SHALLOW_WATER_2010.TEX
Numerical Solution of the Shallow Water Equations

SHALLOW_WATER_2010.TEX is the LaTeX source for two lectures on approximating the shallow water equations, focussing on a simple code by Cleve Moler, and addressed to the class on Mathematical Modeling of Oceanic and Atmospheric Flows, at Virginia Tech, March 22/24 2010.

A PDF version is available as ../../presentations/shallow_water_2010.pdf.

The main file is:

• shallow_water_2010.tex, the main LaTeX file.

The following files were included into the main file:

• cfl_h.png, a plot of H, as the CFL condition is violated.
• cfl_uh.png, a plot of UH, as the CFL condition is violated.
• fd_grid.png, a typical finite difference grid.
• fem_grid.png, a typical finite element grid.
• first_order_0.png, a first order stencil, image 0.
• first_order_1.png, a first order stencil, image 1.
• first_order_2.png, a first order stencil, image 2.
• first_order_3.png, a first order stencil, image 3.
• first_order_4.png, a first order stencil, image 4.
• first_order_5.png, a first order stencil, image 5.
• first_order_6.png, a first order stencil, image 6.
• first_order_7.png, a first order stencil, image 7.
• h099.png, a plot of H after 99 time steps.
• lw0.png, the Lax-Wendroff stencil, image 0.
• lw1.png, the Lax-Wendroff stencil, image 1.
• lw2.png, the Lax-Wendroff stencil, image 2.
• lw3.png, the Lax-Wendroff stencil, image 3.
• lw4.png, the Lax-Wendroff stencil, image 4.
• lw5.png, the Lax-Wendroff stencil, image 5.
• lw6.png, the Lax-Wendroff stencil, image 6.
• lw7.png, the Lax-Wendroff stencil, image 7.
• lw8.png, the Lax-Wendroff stencil, image 8.
• lw9.png, the Lax-Wendroff stencil, image 9.
• shallow_water_2d.png, a snapshot from Moler's 2D shallow water code.
• sixbysix_1.png, a simple 6x6 grid of nodes.
• sixbysix_2.png, a simple 6x6 grid, showing the induced regions.
• sixbysix_3.png, a simple 6x6 grid, showing where fluxes are estimated at midside nodes.
• sixbysix_4.png, the 6x6 grid, turned on its side so that time is displayed in the vertical direction, suggesting how time steps are taken.
• slinky.png, a snapshot from the slinky movie.
• tuning_fork.png, an image suggesting the propagation of acoustic waves from a tuning fork.
• uh099.png, a plot of UH after 99 time steps.
• water_wave.png, a snapshot from the shallow water wave movie.

Some animations were also used as illustrations.

• shallow_water_wave.mov, an animation of the 1D shallow water equations with a sequence of waves entering from the left, showing the gradual movement of physical particles to the right.
• slinky.mov, an animation of a slinky suspended on strings, suggesting the transmission of a signal and the temporary storage of energy by compression.

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