I am dealing with the problem to generate the two-dimensional volume advection problem in which the scalar field has been considered as a hollow square at an angle of 26 57 to the x-axis (The value f = 0 (gas) and f = 1 (fluid))
The problem described in your question is used often for testing of numerical schemes for solution of the advection equation. The square mass distribution is placed in the solid body rotation velocity field; typically one and five rotations are used in such tests.
You can therefore use any good advection program to solve your problem. The main problems to consider are the resolution of the grid system and the flux correction procedure to eliminate the Gibbs noise which is inherently present when advecting discontinuous profiles.
The nice review of the fortran programs which are potentially relevant is presented at
Alternatively you can represent the initial condition by a large number of Lagrangian particles, and after calculating their forward in time trajectories, obtain a solution by remapping the new positions to a fixed grid.
The choice of the methodology depends on the specific considerations of your problem, I do not know all details. The Lagrangian methodology, however, is simpler to use and with proper remapping procedure can lead to more accurate solution with a given number of degrees of freedom.