README.rst

Sloshing water in a parabolic bowl

MODIFIED FOR DCLAW TESTING

Waves in a parabolic bowl with a flat surface sloshing around. An exact analytic solution is known in which the surface stays flat.

To create the topo file before running the code:

make topo

In this code, x and y are in meters (coordinate_system=1 in setrun.py).

Topography: B(x,y) = h_0((x^2 + y^2)/a^2 -1),

Depth: h(x,y,t) = \max\left(0,~~ (\sigma h_0/a^2)(2x\cos(\omega t) + 2y\sin(\omega t) - \sigma) - B(x,y)\right)

Velocities: u(x,y,t) = -\sigma \omega \sin(\omega t),\qquad v(x,y,t) = \sigma \omega \cos(\omega t).

where \omega = \sqrt{2gh_0} / a.

The period of oscillation is T = 2\pi / \omega.

The following parameters are currently hardwired several places:

a = 1, ~~\sigma = 0.5, ~~h = 0.1,~~g = 9.81

This should be cleaned up: better to put them in a setprob.data file that is read in where needed.

W. C. Thacker, Some exact solutions to the nonlinear shallow water wave equations, J. Fluid Mech. 107 (1981), 499-508.
J.M. Gallardo, C. Pares, and M. Castro, On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas, J. Comput. Phys. 227(2007) 574-601.
Y. Xing, X. Zhang and C.-W. Shu, Positivity preserving high order well balanced discontinuous Galerkin methods for the shallow water equations , Advances in Water Resources 33 (2010), pp. 1476-1493.

This test problem has been used in several other papers too.