DSC Kernel Using In Numerically Solving PDE

The traditional methods for numerically solving PDE include local methods(suchas finite difference or finite element method) and global methods(such as spectrummethod). Recently, Wei and Hoffman et.al. have proposed a new method called Dis-crete Singular Convolution method, which both has the high accuracy of global meth-ods and ?exibility of local methods. Different DSC kernels and relative methods havebeen successfully used in many realms such as hydrodynamics and beam analysis.DSCkernel can be regard as a kind of derivative operator but it also has strong tie to wavelettheory.However, DSC algorithm also encounters difficulties such as the oscillation ofthe numerical solution when used in solving non-linear PDE, especially in the hydro-dynamics.Therefore, looking for some robust and reliable methods which can controlthose oscillations by wavelet theory is an important and compelling task.In this thesis, a new DSC kernel with the properties of interpolation,symmetryand compact support is constructed based on the Daubechies wavelet function. More-over, The oscillatory errors produced in the process of DSC method have been ana-lyzed by the convolution properties of distributions and by the context of the theoryof wavelet.Ultimately, a new scheme called DSC-Wavelet algorithm is proposed. Thisnon-oscillation scheme can simply, self-adaptively and accurately solve all kinds ofnon-linear PDEs and efficiently reduce the Gibbs oscillation. At the end of this the-sis,Burger's equation, one kind of Convection-Dominated Diffusion Equations withextremely large values of Reynolds number has been used to test the effect of the newscheme.