Math Colloquium | Parallel Numerical Solution of Boundary-Value Problems of Stiff Differential Equations, April 10

The Mathematics Colloquium series presents “Parallel Numerical Solution of Boundary-Value Problems of Stiff Differential Equations” on Friday, April 10, at 3:05 p.m. in Morton 326.

The presenters are H. Pasic, Ohio University Mechanical Engineering, and William Clark, Ohio University Mechanical Engineering and Mathematics student,

Abstract: Boundary-value problems described by ordinary stiff differential equations are typically solved as initial-value problems combined with shooting methods. More often than not, if the domain is long, the solution gets unbounded before reaching the end point. In that case, the domain is cut into a number of short intervals with guessed missing data between the intervals, and local boundary-value problems are then solved in parallel by reducing the original system to a set of first-order equations and using initial-value-problem solvers. The error introduced by guessing the missing data is then enforced to be zero by numerically solving set of algebraic equations.

Using such an approach can be very expensive. When solving /m /equations of /n/-th order, and using /k /intervals, the size of the aforementioned algebraic system is about /mnk. / When solving 2-nd order equations , H. Pasic proposed a method in which the size of the algebraic system is /m / only by locally matching (in parallel) only two neighboring interval solutions each time. To accelerate the solution, J. Keller of Stanford then developed an algorithm which would match all the intervals simultaneously. Unfortunately, this solution turns out to be unstable because it advances the solution too far in the few initial iterative steps; although, it converges in just one iteration in the case that , irrespective of the number of intervals. It is therefore proposed to combine Pasic’s approach in the first two iteration steps only and then accelerate the solution by using secant method, keeping the method simple and fast.

All of these issues are discussed and demonstrated through examples.

Complete software and solutions of some of the best known test equations are developed and tested by Clark.