In this work, we initially construct an implicit Euler difference scheme for a two-dimensional heat problem, incorporating both local and nonlocal boundary conditions. Subsequently, we harness the power of the discrete Fourier transform and develop an innovative transformation technique to rigorously demonstrate that our scheme attains the asymptotic optimal error estimate in the maximum norm. Furthermore, we derive a series of approximation formulas for the partial derivatives of the solution along the two spatial dimensions, meticulously proving that each of these formulations possesses superconvergence properties. Lastly, to validate our theoretical findings, we present two comprehensive numerical experiments, showcasing the efficiency and accuracy of our approach.
Citation: Liping Zhou, Yumei Yan, Ying Liu. Error estimate and superconvergence of a high-accuracy difference scheme for 2D heat equation with nonlocal boundary conditions[J]. AIMS Mathematics, 2024, 9(10): 27848-27870. doi: 10.3934/math.20241352
In this work, we initially construct an implicit Euler difference scheme for a two-dimensional heat problem, incorporating both local and nonlocal boundary conditions. Subsequently, we harness the power of the discrete Fourier transform and develop an innovative transformation technique to rigorously demonstrate that our scheme attains the asymptotic optimal error estimate in the maximum norm. Furthermore, we derive a series of approximation formulas for the partial derivatives of the solution along the two spatial dimensions, meticulously proving that each of these formulations possesses superconvergence properties. Lastly, to validate our theoretical findings, we present two comprehensive numerical experiments, showcasing the efficiency and accuracy of our approach.
| [1] |
J. Martín-Vaquero, J. Vigo-Aguiar, On the numerical solution of the heat conduction equations subject to nonlocal conditions, Appl. Numer. Math., 59 (2009), 2507–2514. https://doi.org/10.1016/j.apnum.2009.05.007 doi: 10.1016/j.apnum.2009.05.007
|
| [2] |
M. Sapagovas, Z. Joksiene, On the stability of finite-difference schemes for parabolic equations subject to integral conditions with applications to thermoelasticity, Comput. Meth. Appl. Mat., 8 (2008), 362–373. https://doi.org/10.2478/cmam-2008-0026 doi: 10.2478/cmam-2008-0026
|
| [3] |
F. Ivanauskas, V. Laurinavi${\check{c}}$ius, M. Sapagovas, Anatolij Ne${\check{c}}$iporenko, Reaction–diffusion equation with nonlocal boundary condition subject to pid-controlled bioreactor, Nolinear Anal.-Model., 22 (2017), 261–272. https://doi.org/10.15388/NA.2017.2.8 doi: 10.15388/NA.2017.2.8
|
| [4] |
Y. S. Choi, K. Chan, A parabolic equation with nonlocal boundary conditions arising from electrochemistry, Nonlinear Anal., 18 (1992), 317–331. https://doi.org/10.1016/0362-546X(92)90148-8 doi: 10.1016/0362-546X(92)90148-8
|
| [5] |
J. Furter, M. Grinfeld, Local vs. non-local interactions in population dynamics, J. Math. Biology, 27 (1989), 65–80. https://doi.org/10.1007/BF00276081 doi: 10.1007/BF00276081
|
| [6] |
T. Li, A class of non-local boundary value problems for partial differential equations and its applications in numerical analysis, J. Comput. Appl. Math., 28 (1989), 49–62. https://doi.org/10.1016/0377-0427(89)90320-8 doi: 10.1016/0377-0427(89)90320-8
|
| [7] |
Y. Lin, Analytical and numerical solutions f or a class of nonlocal nonlinear asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions, J. Comput. Appl. Math., 88 (1998), 225–238. https://doi.org/10.1016/s0377-0427(97)00215-x doi: 10.1016/s0377-0427(97)00215-x
|
| [8] |
C. V. Pao, Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions, J. Comput. Appl Math., 88 (1998), 225–238. https://doi.org/10.1016/S0377-0427(97)00215-X doi: 10.1016/S0377-0427(97)00215-X
|
| [9] |
G. Avalishvili, M. Avalishvili, B. Miara, Nonclassical problems with nonlocal initial conditions for second-order evolution equations, Asymptot. Anal., 76 (2012), 171–192. https://doi.org/10.3233/asy-2011-1065 doi: 10.3233/asy-2011-1065
|
| [10] |
J. Martín-Vaquero, A. Queiruga-Dios, A. H. Encinas, Numerical algorithms for diffusion-reaction problems with non-classical conditions, Appl. Math. Comput., 218 (2012), 5487–5495. https://doi.org/10.1016/j.amc.2011.11.037 doi: 10.1016/j.amc.2011.11.037
|
| [11] |
J. Martín-Vaquero, S. Sajavičius, The two-level finite difference schemes for the heat equation with nonlocal initial condition, Appl. Math. Comput., 342 (2019), 166–177. https://doi.org/10.1016/j.amc.2018.09.025 doi: 10.1016/j.amc.2018.09.025
|
| [12] |
Y. Lin, Finite difference solutions for parabolic equations with the time weighting initial conditions, Appl. Math. Comput., 65 (1994), 49–61. https://doi.org/10.1016/0096-3003(94)90165-1 doi: 10.1016/0096-3003(94)90165-1
|
| [13] |
R. Čiupaila, M. Sapagovas, K. Pupalaigė, M-matrices and convergence of finite difference scheme for parabolic equation with integral boundary condition, Math. Model. Anal., 25 (2020), 167–183. https://doi.org/10.3846/mma.2020.8023 doi: 10.3846/mma.2020.8023
|
| [14] |
D. E. Jackson, Error estimates for the semidiscrete finite element approximation of linear nonlocal parabolic equations, J. Appl. Math. Stoch. Anal., 5 (1992), 19–27. https://doi.org/10.1155/s1048953392000029 doi: 10.1155/s1048953392000029
|
| [15] |
D. E. Jackson, Iterative finite element approximations of solutions to parabolic equations with nonlocal conditions, Nonlinear Anal.-Theor., 50 (2002), 433–454. https://doi.org/10.1007/bf00276081 doi: 10.1007/bf00276081
|
| [16] |
L. Bougoffa, R. C. Rach, Solving nonlocal initial-boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations by the adomian decomposition method, Appl. Math. Comput., 225 (2013), 50–61. https://doi.org/10.1016/j.amc.2013.09.011 doi: 10.1016/j.amc.2013.09.011
|
| [17] |
M. Tadi, M. Radenkovic, A numerical method for 1-d parabolic equation with nonlocal boundary conditions, Int. J. Comput. Math., 2014 (2014), 1–9. https://doi.org/10.1155/2014/923693 doi: 10.1155/2014/923693
|
| [18] |
Y. Lin, Y. Zhou, Solving the reaction-diffusion equations with nonlocal boundary conditions based on reproducing kernel space, Numer. Meth. Part. D. E., 25 (2009), 1468–1481. https://doi.org/10.1002/num.20409 doi: 10.1002/num.20409
|
| [19] |
S. Shu, L. Zhou, H. Yu, Error estimate and superconvergence of a high-accuracy difference scheme for solving parabolic equations with an integral two-space-variables condition, Adv. Appl. Math. Mech., 10 (2018), 362–389. https://doi.org/10.4208/aamm.OA-2017-0067 doi: 10.4208/aamm.OA-2017-0067
|
| [20] |
S. Shu, L. Zhou, H. Yu, Error estimates and superconvergence of a high-accuracy difference scheme for a parabolic inverse problem with unknown boundary conditions, Numer. Math.-Theory Me., 12 (2019), 1119–1140. https://doi.org/10.4208/nmtma.OA-2018-0019 doi: 10.4208/nmtma.OA-2018-0019
|
Liping Zhou, Yumei Yan, Ying Liu. Error estimate and superconvergence of a high-accuracy difference scheme for 2D heat equation with nonlocal boundary conditions[J]. AIMS Mathematics, 2024, 9(10): 27848-27870. doi: 10.3934/math.20241352
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