报告题目1：Six Correlated Ways of Moving Mesh Adaptations for the Numerical Solution of Nonlinear Quenching Problems
摘要：It was during the last Banff International Research Station (BIRS) Workshop on Adaptive Numerical Methods for Partial Differential Equations with Applications, the topics of different adaptive finite difference procedures were seriously revisited, reevaluated and reinvested. Among various kinds of singular partial differential equation problems, Kawarada problems are particularly attractive to the participants due to their important theoretical and application features.
In a traditional moving mesh approach, mesh adaptations are often configurated based on an equidistribution principle. In such a case, a new mesh is acquired via a monitor function that is equidistributed in some sense. Typical choices of such monitor functions involve the solution or one of its many derivatives. The strategy has been proven to be effective and easy-to-realize in multi-physical applications. However, identifications of optical core monitoring components are proven to be extremely difficult. To this end, in this talk, we consider six different designs of monitoring functions targeting at a highly vibrate nonlinear partial differential equation problem 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. Computational experiments will be presented to illustrate our research and conclusions. Continuing collaborations in the field with CSU colleagues and students will also be pursued.
报告题目2：Contemporary Beam Propagation Approximation Methods via Eikonal Splitting and Beyond
摘要：Our exploration concerns the latest adaptive eikonal splitting methods for solving paraxial Helmholtz equations on high wave numbers. ADI/LOD based splitting procedures are considered. Mesh adaptations are implemented in the transverse and beam propagation directions. Asymptotic stability of the computational procedures is introduced and rigorously investigated. It is shown that the fully discretized oscillation-free Eikonal splitting simulations on nonuniform grids are fast, effective and stable in the asymptotic sense with a stability index one. Simulation experiments are carried out to illustrate our accomplishments and conclusions.