A Class Of Block Lower Bidiagonal Preconditioners With GMRES Method For Systems Of LMF-Based ODE Codes

The numerical approximation of solutions of Differential Equations has been and continues to be one of the principal concerns of Numerical Analysis. Boundary value methods (BVMs), the discretization methods, have been wildly used in ordinary differential equations (ODEs). The advantage in using BVMs over classical initial value methods (IVMs) comes from their unconditional stability properties. In this paper, we consider the linear systems of ODEs by using BVMs. These systems are usually unsymmetric, large and sparse. Recently, a class of circulant preconditioners has been proposed to solve these systems. The circulant preconditioners use the FFT technology to accelerate the performance and they have been successfully utilized in solving many systems. In this article, we propose a new class of block lower diagonal preconditioners for solving these linear systems.Firstly, based on the Crank-Nicolson, least condition number and least squares method, the block lower diagonal preconditioners have been constructed to accelerate the GMRES method, as well as some of their convergent properties have been discussed.Secondly, the computational costs of the block lower diagonal preconditioners and the circulant preconditioners have been analyzed. Compare with the circulant preconditioners, the costs of solving these preconditioned systems with our methods are reduced from O(mnlogn) to O(mn).Finally, numerical experiments performed with these new preconditioners and comparative study between them and the circulant preconditioners demonstrate the efficiency of these new methods.This class of preconditioners has been proved to be highly efficiency for the linear systems of ODEs by using BVM on the grounds of theoretical analysis and numerical results. It is reasonable on theory and effective on computational practice.