Galerkin/Runge-Kutta discretizations for parabolic partial differential equations

Author

Stephen Louis Keeling

Date of Award

8-1986

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Ohannes Karakashian

Committee Members

Nicholas Alikakos, Vassilios Dougalis, Suzanne Lenhart, Steven Serbin, Laurence Bales

Abstract

Efficient, high order Galerkin/Runge-Kutta methods are constructed and analyzed for certain classes of parabolic initial boundary value problems. In particular, the partial differential equations considered are (1) semilinear, (2) linear with time dependent coefficients, and (3) quasilinear. Optimal order error estimates are established for each case. Also, for the problems in which the time stepping equations involve coefficient matrices changing at each time step, a preconditioned iterative technique is used to solve the linear systems only approximatley. Nevertheless, the resulting algorithm is shown to preserve the optimal order convergence rate while using only the order of work required by the base scheme applied to a linear parabolic problem with time independent coefficients. Furthermore, it is noted that special Runge-Kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method.

Recommended Citation

Keeling, Stephen Louis, "Galerkin/Runge-Kutta discretizations for parabolic partial differential equations. " PhD diss., University of Tennessee, 1986.
https://trace.tennessee.edu/utk_graddiss/12276