The density equation is hyperbolic, therefore you can prescribe an initial condition and boundary condition only in a open domain defined by the initial regular curve and the characteristic lines.

A simple example for constant convective velocity (u>0) is

d rho/ dt + u d rho/dx = 0

initial condition: rho(x,0) = rho0(x)

boundary condition: rho(X_bc,t) = g(t)

Hi, I thank you for your kind help. Please allow me to ask for further clarification about your answer.

1. When you write rho(X_bc,t) = g(t), should I understand that, for a given instant, on all points x_bc of the boundary of my domain the value of rho is the same (g only depends on t).

2. How do I determine g from the physics of my problem (i.e. a gas on a 2D rectangular domain, with a field of heat sources and sinks?)

Thank you again,

Giuseppe

1) Not necessarily, I did the example for a 1D problem wherein the boundary is simply a point. In a 2D (and 3D) problem you have that g can be a function also of the space, for example the density distribution along the boundary.

2) If you have pressure and temperature distribution, you can compute the density and prescribe it at the boundary