Research in Mathematics
Strong Stability Preserving Sixth Order Two-Derivative Runge–Kutta Methods
By Gustavo Franco Reynoso
This past summer I joined Professor Sigal Gottlieb and PhD student Zachary Grant in their Computational Mathematics research on “Strong Stability Preserving Sixth Order Two-Derivative Runge-Kutta Methods.” It was a great experience that has helped me understand my abilities and my interests. Before I explain the project, I would like to go back in time to provide some background information about my research.
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.
Left: Portrait of Reynoso at work; right:The first page of a study conducted by Reynoso, Gottlieb, and Grant.
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.
My main job at the beginning was to derive the order conditions needed to design higher order multi-derivative methods. I derived the two derivative Runge-Kutta order conditions up to 6th order using what is known as Butcher trees. Just the one derivative derivation had 37 trees, after including the second derivative, it increased tremendously. Some trees had around 15 sub-derivations; this was a tedious job that taught me a lot on how to be efficient and optimal. After deriving all the order conditions, they had to be included into a code that finds numerically optimal multi-derivative Runge-Kutta methods and tests these methods for accuracy and for the sharpness of the SSP condition on test problems used previously in the SSP field. We were able to find methods that gave us sixth order accurate, and after doing so we found that there are 7th order methods that work as well.
This experience led me to realize how I want to further my education. Thanks to a summer grant from the OUR as well as help from Dr. Gottlieb and Zack Grant, I have decided to pursue a PhD at UMD in Engineering and Applied Science. This will be an amazing experience and I very much look forward to it. To all students out there who have yet to find the beauty hidden in the intricate curiosity that some call research, I recommend that you get involved in research as soon as possible. If you find that you don’t like it, it is easy to get out; but, if you find it luring and attractive, you will feel like you have lost time not doing it earlier. Research is not boring, as many students might think. It is challenging and never definitive or monotonous. You’re always learning something new. Even if you try it once and don’t like it, you could still try it again, because there are so many topics unexplored that you are bound to find something you find interesting.
I’d like to leave you with this quote by the American biochemist and peace activist, Linus Carl Pauling:
“Satisfaction of one’s curiosity is one of the greatest sources of happiness in life.”