[hyperpallium]

Have you seen the Method of Manufactured Solutions (MMS)?

You guess ("manufacture") a solution to the PDE. Then you plug it into the PDE. It won't be correct, so you just add source terms to make it correct. You now have an analytical solution for this PDE with those source terms. You can now run your simulation (with those source terms) and compare it with the "correct" solution.

Unfortunately, you'll introduce discretization error, because your delta x and delta t aren't infinitessimal. But using the order of accuracy of your discretization method, you know how the error should change with delta x and delta t. So you graph error against delta x, on log-log, and see if the curve matches the order of accuracy. Apparently, it shows up even minor bugs in your code really well.

This approach scares me a little, because how do I know my math is correct? I need better appreciation of how the order of accuracy interacts over time as well as space; the behaviour of error summaries; and how log-log plotting works.