A general multigrid framework is discussed for obtaining textbook efficiency to solutions of the compressible Euler and Navier-Stokes equations in conservation law form. The general methodology relies on a distributed relaxation procedure to reduce errors in regular (smoothly varying) flow regions; separate and distinct treatments for each of the factors (elliptic and/or hyperbolic) are used to attain optimal reductions of errors. Near boundaries and discontinuities (shocks), additional local relaxations of the conservative equations are necessary . Example calculations are made for the quasi-one-dimensional Euler equations; the calculations illustrate the general procedure.

Key words. textbook multigrid efficiency, distributed relaxation, Euler equations

Subject classification. Applied and Numerical Mathematics