Initial Conditions

Our problem is non-stationary and suitable initial conditions have to be added. We will prescribe

$$\rho(\cdot,0) =\rho_0,\quad{\bf u}(\cdot,0)={\bf u}_0,\quad p(\cdot,0) = p_0,\qquad \mbox{in } \Omega.$$

From this we can get the initial vector ${\bf w}_0= {\bf w}_0({\bf x})= {\bf w}({\bf x},0)$ for solving the system (15).

Now we have a complete initial-boundary value problem and in the following we will deal with the numerical solution of these equations. We are interested in transonic flow where we can look at the system of the Navier - Stokes equations (15) as at the system of the Euler equations (17) with some ``viscous perturbations'', which is represented by the viscous terms $\ {\bf R}_i,\quad i=1,\dots,d$ because they are small. This is the main reason why we will use the operator splitting method for solving the Navier - Stokes equations. The next section is devoted to solving of the system of the Euler equations.