Stability of Numerical Methods for Solving Systems of Second Order ODE's

摘要

This paper considers the problems of solving a second order linear ordinary equation of the form (x double dot)(t) + C(x dot)(t) + Kx(t) = 0, where x0 and (x dot)0 are given and C and K are N x N square matrices. A proposed solution is given so that the system is stable and the solutions decay. The Houbolt 2-step method is thought to be insufficient to distinguish useful methods. Cases where the identity matrix and the matrices C and K are simultaneously diagonalisable, and are assumed not to be simultaneously block diagonalisable are considered. Use of complex coefficients in the scalar test equation and also the use of matrix test equations gives useful results, but easily leads to instability. Other multi-step methods are examined without finding more restrictions for stability. No algorithmic solution is proposed.