Wavelet Methods For Solving Partial Differential Equations And Application In Computational Combustion

As a mathematics tool, wavelet transform is viewed as mathematics microscope, because it offers multi-resolution analysis in time and frequency domain. Although the theory of wavelet is not to the nines now, the perfect mathematical character are winning more and more favor from scientists and engineering personnel, and it demonstrates the huge potential usefulness of the wavelet technique in large-scale numerical simulations.A novel wavelet adaptive multilevel representation algorithm combined with mapped weighted essentially non-oscillatory schemes has been developed to solve challenging combustion simulation in aerospace engineering. The emphasis of this thesis is an extension of the dynamically adaptive second generation wavelet collocation method for evolution problems. In the new approach which takes advantage of wavelet multilevel decomposition and highly accurate numerical scheme, wavelet decomposition is used for grid adaptation and interpolations while a WEN0M5 finite difference scheme be used for high accurate . The main work of this thesis states bellows:1 The relevant fundamental problem of wavelet theories are introduced. Beginning with some basic theories and important conclusions, the basic characteristics and algorithms of wavelets are discussed. After the study of the basic properties of wavelets, then this thesis gives several example of wavelets, such as Daubechies' orthonormal wavelets and its correlation wavelet function.2 Nonlinear approximation based wavelet method is reviewed. First, the wavelet adaptived approximation is descriptived, followed with its good characteristics ; Next, an adaptive mutisacle processing, the represention of the diffential operators , and Mallat's reconstruction method are discussed;. Then, the class of discrete data for which we can obtain representation in terms of a multiscale decomposition for numerical schemes is presented. Finally, the multiresolution adaptive approach for the solution of partial differential equations (PDE) is developed, and the resulting algorithm is considerably more efficient by using matrix operators. The new method is tested on nonlinear Burgers equation. The present results indicate that the method is competitive with well-established numerical algorithms.3 An wavelet adaptive multilevel representation algorithm based on cubic spline wavelet basis of H₀²(I) has been developed. First, the best approximationproperty of spline wavelets to approximating the functions in Sobloev Spaces is obtained. Next, the parallel algorithm of spline wavelets decomposition is deduced and an adaptive collocation method for solving PDE is proposed. Finally, the method is applied to the solution of one dimensinal Burgers equation ,and the results indicates that the method is very accurate and efficient.4 An adaptive multi-resolution schemes for the computation of hyperbolic conservation laws are presented, which combines WEN05M and second wavelets. The new schemes use the adaptive wavelet collocation as a tool of data compression for the numerical solution in order to reduce the number of numerical fluxes evaluation. A simple modification of the WENO scheme is then modified to be sufficient to give optimal order convergence even near critical points. Finally, two examples utilizing the compressible Euler equations are used to demonstrate the scheme's improved behavior for practical shock capturing problems.5 Chemically reacting with multi-species gas flows which arise in variety of combustion problems is studied using the wavelet-based techniques. First, the compressible Navier-Stokes equations are model for chemically reacting gas flows. Next, Simple expressions for characteristic data, including Jacobian matrix of the convective fluxes, the associated eigenvalues and eigenvectors are derived. Finally, based on multi-species and chemically reacting gas flows numerical approach from Fedkiw and Singh method. The results demonstrats the accuracy and computational efficiency of the new method, and also shows that the new high accuracy numerical method can be usefully extended to the much more complicated problem of chemically reacting gas flows.