Reading Group on Hyperbolic PDEs and Finite Volume Methods

Goals

Students have different backgrounds and interests, so some people may go much further in one direction or another, but here are some things to aim towards understanding:

Mathematical theory of linear and nonlinear systems of hyperbolic PDEs and conservation laws.
- Eigenstructure of Jacobian matrix.
- Shock and rarefaction waves, contact discontinuities.
- Solution to the Riemann problem.
Basic theory of finite volume methods.
- Upwind method, Godunov's method.
- High-resolution methods with limiters, TVD methods.
- Review concepts from 586 such as dissipation, dispersion, Lax-Wendroff method, stability, CFL condition, etc.
Clawpack software
- Setting up problem, defining Riemann solver.
- Plotting solutions.
- Experimenting with different methods.
Applications: Become familiar with several hyperbolic systems, such as
- Linear advection, acoustics, elasticity.
- Nonlinear Burgers' equation, traffic flow, shallow water equations
- Euler equations of compressible gas dynamics.

Outline / schedule

To appear.

First steps

Read Chapters 1 and 2 of the book Finite Volume Methods for Hyperbolic Problems and look at the first set of slides from 2011.
Install Clawpack 4.6.2 and required prerequisites, see the Installation instructions.
One option is to use the Clawpack Virtual Machine if you don't want to worry about installing everything on your laptop.
Note: Some examples from the book have not been converted and require Clawpack 4.3. Click on the "Book" link there for the examples.
The examples included in the current 4.6.2 release can be viewed in the applications gallery.
Test it out and run an example.