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Additional info for Hyperbolic Problems: Theory, Numerics, Applications: Seventh International Conference in Zürich, February 1998 Volume I
40 B. Ben Moussa, N. P. Vila ~I FIGURE 2. Standard Method (left) - Renormalized(right)  S. Benharbit, A. P. Vila, Numerical viscosity, and convergence of finite volume Methods for conservation laws with Boundary conditions, SIAM J. Numer. Anal. , 32 (3) (1995), 775- 796.  B. P. Vila, Convergence of SPH method for scalar nonlinear conservation laws, to appear in SIAM Numerical Analysis. V. Bicknell, The equations of motion of particles in smoothed particle hydrodynamics, SIAM J. Sci. Stat.
129, © 1999 Birkhiiuser Verlag BaseVSwitzerland Large Time Stability of Propagating Phase Boundaries Fumioki Asakura Abstract. We study the Cauchy problem for a 2 x 2-system of conservation laws: Vt - U x = 0, Ut - 0"( v)x = 0 which describe the phase transition. Two constant states satisfying the Maxwell equal-area principle constitute an admissible stationary solution; a small perturbation of these Maxwell states will be our initial data. We shall show that: there exists a global in time propagating phase boundary which is admissible in the sense that it satisfies the Abeyaratne-Knowles kinetic condition; the states outside the phase boundary tend to the Maxwell states as time goes to infinity.
Concluding remarks Starting from the principle of mixed Finite Volume/Finite Element methods, we have constructed a new method for the two-dimensional compressible NavierStokes equations, whose main component lies in our recent staggered grid central finite volume method inspired from the one-dimensional Lax-Friedrichs and Nessyahu-Tadmor difference schemes. From a theoretical viewpoint, Loo-weak convergence of our Finite Volume method has been proved first for a linear hyperbolic equation , using Loo-and weighted total variation estimates, and more recently, for a nonlinear scalar hyperbolic equation, with the help of Young measures.