANALYSIS
NONLINEAR PROBLEMS
NUMERICAL SOLUTION
PARTIAL DIFFERENTIAL EQUATIONS