Исследование задачи Коши для одномерной системы уравнений типа Бюргерса методом слабой аппроксимации

Investigation of the Cauchy problem for a one-dimensional system of the Burgers type equations by the weak approximation method

This paper is concerned with obtaining a system of Burgcrs-typc equations in one limit ease from the system of equations of a two-fluid medium. The system in question differs from the system of equations of a two-fluid medium by the absence of pressure and the incompressibility conditions. For this reason, the problems associated with the Burgcrs-typc system arc called a system without pressure for a two-fluid medium. In the ease when the dissipative function docs not depend on the viscosity coefficients of the medium, we will call the Burgcrs-typc non-viscous system or the Hopf-tvpe system. In the one-dimensional ease, we call it also the Ricmann-typc system of equations, which is a simple quasilincar system of equations. The system of equations of a two-fluid medium and the system of equations of the Burgers type have much in common. For example, the quadratic nonlinear terms - due to the phase velocities - respond to advcctivc terms, corresponding to the dependence of sound on the amplitude of sound waves and linear terms caused by viscosities and the friction coefficient, which arc responsible for the attenuation of the sound waves, whcrcinpropcrtics of solutions arc completely different. With a Burgcrs-typc equation system with disappearing viscosity coefficients and the friction coefficient, both strong (shock waves) and weak discontinuities arc formed, whilethe solutions of a two-fluid system, do not possess such features. However, the scope of applicability of the system proposed is not limited to the examples given, such systems arise in many problems, which is what determines its importance. A study of the system of the Burgcrs-typc equations arising in the nonlinear acoustics is presented. The proposed mathematical model is due to the combination of a conservative nonlinear system with a dissipative term; here, the dissipation is due to both the viscosity of subsystems and the inter-component friction coefficient (analogous to the Darev coefficient), for which equivalent diffusion representations can be effectively used. The Cauehv problem for a one-dimensional system of the Burgcrs-typc equations arising in a two-fluid medium is considered. The system under study is qucusilincar, and analytical research methods do not allow one to obtain solutions to the Cauehv problem. One of the main methods for carrying out theoretical studies of the mathematical models of a two-fluid medium and applying them to solving important practical problems is numerical methods. Consequently, when studying efficient numerical algorithms, one of the main methods for their construction, is the method of weak approximation of differential equations. The weak approximation method, steering a middle course between the differential problem and the corresponding difference model can be used in two versions: as one of the methods for studying the correctness of the problem; as a method for constructing and rigorous mathematical analysis of the corresponding difference splitting schemes. The latter from this point of view arc simple difference approximations of differential problems in fractional steps.