Short Course on DG Methods
Welcome to Discontinuous Galerkin methods
In these lectures, we will give a general introduction to the discontinuous Galerkin(DG) methods for solving hyperbolic conservation laws. We will discuss cell entropy inequalities, nonlinear stability, and error estimates. Issues related to the implementation of the DG method will also be addressed.
- This course will consist of live lectures, with interaction from the students.
- This course familiarizes students with the basics of numerically solving PDEs using discontinuous Galerkin methods.
- Students will become familiar with how mathematics translates into computational code.
The goals of this course are (1) developing an understanding of discontinuous Galerkin approximations (2) basics of implementation. At the conclusion of the course, students will
1. Be able to derive a discontinuous Galerkin (DG) formulation to a PDE.
2. Be able to perform error analysis for DG.
3. Be able to write a working DG code for 1D & 2D hyperbolic equations.
The following books/lecture notes are required and freely available from the library:
1. Jan S. Hesthaven and Tim Warburton, “Nodal discontinuous Galerkin Methods: Algorithms, Analysis, and Applications”, Springer, 2008.
2. Bernardo Cockburn, Chi-Wang Shu, Claes Johnson, Eitan Tadmore, and Alfio Quarteroni, “Advanced Numerical Approximation of Nonlinear Hyperbolic Equations”, Springer, 1997. (DG chapter available at: http://www-users.math.umn.edu/%7Ecockburn/lecture_notes/DG-1.pdf )
3. Chi-Wang Shu, “Discontinuous Galerkin Methods: General Approach and Stability”, lecture notes available at https://www3.nd.edu/~zxu2/acms60790S15/DG-general-approach.pdf
outline of the course:
|1||March 24, 15-17||Introduction, background, notation|
|2||April 7, 15-17||Approximation theory and the linear transport equation|
|3||April 14, 15-17||Runge-Kutta discontinuous Galerkin (RKDG)|
|4||April 21, 15-17||Error Estimates and Dispersion & Dissipation|
|5||April 28, 15-17||Consistency & Stability (=Convergence)|
|6||May 4, 15-17||Nonlinear equations|
|7||May 11, 15-17||Discontinuous solutions|
|8||May 18, 15-17||Multi-dimensional Equations|