Course information

Instructor

• Prof. Randall LeVeque
  • Office: Lewis 328
  • netid for email: rjl
  • Office hours: Tuesday 3-4, Wednesday 11-12, Thursday 11-12

Class meeting times

• MW 3:30 - 4:20 pm in Johnson 026 for lectures.
• F 3:30-4:20 pm in Odegaard Library OUG 141 (This is an Active Learning Classroom where we will work on computing projects.)

Discussion board

Registered students should be able to access the Piazza discussion board.

Canvas page

Registered students can view grades and other materials on the Canvas page

Grading

• Homework: 50%, midterm: 20%, final project: 25%, peer review activities: 5%
• See Homework and project for more information and due dates.

Recommended Background

Prerequisite: Either AMATH 586 or permission of instructor. Applied Math 585-6 or similar background is strongly recommended, i.e. experience in numerical methods for differential equations, along with basic understanding of partial differential equations.

Programming experience is also expected. Experience in Fortran and Python would be most valuable but not required.

Experience with using git and GitHub would also be useful, but not required.

Textbook

• R. J. LeVeque Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.
• The Clawpack Documentation will also be very useful since we will be using this software extensively.

Goals

• Master the basic theory of hyperbolic PDEs and nonlinear conservations laws,
• Understand the development of high-resolution shock-capturing finite volume methods for solving these equations,
• Learn about some applications of hyperbolic problems,
• Gain experience in using the Clawpack software for solving these equations, including how to set up a new problem,
• Learn the basics of Python programming and use of IPython notebooks,
• Become comfortable using Git and GitHub and learn about the development paradigm used for Clawpack and many other open source scientific codes.

Topics

Some of the topics to be covered are listed below. See also Outline and reading assignments.

• Mathematical theory of linear and nonlinear systems of hyperbolic PDEs and conservation laws.

  • Eigenstructure of Jacobian matrix.
  • Shock and rarefaction waves, contact discontinuities.
  • Phase plane analysis – Hugoniot loci and integral curves.
  • Solution to the Riemann problem for linear and nonlinear systems of equations.
  • Entropy functions and admissibility criteria.

• Theory of finite volume methods.

  • Upwind method, Godunov’s method, use of true and approximate Riemann solvers.
  • High-resolution methods with limiters, TVD methods.
  • Review concepts from AMath 586 such as dissipation, dispersion, Lax-Wendroff method, stability, CFL condition, etc.
  • Multi-dimensional finite volume methods on Cartesian and mapped grids
  • Adaptive Mesh Refinement (AMR)

• Use of Clawpack software

  • Setting up a problem, defining a Riemann solver.
  • Plotting solutions.
  • Experimenting with different methods.

• Applications such as

  • Linear advection, acoustics, and elasticity,
  • Nonlinear Burgers’ equation, traffic flow,
  • Shallow water equations,
  • Euler equations of compressible gas dynamics.