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A TVD Type wavelet-galerkin method for Hamilton-Jacobi Equations

Acta Mathematicae Applicatae Sinica 2007年第2期23卷 303-310页

作者： Ling-yan Tang Song-he Song Department of Mathematics and System Science Science School National University of Defense Technology Hunan 410073 China

In this paper, we use Daubechies scaling functions as test functions for the galerkin method, and discuss wavelet-galerkin solutions for the Hamilton-Jacobi equations. It can be proved that the schemes are TVD schemes. Numerical tests indicate that the schemes are suitable for the Hamilton-Jacobi equations. Furthermore, they have high-order accuracy in smooth regions and good resolution of singularities.

关键词： Hamilton-jacobi equation wavelet-galerkin method daubechies wavelet TVD method

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wavelet-galerkin method for the Singular Perturbation Problem with Boundary Layers

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Tsinghua Science and Technology 2000年第4期5卷 365-369,382页

作者：黄忠亿 Department of Mathematical Sciences-Tsinghua University Beijing 100084 China

A wavelet-galerkin method is proposed to solve the singular perturbation problem with boundary layers numerically. Because there are boundary layers in the solution of the singular perturbation problem, the approximation spaces with different scale wavelets and boundary bases are chosen. In addition, the computation of the inner integrals is transformed to an eigenvalue problem. Therefore, a high accuracy method with reasonable computation is obtained. On the other hand, there is an explicit diagonal preconditioning which makes the condition number of the stiff matrix become bounded by a constant. The error estimate of the wavelet-galerkin method and the analysis of the computation complexity are given. The numerical examples show that the method is feasible and effective for solving the singular perturbation problem with boundary layers numerically.

关键词： wavelet wavelet-galerkin method singular perturbation problem boundary layer

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Numerical solution of Helmholtz equation of barotropic atmosphere using wavelets

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Chinese Physics B 2004年第10期13卷 1770-1776页

作者：汪萍戴新刚 KeyLaboratoryofRegionalClimate-EnvironmentforTemperateEastAsia InstituteofAtmosphericPhysicsChineseAcademyofSciencesSTARTRegionalCenterforTemperateEastAsiaBeijing100029China

The numerical solution of the Helmholtz equation for barotropic atmosphere is estimated by use of the waveletgalerkin method. The solution involves the decomposition of a circulant matrix consisting up of 2-term connection coefficients of wavelet scaling functions. Three matrix decompositions, i.e. fast Fourier transformation (FFT), Jacobian and QR decomposition methods, are tested numerically. The Jacobian method has the smallest matrix-reconstruction error with the best orthogonality while the FFT method causes the biggest errors. Numerical result reveals that the numerical solution of the equation is very sensitive to the decomposition methods, and the QR and Jacobian decomposition methods, whose errors are of the order of 10-3, much smaller than that with the FFT method, are more suitable to the numerical solution of the equation. With the two methods the solutions are also proved to have much higher accuracy than the iteration solution with the finite difference approximation. In addition, the wavelet numerical method is very useful for the solution of a climate model in low resolution. The solution accuracy of the equation may significantly increase with the order of Daubechies wavelet.

关键词： wavelet-galerkin method Helmholtz equation matrix decomposition barotropic atmosphere

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