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by Oleg V. Vasilyev, Christopher Bowman
Journal of Computational Physics
http://landau.mae.missouri.edu/~vasilyev/Publications/JCP2000.pdf
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Abstract:
An adaptive numerical method for solving partial differential equations is developed. The method is based on the whole new class of second-generation wavelets. Wavelet decomposition is used for grid adaptation and interpolation, while a new O(N) hierarchical finite difference scheme, which takes advantage of wavelet multilevel decomposition, is used for derivative calculations. The treatment of nonlinear terms and general boundary conditions is a straightforward task due to the collocation nature of the algorithm. In this paper we demonstrate the algorithm for one particular choice of second-generation wavelets, namely lifted interpolating wavelets on an interval with uniform (regular) sampling. The main advantage of using second-generation wavelets is that wavelets can be custom designed for complex domains and irregular sampling. Thus, the strength of the new method is that it can be easily extended to the whole class of second-generation wavelets, leaving the freedom and flexibility to choose the wavelet basis depending on the application.
Citations
|
1190
|
Orthonomal bases of compactly supported wavelets
– Daubechies
- 1988
|
|
290
|
The lifting scheme: A custom-design construction of biorthogonal wavelets
– Sweldens
- 1996
|
|
269
|
Factoring Wavelet Transforms into Lifting Steps
– Daubechies, Sweldens
- 1998
|
|
234
|
The lifting scheme: A construction of second generation wavelets
– Sweldens
- 1998
|
|
175
|
Wavelets on the interval and fast wavelet transforms
– Cohen, Daubechies, et al.
- 1993
|
|
99
|
Symmetric iterative interpolation processes
– Deslauriers, Dubuc
- 1989
|
|
80
|
Wavelets on closed subsets of the real line
– Andersson, Hall, et al.
- 1994
|
|
78
|
Interpolating wavelet transforms
– Donoho
- 1992
|
|
40
|
Multiresolution representations using the autocorrelation functions of compactly supported wavelets
– SAITO, BEYLKIN
- 1993
|
|
39
|
Wavelets on a bounded interval, in: Numerical Methods of Approximation Theory
– Chui, Quak
- 1992
|
|
33
|
Multiresolution algorithms for the numerical solution of hyperbolic conservation laws
– Harten
- 1995
|
|
27
|
Wavelets and filter banks, Wellesley-Cambridge
– Strang, Nguyen
- 1996
|
|
21
|
A wavelet based space-time adaptive numerical method for partial differential equations
– Bacry, Mallat, et al.
- 1992
|
|
17
|
Resolution of the 1D regularized Burgers equation using a spatial wavelet approximation: Algorithms and numerical results
– Liandrat, Tchamitchian
- 1990
|
|
15
|
An adaptive wavelet-vaguelette algorithm for the solution of nonlinear PDEs
– Fröhlich, Schneider
- 1997
|
|
14
|
Adaptive multiresolution schemes for shock computations
– Harten
- 1994
|
|
14
|
A dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain
– Vasilyev, Paolucci
- 1996
|
|
13
|
Wavelets: Theory and Applications
– Louis, Maaß, et al.
- 1997
|
|
12
|
On the adaptive numerical solution of nonlinear partial differential equations in wavelet bases
– Beylkin, Keiser
- 1997
|
|
12
|
A multilevel wavelet collocation method for solving partial differential equations in a finite domain
– Vasilyev, Paolucci, et al.
- 1995
|
|
12
|
Adaptive Multiresolution Collocation Methods for Initial Boundary Value Problems of Nonlinear PDEs
– Cai, Wang
- 1996
|
|
12
|
A fast adaptive wavelet collocation algorithm for multidimensional PDEs
– Vasilyev, Paolucci
- 1997
|
|
11
|
Boundary Conditions for Direct Simulations of Compressible Viscous Flows
– Poinsot, Lele
- 1992
|
|
9
|
Wavelet methods for the numerical solution of boundary value problems on the interval
– Bertoluzza, Naldi, et al.
- 1994
|
|
8
|
Solving hyperbolic PDEs using interpolating wavelets
– Holmstrom
- 1996
|
|
7
|
A wavelet-optimized, very high order adaptive grid and order numerical method
– Jameson
- 1998
|
|
4
|
Adaptive wavelet methods for hyperbolic PDEs
– Holmstrom, Walden
- 1998
|
|
4
|
Filter bank methods for hyperbolic PDEs
– Walden
- 1999
|
|
3
|
Wavelets and Numerical Methods
– Jameson
- 1993
|
|
2
|
Combustion Theory (Addison–Wesley
– Williams
- 1986
|
|
1
|
Wavelets and Operators (translated by
– Meyer
- 1992
|
