A Fast Solution And The Preconditioning Methods For ODEs System

Ordinary differential equation(ODE) has gradually developed into a theoretical discipline, which own its independent research objects over the past 300 years. Many phenomena in nature and engineering technology can be described mathematically with definite problem of ODEs. Many problems can be solved by establishing certain differential equation models,such as the law of universal gravitation, the rule of population development and the changes of market equilibrium price. However, differential equations in the actual production and scientific research are often very complex, and it is impossible to derive the analytical expression of solution in many cases. What’s more, the roots of high order algebraic equations are not easy to get. So it is often unable to realize to derive the analytic solution of ODEs. The numerical method is mainly used in practical computation.Recently, the preconditioning methods for linear equations, which is obtained by three order-linear ODE discretized by Sinc methods, have already studied by Zhongzhi Bai and Zhiru Ren. Our research is mainly based on these former researchers’ results, and it is also the extension of their results. We first introduce the Sinc methods, and then we transform a third-order ODE into a system of two second-order ODEs with which is equivalent by introducing a variable substitution. Sinc method is used to discrete the two-order linear ODEs and the target system of linear equations is obtained whose coefficient matrix is of block two-by-two structure, and each of its blocks is a combination of diagonal and Toeplitz matrices. Combining with existing theoretical knowledge and research results, the following work is done in this paper.According to the structural properties of the coefficient matrix of linear equations generated by the Sinc methods, in order to effectively get the solution of linear equations by Krylov subspace methods, a new preconditioner NP is presented. At the same time, the ILU preconditioners are chosen, and experiments show that the spectral distribution of preconditioning matrix of the two kinds of preconditioners are very dense.Numerical experiments results prove that Sinc methods are of feasibility for solving three-order linear ODEs. And the discretized linear systems can be solved by Krylov subspace methods. The chart shows that the number of iterations is less for the preconditioner NP in the thesis and is nearly halved with the increase of the value of N, compared with the preconditioner P. It can be concluded that the choice of preconditioner NP in the thesis is better.