By D. Randall
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Extra resources for An Introduction to Atmospheric Modeling [Colo. State Univ. Course, AT604]
We return to this point later. e. 108), to infer that λ = 1 – µ ( 1 – cos k∆x + i sin k∆x ) . 109) Note that λ is complex. 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 .
Give a sinusoidal initial condition with a single mode such that exactly four wavelengths ﬁt in the domain. 1. In each case, take enough time steps so that in the exact solution the signal will just cross the domain. Discuss your results. 4. 138) dv = – fu . 139) Note: Solution following the energy method as required does not involve the use of complex variables. 5. 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.
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. We normally apply von Neumann’s method to idealized versions of our equations, in which the various coefﬁcients, such as c, are treated as constants. As a result, von Neumann’s method can “miss” instabilities that arise from variations of the coefﬁcients.
An Introduction to Atmospheric Modeling [Colo. State Univ. Course, AT604] by D. Randall