I am encoutering some difficulties when I try to solve a system of two coupled advection equations in 1D:

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where na and ni are the searched functions; va, vi and beta are constant coefficients.

Firstly, I try solve the simpler system of two non-coupled transport equations, by simpling putting the right hand sides of the equations to 0. For simplicity, I set the initial conditions to be sin(x) and left boundary conditions to be 0. Thus one obtains a pretty straightforward system of two uncoupled transport equations. The solutions obtained for this system seem reasonable:

Then, I introduce some exponential growth along the characteristics. At this point, the solution looses stability and some oscillations are seen in the solutions:

The next step would be to introduce some coupling between the equation. Any suggestions how to prevent the oscillations from arising?