Abstract: We consider a general class of hyperbolic systems made of two equations, an homogeneous one and a nonhomogeneous one, including the usual models (Lagrangian or Eulerian) of hydrodynamics, non linear wave equations (telegraphist), shallow water with topography and friction, etc... First we show that a general solution corresponds to a straight line in the plane phase, and we use this result to build a solution for the two points boundary value problem. Next, this solution is used to generate interface and boundary solvers, in place of the usual Riemann solver. It appears that the resulting solver involves the source term in the general case but is only depending on the sign of this source term in almost all cases. Moreover, it is often simpler (and faster) than the usual Riemann solver for the homogeneous case. The validation of the method is shown first by using this construction of the solution to solve some known problems as the Roll waves or the Dam break with friction, and then by a computational example using the Shallow water model. This study has shown that a (very usual) friction term may develop some instabilitites in some geometrical configurations for slow velocities, even for other numerical schemes.
Conservation Laws Preprint Server <conservation@math.ntnu.no> Last modified: Mon Jun 19 13:19:35 MET DST 2000