Implicit Finite Volume Approximation of Nonlinear Advection-Diffusion Equations
Friday, May 29, 2020
10AM – 11AM
We consider approximation of nonlinear advection-diffusion equations, which express conservation principles for systems exhibiting transport and diffusion processes evolving in time. These systems are often advection dominated, and so display hyperbolic behavior, which means that shocks, or at least steep fronts, can develop in the solution. In the general finite volume framework, the conservation principles are reduced to the mesh element level, and one solves for an approximation to the average value of the solution. Two issues must be addressed by any finite volume scheme. First, one must accurately reconcile the difference between average and point values of the solution. We present weighted essentially non oscillatory methods with adaptive order (WENO-AO) to provide high order accurate reconstructions of point values from averages, when the true solution may have a shock. That is, they limit the oscillatory behavior of the reconstruction near shocks. The second issue to be resolved is the time stepping, which is normally implemented through the method of lines. We present implicit time stepping Runge-Kutta methods. Unfortunately, these can introduce oscillations into the solution, so we propose adaptive methods that can reduce to, say, backward Euler. We then turn to an improvement of the backward Euler method by considering carefully the possibility of a shock in the solution, which results in a self-adaptive theta time stepping scheme.
Todd Arbogast earned his Ph.D. in mathematics from the University of Chicago. He is professor of mathematics, chair of the Computational Sciences, Engineering and Mathematics Graduate Studies Committee, and a founding member and associate director of the Center for Subsurface Modeling at the Oden Institute for Computational Engineering and Sciences at UT Austin. He is the faculty co-adviser of the university’s student chapter of the Society for Industrial and Applied Mathematics. He is the current holder of the W. A. "Tex" Moncrief, Jr. Simulation-Based Engineering and Sciences Professorship I. His research contributes to the development and analysis of numerical algorithms for the approximation of partial differential systems, high performance and parallel scientific computation, and multi-scale mathematical modeling, as applied to fluid flow and transport in geologic porous media. Important applications include petroleum production, groundwater contamination, carbon sequestration, and mantle dynamics.
For those new to the CSEM Program, the Babuška Forum is a seminar series started by Professor Ivo Babuška several years ago to expose students to interesting and curious topics relevant to computational engineering and science with technical content at the graduate student level (i.e. the focus of the series is on main ideas with some technical content). The forum is regularly attended by students and faculty in interdisciplinary fields held together by the common link of computational methods and mathematical modeling. The general idea is to keep the forum approachable to a diverse audience, so a certain level of pedagogy is appreciated. This is what distinguishes the forum from a typical seminar that a faculty member or researcher gives in a conference, which contains more technical material.
Note: Please join this Zoom seminar online with the "Audio Only" function (no video).