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By Randall D.
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Extra resources for An Introduction to Atmospheric Modeling
Minimizing the truncation error is usually easy. Minimizing the discretization error can be much harder. 84) where c∆t µ ≡ -------- . 85) This scheme has the form of either an interpolation or an extrapolation, depending on the value of µ . To see this, refer to Fig. 6. 87) For 0 ≤ µ ≤ 1 we have interpolation. For µ < 0 or µ > 1 we have extrapolation. Note that for the case of interpolation, u jn + 1 will be intermediate in value between u jn– 1 and u jn . For instance, if u jn– 1 and u jn are both ≥ 0 , then u jn will also be ≥ 0 .
A measure of the accuracy of the ﬁnite-difference scheme can be obtained by substituting the solution of the differential equation into the ﬁnite-difference equation. 78) where ε is called “truncation error” of the scheme. 75). n Note that, since u j is deﬁned only at discrete points, it is not differentiable, and so we cannot substitute u jn into the differential equation. Because of this, we cannot measure n how accurately u j satisﬁes the differential equation. 69), we ﬁnd that 1 ∂2 u 1 ∂2 u ε = ----- ∆t -------- + … + c – ----- ∆x --------+… .
7: The amplification factor for the upstream scheme, plotted for three different wave lengths. n uj ∞ ≤ ∑ ( 0 ) imk j∆x n uˆ m e 0 ( λ m ) m = –∞ ∞ ≤ ∑ ( 0 ) imk j∆x n uˆ m e 0 ( λ m ) m = –∞ ∞ = ∑ (0) n uˆ m λ m . 114) If λ m ≤ 1 is satisﬁed for all m , then n uj ≤ ∞ ∑ m = –∞ An Introduction to Atmospheric Modeling (0) uˆ m . 115) 36 Therefore, n uj ∞ will be bounded provided that ∑ Basic Concepts ( 0 ) imk j ∆ x uˆ m e 0 , which gives the m = –∞ initial condition, is an absolutely convergent Fourier series.