Stability criterion for a family of nonlocal difference schemes

We consider finite-difference schemes for the heat equation with nonlocal boundary conditions that contain a real parameter γ . A stability criterion for finite-difference schemes with respect to the initial data was earlier obtained for |γ| ≤ 1. In the present paper, we consider the case in which γ...

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Bibliographic Details
Published in:Differential equations 2010-07, Vol.46 (7), p.973-990
Main Authors: Gulin, A. V., Morozova, V. A., Udovichenko, N. S.
Format: Article
Language:English
Subjects:
Boundary conditions
Criteria
Difference and Functional Equations
Differential equations
Eigenvalues
Eigenvectors
Estimates
Finite difference method
Heat equations
Inequality
Mathematical analysis
Mathematics
Mathematics and Statistics
Numerical Methods
Ordinary Differential Equations
Partial Differential Equations
Stability
Studies
Validity
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Summary:We consider finite-difference schemes for the heat equation with nonlocal boundary conditions that contain a real parameter γ . A stability criterion for finite-difference schemes with respect to the initial data was earlier obtained for |γ| ≤ 1. In the present paper, we consider the case in which γ ∈ (−cosh π ,−1) and the original differential problem is stable, while the stability conditions for the finite-difference schemes substantially depend on γ . We obtain estimates for the energy norm of the solution of the finite-difference problem via the same norm of the initial data and prove the equivalence of the energy norm and the grid L 2 -norm.
ISSN:0012-2661
1608-3083
DOI:10.1134/S0012266110070050