Hybrid Scheme for the Numerical Solution of a Nonlinear Euler's Equation
Author(s):
Yuri Alexandrovich Kostikov
Moscow Aviation Institute (National Research University),
Head of the Department 812,
Candidate of Physical and Mathematical Sciences
jkostikov@mail.ru
Alexandr Mihailovich Romanenkov
Moscow Aviation Institute (National Research University),
Associate Professor of the Department 812,
Candidate of Technical Sciences
romanaleks@gmail.com
Abstract:
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.
Keywords
- Euler's equation
- hybrid scheme
- Riemann problem
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