Convergence is arguably the most important element of any numerical solution. If we cannot demonstrate that the solution approaches a consistent target as the grid size decreases, there is no reason to believe that it is accurate. Though there are more detailed ways to test convergence, here we will use a simple calculation of the convergence factor. Suppose we begin with an initial grid size of 4h. If a method leads to second order accuracy, it is easy to show via a Taylor series expansion that the amplitude of change in the solution between a grid size of h and 2h should be of the order of h-squared, and the same change between grid sizes of 2h and 4h should be of the order of 4h-squared. The convergence factorcan be calculated by taking the ratio of these two errors. For a second order method, which is what we expect here, it should be around 4 at each point; for a first order method it would be around 1, etc.

Convergence of Burgers equation simulation. at 0.2 s:

The convergence factor generally appears to be around 4 in the smooth sections, but is much more scattered elsewhere as expected for second-order accuracy.

Convergence of the ultrarelativistic simulation at 0.2 s:

Here again the convergence factor stays around 4 in selected areas while scattering elsewhere, though there are more breaks from these smooth sections than in the Burger's solution.