In the literature, schemes related to runge-kutta sheme are always tested by solving the model equation as du/dt+df(u)/dx=0 or du/dt+a*du/dx=av*d(du/dx)/dx, while u,f are scalars in a 1 dimensional system. Is there any detailed introduction as how to extend Runge-Kutta scheme to 3 dimensional situation? Furthermore, is Jacobian matrix required in the solution of 3D NS equation?

NS equation: compressible flow, subsonic d(U)/dt+dF(U)/dx + dG(U)/dy + dH(U)/dz = 0 while U, F, G, H are vectors.

Is it correct to directly discretize the convective and diffusive terms using explicit or compact schemes in corresponding dimension whithout calculating the Jacobian matrix and the characteristic values? Thanks very much!!!!!!!!

In general what you do for N-S is calculate df/dx ,dg/dy,dh/dz and then use R-k as du/dt=L ( L=-(df/dx+dg/dy+dh/dz) ).The calculation of jacobian is dependent on the form in which you discretise the N-S.