Module 1: Introduction to Finite Difference Method and Fundamentals of CFD Lecture 6:

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1 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture6/6_1.htm 1 of 1 6/20/ :24 PM The Lecture deals with: ADI Method

2 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture6/6_2.htm 1 of 2 6/20/ :25 PM 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).

3 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture6/6_2.htm 2 of 2 6/20/ :25 PM 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).

4 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture6/6_3.htm 1 of 1 6/20/ :25 PM 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

5 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture6/6_4.htm 1 of 1 6/20/ :25 PM 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.

6 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture6/6_5.htm 1 of 2 6/20/ :25 PM 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)

7 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture6/6_5.htm 2 of 2 6/20/ :25 PM 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.