Abstract: Within the sciences and engineering, researchers use models that involve advective and diffusive processes. These are often formulated as systems of nonlinear advection-diffusion equations. These equations are often advection dominated; moreover, the diffusion may degenerate (to zero). This means that the solutions to the equations can and often do develop steep fronts or even shock discontinuities. We consider approximation of nonlinear advection-diffusion equations using high order finite volume methods in space and time. These methods require two components: (1) reconstruction of a function from knowledge of its element averages and (2) time stepping (in the sense of the method of lines). We first discuss function reconstruction using the weighted essentially non oscillatory (WENO) framework, which allows accurate reconstruction of function values near function discontinuities. We develop a multi-scale, WENO method with adaptive order (WENO-AO), and analyze its performance theoretically and computationally. The idea is to create a high order accurate reconstruction where the function is smooth, and drop the order of the approximation and bias the reconstruction stencil to one side of a discontinuity. We next discuss time stepping procedures of the implicit type, because advection-diffusion equations are stiff. High order Runge-Kutta (RK) methods can fail near discontinuities (i.e., generate oscillations), so we develop an adaptive approach combining the RK method with a simple and stable backward Euler (BE) method. Finally, we discuss approximation of degenerate diffusion within the finite volume WENO framework. Preliminary computational tests on generic and petroleum related problems show the potential of the scheme to handle advection-diffusion processes on unstructured meshes to high order accuracy using relatively long time steps.
Bio: 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 ICES Center for Subsurface Modeling. 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.
Arbogast’s research includes Eulerian-Lagrangian schemes for transport, mixed finite element and mortar techniques for flow, homogenization and modeling of flow through multi-scale fractured and vuggy geologic media, simulation of partially molten materials, and variational multi-scale methods for heterogeneous media.
Arbogast has authored more than 70 scientific and technical publications, and serves on the editorial boards of three scientific journals and technical series. He is the recipient of an ICES Distinguished Research Award, a Moncrief Grand Challenge Faculty Award, and a Frank Gerth III Faculty Fellowship.