An Introduction to Atmospheric Modeling [Colo. State Univ. by D. Randall

By D. Randall

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Given such a choice, the less accurate scheme is definitely better. In general, “good” schemes have the following properties, among others: • High accuracy. • Stability. • Simplicity. • Computational economy. Later we will extend this list to include additional desirable properties. 10 Summary 41 offs. For example, a more accurate scheme is usually more complicated and expensive than a less accurate scheme. We have to ask whether the additional complexity and computational expense are justified by the increased accuracy.

1: In Eq. 3), we use a weighted combination of to compute an “average” value of f n+1 n n–1 n–l , f , f , …, f dq f ≡ ------ over the time interval ( m + 1 )∆t . 2 Non-iterative schemes. 3); this is the case with the backward-implicit scheme, for which β ≠ 0 . The Euler forward scheme uses time level n , so that l = 0 . Note, however, that the Euler forward scheme omits time level n + 1 ; it is an explicit scheme with β = 0 . There are many (in principle, infinitely many) other possibilities, as will be discussed later in this Chapter.

D ----- qˆ = 0 . 19). • ikx ∂u ∂u Advection: The governing equation is ----- + c ----- = 0 . 4 Finite-difference schemes applied to the oscillation equation c Initial conditions for time-stepped variables X, Y, and Z. c The time step is dt, and dt2 is half of the time step. 5 Y=1. Z=0. do n=1,nsteps c Subroutine dot evaluates time derivatives of X, Y, and Z. call dot(X, Y, Z,Xdot1,Ydot1,Zdot1) c First provisional values of X, Y, and Z. X1 = X + dt2 * Xdot1 Y1 = Y + dt2 * Ydot1 Z1 = Z + dt2 * Zdot1 call dot(X1,Y1,Z1,Xdot2,Ydot2,Zdot2) c Second provisional values of X, Y, and Z.

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