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Studies And Applications Of Incremental Unknowns And The Preconditioned Iterative Algorithms

Posted on:2011-03-01Degree:DoctorType:Dissertation
Country:ChinaCandidate:Y WangFull Text:PDF
GTID:1100360305965713Subject:Applied Mathematics
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There are two main parts of work in this paper:The first part of the paper is to study the Wavelet-like Incremental Unknowns (WIUs) preconditioning methods.For a class of porous medium reaction diffusion-type equations, we propose a kind of WIUs-typeθschemes and carefully analyze the stability of these schemes. Numerical results show that theseθschemes based on the WIUs decomposition are efficient.For the anisotropic reaction diffusion equations, with very small parameter∈as a coef-ficient of the partial derivation((?)2u)/((?)x2), the classical WIUs are not usually efficient for solving the equations. Thus, we present a kind of new wavelet-like incremental unknowns based on matrix block in my thesis:Wave-like Block Incremental Unknowns(WBIUs). On the one hand, the new WBIUs possess the characteristics of the classical WIUs, we can use WBIUs to remove automatically some terms in the equivalent finite difference systems by dividing the approximate solution space into two L2 orthogonal subspaces. On the other hand, the new kind of incremental unknowns only introduce WIUs in one direction (in the direction of the largest coefficient), so, it is similar to the one dimensional classical WIUs. Therefore, we can reduce the condition number of the coefficient matrices and we can also save more CPU time with WBIUs comparing with WIUs. Moreover, we set up new inequalities for vector norms and propose WBIUs-type explicit schemes and semi-implicit schemes to solve the equations. The stability of the schemes are proved. Finally, we demonstrate the effectiveness of WBIUs through numerical experiments.The second part is to study several kinds of IUs preconditioning iterative algorithms.Firstly, in Chapter 4, for large sparse non-Hermitian and positive definite linear systems derived from discretization of PDEs, we propose Two-Parameter Preconditioned NSS(TP-PNSS) methods and investigate their variants, e.g., the Inexact Two-Parameter Precondi-tioned NSS (ITP-PNSS) methods. Theoretical analysis shows that the TP-PNSS methods are convergent under some conditions. Moreover, we also present the computational meth-ods of the optimal choice of the two parameters involved in our iterative schemes and the corresponding minimum values for the upper bound of the iterative spectrums. In the final part of Chapter 4, with the help of the IUs, we solve the linear systems. The numerical results demonstrate the effectiveness of the IUs-type iterative methods. Secondly, in Chapter 5, for large sparse systems of nonlinear equations with positive defi-nite Jacobian matrices at the solution point, we propose Newton-TP-PNSS methods through combining TP-PNSS methods and inexact Newton methods. Under proper conditions, we can prove that this class of inexact Newton methods is convergent to the solution of the nonlinear equations. Moreover, two types of local convergent theorems are given. Finally, using incremental unknowns as a preconditioner, the numerical tests show that Newton-TP-PNSS (IU) methods outperform Newton-NSS methods in both iterative steps and iterative time.Finally, in Chapter 6, three versions of modified multiparameter schemes and the cor-responding convergent theorems are presented respectively. Combining with the inexact Newton methods and IUs preconditioning technique, three new types of Newton multipa-rameter algorithms are obtained. The numerical results show that the new algorithms with IUs as a preconditioner can reduce the condition number of the coefficient matrices, thus, they can save a lot of CPU time.
Keywords/Search Tags:finite difference discretization, incremental unknowns, wavelet-like incremental unknowns, wavelet-like block incremental unknowns, condition number of matrix, inertial manifold, θ-schemes, convection diffusion equation
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