The present paper deals with the construction and investigation of economical finite-difference schemes for hyperbolic equations and first-order systems. Numerous economical algorithms of the decomposition method for the above-mentioned class of problems have been suggested so far [1]. Special algorithms that have properties of purely implicit finite-difference methods were constructed in [1] on the basis of absolutely stable second-order finite-difference schemes for hyperbolic equations. Many-component decomposition methods considered in [1, 2] lead to absolutely stable finitedifference schemes of second-order accuracy with minimal constraints on the decomposition operators. These schemes can be classified as methods of total approximation [3–5] and hence do not provide high accuracy, especially in computations for large times. A method of many-component decomposition with complete approximation, which naturally increases the accuracy of decomposition algorithms and has good stabilization properties, was suggested in [6–8]. That method was used in [5–11] for a system of hyperbolic equations. In what follows, we construct and analyze economical additive schemes of complete approximation for multidimensional hyperbolic linear and nonlinear equations and first-order systems. Moreover, just as in [1], the decomposition operators are subjected to minimal restrictions. We write out the evolution equation and the corresponding Cauchy problem in the form