A Compact Difference Scheme for Multi-point Boundary Value Problems of Heat Equations

doi:10.1007/s42967-019-00025-w

Communications on Applied Mathematics and Computation ›› 2019, Vol. 1 ›› Issue (4): 545-563.doi: 10.1007/s42967-019-00025-w

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A Compact Difference Scheme for Multi-point Boundary Value Problems of Heat Equations

Xuping Wang, Zhizhong Sun

School of Mathematics, Southeast University, Nanjing 210096, China

收稿日期:2018-08-15 修回日期:2018-12-14 出版日期:2019-12-30 发布日期:2019-10-16
通讯作者: Zhizhong Sun, Xuping Wang E-mail:zzsun@seu.edu.cn;seuMathWxp@139.com
基金资助:
The research is supported by the National Natural Science Foundation of China (No. 11671081) and the Fundamental Research Funds for the Central Universities (No. 242017K41044).

A Compact Difference Scheme for Multi-point Boundary Value Problems of Heat Equations

Xuping Wang, Zhizhong Sun

School of Mathematics, Southeast University, Nanjing 210096, China

Received:2018-08-15 Revised:2018-12-14 Online:2019-12-30 Published:2019-10-16
Contact: Zhizhong Sun, Xuping Wang E-mail:zzsun@seu.edu.cn;seuMathWxp@139.com
Supported by:
The research is supported by the National Natural Science Foundation of China (No. 11671081) and the Fundamental Research Funds for the Central Universities (No. 242017K41044).

摘要/Abstract

摘要： In this paper, a compact difference scheme is established for the heat equations with multi-point boundary value conditions. The truncation error of the difference scheme is O(τ² + h⁴), where τ and h are the temporal step size and the spatial step size. A prior estimate of the difference solution in a weighted norm is obtained. The unique solvability, stability and convergence of the difference scheme are proved by the energy method. The theoretical statements for the solution of the difference scheme are supported by numerical examples.

关键词: Heat equation, Multi-point boundary value condition, Compact difference scheme, Energy method

Abstract: In this paper, a compact difference scheme is established for the heat equations with multi-point boundary value conditions. The truncation error of the difference scheme is O(τ² + h⁴), where τ and h are the temporal step size and the spatial step size. A prior estimate of the difference solution in a weighted norm is obtained. The unique solvability, stability and convergence of the difference scheme are proved by the energy method. The theoretical statements for the solution of the difference scheme are supported by numerical examples.

Key words: Heat equation, Multi-point boundary value condition, Compact difference scheme, Energy method

中图分类号:

Xuping Wang, Zhizhong Sun. A Compact Difference Scheme for Multi-point Boundary Value Problems of Heat Equations[J]. Communications on Applied Mathematics and Computation, 2019, 1(4): 545-563.

参考文献

1. Alikhanov, A.A.:On the stability and convergence of nonlocal diference schemes. Difer. Equ. 46(7), 949-961 (2010)
2. Ashyralyev, A., Aggez, N.:A Note on the diference schemes of the nonlocal boundary value problems for hyperbolic equations. Numerical Functional Analysis and Optimization 25(5-6), 439-462 (2004)
3. Ashyralyev, A., Gercek, O.:Nonlocal boundary value problems for elliptic-parabolic diferential and diference equations. Discrete Dyn. Nat. Soc. 4, 138-144 (2008)
4. Ashyralyev, A., Gercek, O.:Finite diference method for multipoint nonlocal elliptic-parabolic problems. Comput. Math. Appl. 60(7), 2043-2052 (2010)
5. Ashyralyev, A., Yurtsever, A.:On a nonlocal boundary value problem for semilinear hyperbolic-parabolic equations. Nonlinear Anal. Theory Methods Appl. 47(5), 3585-3592 (2001)
6. Gao, G.H., Sun, Z.Z.:Compact diference schemes for heat equation with Neumann boundary conditions (Ⅱ). Numer. Methods Partial Difer. Equ. 29(5), 1459-1486 (2013)
7. Gordeziani, D., Avalishvili, G.:Investigation of the nonlocal initial boundary value problems for some hyperbolic equations. Hiroshima Math. J. 31(3), 345-366 (2001)
8. Gulin, A.V., Morozova, V.A.:On a family of nonlocal diference schemes. Difer. Equ. 45(7), 1020-1033 (2009)
9. Gulin, A.V., Ionkin, N.I., Morozova, V.A.:Stability of a nonlocal two-dimensional fnite-diference problem. Difer. Equ. 37(7), 970-978 (2001)
10. Gushchin, A.K., Mikhailov, V.P.:On solvability of nonlocal problems for a second-order elliptic equation. Russ. Acad. Sci. Sb. Math. 81(1), 101-136 (1995)
11. Martin-Vaquero, J., Vigo-Aguiar, J.:A note on efcient techniques for the second-order parabolic equation subject to non-local conditions. Appl. Numer. Math. 59(6), 1258-1264 (2009)
12. Martin-Vaquero, J., Vigo-Aguiar, J.:On the numerical solution of the heat conduction equations subject to nonlocal conditions. Appl. Numer. Math. 59(10), 2507-2514 (2009)
13. Sun, Z.Z.:A high-order diference scheme for a nonlocal boundary-value problem for the heat equation. Comput. Methods Appl. Math. 1(4), 398-414 (2001)
14. Sun, Z.Z.:Compact diference schemes for heat equation with Neumann boundary conditions. Numer. Methods Partial Difer. Equ. 29, 1459-1486 (2013)
15. Wang, Y.:Solutions to nonlinear elliptic equations with a nonlocal boundary condition. Electron. J. Difer. Equ. 05, 227-262 (2002)
16. Yildirim, O., Uzun, M.:On the numerical solutions of high order stable diference schemes for the hyperbolic multipoint nonlocal boundary value problems. Appl. Math. Comput. 254, 210-218 (2015)
17. Zikirov, O.S.:On boundary-value problem for hyperbolic-type equation of the third order. Lith. Math. J. 47(4), 484-495 (2007)

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A Compact Difference Scheme for Multi-point Boundary Value Problems of Heat Equations

A Compact Difference Scheme for Multi-point Boundary Value Problems of Heat Equations

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