Read e-book online An Introduction to Atmospheric Modeling PDF

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By Randall D.

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109), and so we find that 0 ≤ µ ≤ 1 is the stability condition. The advantage of the matrix method is that the boundary conditions can be directly included in the stability analysis, as in the example above. 9 The effects of increasing the number of grid points Recall that the upstream scheme can be written as u jn + 1 = u jn ( 1 – µ ) + u jn– 1 µ . 127) λ 2 = 1 + 2µ ( µ – 1 ) ( 1 – cos k∆x ) . 128) so that The stability criterion is 0 ≤ µ ≤ 1. 129) λ ≤ 1. 129) is satisfied, we have Consider what happens when we increase the number of grid points, while keeping the domain size, D , the wind speed, c , and the wave number, k , of the advected signal the same.

Check nonlinear stability, if possible. This can be accomplished, in some cases, by using the energy method. Otherwise, empirical tests are needed. Increased accuracy does not always give a better scheme. For example, consider two schemes A and B, such that scheme A is first-order accurate but stable, while scheme B is second-order accurate but unstable. Given such a choice, the less accurate scheme is definitely better. In general, “good” schemes have the following properties, among others: • High accuracy.

The answer can be seen in Fig. 8. Increasing the resolution leads to less total damping, even though the number of time steps needed to cross the domain increases. 10 Summary Suppose that we are given a non-linear partial differential equation and wish to solve it by means of a finite difference approximation. 8: “Total” damping experienced by a disturbance crossing the domain, as a function J, the number of grid points across the domain. Here we have assumed that D ⁄ L = 2 . follows: • Check truncation error.

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An Introduction to Atmospheric Modeling by Randall D.

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