About

Todd Arbogast is a professor affiliated with the Department of Mathematics and the Center for Subsurface Modeling at the Institute for Computational Engineering and Sciences, The University of Texas at Austin. His research focuses on numerical methods and computational techniques for solving complex problems in applied mathematics, particularly those related to porous media, conservation laws, and multiscale modeling. He has contributed extensively to the development and analysis of finite volume, finite element, and mixed finite element methods, with applications to two-phase flow, Darcy-Stokes systems, and heterogeneous elliptic problems. Arbogast's work includes the design of high-order numerical schemes such as WENO (Weighted Essentially Non-Oscillatory) methods and adaptive order reconstructions for nonlinear conservation laws and advection-diffusion equations. He has also advanced the theory and implementation of multiscale and mortar mixed finite element methods, which address challenges in modeling anisotropic and heterogeneous media. His scholarship encompasses both theoretical analysis and practical computational algorithms, often emphasizing conservation properties, stability, and accuracy in numerical simulations. Arbogast has co-authored a textbook on functional analysis for applied mathematicians and has contributed to interdisciplinary graduate education in computational engineering and science. His research outputs demonstrate a sustained commitment to advancing numerical methods for complex physical and engineering systems, with a particular emphasis on subsurface flow and transport phenomena.

Research topics

• Computer Science
• Mathematics
• Artificial Intelligence
• Geometry
• Mathematical analysis
• Pure mathematics
• Physics
• Applied mathematics
• Algorithm

Selected publications

• A High Order, Finite Volume, Multilevel WENO Scheme Applied to Porous Media

  Lecture notes in computational science and engineering · 2025-01-01

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• Further Studies on the Self-Adaptive Theta Scheme for Conservation Laws

  Journal of Scientific Computing · 2025-05-22

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  Abstract The finite volume, self-adaptive theta (SATh) scheme was defined in Arbogast and Huang, A self-adaptive theta scheme using discontinuity aware quadrature for solving conservation laws , IMA J. Numer. Anal. (2022). The basic scheme evolves both the local space and space-time averages of the solution in time with an implicitly defined theta parameter. Here, the scheme is extended to unstructured meshes in multiple space dimensions, general numerical flux functions, and higher (formally second) order using WENO reconstructions. Theoretical results apply to the one space dimension, upstream weighted case, in the setting of a monotone solution. In this case, if the theta parameter is bounded below by $$\theta _{\min }=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>θ</mml:mi> <mml:mo>min</mml:mo> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , it is shown that SATh is stable, L-stable for the linear problem, total variation diminishing (TVD), and maximum principle preserving (MPP). These results generalize those known previously with the assumption that $$\theta _{\min }=1/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>θ</mml:mi> <mml:mo>min</mml:mo> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . Numerical tests for problems with contact discontinuities, shocks, and rarefactions show that SATh performs better than finite volume schemes using backward Euler time stepping. Moreover, SATh gives solutions about as sharp as when using Crank-Nicolson time stepping, but SATh is non-oscillatory. In cases covered by the theoretical results, SATh combined with a Lax-Friedrichs numerical flux (rather than upstream weighting) appears to be TVD and MPP. SATh is non-oscillatory if $$\theta _{\min }=1/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>θ</mml:mi> <mml:mo>min</mml:mo> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , but if $$\theta _{\min }=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>θ</mml:mi> <mml:mo>min</mml:mo> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and the solution is not monotone, it can develop oscillations. The higher order SATh scheme converges to order two and compares favorably with CN, but is less oscillatory.

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• Direct serendipity finite elements on cuboidal hexahedra

  Computer Methods in Applied Mechanics and Engineering · 2024-10-31 · 2 citations

  articleOpen access1st authorCorresponding

  We construct direct serendipity finite elements on general cuboidal hexahedra, which are H 1 -conforming and optimally approximate to any order. The new finite elements are direct in that the shape functions are directly defined on the physical element. Moreover, they are serendipity by possessing a minimal number of degrees of freedom satisfying the conformity requirement. Their shape function spaces consist of polynomials plus (generally nonpolynomial) supplemental functions, where the polynomials are included for the approximation property and supplements are added to achieve H 1 -conformity. The finite elements are fully constructive. The shape function spaces of higher order r ≥ 3 are developed first, and then the lower order spaces are constructed as subspaces of the third order space. Under a shape regularity assumption, and a mild restriction on the choice of supplemental functions, we develop the convergence properties of the new direct serendipity finite elements. Numerical results with different choices of supplements are compared on two mesh sequences, one regularly distorted and the other one randomly distorted. They all possess a convergence rate that aligns with the theory, while a slight difference lies in their performance.

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• Functional Analysis for the Applied Mathematician

  2024-12-03 · 1 citations

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  Functional Analysis for the Applied Mathematician is a self-contained volume providing a rigorous introduction to functional analysis and its applications. Students from mathematics, science, engineering, and certain social science and interdisciplinary programs will benefit from the material. It is accessible to graduate and advanced undergraduate students with a solid background in undergraduate mathematics and an appreciation of mathematical rigor. Students are called upon to actively engage with the material, to the point of proving some of the basic results or their straightforward generalizations, both within the text and within the generous set of exercises. Features: Replete with exercises and examples Suitable for graduate students and advanced undergraduates Develops the basics of functional analysis, exploring the interplay between algebraic linear space theory and topology Presents a variety of applications, often dealing with partial differential equations and their numerical approximation Doubles as a reference book with an extensive index listing the concepts and results

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• A finite volume multilevel WENO scheme for multidimensional scalar conservation laws

  Computer Methods in Applied Mechanics and Engineering · 2024-02-02 · 6 citations

  articleOpen access1st author

  We develop a general framework for solving scalar conservation laws using finite volume weighted essentially non oscillatory (WENO) techniques on general computational meshes in multiple space dimensions. We address two fundamental issues. First, polynomial approximations on general stencils of mesh cells can be of poor quality, even for what appear to be geometrically nice stencils. We present a robust and efficient procedure for producing accurate stencil polynomial approximations. Bad stencils are identified by considering the condition number of the linear system used to define the stencil polynomial. Second, we develop a novel and efficient finite volume, multilevel WENO (ML-WENO) reconstruction that is flexible enough to be applied effectively in a variety of settings and with essentially any reasonable set of stencils. It combines stencil polynomial approximations of various degrees with a nonlinear weighting biasing the reconstruction away from both inaccurate oscillatory polynomials of high degree (i.e., those crossing a shock or steep front) and smooth polynomials of low degree, thereby selecting the smooth polynomial(s) of maximal degree of approximation. We conduct numerical tests showing poor quality mesh stencils, the behavior of the reconstruction for both smooth and discontinuous functions, and applications to scalar conservation laws.

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• Spectral Theory and Compact Operators

  2024-12-03

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• The Fourier Transform

  2024-12-03

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• Preliminaries

  2024-12-03

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• Hilbert Spaces

  2024-12-03

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• The Calculus of Variations

  2024-12-03

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Recent grants

• CMG Research: Multi-scale Flow and Transport Modeling of Large-vug Cretaceous Carbonates

  NSF · $677k · 2004–2008

• Implicit Weighted Essentially Non-Oscillatory (WENO) Schemes for Advection-Diffusion-Reaction Systems

  NSF · $250k · 2019–2023

• Direct Finite Elements on Convex Polygons and Polyhedra

  NSF · $270k · 2021–2024

• Fully Locally Conservative Characteristic Methods for Transport Problems

  NSF · $259k · 2007–2011

• Numerical algorithms for nonlinear subsurface flow and transport

  NSF · $365k · 2014–2018

Frequent coauthors

• Chieh‐Sen Huang

  National Sun Yat-sen University

  21 shared

• Jim Douglas

  14 shared

• Mary F. Wheeler

  11 shared

• Ivan Yotov

  11 shared

• Zhangxin Chen

  11 shared

• Mary F. Wheeler

  10 shared

• Jerry L. Bona

  9 shared

• Clint Dawson

  8 shared

Education

• M.S. and Ph.D., Mathematics

  University of Chicago

  1987

• B.S., Mathematics and Physics

  University of Minnesota System

  1981

Awards & honors

• Fellow of the American Mathematical Society (2012)
• Fellow of the Society for Industrial and Applied Mathematics…