Differentiate difference approximations for nonlinear initial-value problems

Abstract

Part II of the present paper continues the treatment of differnce approximations for nonlinear initial-value problems (for part I see [20]). by virtue of the characterizations of stability and continuous convergence in part I, one obtains a series of equivalent conditions for weak stability and continuous convergence of certain order. these characterizations include locally uniform Lipschitz condision and discretization error estimates. The fundamental differentiability condition (D) and boundedness condition (B_P) of part I now assume the form (D and (b^p), resp., for some constants ?≧0, ≧0. For semi-homogeneous methods the concept of uniform continuous convergence with respect to all initial-times is proved to be the proper concept in the sense of being necessary and sufficient for stability in the case of spedial norms and special orders. finally, examples fo hyperbolic and parabolic intial-value problems will be treated