By D. Randall
Read or Download An Introduction to Atmospheric Modeling [Colo. State Univ. Course, AT604] PDF
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Additional resources for An Introduction to Atmospheric Modeling [Colo. State Univ. Course, AT604]
Normally this is done by means of a Taylor series expansion. We are concerned with the lowest powers of the space and time grid-interval in the expansion of the independent variables. • Check linear stability for a simplified (linearized, constant coefficients) version of the equation. Von Neumann's method is often used here. • Check nonlinear stability, if possible. This can be accomplished, in some cases, by using the energy method. Otherwise, empirical tests are needed. Increased accuracy does not always give a better scheme.
It is better than would be expected from the general rule, stated earlier, for explicit schemes. The leapfrog scheme has been very widely used, but it has some serious disadvantages, as will be discussed later. 2 Non-iterative schemes. 49 m = 1, l = 1 αn – 1 Here there is no gain in accuracy. The highest accuracy (second order) is obtained for = 0 (the leapfrog scheme). m = 1 , l > 1 (Nystrom schemes) We can increase the order of accuracy by choosing appropriate α's if l > 1 . Schemes with m > 1 are not of much interest and will not be discussed here.
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! 3! ( 3∆t ) ( 3∆t ) + α n – 3 f – 3∆tf′ + ---------------- f′′ – ---------------- f′′′ + … 2! 3! +… 2 3 ( l∆t ) ( l∆t ) + α n – l f – l∆tf′ + --------------- f′′ – --------------- f′′′ + … 2! 3! 4) where ε is the truncation error and a prime denotes a time derivative.