Precise time integration for linear two-point boundary value problems

Abstract

The precise time integration (PTI) method is proposed to solve the linear two-point boundary value problem (TPBVP). By employing the method of dimensional expanding, the non-homogeneous ordinary differential equations (ODEs) can be transformed into homogeneous ones and then the original PTI algorithms can be applied directly to the TPBVP. The PTI consists of two methods: the method of matrix exponential and the method of Riccati equations, both of which utilize the merit of 2 N algorithm and guarantee high precise numerical results. The method of matrix exponential follows the similar scheme proposed for initial value problem of ODEs, and uses the matrix exponential to link boundary conditions between the two points. The method of Riccati equations employs the Riccati equations to express the relationships of two point boundary conditions. And then in terms of the relationships of boundary conditions at two points, the full conditions at initial point can be obtained and then the TPBVP can be transformed into initial value problem and then solved by direct time marching scheme. With some modifications, the above algorithms can be directly extended to the infinite-interval problem and the variant coefficient ODEs problem. In the program implementation, the object-oriented (OO) design of PTI is proposed to demonstrate the applicability and easy maintenance of OO techniques in numerical computation. Finally, four selected numerical examples are given to show the high precision characteristics of PTI.