Delay-dependent stability of symmetric Runge-Kutta methods for second order delay differential equations with three parameters

The paper is concerned with the delay-dependent stability analysis of symmetric Runge-Kutta methods, which include the Gauss methods and the Lobatto IIIA, IIIB and IIIS methods, for the second order delay differential equations with three parameters. By using the root locus technique, the root locus curve is given and the numerical stability region of symmetric Runge-Kutta methods is obtained. It is proved that, under some conditions, the analytical stability region is contained in the numerical stability region. Numerical examples confirming the theoretical results are presented.