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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. 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.

103) is a valid solution only for c = constant. 100) ( n ) ikj∆x n ]. 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.

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. Normally this is done by means of a Taylor series expansion. We are concerned with the lowest powers of the space and time grid-interval in the expansion of the independent variables.

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

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