Document Type

Presentation Abstract

Presentation Date

4-22-2019

Abstract

Multistep methods provide a computationally efficient way to approximate solutions to differential equations. In general, there is a three-way tradeoff between the accuracy, stability, and computational cost of numerical methods. The stability domain is a picture in the complex plane that shows the problems and stepsizes for which a given numerical method will give stable solutions (i.e., roundoff will not grow exponentially).

I will discuss the development and analysis of novel multistep methods created by introducing parameters that are allowed to vary. Dahlquist's First Stability Barrier puts a cap on the maximum order of a stable method; we seek to maximize the order while maintaining stability. Applying Taylor series gives a linear system for the unknown coefficients of the multistep method. Requiring stability gives bounds on the domains of the free parameters; varying the parameters within this domain results in changes in the size and shape of the stability domain, allowing us to produce methods that work better for a given differential equation, thus creating “designer” numerical methods. I will also discuss staggered methods, some theoretical results, and some real-world applications of our methods.

Here the abstract includes two images described as: Stability domains for two second order Adams-Bashforth/Moulton predictor-corrector methods.

Download attached PDF for the complete abstract with images.

Additional Details

Monday, April 22, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

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