1.1.8.3. The Riemann Problem

The Riemann Problem is a family of problems, (which includes the 1D Euler Equations).

There are two components for the Riemann problem:

  1. A conservation law (e.g. the 1D Euler Equations)

  2. Piecewise constant initial data with a single jump discontinuity

e.g. for Euler Equations:

For \(x \lt x_0\) \(\quad \mathbf{U}(x, t_0) = \mathbf{U}_L\)

For \(x \gt x_0\) \(\quad \mathbf{U}(x, t_0) = \mathbf{U}_R\)

\(\mathbf{U}_L\) and \(\mathbf{U}_R\) are constant vectors

Take \(x_0 = 0\) and \(t_0=0\)

1.1.8.3.1. Usefulness of the Riemann Problem

It has an exact analytical solution for the Euler Equations (and any scalar conservation laws, or any linear system of equations)

1.1.8.3.2. Features of the Riemann Problem

  1. Solution is self-similar - solution stretches in space and time, but doesn’t change shape.

  • \(\mathbf{U}(x, t_1)\) and \(\mathbf{U}(x, t_2)\) are “similar”

  • In other words, it really depends on a single variable \(x / t\)

  • Or the solution is constant along a characteristic line \(x = ct\) (c = constant) on x-t plane

  • Generally, self-similarity means only one independent variable like: \(x / t\) or \(t / \sqrt{x}\), similar to the Blasius solution from boundary layer theory.

  1. There is an analytical solution \(\Rightarrow\) Riemann problem useful to test numerical schemes

  2. Riemann problem appears as part of numerical formulation of several CFD methods, e.g. “wave capturing” methods \(\Rightarrow\) “Riemann solvers” are at the heart of the method - may need to solve these multiple times - major cost of the solution