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CONVERGENCE SPEED OPTIMIZATION IN METHODS OF APPROXIMATE SOLVING OF INITIAL VALUES PROBLEM FOR THE SYSTEM OF QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER

In the theory of systems of quasilinear partial differential equations of the first order, the main questions are the solvability of initial values problem and justification of the approximate methods. This is due to problems in gas dynamics and hydromechanics. In the second half of the previous century attempts were made to construct a correct theory of solvability of problems or the systems of quasilinear partial differential equations of the first order. The necessity of the correct way of introductions the nothions of a generalized solution of initial values problems is connected with this. In this paper a class of systems of quasilinear partial differential equations of the first order is singled out for which the concept of a generalized solution is introduced. A method for constructing approximate methods for solving initial values problem is proposed. We obtained estimates of the convergence speed in approximate methods and proved the existence and uniqueness of the solution of initial values problem for systems of quasilinear partial differential equations of the first order of a certain form.