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104) (n) Then uˆ is the amplitude of the wave at time-level n . Note that the shortest resolvable wave, with L = 2∆x , has k∆x = π , while longer waves have k∆x < π , so there is never any need to consider k∆x > π . Define λ , which may be complex, by (n + 1) (n) ≡ λuˆ . 105) (n + 1) (n) = λ uˆ Then uˆ . ” As shown below, we can work out the form of λ for a particular finite-difference scheme. In general λ depends on k , so we could write λ k , but usually we suppress that urge for the sake of keeping the notation simple.

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.

With the approach outlined above, schemes of higher order accuracy are made possible by bringing in more time levels. It is also possible to obtain schemes of higher accuracy in other ways. This will be explained later. We now survey a number of time-differencing schemes, without specifying f . In this analysis, we can determine the order of accuracy of each scheme. We cannot decide whether a scheme is stable or unstable, however, unless f is specified. Later in this Chapter we investigate what happens with two particular choices of f , and later in the course we will consider additional choices for f .

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An Introduction to Atmospheric Modeling [Colo. State Univ. Course, AT604] by D. Randall

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