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Undergraduate Research at UMass Dartmouth
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When I first started taking Computational Mathematics
curriculum courses back in 2012, I never
thought research is what I wanted to do. In 2012 I
joined a class called CSUMS that was centered on
independent undergraduate research. Even though
I enjoyed the class, research was not on my mind.
Eventually, I started taking higher level classes and
realized that research was the base of everything I
did, whether it be in my Civil Engineering classes or
in my Math classes. Subsequently, I decided to do
research independent of classwork.
This past summer I approached Dr. Gottlieb to see
if she would let me join her research group. She
warmly accepted and started to instruct me in the
topics I needed to learn. This was just the start.
Shortly thereafter an OUR summer grant enabled
me to work with Dr. Gottlieb on a research titled
“Strong Stability Preserving Sixth Order Two-Derivative
Runge-Kutta Methods.” Hyperbolic partial
differential equations (PDEs) describe a wide-range
of physical phenomena in a variety of fields, such as
aeronautics, oceanography, and astrophysics. These
equations describe solutions that have wave-like
behavior, such as fluid flows and gravitational
waves. In many cases, the physical behavior of this
phenomenon and the related solutions to the hyperbolic
PDE develop sharp gradients or discontinuities.
In such cases, the numerical methods used to
approximate the solutions in space and evolve them
forward in time need to be very carefully designed
so they can handle the discontinuities and remain
stable and accurate.
The design of high order Strong Stability Preserving
(SSP) time-stepping methods that are advantageous
for use with spatial discretizations and that have
nonlinear stability properties needed for the solution
of hyperbolic PDEs with shocks, has been an
active area of research over the last two decades. In
particular, the focus has been to design high order
methods with large allowable time-step. SSP methods
in the multistep and Runge-Kutta families have
been developed. However, these methods have order
barriers and time-step restrictions. The focus of this
project was to develop new SSP time discretizations
by further exploring the class of multi-derivative
Runge-Kutta methods.