Spectral Deferred Correction Methods For Delay Differential Equations

Delay differential equations (DDEs) arise widely in Automatic Control, Biology, Medical Science, Aviation, Economics and so on. It is very important to study the solving methods( especially the numerical solving methods ). The construction of efficient and stable methods for solving ODEs initial value problems has been considered as a mature subject. Most of the existing methods for DDEs are based on the methods for ODEs. However, the disadvantages of existing schemes are also becoming obvious. These include: ( I ) the step-sizes of the solvers are often constrained by stability properties, especially for stiff problems; ( II )higher order versions of existing solvers either lack desired stability properties or are extremely complicated to solve and may experience efficiency problems. To develop efficient algorithms for differential equations keeping both higher order accuracy and good stability properties is the computational mathematics worker's goal.Alock Dutt et al introduced the spectral deferred correction ( SDC ) strategy for the classical ODEs in 2000. This method couples the Picard integral equation formulations, the classical defect and deferred correction methods and Gauss Legendre orthogonal polynomials and the corresponding Gaussian quadrature rules. It ensures the method's higher accuracy by deferred correction and Gaussian quadrature rules. The algorithm is driven with lower order ( but good stability properties ) methods at the beginning ,and always used at each step until the algorithm has ended, can ensure the very good stability properties. The numerical tests show the algorithm is very excellent.This paper introduced SDC methods for solving the DDEs. The SDC methods for DDEs is constructed , and its convergence and stability is researched too. The numerical tests show the algorithm is very excellent. I believe this paper can enrich the numerical theories of solving delay differential equations and it will bring SDC methods to application widely and more deeply research.