The integrals of the equations of motion

Aim: analytical development of the method of total differentials, created earlier for the numerical solution of hyperbolic systems of partial differential equations of the first order and the construction of a complete system of integrals of the equations of motion.

Methodology. The method consists in the fact that the systems of first-order partial differential equations are reduced to relationships between total differentials of gas-dynamic variables along different directions using linear transformations. The integration procedure is applied to the resulting transformed systems and the analysis and synthesis of the obtained results is performed.

Results. A complete system of integrals is obtained for the Euler system of gas dynamics equations. Based on their analysis, a new result was obtained on the influence of plane waves and on the hypersurfaces formed by the intersection points of these waves, covering the characteristics. A new idea is obtained about the causes of numerical instability of solutions of general gas dynamics equations by the method of characteristics. A new understanding of the causes of numerical instability of solutions to general gas dynamics equations is obtained using the method of characteristics. Based on the application of the method to the Laplace equation, the Cauchy integral theorem is obtained.

Research implications. Complete integrals of Euler equations give an idea of how its solution is generally structured. The expressions obtained in the work can also be used to construct difference schemes of high resolution and high order of approximation. Such schemes are most suitable for describing transition regimes of flow of viscous heat-conducting gas.

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