lecture 16: convection and diffusion (cont’d). last time … we l looked at cds/uds schemes to...
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Lecture 16: Convection and Diffusion (Cont’d)
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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 (Cont’d)
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
x
Pe
-Pe
Pe=0
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 (Cont’d)
Area vectors
Flux*Area:
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 S=0, 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 (Cont’d)
Instead of using the exact curve for aE/De, use three
tangents
Similar manipulation for other coefficients
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Hybrid Difference Scheme (Cont’d)
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
exp( / )A UX BY C X
Y
U
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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
(i,j)(i-1,j) (i+1,j)
(i,j+1)
(i, j-1)
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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