Some examples

Here are a few examples of solutions to hyperbolic partial differential equations. For some other examples, see Jupyter notebooks, Riemann problems and solutions, and the Clawpack Gallery.

Advection

The advection equation \(q_t +uq_x=0\) with constant advection velocity \(u\) is the simplest example of a hyperbolic PDE. The solution simply advects with shape unchanged.

This animation below shows an example when the velocity is \(u=1\) and periodic boundary conditions are used on a domain of length 1. The computed solution was obtained using one of the high-resolution finite volume methods to be studied in this class, using the code from $CLAW/classic/examples/advection_1d_example1:



Once Loop Reflect

Burgers’ equation

Nonlinear scalar equations lead to shock formation, as illustrated by Burgers’ equation \(u_t + \left(\frac 1 2 u^2\right)_x = 0\). This is an animation of Figure 11.8 in [FVMHP], computed using the code in $CLAW/apps/fvmbook/chap11/burgers.



Once Loop Reflect