The Preconditioning Technique For Delay Differential Equations

This paper studies the preconditioning methods for the discrete systems of delay bound-ary value problems, delay parabolic partial differential equations, constant coefficient par-tial differential-algebraic equations and singular perturbation delay partial differential equa-tions. Boundary value method (BVMs) are applied to solve the continuous problems. Theseobtained discrete algebraic systems can be written in the form My = b, where M is usuallya large sparse non-symmetric matrix with special form. For solving the algebraic equationeffectively, we construct circulant preconditioners to the original equation in order to ac-celerate the convergence rate, reduce the number of iterations and shorten the computingtime.Boundary value methods are a class of more recently and effective numerical methodfor solving differential equations, they are known as the"the third method between linearmultistep methods and Rung-Kutta method". They are widely used to solve many prob-lems such as initial problems, two point boundary value problems, differential algebraicsystems and partial differential equations. Compared to linear multi-step methods, BVMscan possess both high accuracy and good stability properties.This paper is organized as follows. First of all, we give a brief introduction to therelated background knowledge of iterative methods for sparse linear system and reviewthe developments on the circulant preconditioning technique. Then we apply circulantpreconditioning technique to delay boundary value problems and compare the convergencespeed of different preconditioners. Furthermore, we construct some preconditionersfor the discrete systems derived from BVMs discretization of delay parabolic partialdifferential equations (DPPDAEs), delay partial differential-algebraic equations (DPDAEs)and singular perturbation delay partial differential equations (SPDPDEs). We show that ourcirculant preconditioners are nonsingular and the spectrum of the preconditioned matricesare clustered. Therefore, the GMRES method will converge fast when the methods areapplied to solve the preconditioned systems. Numerical illustrations are reported andconfirm the efficiency of our preconditioning methods.