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the stability problem. A comparison of various methods is discussed by Johnson [15] , and Zienkiewicz and Taylor[16] . Main interest of our work is to remove numerical instability by adaptive grid generation (very fine grid at the critical zone) and show the effect of filtering by using wavelets without sacrificing the accuracy of the results. In the paper, two wavelet based methods are presented and the results are compared with some recent finite difference methods.

In the first method, linear advection–dispersion equation is solved by using wavelet-Galerkin method. For calculation of inner product, Newton–cotes method is used which can be replaced by recently developed highly efficient methods [17,18]. The basic idea behind the adaptive solution is simply based on the analysis of wavelet coefficients, which gives information about the region where sharp change starts or ends. At any time step only local matrix reflecting the local changes in the solution, is solved. The method uses efficient data structure of uniform grid and periodic basis function to evaluate the en-tries of the stiffness matrix. In the second method, the finest scale finite element solution space is projected onto the scaling and wavelet spaces resulting in the decomposition of high- and low-scale components. Repetition of such a projection results in multi-scale decomposition of the fine scale solution. In the proposed wavelet projection method, the fine scale solution can e
obtained by any other numerical method also. Subsequently the properties of the wavelet functions are exploited to eliminate the nodes from the smooth region where the wavelet coefficients will not exceed a preset tolerance. This wave-let-based multi-scale transformation hierarchically filters out the less significant part of the solution, and thus provides an effective framework for the selection of significant part of the solution. In this process, the ‘big’ coefficient matrix at the finest level will be calculated once for complete domain whereas the ‘small’ adaptively compressed coefficient matrix for a priory known localized dynamic zone of high gradient, which will be considerably less expensive to solve, will be used for the solution in every step of the solution. Similar technique is used in the software QUADFLOW [19] using finite volume method.

References
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[4] G. Strang, T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, Wellesley, MA, 1996.
[5] J. Liandrat and Ph. Tchamitchian, Resolution of the 1D regularized Burgers equation using a spatial wavelet approximation, Report No: NASA CR –187480, NASA Langley Research Centre, Hamptonva, (1990).
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[8] S. Qian, J. Weiss, Wavelets and the 本论文由英语论文网提供整理，提供论文代写，英语论文代写，代写论文，代写英语论文，代写留学生论文，代写英文论文，留学生论文代写相关核心关键词搜索。