Библиографическое описание:A Conservative Semi-Lagrangian Method for a Continuity Equation with Various Time Steps in Different Parts of the Computational Domain : научное издание / A. V. Vyatkin, A. D. Maltsev. - Текст : непосредственный // Theoretical Foundations of Chemical Engineering. - 2024. - Т. 58, № 4. - P. 1478-1487. - This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation in the framework of the establishment and development of regional Centers for Mathematics Research and Education (agreement no. 075-02-2025-1606). - ISSN 0040-5795. - ISSN 1608-3431, DOI 10.1134/S004057952560041X.
Аннотация:The flow of gas flowing out of the rocket nozzle is described by a system of Navier–Stokes equations. The solution of these equations is typically achieved through the implementation of numerical methods. However, contemporary numerical methods are not yet capable of simulating gas flow in its entirety. This limitation stems from the complicated physical processes that occur during gas flow behind a nozzle and the limitations of computational capabilities. There are at least two approaches to address this complexity: the development of numerical methods that are characterized by less computational complexity and the enhancement of computing system performance. In this context, many research efforts are focused on the development of numerical methods that demand reduced computational resources in comparison to existing methods. Concurrently, these methods are designed to facilitate the identification of solutions without compromising accuracy. In 1959, Aksel C. Wiin-Nielsen presented a novel numerical method, designated as the trajectory method, which was employed to address the challenge of weather forecasting. Subsequently, in 1966, K.M. Magomedov employed a comparable approach to develop a numerical algorithm aimed at identifying a numerical solution of a problem pertaining to the simulation of three-dimensional gas flow. In 1982, O. Pironneau developed this method to construct a numerical solution of the two-dimensional Navier–Stokes equations. Presently, these methods are undergoing extensive development and they have acquired a common appellation, namely semi-Lagrange or Eulerian–Lagrange methods. In order to employ this approach, first, it is necessary to decompose the Navier–Stokes equations into three components. The first component is the convective part, representing the hyperbolic part of the equations. The second component is the elliptic part. The third component is the part of known quantities, which is written on the right-hand side of the equations. When solving the Navier–Stokes equations, semi-Lagrange methods are employed to approximate the convective part. This part includes all the terms of the continuity equation. In order to develop a numerical method, it is necessary to seek a solution to the continuity equation. Conservative versions of semi-Lagrange methods are founded on the Gauss–Ostrogradsky theorem (in the foreign literature known as the divergence theorem). This approach enables the fulfillment of the conservation law for the numerical solution of the problem in norm of the <i>L</i><sub>1</sub> space. The objective of the present study is to develop a numerical algorithm that uses distinct time steps in different parts of the computational domain. The implementation of this approach is expected to yield three significant outcomes: first, the convergence of the numerical solution to the exact solution will be achieved; second, the computational complexity of the method will be reduced; and third, the conservation law will be fulfilled without the use of correction (weight) coefficients. The construction of the algorithm necessitated the division of the one-dimensional computational domain into two distinct subdomains, with the implementation of different time steps in each. The primary challenge in developing the algorithm pertains to the identification of a numerical solution at the boundaries of the two subdomains. The one-dimensional (in space) continuity equation serves as a diagnostic criterion for the development of algorithms. It demonstrates the feasibility of formulating algorithms that exhibit the aforementioned properties. Subsequent studies will extend this algorithm to address two-dimensional and three-dimensional problems. In the context of simulating real physical problems, this approach will facilitate a more precise simulation of gas flow, thereby circumventing the artificial blurring that is associated with the computation of integrals at the lower temporal layer in certain regions of the computational domain with a high level of change in the numerical solution.
