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Department of Mathematics
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A discontinuous Galerkin method for higher-order ordinary differential equations

Advisor

Mahboub Baccouch

Description

Discontinuous Galerkin methods (DG methods) form a class of numerical methods for solving ordinary and partial differential equations. They combine features of the finite element and the finite volume framework and have been successfully applied to hyperbolic, elliptic, parabolic and mixed form problems arising from a wide range of applications. DG methods have in particular received considerable interest for problems with a dominant first-order part, e.g. in electrodynamics, fluid mechanics and plasma physics.

Discontinuous Galerkin methods were first proposed and analyzed in the early 1970s as a technique to numerically solve partial differential equations. In 1973 Reed and Hill introduced a DG method to solve the hyperbolic neutron transport equation. The DG method has found rapid applications in such diverse areas as aeroacoustics, electro-magnetism, gas dynamics, granular flows, magnetohydrodynamics, meteorology, modeling of shallow water, oceanography, oil recovery simulation, semiconductor device simulation, transport of contaminant in porous media, turbomachinery, turbulent flows, viscoelastic flows and weather forecasting, among many others.

In this project, we propose a discontinuous finite element method to solve ordinary differential equations. First, we will give a general introduction to the DG methods for solving higher-order differential equations; including second-order initial and boundary value problems. The important ingredient of the design of DG schemes, namely the adequate choice of numerical fluxes, will be explained in detail. Issues related to the implementation of the DG method will also be addressed. Finally, we present several computational examples to validate our theory.

Tasks

  1. Give a general survey on discontinuous Galerkin methods for solving differential equations.
  2. Present a few recent developments on the design, analysis and application of these discontinuous Galerkin methods.
  3. Apply the discontinuous Galerkin method to solve initial-value problems.
  4. Apply the discontinuous Galerkin method to solve boundary-value problems.
  5. Write up results and present at Math Awareness Month.

Prerequisites

Math 2050 and Math 2350 are required. You will be using Matlab.