Lecture 16: Convection and Diffusion (Cont

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Lecture 16 Convection and Diffusion (Contd)
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Last Time

• We
• Looked at CDS/UDS schemes to unstructured meshes
• Look at accuracy of CDS and UDS schemes
• Look at false diffusion in UDS using model
  equation

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This Time

• We will use model equation to look at behavior of
  CDS scheme
• Look at some first-order schemes based on exact
  solutions to the convection-diffusion equation
• Exponential scheme
• Hybrid scheme
• Power-law scheme

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CDS Model Equations

• Pure convection equation
• Apply CDS
• Expand in Taylor series

Do same type of expansion in y direction
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Model Equation (Contd)

• Subtract to obtain
• Do same in y direction
• Substitute into discrete equation

Dispersion Term
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Discussion

• Model equation for CDS has extra third-derivative
  (dispersive) term
• This type of odd-derivative term tends to cause
  spatial wiggles
• Note that truncation error for CDS is O( ?x2 )
• Thus, UDS is dissipative and CDS is dispersive

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First-Order Schemes Based on Exact Solutions

• 1D Convection-diffusion equation

-Pe
?
Pe0
Pe
x
What are the limits of this equation for
different Pe?
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Exponential Scheme

• Use 1-D exact solution as profile assumption in
  doing discretization
• Consider convection-diffusion equation
• Integrate over control volume

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Exponential Scheme (Contd)

• Area vectors
• FluxArea
• Use exact solution to write convection and
  diffusion terms

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Exponential Scheme Discrete Equations

• Both convection and diffusion terms estimated
  from exact solution
• If S0, we would get the exact solution in 1D
  problems
• But obviously not exact for non-zero S,
  multi-dimensional problems
• Discretization has boundedness, diagonal
  dominance
• Only first-order accurate

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Approximations to Exponential Scheme

• Exponentials are expensive to compute
• Approximations to the exponential profile
  assumption have been used to offset the cost.
• Hybrid difference scheme
• Power-law scheme
• Both these approximations are also only
  first-order accurate

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Hybrid Difference Scheme

• Consider the aE coefficient in exponential scheme
• Limits with respect to Pe

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Hybrid Difference Scheme (Contd)

• Instead of using the exact curve for aE/De, use
  three tangents
• Similar manipulation for other coefficients

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Hybrid Difference Scheme (Contd)

• Guaranteed bounded solutions
• Satisfies Scarborough criterion
• O(?x) accurate

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Power-Law Scheme

• Employs fifth-order polynomial approximation to
• Similar approach to other coefficients
• Scheme is bounded and satisfies the Scarborough
  criterion
• Is O(?x) accurate

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Multi-Dimensional Schemes

• Exact solutions have been used as profile
  assumptions in multi-dimensional situations
• Control volume-based finite element method of
  Baliga and Patankar (1983)
• This form is the solution to
• the 2D convection-diffusion equation

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Multi-Dimensional Schemes

• Finite analytic scheme (Chen and Li, 1979)
• Write 2D convection diffusion equation with
  source term for element
• Fix coefficient using (i,j) values
• Find analytical solution using separation of
  variables
• Use exact solution for profiles assumptions

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Closure

• In this lecture, we
• Looked at the model equation for CDS
• Shown dispersive nature of model equation
• Looked at differencing schemes based on exact
  solution to 1D convection-diffusion equation
• Looked at schemes which are approximations to the
  exponential scheme
• Looked at multidimensional schemes based on exact
  solutions