Accelerating Iterative And Parallel Algorithms Of Some Difference Schemes For Solving Parabolic And Hyperbolic Equations

With the development of computer, numerical solution for partial differential equations has also been a huge development. Difference method is a primary method for solving partial differential equations. As well known, the explicit difference scheme is suitable for parallel computation, but it has the limitation of stability, and it is often restricted especially in the treatment of high-dimensional problems. The implicit difference scheme is absolutely stable generally, but it is necessary to solve different linear systems at each level of time.In chapter one of the paper, we construct a parallel algorithm of accelerative iteration for solving compact scheme of one-dimensional parabolic differential equations at first. We split the coefficient matrix of the linear systems with respect to a compact difference scheme, and then we use iterative methods to solve the subsystems one by one. The convergence of the algorithm and the property of asymptotic convergence are proved. For the two-dimensional parabolic partial differential equation, we discuss the compact and alternating direction implicit scheme. The parallel algorithm of accelerative iteration is constructed.In the chapter two, we mainly study the hyperbolic partial differential equation. The initial and boundary value problem of the wave equation is used as an example to construct the parallel algorithm of accelerative iteration with respect to the classical implicit difference scheme and the compact difference scheme. Then the alternating direction implicit scheme and its parallel algorithm of accelerative iteration for two-dimensional hyperbolic partial differential equation are also discussed.The numerical examples are carried out at last. The results of numerical examples show that the analysis is correct and the algorithm is feasible and efficient.