Rational approximations of trigonometric matrices with application to second-order systems of differential equations

摘要

We consider the direct treatment of the second-order system of equations y” (t)+ Ay(t) = tf;(t), such as might arise in finite-element or finite-difference semidiscretizations of the wave equation. We develop the exact solution and some three-term recurrences involving trigonometric matrices. We approximate these trigonometric matrices by rational approximants of Padé type and thus develop a two-parameter family of approximation schemes. We analyze the stability behavior and computational complexity of members of this family and isolate four schemes for numerical experimentation, the results of which we tabulate. We single out as particularly effective the classical Stormer-Numerov method and also a new sixth-order scheme.