Strang-Type Preconditioners for Solving Linear Systems from Delay Differential Equations

We consider the solution of delay differential equations (DDEs) by using boundary value methods (BVMs). These methods require the solution of one or more nonsymmetric, large and sparse linear systems. The GMRES method with the Strang-type block-circulant preconditioner is proposed for solving these linear systems. We show that if a Pk1,k2-stable BVM is used for solving an m-by-m system of DDEs, then our preconditioner is invertible and all the eigenvalues of the preconditioned system are clustered around 1. It follows that when the GMRES method is applied to solving the preconditioned systems, the method may converge fast. Numerical results are given to illustrate the effectiveness of our methods.

[1]  T. Chan.  An Optimal Circulant Preconditioner for Toeplitz Systems , 1988 .

[2]  Xiao-Qing Jin,et al.  Developments and Applications of Block Toeplitz Iterative Solvers , 2003 .

[3]  L. Torelli,et al.  Stability of numerical methods for delay differential equations , 1989 .

[4]  Daniele Bertaccini.  Reliable preconditioned iterative linear solvers for some numerical integrators , 2001 .

[5]  L. Brugnano,et al.  Solving differential problems by multistep initial and boundary value methods , 1998 .

[6]  Eugene E. Tyrtyshnikov,et al.  Circulant preconditioners with unbounded inverses , 1995 .

[7]  Fabio Di Benedetto,et al.  Analysis of Preconditioning Techniques for Ill-Conditioned Toeplitz Matrices , 1995, SIAM J. Sci. Comput..

[8]  T. Mori,et al.  Simple stability criteria for single and composite linear systems with time delays , 1981 .

[9]  Raymond H. Chan,et al.  Strang‐type preconditioners for systems of LMF‐based ODE codes , 2001 .

[10]  Daniele Bertaccini,et al.  A Circulant Preconditioner for the Systems of LMF-Based ODE Codes , 2000, SIAM J. Sci. Comput..

[11]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[12]  John R. Gilbert,et al.  Sparse Matrices in MATLAB: Design and Implementation , 1992, SIAM J. Matrix Anal. Appl..

[13]  Raymond H. Chan,et al.  Conjugate Gradient Methods for Toeplitz Systems , 1996, SIAM Rev..

[14]  Erik Noldus,et al.  A way to stabilize linear systems with delayed state , 1983, Autom..

[15]  Michael K. Ng,et al.  Skew-Circulant Preconditioners for Systems of LMF-Based ODE Codes , 2000, NAA.

[16]  Tian Hong-jiong,et al.  The numerical stability of linear multistep methods for delay differential equations with many delays , 1996 .