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

By D. Randall

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This is bad, though, because if-tests can cause slow execution on certain types of computers, and besides, if-tests are ugly. 82) 28 n+1 n n n n Basic Concepts n  u j – u j – 1  uj + 1 – uj  – uj uj - + c -  ---------------------- = 0 . 83) This form avoids the use of if-tests and is also convenient for use in pencil-and-paper analysis, discussed later. In summary: Truncation error measures the accuracy of an approximation to a differential operator or operators. It is a measure of the accuracy with which a differential equation has been approximated.

Program the upstream scheme on a periodic domain with 100 grid points. Give a sinusoidal initial condition with a single mode such that exactly four wavelengths fit in the domain. 1. In each case, take enough time steps so that in the exact solution the signal will just cross the domain. Discuss your results. 4. 138) dv = – fu . 139) Note: Solution following the energy method as required does not involve the use of complex variables. 5. Work out the form of the most compact second-order accurate approximation for 2 d f   2  on a non-uniform grid.

Give a sinusoidal initial condition with a single mode such that exactly four wavelengths fit in the domain. 1. In each case, take enough time steps so that in the exact solution the signal will just cross the domain. Discuss your results. 4. 138) dv = – fu . 139) Note: Solution following the energy method as required does not involve the use of complex variables. 5. Work out the form of the most compact second-order accurate approximation for 2 d f   2  on a non-uniform grid. Also find the simpler form that applies when the dx  j grid is uniform.

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