Asymptotic stability of Runge-Kutta methods for the pantograph equations

This paper considers the asymptotic stability analysis of both exact and numerical solutions of the following neutral delay differential equation with pantograph delay. {x'(t) + Bx(t) + Cx'(qt) + Dx(qt) = 0, t > 0, x(0) = x(0), where B, C, D is an element of C-dxd, q is an element of (0, 1), and B is regular. After transforming the above equation to non-automatic neutral equation with constant delay, we determine sufficient conditions for the asymptotic stability of the zero solution. Furthermore, we focus on the asymptotic stability behavior of Runge-Kutta method with variable stepsize. It is proved that a L-stable Runge-Kutta method can preserve the above-mentioned stability properties.