# incompressible.problems package¶

## incompressible.problems.converge module¶

Initialize a smooth incompressible convergence test. Here, the velocities are initialized as

\begin{align}\begin{aligned}u(x,y) = 1 - 2 \cos(2 \pi x) \sin(2 \pi y)\\v(x,y) = 1 + 2 \sin(2 \pi x) \cos(2 \pi y)\end{aligned}\end{align}

and the exact solution at some later time t is then

\begin{align}\begin{aligned}u(x,y,t) = 1 - 2 \cos(2 \pi (x - t)) \sin(2 \pi (y - t))\\v(x,y,t) = 1 + 2 \sin(2 \pi (x - t)) \cos(2 \pi (y - t))\\p(x,y,t) = -\cos(4 \pi (x - t)) - \cos(4 \pi (y - t))\end{aligned}\end{align}

The numerical solution can be compared to the exact solution to measure the convergence rate of the algorithm.

incompressible.problems.converge.finalize()[source]

print out any information to the user at the end of the run

incompressible.problems.converge.init_data(my_data, rp)[source]

initialize the incompressible converge problem

## incompressible.problems.shear module¶

Initialize the doubly periodic shear layer (see, for example, Martin and Colella, 2000, JCP, 163, 271). This is run in a unit square domain, with periodic boundary conditions on all sides. Here, the initial velocity is:

              / tanh(rho_s (y-0.25))   if y <= 0.5
u(x,y,t=0) = <
\ tanh(rho_s (0.75-y))   if y > 0.5

v(x,y,t=0) = delta_s sin(2 pi x)

incompressible.problems.shear.finalize()[source]

print out any information to the user at the end of the run

incompressible.problems.shear.init_data(my_data, rp)[source]

initialize the incompressible shear problem