A Comparison Of Two Kinds Of Interprolation Procedures Of Runge-Kutta Methods For Stiff Delay Differential Equations

Runge-Kutta methods are usually used to solve stiff ordinary differential equations and stiffdelay differential equations There are two kinds of commonly used interpolation procedures todeal with the delay One is to construct a regular piecewise Lagrange interpolation operator(?)^h(t;ψ,y₁,…,y_n+1)；The other is to construct the interpolation operator with the values y_n andthe internal stage values Yⁿof the Runge-Kutta method The obvious merit of the latter is thatthere needs no any starting values for the method to start and the obvious defect is that thereneeds additional storage capacity for the values Yⁿ.However．SO far．we have never seen anywork in literatures for the comparison of the two kinds of interpolation procedures of Runge-Kuttamethods in the computational precision and efficiencyFor the constant coefficient linear stiff delay differential systems,by soiile theoretical analysisand many numerical experiments,that the order of B-convergence of Runge-Kutta methods withthe first type of interpolation procedure can reach its order in the classical sense is correct anddemonstrated by the authors for the implicit mid-point methodOn the other hand,the authors point out that Runge-Kutta methods with the second type ofinterpolation procedure doesn't possess the above advantages,and obtain the following conclusionby theory analysis and numerical tests:For the constant coefficient linear stiff delay differential systems and nonlinear stiff delaydifferential systems with Jacobi matrix changing slowly,the computational precision and efficiencyof a Runge-Kutta method with the first type of interpolation procedure are usually higher thanthat of the same Runge-Kutta method with the second type of interpolation procedure providedthat the stage order s is not too little,moreover,the former is much higher with the stage orders changing largerThe conclusion has a theoretical significance to soiile extent．and provide standards andreferences to choose efficient numerical methods for the practical computation...