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75). ” j Here the summation obviously must be taken over a finite number of grid points. This is not an important limitation because in practice we are always dealing with a finite n number of grid points. If the sum is bounded, then each u j must also be bounded. n+1 Whereas in the direct method we checked max ( j ) u j check n 2 ∑ ( uj ) , here in the energy method we . The two approaches are therefore somewhat similar. 91) . j For simplicity, suppose that u is periodic in x , and consider a summation over one complete cycle.

M = 0 , l > 0 (Adams-Bashforth schemes) Better accuracy can be obtained by proper choice of the α 's, if we use l > 0 . For example, consider the case l = 1 . 10) 2 1 ε = ∆tq′′  α n – 1 + --- + O ( ∆t ) . 11) and the truncation error is An Introduction to Atmospheric Modeling 48 A Survey of Time-Differencing Schemes for the Oscillation and Decay Equations 1 If we choose α n – 1 = – --- , the scheme has second-order accuracy. This is the second-order Adams-Bashforth scheme. 9) involves two different values of f , we only have to evaluate f once per time step, if we simply save one “old” time level of f for later use on the next time step.

As we increase the resolution, J increases. 137) to increase, which strengthens the damping. Which effect dominates is not obvious. 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.

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An Introduction to Atmospheric Modeling by Randall D.

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