A class of Rosenbrock-type schemes for second order nonlinear systems of ordinary differential equations

This dissertation deals with the numerical solution of second-order nonlinear systems of ordinary differential equations. The methods developed are a class of generalized Rosenbrock-type schemes which have the advantage that they do not require the solutions of nonlinear systems of equations. An s-stage scheme requires the solution of 2s linear systems at each time step, with the same real matrix. These schemes, when applied to a linear time-invariant system u_tt + Au = 0 , reproduce the schemes given by Baker and Bramble that are derived from rational approximation to e^-z , and thus can be chosen to be unconditionally stable for appropriate choice of parameters.

Besides obtaining the order conditions by brute force expansion for two-stage fourth-order scheme, this dissertation also developes the order conditions for these generalized Rosenbrock-type methods based on the theory of Butcher series. Besides the usual linear stability analysis, the concept of P-stability is also extended to the one-step methods of Baker and Bramble. A new criterion of SN-stability for second-order nonautonomous problems is introduced.

These newly developed schemes are implemented and compared with certain well-known standard methods for accuracy, stability and the cost of implementation.