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Computer Research and Modeling, 2016, Volume 8, Issue 5, Pages 721–753
DOI: https://doi.org/10.20537/2076-7633-2016-8-5-721-753
(Mi crm24)
 

This article is cited in 5 scientific papers (total in 5 papers)

NUMERICAL METHODS AND THE BASIS FOR THEIR APPLICATION

On the construction and properties of WENO-schemes order five, seven, nine, eleven and thirteen. Part 1. Construction and stability

N. M. Evstigneev

Federal Research Center ‘Informatics and Control’ of Russian Academy of Science, 9 pr. 60-letiya Oktyabrya, Moscow, 117312, Russia
Full-text PDF (748 kB) Citations (5)
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Abstract: Currently, different nonlinear numerical schemes of the spatial approximation are used in numerical simulation of boundary value problems for hyperbolic systems of partial differential equations (e. g. gas dynamics equations, MHD, deformable rigid body, etc.). This is due to the need to improve the order of accuracy and perform simulation of discontinuous solutions that are often occurring in such systems. The need for non-linear schemes is followed from the barrier theorem of S. K. Godunov that states the impossibility of constructing a linear scheme for monotone approximation of such equations with approximation order two or greater. One of the most accurate non-linear type schemes are ENO (essentially non oscillating) and their modifications, including WENO (weighted, essentially non oscillating) schemes. The last received the most widespread, since the same stencil width has a higher order of approximation than the ENO scheme. The benefit of ENO and WENO schemes is the ability to maintain a high-order approximation to the areas of non-monotonic solutions. The main difficulty of the analysis of such schemes comes from the fact that they themselves are nonlinear and are used to approximate the nonlinear equations. In particular, the linear stability condition was obtained earlier only for WENO5 scheme (fifth-order approximation on smooth solutions) and it isa numerical one. In this paper we consider the problem of construction and stability for WENO5,WENO7, WENO9, WENO11, and WENO13 finite volume schemes for the Hopf equation. In the first part of this article we discuss WENO methods in general, and give the explicit expressions for the coefficients of the polynomial weights and linear combinations required to build these schemes. We prove a series of assertions that can make conclusions about the order of approximation depending on the type of local solutions. Stability analysis is carried out on the basis of the principle of frozen coefficients. The cases of a smooth and discontinuous behavior of solutions in the field of linearization with frozen coefficients on the faces of the final volume and spectra of the schemes are analyzed for these cases. We prove the linear stability conditions for a variety of Runge-Kutta methods applied toWENO schemes. As a result, our research provides guidance on choosing the best possible stability parameter, which has the smallest effect on the nonlinear properties of the schemes. The convergence of the schemes is followed from the analysis.
Keywords: WENO schemes, nonlinear schemes, stability of numerical schemes, hyperbolic systems of differential equations, Hopf equation.
Funding agency Grant number
Russian Foundation for Basic Research 14-07-00123
This work is supported by Russian foundation for basic research (grant No. 14-07-00123).
Received: 01.07.2016
Revised: 10.08.2016
Accepted: 01.09.2016
Document Type: Article
UDC: 519.633.2:519.633.6
Language: Russian
Citation: N. M. Evstigneev, “On the construction and properties of WENO-schemes order five, seven, nine, eleven and thirteen. Part 1. Construction and stability”, Computer Research and Modeling, 8:5 (2016), 721–753
Citation in format AMSBIB
\Bibitem{Evs16}
\by N.~M.~Evstigneev
\paper On the construction and properties of WENO-schemes order five, seven, nine, eleven and thirteen. Part 1. Construction and stability
\jour Computer Research and Modeling
\yr 2016
\vol 8
\issue 5
\pages 721--753
\mathnet{http://mi.mathnet.ru/crm24}
\crossref{https://doi.org/10.20537/2076-7633-2016-8-5-721-753}
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  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Computer Research and Modeling
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