Parabolic PDEs

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Parabolic PDEs Generally involve change of
quantity in space and time Equivalent to our
previous example - heat conduction
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Can use the same grid idea, only boundaries are
different
time (j index)
space (i index)
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Need initial conditions - green dots and boundary
conditions - red dots
One side of grid is always open
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• Because parabolic PDEs are open-ended in time,
  can have instability
• Need to address this in solution methods
• Explicit
• Implicit

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Explicit solutions Use finite differences
tj1
tj
xi
xi-1
xi1
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Substitute into equation and rearrange
Explicit solution for this parabolic PDE
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Example Given the initial condition and boundary
conditions below, solve for heat distribution
over time See matlab code
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Stability conditions It can be shown that the
condition for stability is
or
look at what happens when you play with lambda
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• Other points
• Can include derivative boundary conditions -
  introduce an imaginary point as in ellipitical
  example

Use finite difference for derivative to eliminate
imaginary point, introduce derivative into
propagation equation
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We get
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Can also use higher order temporal approximations
for time term (and for space term)
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• Implicit methods
• avoid stability problems
• One example implicit method - use next time step
  to approximate spatial derivative

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Substituting,
Three unknowns
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Write these equations for all interior nodes -
you get enough equations Set up in matrix and get
a tridiagonal matrix Unconditionally stable -
although accuracy is degraded at larger ?t