# Richardson Extrapolation

This homework could get a little ugly, but I've chosen an example that
illustrates the process without too many side calculations. If I
ask you to study sensitivity to the spatial mesh in a flow you might
start with a volume:

-------------------------

| x |

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and split it in two.

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| x | x |

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The problem you've got is that answers for state variables
like temperature and pressure from the fine mesh never align with those
for the coarser mesh. To apply what you learned about Richardson
extarpolation, you need to interpolate results on the fine mesh to get
values aligned with those on the coarser mesh. If you don't chose
an interpolation method with an order of accuracy higher than the
original difference method, you'll get results from a three mesh
Richardson that tell you about your interpolation and not about the
underlying difference method.

To avoid this complication I want you to study the spatial accuracy of
the wall conduction solution. It evaluates temperatures at the
volume edges, so cutting the mesh spacing in half always leaves a fine
mesh node aligned with each coarser mesh node. The simplest
conduction mesh has a temperature node at each surface of the metal:

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x x

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to split the mesh spacing in two, one temperature node is added in the
middle:

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x x x

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The next halving gives:

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x x x x x

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### The Problem:

A one meter long section of
pipe has an inside diameter 0.25m. The of 0.01m thick wall is Inconel
600, and should be assumed to be perfectly insulated on its outer
surface. Liquid enters at 300K and 50 m/s and exits to a pressure of
2.0e5 Pa. Assume no wall friction or irrecoverable losses. The
initial metal temperature is 400K. The initial water temperature
is 300K. Use a Semi-Implicit method for the fluid and only one
fluid volume. Study the sensitivity to the radial conduction mesh
spacing of the temperature at the insulated surface of the metal after
20 seconds of transient time. Start with just one temperature
node at each metal surface (2 temperature nodes, one metal volume), and
refine the mesh spacing by factors of 1/2 until you can demonstrate a
discretization error of less than 0.05K. Use the three mesh
Richardson extrapolation to predict the spatial order of accuracy (p),
and apply that value of p to the standard Richardson error formula to
predict error on your finest mesh. After the you've got 3 mesh
refinements, you will apply this procedure to the 3 finest meshes to
estimate error, until you get a mesh with discretization error below
0.05K. I recommend that you use a spreadsheet for these
calculations.

With your final spatial mesh use the same procedure with fixed spatial
mesh and successively halved time steps until you demonstrate the order
of accuracy in time, and a time discretization error less than 0.05K
for the outer surface temperature at 20 seconds.

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