Introduction to Finite Differences

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Introduction to Finite Differences

• Partial derivatives appearing in the conservation
  equations are replaced by algebraic difference
  quotients. This leads to algebraic equations for
  the dependent variables at specified grid points.
• There are 3 types of differences
• Forward Difference
• Backward Difference
• Central Difference
• These are generated using Taylors series
  approximations.

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Forward Differences
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where (Dx) xi1 - xi
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Backward Differences
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where (Dx) xi - xi-1
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Central Differences
and
Subtracting,
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• We will introduce finite difference techniques
  with the non-reacting, i.e. compositionally
  frozen flow (quasi-1D) as an illustration
• These are of the form
• where for example Gr, a1/A, and FruA
• for the continuity equation

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• We illustrate a most important concept known as
  Artificial Diffusion or Numerical Diffusion using
  a particular technique developed by Lax, which is
  outlined below
• Central difference the spatial derivative to
  obtain
• Forward difference in time
• Replace based on its neighbors values, i.e.

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• Thus, using Laxs method, the equation becomes

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Subtracting Gin from both sides, and dividing by
Dt,
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• This can also be written as
• In the limit as Dt, Dx ? 0, we have
• There is the appearance of the first term on the
  right hand side, which was NOT there in the
  original PDE.
• Discretized operator is different from original
  PDE!

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• Note that is (can be) hyperbolic. But,
• is parabolic.
• The second order derivative resembles the viscous
  term in the Navier-Stokes equations which govern
  flow with friction (as we shall see later), but
  here it was obtained as a consequence of the
  particular differencing scheme used. Because of
  the resemblance, the coefficient of this
  additional term, , is called the Artificial
  Viscosity.

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• The presence of the term leads to changes in
  amplitude and phase of the solution, and can be
  de-stabilizing.
• Therefore, to counter such differences between
  the PDE and its discretized form, second or
  higher order terms have to be added to the PDE or
  its discretized form.
• Addition of Artificial Dissipation can lead to
  inaccuracies and therefore, the final solution
  must be checked for varying levels of
  dissipation.

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• This idea of adding dissipative terms to the
  governing equation explicitly to stabilize and
  dampen numerical oscillations was due to Von
  Neumann Richtmeyer.
• Be aware that in the forthcoming numerical
  techniques, you will need to add artificial
  dissipation terms to the governing equations.

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• This Lax method is an example of an explicit
  method. All quantities are known at time level n
  and the quantities at time level n1 can be
  solved for directly or explicitly, i.e. no
  iteration is necessary.
• Example
• Laxs method for yielded
• In contrast, an implicit form would be
• Since and will depend on , usually
  in a non-linear manner.

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• In general, explicit methods are easier to
  program than implicit methods, but suffer from
  possible instability problems and stiffness.
• Explicit MacCormack Scheme
• Suppose we want to solve , where f and y are
  some non-linear functions of the dependent
  variables (such as r, u, T), and a could depend
  on x.

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• MacCormack proposed the following
  predictor-corrector method
• Predictor
• Note that is forward differenced in time
  and is forward differenced in space.
• Corrector Step

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• Where and depend upon the predicted values
  . This corrector step is obtained from
• or

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We will now apply this method to our quasi 1D
problem.