Introduction to CFD Assignment 6 — Numerical Error  
You will need the warp package attached for Problems #1 and #2.
Question #1
For the rarefaction.wrp case and a mesh of $41\times41$ nodes estimate the solution convergence error as a function of the maximum residual. The solution convergence error is to be estimated by monitoring $\delta_x P$ at $x=0,~y=0$. For this purpose, make a table as follows:
$(R_\Delta)_\max$$\delta_x P$Estimate of relative error on $\delta_x P$
Choose enough values of $(R_\Delta)_\max$ so that you can make a plot from 1-2 orders of magnitude of convergence till 8-10 orders of magnitude of convergence. Explain clearly how all the values in the table are obtained (show the calculations for each number).
Question #2
Using an optimal value of $(R_\Delta)_\max$ as determined in Question #1 above, do a grid convergence study to assess the grid-induced error for the rarefaction.wrp test case. The grid convergence study should be done with a grid refinement ratio $r=\sqrt{2}$ with the coarsest mesh being composed of $21\times 21$ nodes and with at least 4 different grid levels (or more if the solution is not within the asymptotic range of convergence). As much as possible and as accurately as possible for each grid level, estimate:
(a)  the effective order of accuracy $p$
(b)  $\delta_x P$ at $x=0,~y=0$
(c)  the grid convergence index (GCI) for $\delta_x P$ at $x=0,~y=0$
(d)  an estimate of the exact solution $\partial_x P$ at $x=0,~y=0$
Present your results in tabular form and explain clearly how each entry was obtained (show the calculations and formulas used). Also determine at which mesh level the solution is within the asymptotic range of convergence.
Question #3
Starting from Taylor series expansion of a first derivative, show that the following holds: $$ \epsilon^{\rm disc}_{\rm f} = \bigg((\delta_x \phi)_{\rm f}- (\delta_x \phi)_{\rm c}\bigg) \left/ \left(1- \left( \frac{\Delta x_{\rm c}}{\Delta x_{\rm f}}\right)^p \right) \right. $$ with $$ \epsilon^{\rm disc}_{\rm f} \equiv (\delta_x \phi)_{\rm f} - \partial_x \phi $$ Outline all assumptions and limitations if any.
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