In this paper boundary procedures are discussed for a new second order accurate method, developed in Morris (1972) and Zwas, Eilon & Gottlieb (1972), for non-linear hyperbolic systems in two space variables. This method is a multilevel scheme of the same type as those of Strang, (1964, 1968). It is shown that the straightforward method of incorporating boundary data gives, in general, only locally first order accurate values. A boundary procedure which preserves local second order accuracy is developed. The method is also extended to systems in many space variables. The results of some numerical experiments are reported. © 1976 Academic Press Inc. (London) Limited.