The semigroup stability of the difference approximations for initial-boundaryv alue problems

For semidiscrete approximations and one-step fully discretized approximations of the initial-boundary value problem for linear hyperbolic equations with diagonalizable coefficient matrices, we prove that the Kreiss condition is a sufficient condition for the semigroup stability (or l2 stability). Also, we show that the stability of a fully discretized approximation generated by a locally stable Runge-Kutta method is determined by the stability of the semidiscrete approximation.