Chad Vidden

Research Interests


    My general research interest is in developing and analyzing efficient numerical algorithms for solv-
ing partial differential equations (PDEs). More specifically, I am currently working with discontinuous
Galerkin (DG) methods which are a class of finite element methods. DG methods were originally introduced in 1973 by Reed and Hill as a way to solve the hyperbolic neutron transport equation.
A major development of the DG method for nonlinear hyperbolic conservation laws was carried out in the 1990s. Since then, these methods have gained popularity and have been developed for many different classes of PDEs. Because of its nice features, these robust and accurate methods have quickly attracted the interest of the scientific community and are currently finding use in very diverse applications such as electromagnetism, magneto-hydrodynamics, meteorology, oceanography, semiconductor device simulation, weather forecasting, turbulent flows, among many others.
Application of these methods to diffusion problems has been considered, though existing works have drawbacks and a complete mathematical theory is lacking. My thesis research addresses this.

Here is a good introduction to Discontinuous Galerkin methods by Chi-Wang Shu


Papers


  • C. Vidden, J. Yan, Direct Discontinuous Galerkin Method with Symmetric Structure for Diffusion Equations. Submitted to SIAM Journal on Numerical Analysis Fall 2011.
  • C. Vidden, H. Liu, J. Yan, Admissibility Analysis for the Direct Discontinuous Galerkin Method and its Variations. To be submitted Fall 2011.
  • C. Vidden, J. Yan, Positivity Preserving (Maximum Principle Satisfying) DDG Methods for Convection Diffusion Problems. In preparation.
  • X. Garcia, J. Kunze, T. Rudelius, A. Sanchez, S. Shao, E. Speranza, C. Vidden, Invariant Measures for Hybrid Stochastic Systems. Submitted to Systems and Control Letters Fall 2011.
  • T. Pluta, M. Temba, B. Wu, K. Jungjaturapit, R. Rastegar, A. Roitershtein, C. Vidden, Trading Cookies in a Gambler's Ruin Scenario. Submitted to the Involve Fall 2011.
  • C. Vidden, Mathematical Modeling and Simulation of Multialleic Migration-Selection Models. Published in Minnesota State University Mankato Journal of Undergraduate Research, Spring 2007.


Seminar and Conference Talks


  • Recovery of Normal Derivatives from the Piecewise L2 Projection, Focused Research Interaction Seminar, Ames, IA, September 22, 2011.
  • Direct Discontinuous Galerkin Method for Nonlinear Diffusion Equations with Symmetric Structure, 11th US National Congress on Computational Mechanics, Minneapolis, MN, July 25-28, 2011.
  • Trading Cookies in a Gambler's Ruin Scenario, Loras College Undergraduate Mathematics Seminar, Dubuque, IA, April 6, 2011.
  • Introduction to Hybridizable Discontinuous Galerkin Methods, Focused Research Interaction Seminar, Ames, IA, February 9, 2011.
  • Discontinuous Galerkin Methods for Stokes Flow Problems, Focused Research Interaction Seminar, Ames, IA, September 30, 2010.
  • A Variation of the Direct Discontinuous Galerkin Method for Diffusion Problems with Symmetric / Nonsymmetric Structure, Midwest Numerical Analysis Day 2010, Ames, IA, April 24-25, 2010.
  • Central Discontinuous Galerkin Methods, Focused Research Interaction Seminar, Ames, IA, Spring 2010.
  • Mathematical Modeling and Simulation of Multialleic Migration-Selection Models, Undergraduate Research Conference, Mankato, MN, April 23-24, 2007.


Conferences Attended


  • 5th Annual Mathematical Field of Dreams Conference, Arizona State University, Tempe AZ, October 14-16, 2011.
  • 11th US National Congress on Computational Mechanics, Minneapolis, MN, July 25-28, 2011.
  • 6th Midwest Undergraduate Mathematics Symposium, Simpson College, Indianola, IA, April 3-4, 2011.
  • AMS Sectional Meeting, University of Iowa, Iowa City, IA, March 18-20, 2011.
  • Institute for Mathematics and its Applications Workshop: Numerical Solutions of Partial Differential Equations, Novel Discretization Techniques, University of Minnesota, Minneapolis, MN, November 1-5, 2010.
  • Iowa Summer Undergraduate Research Symposium, Iowa State University, Ames, IA, July 28- 29, 2010.
  • Iowa PDE Conference, University of Iowa, Iowa City, IA, April 30-May 2, 2010.
  • Midwest Numerical Analysis Days 2010, Iowa State University, Ames, IA, April 24-25, 2010.
  • Iowa PDE Conference 2009, Iowa State University, Ames, IA, October 10, 2010.
  • Minnesota State University Undergraduate Research Conference, Minnesota State University - Mankato, Mankato, MN, April 23-34, 2007.


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