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Navy Ocean Circulation and Tide Models Activity

Objective: In this exercise, you will experience first-hand the trade-offs between computational stability, numerical resolution, and computer resource limitations.

Activity: Given the one-dimensional advection equation,

where u = u(x,t), (u is only a function of x and t), and c is a constant,(|c| = 2 m/s).

Solve the equation in a 100 km domain for 5 days such that

  1. The scheme is computationally stable (Hint: the CFL condition requires |c| t / x < 1.)
  2. The horizontal structure in u is resolved.
  3. Computer (cpu) resources are optimized (i.e. the resolution is not too high to be wasteful).

Initially, the amplitude of u varies from 0.1 to -0.1 m/s over the spatial domain which is x = 0 to x = 100 km. The figure below shows this initial condition.


The boundary conditions are periodic in x.

Now, run this simple model yourself!

  Enter the phase speed of the wave, c, in meters/second:
  Enter the grid spacing, x, in meters:
  Enter the time step, t, in seconds:
     

Try varying the x and t until you feel you have adequately resolved the spatial structure, the CFL criterion is not exceeded, and you have achieved good efficiency (number of computations < 1000). Record your attempts and then fill in and submit the table below. Do one case where the CFL criterion is exceeded.

x (m) t (s) # time steps # grid points # computations (or error message)
CFL Criteria Exceeded

With x = 20,000 m, what is the largest t (in seconds) that satisfies the CFL criterion?

Now, say the phase speed, |c|, is 200 m/s. For a value of x = 20,000 m, what is the largest acceptable t?

Your name:

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