On the stability of a finite-difference scheme for nonlocal parabolic boundary-value problems. (English) Zbl 1206.65219
Summary: We deal with the stability analysis of difference schemes for a one-dimensional parabolic equation subject to integral conditions. It is based on the spectral structure of the transition matrix of the difference scheme. The stability domain is defined by using the hyperbola which is the locus of points where the transition matrix has trivial eigenvalues. The stability conditions obtained are much more general compared with those known in the literature. We analyze three separate cases of nonlocal integral conditions and solve an example illustrating the efficiency of the technique.
MSC:
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35K05 | Heat equation |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
nonlocal integral conditions; parabolic equations; finite-difference schemes; stability; numerical exampleReferences:
| [1] | K.E. Atkinson, Introduction to Numerical Analysis, John Wiley & Sons, New York, 1978. · Zbl 0402.65001 |
| [2] | B.I. Bandyrskii, I. Lazurchak, V.L. Makarov, and M. Sapagovas, Eigenvalue problem for the second order differential equation with nonlocal conditions, Nonlin. Anal. Model. Control, 11(1):13–32, 2006. · Zbl 1112.65071 |
| [3] | N. Borovykh, Stability in the numerical solution of the heat equation with nonlocal boundary conditions, Appl. Numer. Math., 42:17–27, 2002. · Zbl 1003.65102 · doi:10.1016/S0168-9274(01)00139-8 |
| [4] | B. Cahlon, D.M. Kulkarni, and P. Shi, Stepwise stability for the heat equation with a nonlocal constrain, SIAM J. Numer. Anal., 32(2):571–593, 1995. · Zbl 0831.65094 · doi:10.1137/0732025 |
| [5] | R. Čiegis, A. Štikonas, O. Štikonien\.e, and O. Suboč, A monotonic finite-diference scheme for a parabolic problem with nonlocal conditions, Differ. Equ., 38(7):1027–1037, 2002. · Zbl 1030.65092 · doi:10.1023/A:1021167932414 |
| [6] | R. Čiupaila, Ž. Jesevičiūt\.e, and M. Sapagovas, On the eigenvalue problem for one-dimensional differential operator with nonlocal integral condition, Nonlinear Anal. Model. Control, 9(2):109–116, 2004. · Zbl 1070.47036 |
| [7] | G. Ekolin, Finite difference methods for a nonlocal boundary value problem for heat equation, BIT, 31:245–261, 1991. · Zbl 0738.65074 · doi:10.1007/BF01931285 |
| [8] | G. Fairweather and J.C. Lopez-Marcos, Galerkin methods for a semilinear parabolic problem with nonlocal conditions, Adv. Comput. Math., 6:243–262, 1996. · Zbl 0868.65068 · doi:10.1007/BF02127706 |
| [9] | S.K. Godunov and V.S. Ryabenjkii, Difference Schemes Introduction to the Theory, Moscow, Nauka, 1977 (in Russian). |
| [10] | A.V. Gulin, N.I. Ionkin, and V.A. Morozova, Stability of a nonlocal two-dimensional finite-difference problem, Differ. Equ., 37(7):970–978, 2001. · Zbl 1004.65091 · doi:10.1023/A:1011961822115 |
| [11] | A.V. Gulin, N.I. Ionkin, and V.A. Morozova, Stability criterion of difference schemes for the heat conduction equation with nonlocal boundary conditions, Comput. Methods Appl. Math., 6(1):31–55, 2006. · Zbl 1101.65091 |
| [12] | A.V. Gulin, N.I. Ionkin, and V.A. Morozova, Study of the norm in stability problems for nonlocal difference schemes, Differ. Equ., 42(7):974–984, 2006. · Zbl 1126.65082 · doi:10.1134/S0012266106070068 |
| [13] | A.V. Gulin and V.A. Morozova, On the stability of nonlocal finite-difference boundary value problem, Differ. Equ., 39(7):962–967, 2003. · Zbl 1063.65090 · doi:10.1023/B:DIEQ.0000009192.30909.13 |
| [14] | Y. Liu, Numerical solution of the heat equation with nonlocal boundary conditions, J. Comput. Appl. Math., 110(1):115–127, 1999. · Zbl 0936.65096 · doi:10.1016/S0377-0427(99)00200-9 |
| [15] | S. Pečiulyt\.e, O. Štikonien\.e, and A. Štikonas, Sturm-Liouville problem for stationary differential operator with nonlocal integral boundary conditions, Nonlin. Anal. Model. Control, 11(1):47–78, 2006. |
| [16] | R.D. Richtmyer and K.W. Marton, Difference Methods for Initial-Value Problems, Second Edition, John Wiley & Sons, 1967. · Zbl 0155.47502 |
| [17] | A.A. Samarskii, The Theory of Difference Schemes, Moscow, Nauka, 1977 (in Russian); Marcel Dekker, Inc., New York and Basel, 2001. · Zbl 0368.65031 |
| [18] | A.A. Samarskii and A.V. Gulin, Numerical Methods, Moscow, Nauka, 1989 (in Russian). |
| [19] | M.P. Sapagovas, The eigenvalue of some problems with a nonlocal condition, Differ. Equ., 38(7):1020–1026, 2002. · Zbl 1072.65119 · doi:10.1023/A:1021115915575 |
| [20] | M.P. Sapagovas, On stability of finite-difference schemes for one-dimensional parabolic equations subject to integral condition, Zh. Obchysl. Prykl. Mat., 92:70–90, 2005. |
| [21] | M.P. Sapagovas and A.D. Štikonas, On the structure of the spectrum of a differential operator with a nonlocal condition, Differ. Equ., 41(7):1010–1018, 2005. · Zbl 1246.35139 · doi:10.1007/s10625-005-0242-y |
| [22] | A. Štikonas, The Sturm-Liouville problem with a nonlocal boundary condition, Lith. Math. J., 47(3):336–351, 2007. · Zbl 1266.34042 · doi:10.1007/s10986-007-0023-9 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
