**Numerical and Computational Challenges
in Science and Engineering Program**

## Ian Gladwell

Department of Mathematics, Southern Methodist University

*LECTURES*

Friday, October 26, 2001, 2:00 pm, room 230

*Variable step methods for model Hamiltonian Systems*

We consider the implementation of a variety of variable step size methods
for partitioned Hamiltonian systems. Some of these methods are Hamiltonian
preserving. To test the implementations we use some commonly occurring
model partitioned Hamiltonian systems. We study the impact of the step
size adaptivity strategy on the cost and quality of solutions of problems
of a range of difficulties. Particularly, we are concerned with the
effects of iteration and accumulated roundoff error, and of limited
precision.

This is joint work with Valeria Antohe.

____________________________

Friday, September 28, 2001, 10:00 am, room 230

*SOLVING BORDERED ALMOST BLOCK DIAGONAL LINEAR SYSTEMS*

Almost block diagonal (ABD) linear systems arise in the discretization
of boundary value problems in differential equations when there are
separated boundary conditions, e.g. for problems with Dirichlet

boundary conditions. When the boundary conditions are non-separated,
e.g. for periodic boundary conditions, the corresponding system is bordered
almost block diagonal (BABD).

We show by example why the standard Gaussian elimination algorithm
when used to solve a BABD system might fail, and we consider alternative
algorithms which work with the BABD structure. We discuss the performance
of the MATLAB dense and sparse linear system library functions on this
BABD problem.

Finally, we show how to convert a BABD system to a larger ABD system
and discuss the use of standard and specially designed ABD algorithms
for this transformed system.

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