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Extra resources for An Introduction to Atmospheric Modeling

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The truncation error is 2 1 ∆tq′′  --- – β + O ( ∆t ) When β = 1, α n = 0 , the scheme is called the backward 2  (implicit) scheme. It has first-order accuracy. It can be said to correspond to l = – 1 . 1 Higher accuracy is obtained for β = α = --- , which gives the “trapezoidal” (implicit) 2 scheme. It has second-order accuracy, as we expect from the general rule for implicit schemes. m = 0 , l > 0 (Adams-Moulton schemes) These are analogous to the Adams-Bashforth schemes, except that β ≠ 0 .

See text for details. extrapolation, u jn + 1 will lie outside the range of u jn– 1 and u jn . We use both interpolation and extrapolation extensively in this course. 8 Stability n We now investigate the behavior of the discretization error u j – u ( j∆x, n∆t ) as n increases, for fixed ∆x and ∆t . Does the error remain bounded for any initial condition? If so the scheme is said to be stable; otherwise it is unstable. In most physical problems the true solution is bounded, at least for finite t, so that the solution of the scheme is bounded if the scheme is stable.

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! 3! 4!  2 3 ∆t ∆t = β f + ∆tf′ + -------- f′′ + -------- f′′′ + … 2! 3! + αn f + αn – 1 2 3 ∆t ∆t f – ∆tf′ + -------- f′′ – -------- f′′′ + … 2! 3! 2 3 2 3 ( 2∆t ) ( 2∆t ) + α n – 2 f – 2∆tf′ + ---------------- f′′ – ---------------- f′′′ + … 2!

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