Grid-characteristic methods. 55 years of developing and solving complex dynamic problems

Abstract

The development of a computational method is not a simple matter and boils down to replacing the differential operator with a difference one. To construct it, it is necessary to correctly set a mathematical problem that is adequate to the physical one under consideration. In addition, the algorithm must meet some other requirements. Therefore, to create a numerical algorithm requires not only ingenuity and imagination, but also a deep understanding of the reasons why these requirements are caused.Systems of partial differential equations of hyperbolic type are used to describe the unsteady behavior of continuous media. To solve these problems, characteristic methods were developed in such a way as to take into account the corresponding properties of hyperbolic equations and to be able to build a so-called characteristic irregular grid adapting to the solution of the problem. Methods of end-to-end counting have been developed that take into account the properties of systems of hyperbolic equations — inverse methods of characteristics or grid-characteristic methods.In grid-characteristic methods, a regular computational grid is used, not a solvable initial system is approximated on it, but compatibility conditions along characteristic lines with interpolation of the desired functions at the points of intersection of characteristics with a coordinate line on which the data is already known. The obtained characteristic form of the gas dynamics equations makes it possible to understand how to set the boundary conditions correctly.The construction of a numerical method is not a simple matter and is not reduced to the formal replacement of derivatives by approximating their difference relations (for example, using finite differences). When developing the method, it is necessary to take into account the physical side of the problem being solved. At the same time, the method must meet certain requirements, the understanding of which is necessary during its development.