UNDERCOMPRESSIVE DISCONTINUITIES OF A HYPERBOLIC SYSTEM OF CONSERVATION LAW EQUATIONS: FINITE-DIFFERENCE SCHEMES

Capa

Citar

Texto integral

Acesso aberto Acesso aberto
Acesso é fechado Acesso está concedido
Acesso é fechado Acesso é pago ou somente para assinantes

Resumo

A class of finite-difference schemes with well-controlled dissipation is used to solve equations describing long longitudinal–torsional waves in elastic rods. The governing system of equations is a hyperbolic system of conservation laws whose solutions may include undercompressive discontinuities (nonclassical discontinuities). It is well known that such solutions depend on the choice of a regularizing dissipative operator distinguishing a unique solution of the problem. In the scheme with well-controlled dissipation, the dissipative operator defined by its first differential approximation coincides up to small higher order terms with the operator used to define the solution in the continual formulation. The class of schemes under discussion has been poorly studied to date. Numerical experiments are presented that demonstrate the efficiency of this approach.

Sobre autores

R. Polekhina

Steklov Mathematical Institute, Russian Academy of Sciences; Federal Research Center Keldysh Institute of Applied Mathematics, Russian Academy of Sciences

Moscow, Russia; Moscow, Russia

A. Chugainova

Federal Research Center Keldysh Institute of Applied Mathematics, Russian Academy of Sciences

Email: anna_ch@mi-ras.ru
Moscow, Russia

Bibliografia

  1. Куликовский А.Г., Чугайнова А.П. Продольно-крутильные волны в нелинейно-упругих стержнях // Тр. МИАН. 2023. Т. 322. С. 157–166.
  2. Dafermos C.M. Hyperbolic Conservation Laws in Continuum Physics. Berlin Heidelberg: Springer-Verlag, 2010.
  3. Lax P.D. Hyperbolic systems of conservation laws// Comm. Pure Appl. Math. 1957. V. 10. P. 537–566.
  4. Куликовский А.Г., Свешникова Е.И. Нелинейные волны в упругих средах. М.: Московский лицей, 1998. C. 412.
  5. Гельфанд И.М. Некоторые задачи теории квазилинейных уравнений // Успехи матем. наук. 1959. Т. 14.№2. С. 87–158.
  6. Beljadid A., LeFloch P.G., Mishra S., Pares C. Schemes with well-controlled dissipation. Hyperbolic systems in nonconservative form // Comm. Comput. Phys. 2017. V. 21.№4. P. 913–946.
  7. Hayes B.T., Lefloch P.G. Nonclassical shocks and kinetic relations: Finite difference schemes // SIAM J. Numeric. Analys. 1998. V. 35.№6. P. 2169–2194.
  8. Chalons C., LeFloch P.G. Computing undercompressive waves with the randomchoice scheme // Nonclassical shock waves. Interfaces and Free Boundaries. 2003. V. 5.№2. P. 129–158.
  9. Shargatov V.A., Chugainova A.P., Kolomiytsev G.V., Nasyrov I.I., Tomasheva A.M., Gorkunov S.V., Kozhurina P.I. Why stable finite-difference schemes can converge to different solutions: analysis for the generalized hopf equation // Computation. 2024. V. 12.№4. P. 76.
  10. Куликовский А.Г., Чугайнова А.П. О структурах неклассических разрывов в решениях гиперболических систем уравнений // УМН. 2022. Т. 77.№1. С. 55–90.
  11. Ахиезер А.И., Любарский Г.Я., Половин Р.В. Об устойчивости ударных волн в магнитной гидродинамике // ЖЭТФ. 1959. Т. 35.№3. С. 731–737.
  12. Полехина Р.Р., Савенков Е.Б. Применение схемы с хорошо контролируемой дисспацией для решения уравнений модели Капилы // Дифференц. ур-ния. 2024. Т. 60.№7. С. 937–953.
  13. Cockburn B., Chi-Wang Shu. The Runge-Kutta local projection-discontinuous-Galerkin finite element method for scalar conservation laws // ESAIM: Math. Model. and Numeric. Analys. 1991. Т. 25.№3. С. 337–361.

Arquivos suplementares

Arquivos suplementares
Ação
1. JATS XML
Baixar

Declaração de direitos autorais © Russian Academy of Sciences, 2025