Структура устойчивого многообразия полностью неявных схем

E.Yu. Vedernikova; A.A. Kornev

The structure of a stable manifold for fully implicit schemes

Authors

E.Yu. Vedernikova
A.A. Kornev

Keywords:

stationary distribution

product of matrices

integral equation

Jacobi equation

Abstract

An analog of the Hadamard-Perron theorem on the existence of a local stable manifold in a neighborhood of a fixed hyperbolic-type point for implicit mappings is proved. This result allows one to constructively study the structure of a manifold for a finite-difference approximation in time in the case of quasilinear parabolic-type equations and to prove that, in terms of the integral metric, the manifold of the nonlinear problem exists in an unbounded ellipsoid. Several theoretical estimates are given. A number of numerical results are discussed.

Downloads

PDF (Русский)

Published

2013-01-28

Issue

Vol. 14 (2013): Issue 1.

Section

Section 1. Numerical methods and applications

Author Biographies

E.Yu. Vedernikova

Lomonosov Moscow State University,
Faculty of Mechanics and Mathematics
• PhD Student

A.A. Kornev

Lomonosov Moscow State University,
Faculty of Mechanics and Mathematics
• Professor

References

Аносов Д.В. Геодезические потоки на замкнутых римановых многообразиях отрицательной кривизны // Тр. матем. ин-та им. В.,А. Стеклова. 1967. 90, № 5. 3-210.
Kostin I.N. Rate of attraction to a non-hyperbolic attractor // Asymptotic Analysis. 1998. 16, N 3/4. 203-222.
Ладыженская О.А., Солонников В.А. О принципе линеаризации и инвариантных многообразиях для задач магнитной гидродинамики // Зап. научн. семин. ЛОМИ. 1973. 38. 46-93.
Лебедев В.И. Функциональный анализ и вычислительная математика. М.: Физматлит, 2005.
Иванчиков А.А., Корнев А.А., Озерицкий А.В. О новом подходе к решению задач асимптотической стабилизации // Журн. вычислит. матем. и матем. физики. 2009. 49, № 12. 2167-2181.
Fursikov A.V. Local existence theorems with unbounded set of input data and unboundedness of stable invariant manifolds for 3D Navier-Stokes equations // Discrete and Continuous Dynamical Systems. Series S. 2010. 3, № 2. 269-290.
Fursikov A.V., Kornev A.A. Feedback stabilization for Navier-Stokes equations: theory and calculations // Mathematical Aspects of Fluid Mechanics. Lecture Notes Series. Cambridge: Cambridge University Press, 2012. 130-172.
Чижонков Е.В. Об операторах проектирования для численной стабилизации // Вычислительные методы и программирование. 2004. 5, № 1. 161-169.
Милютин С.В., Чижонков Е.В. О двух методах приближенного проектирования на устойчивое многообразие // Вычислительные методы и программирование. 2007. 8, № 1. 177-182.

How to cite

Vishnevsky D.M., Lisitsa V.N. and Reshetova G.V. Numerical simulation of seismic wave propagation in media with viscoelastic intrusions // Numerical Methods and Programming. 2013. 14, No 1. 155–165.

TEX CODE:

{\it Vishnevsky~D.M., Lisitsa~V.N.~and~Reshetova~G.V.}
Numerical simulation of seismic wave propagation in media with viscoelastic intrusions~//
Numerical Methods and Programming. 2013.
\textbf{14}, No~1. 155--165.

Vishnevsky D. , Lisitsa V.  and Reshetova G. ,  (2013) “Numerical simulation of seismic wave propagation in media with viscoelastic intrusions,” Numerical Methods and Programming, vol. 14, no. 1, pp. 155–165.

TEX CODE:

Vishnevsky~D.~, Lisitsa~V.~ and~Reshetova~G.~, (2013)
``Numerical simulation of seismic wave propagation in media with viscoelastic intrusions,''
{\it Numerical Methods and Programming}, vol.~14, no.~1, pp.~155--165.

D.  Vishnevsky, V.  Lisitsa and G.  Reshetova,  “Numerical simulation of seismic wave propagation in media with viscoelastic intrusions,” Numerical Methods and Programming 14, no. 1 (2013): 155–165

TEX CODE:

D.~ Vishnevsky, V.~ Lisitsa~and~G.~ Reshetova,
``Numerical simulation of seismic wave propagation in media with viscoelastic intrusions,''
{\it Numerical Methods and Programming}, no.~1 (2013): 155--165

Vishnevsky D. , Lisitsa V.  and Reshetova G.  Numerical simulation of seismic wave propagation in media with viscoelastic intrusions. Numerical Methods and Programming. 2013;14(1):155–165.(In Russ.).

TEX CODE:

Vishnevsky~D.~, Lisitsa~V.~ and~Reshetova~G.~
Numerical simulation of seismic wave propagation in media with viscoelastic intrusions.
Numerical Methods and Programming. 2013;14(1):155--165.(In Russ.).