Module 1: Introduction to Finite Difference Method and Fundamentals of CFD Lecture 7:

Transcription

1 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture7/7_1.htm 1 of 1 6/20/ :26 PM The Lecture deals with: Errors and Stability Analysis

2 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture7/7_2.htm 1 of 1 6/20/ :26 PM Introduction There is a formal way of examining the accuracy and stability of linear equations, and this idea provides guidance for the behavior of more complex non-linear equations which are governing the equations for flow fields. Consider a partial differential equation, such as Eq. (3.3). The numerical solution of this equation is influenced by the following two sources of error.

3 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture7/7_3.htm 1 of 1 6/20/ :26 PM Discretization: This is the difference between the exact analytical solution of the partial differential Eq. (3.3) and the exact (round-off free) solution of the corresponding finite-difference equation (for example, Eq.(3.4). The discretization error for the finite-difference equation is simply the truncation error for the finite-difference equation plus any error introduced by the numerical treatment of the boundary conditions.

4 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture7/7_4.htm 1 of 2 6/20/ :26 PM Round-off: This is the numerical error introduced for a repetitive number of calculations in which the computer is constantly rounding the number to some decimal points. If A= analytical solution of the partial differential equation, D= exact solution of the finite-difference equation N=numerical solution from a real computer with finite accuracy Then, Discretization error = A D = Truncation error + error introduced due to treatment of boundary condition Round-of error or, (7.1) where, is the round-off error, which henceforth will be called error for convenience. The numerical solution N must satisfy the finite difference equation. Hence from Eq. (3.4) (7.2) By definition, D is the exact solution of the finite difference equation, hence it exactly satisfies (7.3) Subtracting Eq. (2.44) from Eq. (2.43) (7.4) From Equation (7.4) we see that the error also satisfies the difference equation. If errors are already present at some stage of the solution of this equation, then the solution will be stable if the 's shrink, or at least stay the same, as the solution progresses in the marching direction, i.e from step n to n+1. If the 's grow larger during the progression of the solution from step n to n+1, then the solution is unstable. Finally,

5 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture7/7_4.htm 2 of 2 6/20/ :26 PM it stands to reason that for a solution to be stable, the mandatory condition is (7.5) For Eq. (3.4), let us examine under what circumstances Eq. (7.5) hold good.

6 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture7/7_5.htm 1 of 2 6/20/ :27 PM Assume that the distribution of error along the x- axis is given by a Fourier series in x and the time-wise distribution is exponential in t, i.e, (7.6) where I is the unit complex number and k the wave number. Since the difference is linear, when Eq. (7.6) is substituted into Eq. (7.4), the behavior of each term of the series is the same as the series itself. Hence, let us deal with just one term of the series, and write (7.7) Substitute Eq. (7.7) into (7.4) to get (7.8) Divide Eq. (7.8) by or, (7.9) Recalling the identity Eq. (7.9) can be written as or,

7 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture7/7_5.htm 2 of 2 6/20/ :27 PM (7.10) From Eq.(7.7), we can write (7.11) Combining Eqns. (7.10), (7.11) and (7.5), we have (7.12) Eq. (7.12) must be satisfied to have a stable solution. In Eq (7.12) the factor is called the amplification factor and is denoted by G.

8 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture7/7_6.htm 1 of 2 6/20/ :27 PM Evaluating the inequality in Eq. (7.12), the two possible situations which must hold simultaneously are Thus, Since is always positive, this condition always holds. The other condition is Thus, For the above condition to hold (7.13) Eq. (7.13) gives the stability requirement for which the solution of the difference Eq. (3.4) will be stable. It can be said that for a given the allowed value of must be small enough to satisfy Eq. (7.13). For the error will not grow in subsequent time marching steps in t, and the numerical solution will proceed in a stable manner. On the contrary, if then the error will progressively become larger and the calculation will be useless. The above mentioned analysis using Fourier series is called as the Von Neumann stability analysis. Congratulations! You have finished Lecture 7. To view the next lecture select it from the

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