Conditional contractivity of Runge–Kutta methods for nonlinear differential equations with many variable delays

This paper deals with conditional contractivity properties of Runge–Kutta (RK) methods with variable step-size applied to nonlinear differential equations with many variable delays (MDDEs). The concepts of CRNm(ω, H)- and BNf(μ, ℏ)-stability are introduced. It is shown that the numerical solution produced by a BNf(μ, ℏ)-stable Runge–Kutta method with an appropriate interpolation is contractive. In particular, these results are also novel for nonlinear differential equations with many constant delays or single variable delay. To obtain BNf(μ, ℏ)-stable methods, (k, l)-algebraically stable Runge–Kutta methods are also investigated.