Exponential, Non-Exponential Splitting Methods and Their Applications for Solving Kawarada Equations
Department of Mathematics
Center for Astrophysics, Space Physics and Engineering Research (CASPER)
This talk consists of two interactive components. First, we will pay an attention to classical splitting methods, such as the non-exponential ADI and exponential LOD methods, and explore their modernizations. Then we will focus at interesting issues involving the design and analysis of highly-effective and highly-efficient finite difference methods for solving singular Kawarada equations which are fundamental in numerical combustion applications. We will outline the physical background of the quenching phenomena involved. Adaptive splitting approaches will be introduced. Numerical analysis on their monotonicity, convergence and stability will be discussed. We will also present ideas of the latest exponential evolving grid development inspired by moving grid strategies which can be extended for solving multiphysics equations with similar singularities from studies of biophysics, oil pipeline decay detections, cancer treatments, and laser-materials interactions. Certain stochastic inferences will be mentioned. Potentials of further investigations and collaborations will be discussed.
Dr. Sheng received his BS and MS in Mathematics from Nanjing University in 1982, 1985, respectively. Then he acquired his Ph.D. from the University of Cambridge under the supervision of Professor Arieh Iserles. After his postdoctoral research with Professor Frank T. Smith, FRS, in University College London, he joined National University of Singapore in 1990. Since then, Dr. Sheng was on faculty of several major universities till his joining Baylor University, which is one of known research institutions and the second biggest private university in the United States.
Dr. Sheng has been interested in splitting and adaptive numerical methods for solving
linear and nonlinear partial differential equations. He is also known for the Sheng-Suzuki theorem in numerical analysis. He has published over 110 refereed articles as well as several joint research monographs. He has been an Editor-in-Chief of the SCI journal, International Journal of Computer Mathematics, published by Taylor and Francis Group since 2010. He gives invited presentations, including keynote lectures, in international conferences every year. Dr. Sheng's projects have been supported by several U.S. research agencies. He currently advises 2 doctoral students and 1 postdoctoral research fellow.