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

By D. Randall

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This is the situation shown in the ﬁgure. In general, there is no hope of obtaining smaller discretization error, no matter how c∆t small ∆x and ∆t become, so long as -------- is unchanged, because the true solution ∆x u ( j∆x, n∆t ) depends only on the initial value of u at the single point ( x 0 , 0) which n cannot inﬂuence u j . You could change u ( x 0, 0 ) [and hence u ( j∆x, n∆t ) ], but the n computed solution u j would remain the same. In such a case, the error of the solution usually will not be decreased by reﬁning the grid.

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. 3), and expand into a Taylor series around t = n∆t . We get  1 ∆t ∆t ∆t ------------------------   q + ∆tq′ + -------- q′′ + -------- q′′′ + -------- q′′′′ + …  ( 1 + m )∆t   2! 3! 4! 2 3 4 2 3 4  ( m∆t ) ( m∆t ) ( m∆t ) – q – ( m∆t )q′ + -----------------q′′ – -----------------q′′′ + -----------------q′′′′ – …  2!

134) The time required for the air to ﬂow through the domain is D T = ---- . 135) Let N be the number of time steps needed for the air to ﬂow through the domain, so that T N = ----∆t D = -------c∆t D = ---------µ∆x J = --- . 136) The total amount of damping that “accumulates” as the air moves across the domain is measured by λ N = ( λ 2 )N / 2 J  kD -----=  1 – 2µ ( 1 – µ ) 1 – cos  ------- 2µ . 136). As we increase the resolution, J increases. 137) to increase, which strengthens the damping.