A proof that the discrete singular convolution (DSC)/Lagrange-distributed approximating function (LDAF) method is inferior to high order finite differences. (English) Zbl 1092.65020
The author discusses the approximation of the derivatives of a function by means of finite differences. The discrete singular convolution (DSC) and Lagrange-distributed approximating function (LDAF) methods are investigated by the author, as a single algorithm. Sum-accelerated pseudospectral methods and so called “sinc” basis are successfully used. By means of Fourier analysis and error theorems, it is shown that the DSC method is, sometimes, worse than the standard finite differences, in differentiating the exponential function. This does not mean that DSC is spectrally inferior to finite differences (in the sens of giving a poorer approximation to the eigenvalue of the first derivative operator ) for all wave numbers and all choices of the DSC width parameter. For some values of this parameter, DSC errors are slightly smaller than finite differences for wavelengths between two and four times the grid spacing. The conclusion of the author is that the DSC/LDAF method is never the method of choice for approximating derivatives.
Added in 2009: See also the comment by Wei and Zhao in J. Comput. Phys. 226, No. 2, 2389–2392 (2007; Zbl 1161.65359).
Added in 2009: See also the comment by Wei and Zhao in J. Comput. Phys. 226, No. 2, 2389–2392 (2007; Zbl 1161.65359).
Reviewer: Iulian Coroian (Baia Mare)
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
Keywords:
Pseudospectral method; Lagrange-distributed approximating function method; Discrete singular convolution method; Nonstandard finite differences; Spectral differences; Algorithm; Sinc basisCitations:
Zbl 1161.65359Software:
MatlabReferences:
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