Implicit time dependent methods for inviscid and viscous compressible flows

Linear multistep methods; predictor-corrector methods; centered implicit dissipation schemes; and characteristic modeling schemes for compressible flows are introduced. The multistep algorithm is unconditionally stable and the error on the converged steady solution does not depend on the time step used, but the short wavelengths are not damped and therefore the scheme is not dissipative in the sense of Kreiss. Dissipation is provided by appending explicit and implicit artificial viscosity. The predictor-corrector scheme is centered in space for a linear equation intrinsically dissipative without artificial viscosity and requires the solution of two block bidiagonal systems for each space variable. The implicit schemes are single step schemes with centered differencing in space. The time differencing produces the appropriate dissipation. For Euler equations, characteristic schemes produce the dissipation by noncentered discretization in space.