Passive Runge-Kutta methods -- properties, parametric representation, and order conditions (Q1415405)

Motivated by the need for numerical simulation of {lossless} Kirchoff networks, the ideas of passive and lossless Runge-Kutta methods are introduced. For a method represented by the arrays \((b^T, A, c)\), the characteristic impedence matrix is defined to be \(Z(\psi) = \psi^{-1} eb^T+2A-eb^T\). A method is passive if \(Z(\psi)+Z^H(\psi) \geq 0\) for all \(\psi\) with positive real part and lossless if, in addition \(Z(\psi)+Z^T(-\psi)=0\). These definitions are equivalent to algebraic stability and symplecticity, respectively, if and only if the components of \(b^T\) are all equal. A consequence of a method being lossless, is that it cannot be L-stable. A parametric representation of passive methods is derived as a means of identifying good methods. By extending the analyses of \textit{M. Sofroniou} and \textit{W. Oevel} [SIAM J. Numer. Anal. 34, No. 5, 2063--2086 (1997; Zbl 0889.65079)] and of \textit{J. M. Sanz-Serna} and \textit{L. Abia} [SIAM J. Numer. Anal. 28, No. 4, 1081--1096 (1991; Zbl 0785.65085)], the order conditions are simplified, in the case of lossless and passive methods. This leads to the derivation of particular lossless methods as well as passive methods which are also diagonal implicit or L-stable or both diagonally implicit and L-stable.