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

Author

Sulbha Goyal

Date of Award

12-1983

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Steven M. Serbin

Committee Members

J. S. Bradley, O. A. Karakashian, A. J. Baker

Abstract

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.

Recommended Citation

Goyal, Sulbha, "A class of Rosenbrock-type schemes for second order nonlinear systems of ordinary differential equations. " PhD diss., University of Tennessee, 1983.
https://trace.tennessee.edu/utk_graddiss/13057