In this paper, we consider linear hyperbolic initial boundary value problems on mulidimensional domains. We assume that the system is symmetric hyperbolic, with maximal dissipative boundary conditions, the boundary is either characteristic of constant multiplicity either noncharacteristic. We show that this problem can be seen as a limit when epislon converge to 0^+ of parabolic initial boundary value problem. The parabolic operators are obtained from the hyperbolic operator by adding a viscosity epsilon E, where E is a well chosen elliptic and dissipative second oder operator. We prescribe a Dirichlet boundary condition for these parabolic pertubations. In particular, we treat the case of "conservative" boundary conditions. This answers to a question raised by J.Rauch in a paper in Séminaire Goulaouic-Schwartz (1978/1979). We also give a topological description of the set of the convenient symmetric viscosities for the Maxwell's system with "incoming wave" condition.