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Work out the form of the most compact second-order accurate approximation for 2 d f   2  on a non-uniform grid. Also find the simpler form that applies when the dx  j grid is uniform. An Introduction to Atmospheric Modeling Revised April 22, 2004 4:41 pm CHAPTER 3 A Survey of Time-Differencing Schemes for the Oscillation and Decay Equations Copyright 2004 David A. 1 Introduction In atmospheric dynamics, the governing equations are usually non-linear partial differential equations. Some knowledge of finite-difference approximations to ordinary differential equations (especially first order) is needed, however.

109), we obtain λ 2 = 1 + 2µ ( µ – 1 ) ( 1 – cos k∆x ) . 110) reduces to 2 λ 2 1 = --- ( 1 + cos k∆x ) . 111), the amplification factor λ depends on the wave number, k. , the various curves shown in Fig. 7 can be constructed. We see clearly that this scheme damps for 0 < µ < 1 and is unstable for µ < 0 and µ > 1 . , on n ). Why not? The reason is that our “coefficient” c , has been assumed to be independent of x and t . Of course, in realistic problems the advecting current varies in both space and time.

The appendix to this chapter provides a proof that the scheme really has fourth-order accuracy. 4 Finite-difference schemes applied to the oscillation equation In order to test the stability of the finite-difference schemes discussed in the previous section, we must specify the form of the function f ( q, t ) . As a first example, consider the oscillation equation: dq ------ = iωq , q complex, ω real. 19) iωt The exact solution is q = qˆ ( t )e , where qˆ is the “initial” value of q , at t = 0 . , d ----- qˆ = 0 .

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An Introduction to Atmospheric Modeling by Randall D.


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