ISSN 2079-6900 (Print) 
ISSN 2587-7496 (Online)

Middle Volga Mathematical Society Journal

DOI 10.15507/2079-6900.24.202203.304-316

Original article

ISSN 2079-6900 (Print)

ISSN 2587-7496 (Online)

MSC2020 65L11

Numerical solution of a singularly perturbed boundary value problem of supersonic flow transformed to the modified best argument

E. D. Tsapko

Moscow Aviation Institute (National Research University) (Moscow, Russian Federation)

Abstract. When solving problems of aerodynamics, researchers often need to numerically solve singularly perturbed boundary value problems. In some cases, the problem can be reduced to solving a boundary value problem for an ordinary differential equation. Then it is possible to apply various numerical methods such as the grid method, the shooting method, as well as a number of projection methods, which, in turn, can form the basis of the finite element method. The grid method requires solving a system of algebraic equations, that are often nonlinear, which leads to an increase in the calculation time and to the difficulties in convergence of the approximate solution. According to the shooting method, the solution of boundary value problem is reduced to solving a certain set of Cauchy problems. When solving stiff Cauchy problems, implicit schemes are used as a rule, but in this case the same difficulties arise as for the grid method. The transformation of the problem to the best argument $\lambda$, calculated tangentially along the integral curve, makes it possible to increase the efficiency of explicit numerical methods. However, in cases where the growth rate of integral curves is close to exponential, the transformation to the best argument is not efficient enough. Then the best argument is modified in such a way as to smooth out this flaw. This paper investigates the application of modified best argument to the solution of the boundary value problem of an aerodynamic flow movement in case when the gas is injected at supersonic speed into a channel of variable cross-section.

Key Words: singularly perturbed problem, ordinary differential equation, boundary value problem, method of solution continuation, best argument, modified best argument, supersonic flow

For citation: E. D. Tsapko. Numerical solution of a singularly perturbed boundary value problem of supersonic flow transformed to the modified best argument. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 24:3(2022), 304–316. DOI:

Submitted: 03.07.2022; Revised: 12.08.2022; Accepted: 24.08.2022

Information about the author:

Ekaterina D. Tsapko, Postgraduate Student, Department of Mechatronics and Theoretical mechanics, Moscow Aviation Institute (National Research University) (4 Volokolamskoe Av., Moscow 125993, Russia), ORCID:,

The author have read and approved the final manuscript.

Conflict of interest: The author declare no conflict of interest.

Creative Commons Attribution 4.0 International License This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License.