Solution structure and high order numerical methods for time-dependent problems

Overview

The course discusses the solution structure and high order structure-preserving numerical methods for several time-dependent problems, modeled by partial differential equations balanced with diffusion, convection, and various interactions. The main topic focuses on the model structure and the construction of stable, accurate approximate algorithms for these problems, and the interplay between analytical theory and computational aspects of such algorithms with real applications. Here is a list of topics to be discussed: (1) Optimization, entropy and differential equations (2) Critical threshold in nonlinear balance laws (3) Direct DG methods for diffusion with drifts #1 Optimization, Entropy and Differential Equations Interesting things are often understood in different ways. In this lecture, I choose three topics such as optimization, entropy, and differential equations, and show by examples how these concepts are closely related in mathematically formulating the same problem, but each has a distinct flavor of its own. The examples include Nestorov’s accelerated scheme, Newton’s apple, Osher’s level set, and Kompaneets’ equation for photon transport. #2 Critical Thresholds in Nonlinear Balance Laws In many mathematical models of physical reality, coherent structures are formed and maintained by a balance of competing influences. We study how such competing effects lead to critical solution behavior. Our approach is to identify critical thresholds of initial data that lead to either global regularity or finite time breakdown in solutions to a class of nonlinear balance laws. In this lecture, I shall assess the state of the art, and gaps that need filling in the investigation of critical threshold phenomena in hyperbolic balance laws. #3 Direct DG methods for diffusion with drifts Discontinuous Galerkin (DG) methods for hyperbolic problems have been extensively developed. In contrast, the DG formulation is far less certain for diffusion, for which the viscous flux requires the evaluation of solution derivatives at the element interfaces. A number of methods have been proposed in the literature to address this issue. In this lecture, I shall focus on the direct DG method, by showing the remarkably stable and accurate properties of the method, and a selected set of applications (including the drift-diffusion equations and Navier-Stokers equations).

Bio: Dr. Hailiang Liu is a Mathematics Professor at the Iowa State University (ISU) and the Holl Chair in Applied Mathematics from 2002-2012. He received his Master degree in Applied Mathematics from Tsinghua University of China in 1988, and Ph.D. degree from the Chinese Academy in 1995; while he held professorship positions at Henan Normal University from 1989-1996. He received an Alexander von Humboldt-Research Fellowship in 1996 that allowed him to conduct research in Germany from 1997-1999. He joined UCLA as a CAM Assistant Professor from 1999-2002. He then came to Iowa State University as an Associate Professor in 2002, moving up to Full Professor in 2007. Liu’s primary research interests include analysis of applied partial differential equations, the development of novel, high order algorithms for the approximate solution of these problems, and the interplay between analytical theory and computational aspects of such algorithms with applications to shock waves, kinetic transport, level set closure, propagation of critical thresholds and recovery of high frequency wave fields. Liu serves on the editorial board of the JMAA journal and has given many invited lectures, including the invited addresses in the international conference on hyperbolic problems in 2002 and 2018. Liu published more than 120 research papers, mostly in Numerical Analysis and Applied Partial Differential Equations.