Introductory notebook outline

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David Ketcheson edited this page Mar 20, 2017 · 2 revisions

Introductory notebook to set the stage

What is a Riemann problem?

First order hyperbolic PDE in 1D together with piecewise constant initial conditions
Solution consists of set of waves propagating at finite speeds

Why is this important

Many important physical systems can be described by Hyperbolic PDEs; for example: sound waves, light waves, water waves, traffic jams, fluid dynamics
Discontinuities arise naturally in the solution of these systems
For a broad class of problems (even nonlinear PDEs) the Riemann solution is a similarity solution
Solution q(x,t) = Q(x/t) (use tilde or capital?) can often be computed analytically
Understanding Riemann solutions is a fundamental tool in analyzing the mathematical structure of solutions, and interpreting physical phenomena.
For many physical problems the waves are one of three types: shocks, rarefactions, or contact discontinuities
Riemann solvers are a key building block for numerical methods, including in multiple dimensions

What is a hyperbolic equation?

Define for linear and nonlinear systems

Riemann solution for a linear system

Define in terms of eigenvectors and eigenvalues of A
Use acoustics as a first example
Javascript to move states in phase space and show solution

Riemann solution for a nonlinear system

Shallow water equation
Briefly explain that each wave can now be shock or rarefaction
Javascript to move states in phase space and show solution

Approximate Riemann solutions

For numerical methods a cheap-to-compute approximate solution may suffice
Example: shallow water with small jumps well approximated by acoustics

Generalizations:

Briefly mention more complex situations, e.g.
- Heterogeneous (piecewise constant) media: q_t + A(x)q_x = 0 or q_t + f(q,x)_x = 0
- Nonconvex flux functions
- Source terms / balance laws with source replaced by delta function at x = 0
- Varying source term on each side? (E.g. radial source term in kitchen sink)
- 2D Riemann problems (quadrants)