Theories And Applications Of Highly Accurate Alternating Direction Implicit (ADI) Difference Methods

Partial differential equations (PDEs) are frequently encountered in science, engineerand society. However, it is difficult to get exact solutions, or to give practical expressions ofexact solutions. Hence, for obtaining good approximate solutions, it is necessary to devel-op high performance numerical algorithms for PDEs. Alternating direction implicit (ADI)methods can reduce the solution of a multidimensional problem to series of independentone-dimensional problems, and thus save time cost. High-order compact (HOC) schemesis a kind of difference methods, which can use less nodes to attain high accuracy. Combi-nations of ADI methods with HOC schemes yield HOC ADI methods, which preserve theadvantages of HOC schemes and high performance of ADI methods, and thus have attract-ed much attention over the years. Taking some PDEs for example, we study the theoreticalproperties of some HOC ADI methods in this dissertation.In chapter1, we introduce backgrounds and developments of numerical methods, andgive research motivation, main works, notations associated to spatial and temporal grids,and some lemmas.In chapter2, a conditionally stable HOC ADI method for a linear hyperbolic equationwith two spatial variables is derived. Its stability criterion is determined by using von Neu-mann method. It is shown through a discrete energy method that this method can attainfourth-order accuracy in both time and space with respect to H~1-and L~∞-norms providedthe stability is fulfilled. Although the method is conditionally stable, stable restriction is notbad. Numerical solutions can fast converge to exact solution only if that spatial and tempo-ral mesh-sizes are suitably chosen according to stability criterion. In addition, extension ofHOC ADI method to3D case is also discussed, and theoretical results are similarly given.In chapter3, a new three-level HOC ADI method for a nonlinear wave equation isproposed. Basing on a fourth-order approximation to the exact solution at the first timelevel, convergence orders of HOC ADI method in different norms are obtained. A class ofRichardson extrapolation formulas is developed to improve computational efficiency. By in-troducing two auxiliary problems, we give the convergence rates of extrapolation solutions.Chapter4focuses on the construction and extrapolation of a family of three-level HOC ADI methods for a damped wave equation. Applying the methods similar to those usedin Chapter3, the convergence orders of numerical solution and extrapolation solution indifferent norms are obtained.Chapter5is primarily aimed at the analysis of a high-order compact multi-step frac-tional steps (HOC MFS) method for a linear parabolic equation. A combination of lemmasproposed with energy method can prove the discrete solution is unconditionally convergentwith an order of O(△t~2+h_x~4+h_y~4) in H~1-, L~2-and L~∞-norms. Moreover, a class of Richard-son extrapolation methods is established to obtain the extrapolation solution of order four inboth time and space with respect to H~1-, L~2-and L~∞-norms. By using a new transforma-tion, HOC MFS method is easily generalized to convention diffusion equation with constantcoefficients.In Chapter6, we propose a HOC MFS method with three-level for a two-dimensional(2D) nonlinear viscous wave equation. First, we introduce two auxiliary functions to trans-form this equation into an equivalent form similar to parabolic equation. Then, HOC MFSmethod and corresponding extrapolation algorithms for the equivalent form are established.Convergence analysis of the algorithms is carried out in detail.In Chapter7, a Crank-Nicolson HOC ADI method for a three-dimensional (3D) non-linear viscous wave equation is derived. We need to solve the system of linear tridiagonalalgebra equations in x-and y-directions, and need to solve the system of nonlinear algebraequations in z-direction, respectively. By the discrete energy method, it is shown that thismethod has convergence order of O(△t~2+h_x~4+h_y~4) in H~1-norm. A two-grid ex-trapolation method is developed to improve temporal accuracy. By introducing an auxiliaryproblem, we can prove the convergence rate of extrapolation solution.Chapter8is devoted to the conclusions of main contributions, main results and someresearches in future.Numerical results testify the performance of the algorithms and support theoreticalanalysis. In addition, the algorithms proposed in this dissertation can be easily extended toother PDEs.