Eulerian method for computing multivalued solutions of the Euler-Poisson equations and applications to wave breaking in klystrons

The Eulerian method for evaluation of computing multivalued solutions to the euler-Poisson equations, applied to wave breaking in klystrons, was described. An Eulerian formulation which was capable of computing multivalued solutions was also derived from a kinetic description of the Euler-Poisson system and a moment closure. The Eulerian moment equations were computed for velocity modulated electron beam, which was observed by prior Lagrangian theories to break in finite time and form multivalued solutions. The results show that the Lagrangian formulation was used for explicit computation of wavebreaking time and location for velocity boundary conditions.