# What are the good testing problems for hyperbolic equation?

I read the whole list of this question: Where can one obtain good data sets/test problems for testing algorithms/routines? But the answers are in different areas and I want to ask a specific area.

I am looking for resting problems for wave equation, telegraph equation or Klein-Gordon equation, etc. What they have in common is that they are second-order linear hyperbolic partial differential equations. By testing problem, I mean those can show how well my numerical algorithms behaved, which is exactly the meaning in the similar question I listed above.

• You can use the method of manufactured solutions to come up with test problems for any equation. – Aurelius Oct 30 '14 at 22:57
• How many spatial dimensions are you interested in? – David Ketcheson Nov 2 '14 at 7:27
• Since these equations can be solved (in simple geometries) exactly by classical Fourier analysis, it seems trivial to come up with test problems. – David Ketcheson Nov 2 '14 at 7:29

## 1 Answer

The first two can be decomposed into a system of first-order advection equations of which there are a number of standard test problems often involving both smooth and discontinuous initial conditions and all of which can be solved analytically. For instance this solution is to the linearized acoustics equations. The choice between one or the other is mainly based on what you want to do, a spectral methods based code for instance would not behave well for discontinuous initial conditions while codes that tend to be able to capture discontinuous solutions do not tend to do smooth solutions as well, hence using both initial conditions.

For the Klein-Gordon equation this link will give you a way to construct analytical solutions and test them. I am not certain about a particular solution that would test a code to its limits though.

• The first sentence is true only in one dimension. – David Ketcheson Nov 2 '14 at 7:28