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Circular advection

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The scalar advection equation

\frac{\partial u}{\partial t} + \frac{\partial}{\partial x}(yu) - \frac{\partial}{\partial y}(xu)= 0

models advection along concentric circles. The characteristics are given by

\frac{dx}{dt} = y, \quad \frac{dy}{dt} = -x

and they are concentric circles about the origin going in clockwise direction. Any given initial condition is advected in circles without change of shape or magnitude. This is a good test case for assessing the numerical viscosity in a scheme.

Advection of a hill

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