Incremental Unknowns And Wavelet-like Incremental Unknowns Methods Applied In PDEs

Incremental unknowns of different types were proposed as a means to develop numerical schemes in the context of finite difference discretizations.This thesis mainly concerns about incremental unknowns in several aspects as follows.Firstly,this work is to set up the explicit matrix framework appropriate to three-dimensional partial differential equations by means of the incremental unknowns method.Multilevel schemes of the incremental unknowns are presented in the three space dimensions,and through numerical experiments,we confirm that besides solving Poisson equation,the hierarchical preconditioning based on the incremental unknowns can be applied in a more general form.Secondly,we use the incremental unknowns method in conjunction with several iterative methods to approximate the solution of the nonsymmetric and positive-definite linear systems generated from a multilevel discretization of three-dimensional convection-diffusion equations.By theoretical analysis,condition numbers of incremental unknowns matrices associated with the stationary convection diffusion equations and number of iterations needed to attain an acceptable accuracy are estimated.Numerical results are performed with two-level approximations, which demonstrate that the incremental unknowns method when combined with some iterative methods is very efficient.Thirdly,aθ-scheme using two-level incremental unknowns is presented for solving time-dependent convection-diffusion equations in 2-dimensional case.The IMG algorithm(Inertial Manifold-Multigrid algorithm)including the second-order incremental unknowns is convergent.The incremental unknowns method based on theθ-scheme needs a stability condition as 0≤θ＜1/2 and is unconditionally stable as 1/2≤θ≤1.By the GMRES method in the iteration at each time step,numerical results of the convection-diffusion equations are investigated and confirm that oscillations can be controlled by the incremental unknowns method.Fourthly,a modified Crank-Nicolson scheme based on one-sided difference approximation is proposed for solving time-dependent convection dominated diffusion equations in two space variables.A one-sided difference approximation is used for convection terms and a second-order central difference approximation for diffusion term.The numerical scheme is consistent,unconditionally stable. For the fully-discrete time scheme a priori L~2 error estimation is derived.With the use of incremental unknowns as a preconditioner,numerical results confirm stability and efficiency of the modified Crank-Nicolson scheme.And then,we propose an implicit finite difference scheme for the similar problems.For convection dominated diffusion equations with constant coefficients, the implicit scheme is consistent,unconditionally stable and its L~2 error estimation is derived.Moreover,the Burgers' type equation is discretized similarly with the nonlinear term being linearized as a standard semi-implicit scheme, which is conditionally stable.By preconditioning technique of incremental unknowns method,the implicit scheme or the semi-implicit scheme is an efficient one that avoids numerical oscillations and has good performance in stability and accuracy.At the same time,a class of wavelet-like incremental unknowns(WIUs)is presented and has some good properties.On the one hand,the approximation solution space can be decomposed into two L~2-orthogonal subspaces by WIUs, which will cancel automatically some terms in a coupled system.On the other hand,the transfer matrix from the WIUs to the nodal unknowns is orthogonal, which will be convenient for computing.Convergence of the wavelet-like incremental unknowns method is proved when applied in the inertial manifold multigrid algorithms(IMG)or nonlinear Galerkin methods.Finally,we present a wavelet-like incremental unknowns(WIUs)method for the two-dimensional/three-dimensional reaction-diffusion equations with a polynomial nonlinearity and prove that the WIUs are small as expected and the WIUs decompositions are L~2 orthogonal.We obtain the sufficient stability conditions of the Euler explicit schemes and those of the semi-implicit schemes based on the WIUs method,which are improved when compared with stability conditions of the corresponding standard algorithms.Numerical results point out efficiency of the WIUs method in 2D case.Furthermore,stability condition in the three-dimensional case can be derived similarly.