Concept of Computational Fluid Dynamics Material

Module 1: Introduction to Finite Difference Method and Fundamentals of CFD
Lecture 6:
The Lecture deals with:
ADI Method
Objectives_template file:///D:/chitra/nptel_phase2/mechanical/cfd/lecture6/6_1.htm
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Lecture 6:
ADI Method
The difficulties described in the earlier section, which occur when solving the
two-dimensional equation by conventional algorithms, can be removed by alternating
direction implicit (ADI) methods. The usual ADI method is a two-step scheme given by
(6.1)
and
(6.2)
The effect of splitting the time step culminates in two sets of systems of linear algebraic
equations. During step 1, we get the following
or
(6.3)
Now for each “j” rows( j = 2,3...)we can formulate a tridiagonal matrix, for the varying i
index and obtain the values from i=2 to (imax-1) at (n+1/2) level Fig.6.1 (a).
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Figure 6.1(a)
Similarly, in step-2, we get
or
Now for each “i” rows ( i = 2,3...) we can formulate a tridiagonal matrix for the varying j
index and obtain the values from j =2 to (jmax-1) at nth level Fig. 2.5 (b).
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Lecture 6:
With a little more effort, it can be shown that the ADI method is also second- order
accurate in time. If we use Taylor series expansion around on either direction, we
shall obtain
and
Subtracting the latter from the former, one obtains
or
(6.5)
The procedure above reveals that the ADI method is second-order accurate with a
truncation error of
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Lecture 6:
Major advantages and disadvantages of explicit and implicit methods
| Advantages | | Disadvantages |
Explicit
Method
The solution algorithm is simple
to set up
for a given must be less
than a specific limit imposed by
stability constraints.
This requires many time steps to
carry out the calculations over a
given interval of t.
Implicit
Method
Stability can be maintained over
much larger values of .
Fewer time steps are needed to
carry out the calculations over a
given interval.
More involved producer is needed
for setting up the solution
algorithm than that for explicit
method.
Since matrix manipulations are
usually required at each time
step, the computer time per time
step is larger than that of the
explicit approach.
Since larger can be taken, the
truncation error is often large,
and the exact transients (time
variations of the dependent
variable for unsteady flow
simulation) may not be captured
accurately by the implicit scheme
as compared to an explicit
scheme.
Apparently finite-difference solutions seem to be straightforward. The overall procedure is
to replace the partial derivatives in the governing equations with finite difference
approximations and then finding out the numerical value of the dependent variables at
each grid point. However, this impression is needed incorrect! For any given application,
there is no assurance that such calculations will be accurate or even stable! We shall soon
discuss about accuracy and stability.
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Lecture 6:
Polynomial-fitting Approach:
By assuming a polynomial variation of the field variable in the neighborhood of the point
of interest, it is possible to obtain the difference expressions for the derivatives. For
instance, degree polynomial can be fitted between n nodes, for the field variable
and this polynomial can be used for evaluating upto the derivative.
Using two points, say i and i+1 , a linear variation can be assumed for the variable
and this leads to
(6.6)
For linear variation between i and
(6.7)
Similarly, the central difference expression
(6.8)
can be obtained by using a linear variation between and
For parabolic variation between points, and one can set:
(6.9)
where , a, b, c, are obtained from
(6.10)
(6.11)
(6.12)
In matrix form,
(6.13)
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Inversion of Eqn. (6.13) leads to the values of a, b, c in terms of and . Having
obtained the values of these coefficients, the derivative can be evaluated as:
(6.14)
where,
The second derivative at i can also be evaluated from the polynomial expression of (2.32)
and this is given by:
(6.15)
The polynomial fitting procedure can thus be extended for obtaining difference
expressions for higher order derivatives also. By considering neighboring point in
addition to i, derivatives upto th order can be calculated. The polynomial fitting
technique is very useful when the boundary conditions of the problem are of a very
complex nature and involve various derivatives of the unknown dependent variable.
Congratulations! You have finished Lecture 6. To view the next lecture select it from the
left hand side menu of the page or click the next button.
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Concept of Computational Fluid Dynamics Material

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