Order of Convergence of the Method

Throughout this report and subsequent ones we will refer to our web site:

http://bh0.physics.ubc.ca/Students/p55505g2/

We provide a basic description of high resolution capturing shock methods in Appendix A. There are at least two excellent descriptions of Godunov methods ([11],[1] and [10]), and we decided to focus on our results rather in a detailed description of the methods.

There are some subtle things about convergence and TVD schemes[9] As a general rule, a first order reconstruction will give a second order in space scheme. A second order reconstruction will give a third order scheme, etc.

However, when the data are discontinuous, then the scheme reduces to first order accuracy. Moreover, the notion of convergence near a discontinuous solution is problematic. TVD schemes with first order reconstruction also reduce to first order accuracy at extrema (minima and maxima) of functions, even if the data are smooth. This is because the first derivative changes signs on either side of the extremum, which the TVD scheme interprets as point-to-point oscillations.

To test the convergence of our code we must run very "weak" and smooth initial data. Then run at three resolutions and look at function $q$ at each resolution, $q\_h, q\_{2h}$ and $q\_{4h}$. We calculate

$$C\cdot\frac{(q\_(2h) - q\_(h))}{(q\_(4h) - q\_(2h))}$$ (5)

where $C$ is either 2 or 4. Note: these are not norms, just the differences of the functions evaluated at points where they coincide. I then superpose the functions in xvs. If $C=2$, then convergence is first order where the lines coincide. If $C=4$, then convergence is second order where the lines coincide.

The evolution of the gaussian profile using the three steppings available and linear and constant reconstruction is shown in the web site. The piecewise constant reconstruction produces a decay as the evolution goes by,