Abstract:

The contribution of Sergei Konstantinovich Godunov to the development of numerical methods is difficult to overestimate. One of the authors, Abuziarov M.H., participated in the work of the international symposium "The Godunov Method in Gas Dynamics"at the University of Michigan (An Arbor USA) in May 1997, organized by NASA USA in honor of Godunov. At this symposium, Sergei Konstantinovich was presented as the most outstanding applied mathematician of the 20th century, and a special tour of NASA laboratories in the USA was organized for him. The scheme originally proposed by Godunov for solving the equations of gas dynamics has also found wide application for solving elastic-plastic problems of continuum mechanics. This finite-volume flow scheme of the predictor-corrector type is based on solving the Riemann problem at the predictor step at the cell boundaries under the assumption of a piece-wise constant distribution of parameters in these cells. The main advantages of the scheme, such as monotonicity, the possibility of identifying impact and contact discontinuities, the use of Lagrangian and Eulerian approaches, and the simplicity of implementing boundary conditions are the consequences of solving this problem. At the same time, solving the Riemann problem using a piecewise constant distribution of parameters in cells is the source of the main disadvantage of the scheme - significant scheme viscosity, while the corrector step of the scheme is sufficient for the second-order of approximation. In this paper, the authors proceed from the linear distribution of flow parameters between the centers of neighboring cells, and from the analysis of the differential approximation of the scheme, by the appropriate choice of these parameters for solving the Riemann problem, they increase the order of approximation of the scheme to the second in the region of smooth solutions while maintaining monotonicity on discontinuous ones. The appropriate choice of these parameters increases the order of approximation of the scheme to the second in space and time on a moving nonuniform grid for both the Lagrangian and Eulerian cases on a compact stencil, for both gas-dynamic and elastic-plastic flows. Monotonicity near discontinuous solutions is ensured by switching to the predictor step of the first-order accuracy scheme. The coordinates of the interpolation points of the flow parameters have an obvious physical meaning - these are the boundaries of the dependence region of the Riemann problem solution for the moving coordinate of the face center at half the time integration step. This scheme modification is practically the same for both gas-dynamic and elastic-plastic flows. In contrast to gas-dynamic problems, for elastic-plastic flows, the corresponding Riemann invariants are interpolated at the boundaries of the dependence regions. The quality of the scheme is illustrated by test problems.