Block-Circulant Preconditioned Methods For Ordinary And Neutral Differential-Algebraic Systems

We study the solution of linear constant coefficient differential-algebraic equations(DAEs) and delay differential-algebraic equations(DDAEs) by using boundary value methods (BVMs). BVMs are a relatively recent class of methods for the numerical treatment of a wide variety of differential equations, and these methods are based on the linear multistep formulae(LMF). Their application transforms a (delay) differential-algebraic problem into a discrete one, represented by a unsymmetric linear system that are often large and sparse. We use GMRES method to solve the linear system. In order to speed up the convergence rate of GMRES method, we construct the corresponding block-circulant preconditioners.In the first chapter, we give a brief introduction to the related background knowledge of iterative methods for sparse linear system and review the developments on the circulant preconditioners since 2000.In the second chapter , we use BVMs to solve the DAEs in a direct way. We note that the coefficient matrix of linear system arising from the discretization of DAEs is block Toeplitz matrix with low rank perturbation. Therefore, we construct the Strang-type block-circulant preconditioners. We show that, when an Ak 1 , k2-stable BVM is used for solving a system of DAEs, our preconditioner is nonsingular and the spectrum of the preconditioned matrix is clustered. Therefore, the GMRES method will converge fast if the method is applied to solving the preconditioned systems. Numerical experiments are given to illustrate the effectiveness of our methods.In the third chapter, we use BVMs with GMRES method to solve a class of DDAEs. We construct the corresponding block-circulant preconditioners. Theoretical analysis shows that our preconditioners are invertible and efficient. Finally, we give a numerical example to confirm the theoretical results.