The Numerical Stability Of Rosenbrock Methods For Delay Differential Equations With Many Delays

As we know, many problems in physics, engineering, biology and economics can be modeled by systems of ordinary differential equations (ODEs). Actually, in many realistic models, we should know some past states of these systems. So it is converted to delay differential equations (DDEs), it is widely used bionomics, environment science and electrodynamics. At the first, people always believe that there is no any difference between the numerical treatment to ODEs and DDEs. But that's not the fact. Actually, the analysis of the stability behavior of a numerical method for DDEs is more complex than for ODEs. Furthermore, few analytic expression of the theoretic solution of DDEs can be obtained, so the numerical treatment to DDEs becomes very necessary.Runge-Kutta method and Linear Multi-step method are effectively numerical methods for solving delay differential equations. Rosenbrock H.H. gave the Rosenbrock method in 1963. It is also an effectively numerical method for stiff ordinary differential equations. Among the methods which already gave satisfactory results for stiff equations, Rosenbrock methods are easier to program.The purpose of this paper is to study the stability behavior ofnumerical solution of systems of delay differential equations with many delays. Based on Lagrange interpolation, by the conditions of asymptotic stability of theory solution, we give the sufficient and necessary conditions of GPm-stability & GPmL -stability of Rosenbrock method for DDE (DDEs) with many delays, then we prove that it is GPm-stable for DDE only and only if it is v4-stable for ODE, and it is GPmL-stable for DDE only and only if it is L-stable for ODE.