MACS471 Lecture 8

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MA/CS471Lecture 8

• Fall 2003
• Prof. Tim Warburton
• timwar_at_math.unm.edu

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Today
Today we are going to discuss implementation of
the simplestpossible partial differential
equations Example domain discretization
(L-shape domain with 3 cells)
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Traveling Wave Solutions in 1D and 2D
This PDE is the two-dimensional analogue to the
PDE we saw last time
1D
2D
Solutions
t0 Later t
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In Words

• The initial condition is translated with
  velocity
• i.e. the density does not change shape it
  simply translates with a constant velocity.

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Conservation Law

• We first divide the full domain into
  quadrilateral cells. For each cell e the
  following conservation law holds
• i.e. the total density change in a cell e is
  equal to the flux of density through the boundary
  of e
• -- or the rate of change of material in the
  cell e is equal to the amount translated
  through the boundary.

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Boundary Conditions

• Where do we need to apply boundary conditions
• Hint which way is the solution translating?

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Boundary Conditions

• Where do we need to apply external boundary
  conditions ?
• i.e. wherever where
  is the outwards facing normal for the cell e
  at the face f

e
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Definition of Cell Average and Area
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Use Upwinding At Boundary of e

• The tau variable acts as a switch.
• If tau1 at a face then rho is approximated by
  the local cell average at the face
• If tau0 at a face then rho is approximated by
  the neighbor cell average.

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Upwind Switch
rhoe
rhoe,1

• In this case so
• i.e. for the surface flux term we should use the
  cell average density from the neighbor cell.

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Euler Forward In Time

• For each cell we now have a discrete space in
  time and space which will compute
  approximations the cell average density at a
  given time level.
• We do need to specify an initial value at each
  cell for the cell average density
• We also need to specify boundary conditions at
  inflow edges.

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Summary of Scheme
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The time rate ofchange of total density in the
cell e
The flux through the boundary of the four faces
of cell e
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Implementation

• What do we need for the implementation
• (1) A list of vertex locations (x0,y0,x1,y1)
• (2) For each cell e a list of the four vertices
• (3) A routine to calculate the area of each cell
  Ae
• (4) A routine to calculate the length of each
  cell edge Ae,f
• (5) A routine to calculate the outwards facing
  normal to each cell edge
• (6) A routine to calculate the inflow switch tau
  for each cell edge.
• (7) A routine to calculate the initial density
  profile.
• (8) A routine to calculate dt in the following
  way
• (9) A routine to figure out which cells connect
  to each cell.

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Details

• Compute cell area by dividing each cell into two
  triangles, find their areas, and sum up.
• Do not assume cells are right-angled
  quadrilaterals (can be deformed).
• To compute normal to a face

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Project 2 continued

• Create a serial version of the 2d finite volume
  scheme for the above one way wave equation.
• Make it parallel.

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