The Numerical Stability Analysis Of Linear Multistep Methods For Delay Differential Equations

Stiff delay differential equations (DDEs) are widely used in the fields of physics, biology, medicine, engineering cybernetics and many other science and engineering fields, so it turns very essential to make research on the numerical solution of stiff DDEs. The existing literature has done a series of research regarding stiff DDEs and the stability of numerical methods, but the majority are directed at the inner product space of DDEs. So far, results of stability theory of DDEs in Banach spaces are still very few. This paper is devoted to a study of nonlinear stability and asymptotic stability properties of a class of linear multistep methods when applied to nonlinear stiff DDEs in Banach spaces.In the first chapter, we briefly introduce some examples of DDEs in different areas and the development of the stability for theoretical and numerical solutions of stiff DDEs. Then, we review the stability properties for theoretical and numerical solutions of stiff DDEs in Banach spaces. And some new problems are introduced.In the second chapter, for a class of linear multistep methods applied to initial value problems of DDEs, when the interpolation operator satisfies certain limited conditions, we study the nonlinear stability of the methods for the two test problem classes D (α,λ~* ,β) and D (α,λ~* ,δ,β) in Banach spaces respectively.In the third chapter, when the interpolation operator of the linear multistep methods which is discussed in the second chapter is linear interpolation, for the two test problem classes D (α,λ~* ,β) and D (α,λ~* ,δ,β) in Banach spaces, we study the asymptotic stability of the methods respectively.The stability results in the second chapter and the third chapter are independent of the interval length, and by use of a special class of linear multistep methods the stability results show reasonable. Especially, the implicit Euler method unconditionally preserves the contractivity and asymptotic stability of the model equations.In the fourth chapter, some numerical experiments are given to illustrate that the stability results are effective.