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hw3 - Plot compared to the exact solution which is the same...

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Numerical Methods for Coastal and Oceanographic Engineering Slinn Spring 2004 Home Work Problem # 3. Due March 4, 2004. This problem set will give you experience with, 1) periodic boundary conditions, 2) the compact scheme, 3) higher order explict time step methods, 4) fast Fourier transform (FFT) methods, and 5) numerical diffusion and dissipation. Again we examine the one way wave equation, 0 = ∂ ∂ + ∂ ∂ x u C t u . 1. Solve using the third order Adams-Bashforth time step method. Use a Courant number of 0.2 and initial conditions u , with C = 1 m s 2 ) 2 ( 5 3 ) 0 , ( − − = x e x -1 , on the domain, 0 < x < 10 m. Use periodic boundary conditions, and plot solutions with t = 10 s and t = 20 s. Use three values of ∆ x, with 64, 128, and 256 grid points. Use 3 types of spatial differences; 1) second-order central differences, 2) 4 th order compact scheme, 3) pseudo-spectral methods using FFT’s to calculate the spatial derivatives.
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Unformatted text preview: Plot compared to the exact solution, which is the same as the initial conditions. 2. Add weak spatial smoothing to each solution, consistent with the type of spatial derivatives being used, plot compared to the exact solution. A) second order spatial smoothing with the coefficient chosen to damp the 2 ∆ x wave. (use 64 and 256 grid points) B) 4 th-order compact smoothing with β = 0, α = 0.4, a = 0.925, b = 0.9, c = -0.025 (see page 57 of thesis notes). Use 64 and 128 grid points. C) Spectral cutoff filter, for de-aliasing k cutoff = 2/3 k max . Use 32 and 64 grid points only. A periodic tri-diagonal matrix solver and a FFT package is provided for the assignment. The FFT package provided uses only powers of 2 so the number of grid points needs to be 128, 256, 512,… a power of 2 (or 129, 257, 513,…) depending on method of periodicity. Plot the results with Tecplot....
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  • Spring '07
  • Slinn
  • Power of two, grid points, Fast Fourier transform, periodic boundary conditions, Coastal and Oceanographic Engineering
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