Mathematics professor awarded Air Force grant
UMass Dartmouth Mathematics Professor Dr. Sigal Gottlieb has received a $271,299 grant from the Air Force Office of Scientific Research (AFOSR) to improve current numerical methods used in simulations in the design of planes. The study of aeronautics involves simulations of modeling unknown, multivariable equations. These simulations enable aircraft designers to understand factors such as the impact of different wing shapes on the plane's fuel efficiency and stability of the aircraft, and thereby design better planes. For this and similar reasons, the AFOSR is interested in the design of numerical methods that are tailored to the types of equations that describe complex air flows.
Those numerical methods are the basis for the algorithms that are used in these simulations. The challenge for researchers in designing and proving these methods is in the rapidly changing quantities (called sharp gradients) and even shock waves. For example, when the plane crosses the speed of sound there is a shock wave that will rattle your windows if you live nearby. These are hard to simulate reliably.
Current methods used prove to be stable, but require small steps to be taken by designers when evolving the solution forward in order to account for all the variables impacting the plane's design. Professor Gottlieb's research involves designing and proving stability for numerical methods that can be used that better simulate the constantly evolving equations with sharp gradients shocks. There are two aspects to this - the numerical method used to approximate the behavior of the plane in space and the numerical method used to account for this behavior moving forward in time.
"Designing these methods and mathematically proving that they will give reliable solutions can be difficult," Dr. Gottlieb said. "I am working on getting methods that are more accurate than those previously designed, and that allow us to take larger steps forward when evolving the solution over time, because tiny steps are not efficient."
In her own research activities, Dr. Gottlieb develops and analyzes numerical methods that reliably compute how complex flows evolve in time. Her work has been funded by the National Science Foundation, AFOSR and by a grant from King Abdullah University of Science & Technology (KAUST). She was the opening plenary speaker on the topic "Strong Stability Preserving Time Discretization" at this year's International Conference on Spectral and High Order Methods (ICOSAHOM) and has co-authored two books: "Spectral Methods for Time Dependent Problems" (Cambridge) and "Strong Stability Preserving Runge--Kutta and Multistep Methods" (World Scientific Press). Professor Gottlieb leads a computational mathematics research group specializing in numerical solution of hyperbolic partial differential equations. She is currently supervising the research activities of several graduate students in mathematics.
Dr. Gottlieb is also the founding Director of UMass Dartmouth's Center for Scientific Computing and Visualization Research. The CSCVR promotes the mission of UMass Dartmouth by providing undergraduate and graduate students with high quality discovery-based educational experiences that transcend the traditional boundaries of academic field or department, and foster collaborative research in the computational sciences within the University and with researchers at other universities, National Labs, and industry.