Many finite difference methods that involve spatial adaptation employ an equidistribution principle. In these cases, a new mesh is constructed such that a given monitor function is equidistributed in some sense. Typical choices of the monitor function involve the solution or one of its many derivatives. This constructive strategy has been proven to be extremely effective and easy-to-use in multiphysical computations. However, selections of core monitoring functions are often challenging and crucial to the computational success. This talk concerns several different designs of the monitoring function that targets a highly nonlinear partial differential equation that exhibits both quenching-type and degeneracy singularities. While the first a few monitoring designs to be discussed are within the so-called direct regime, the rest belong to a newer category of the indirect type, which requires the priori knowledge of certain important solution features or characteristics. Simulated examples will be presented to illustrate our study and conclusions. Further research initiatives with Macau colleagues will be discussed.
Prof. Sheng joined the Baylor faculty in August 2005. Prior to coming to Baylor he held a research position in University of London (1989-1990), a visiting professor position in Baylor University (2003) and faculty positions in National University of Singapore (1990-1995), University of Louisiana (1996-2001) and University of Dayton (2001-2005). He was a recipient of the J. T. Knight Prize in Mathematics (1987) and Lundgren Research Award (1989). Dr. Sheng was an invited research participant of the Isaac Newton Institute for Mathematical Sciences, Cambridge, England (2007). He was a U.S. Air Force SFFP Research Fellow (2005-2007). Dr. Sheng directed two doctoral dissertations, 8 Master of Science theses and a number of undergraduate research theses. He is married to Helen. They have sons Andy and Dan. He enjoys reading, painting, traveling and spending time with the family.
Prof. Sheng’s research is in computational mathematics. In particular, he is interested in splitting and adaptive methods for solving singular partial differential equations. He has been involved in cross-disciplinary projects in scientific and engineering computations. Prof. Sheng has been active in his research fields and community. He is on editorial boards of several scholarly journals and special research issues. His projects have been supported by the United States Air Force Research Laboratory and Department of Defense.