ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Hybrid Scheme for the Numerical Solution of a Nonlinear Euler's Equation


Yuri Alexandrovich Kostikov

Moscow Aviation Institute (National Research University),
Head of the Department 812,
Candidate of Physical and Mathematical Sciences

Alexandr Mihailovich Romanenkov

Moscow Aviation Institute (National Research University),
Associate Professor of the Department 812,
Candidate of Technical Sciences


In this paper, we consider a hybrid difference scheme for constructing an approximate solution of a nonlinear system of differential equations with a discontinuous initial condition. As a model problem, the system of Euler's ordinary differential equations, which is a conservative on real line, is used. To find an approximate solution, we use a combination from the Lax-Friedrichs and Lax-Wendroff methods and switching between them. The switching is necessary, since the higher order accuracy method (the Lax-Wendroff method) leads to the appearance of the Gibbs effect, which is an unwanted non-physical artifact in the numerical solution. Therefore, in the areas where this effect occurs, it is necessary to perform the switching to another method. The authors propose a criterion for the switching. The script describing the hybrid scheme was implemented in Matlab programming system. The proposed algorithm operates with vector quantities and does not use specific expressions for the coordinates of the vectors under consideration. As a sequence, the algorithm is independent of a given system and may be applied to other problems of a similar type without significant changes. The ideas used to obtain these methods are demonstrated and the scope of their applicability is indicated.



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