In this paper, a finite-difference method for solving boundary initial value problem of nonlinear system equations of hyperbolic type in a class of discontinuous functions is suggested. In order to obtain the numerical solution of the Main problem in a class of discontinuous functions the auxiliary problem is introduced. The degree of smoothness of the solution of the auxiliary problem is higher than of smoothness of the solution of the main problem. Furthermore, the suggested auxiliary problem lets us write out effective and-higher order numerical algorithms. The solutions obtained from these algorithms represent the physical nature of the problem with a high accuracy. Some numerical experiments are carried out by using the auxiliary problem. (C) 2003 Elsevier Inc. All rights reserved.