M-matrices and convergence of finite difference scheme for parabolic equation with integral boundary condition
Abstract
In the paper, the stability and convergence of difference schemes approximating semilinear parabolic equation with a nonlocal condition are considered. The proof is based on the properties of M-matrices, not requiring the symmetry or diagonal predominance of difference problem. The main presumption is that all the eigenvalues of the corresponding difference problem with nonlocal conditions are positive.
Keyword : finite difference method, nonlocal boundary condition, convergence, M-matrices
How to Cite
Čiupaila, R., Sapagovas, M., & Pupalaigė, K. (2020). M-matrices and convergence of finite difference scheme for parabolic equation with integral boundary condition. Mathematical Modelling and Analysis, 25(2), 167-183. https://doi.org/10.3846/mma.2020.8023
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
A. Berman and R.J. Plemmons. Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia, 1994. https://doi.org/10.1137/1.9781611971262
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. https://doi.org/10.1137/0732025
J.R. Cannon. The solution of the heat equation subject to specification of energy. Q. Appl. Math., 21(2):155–160, 1963. https://doi.org/10.1090/qam/160437
J.R. Cannon, Y. Lin and A.L. Matheson. The solution of the diffusion equation in two-space variables subject to specification of mass. Appl. Anal., 50(1):1–15, 1993. https://doi.org/10.1080/00036819308840181
Y.S. Choi and K.Y. Chan. A parabolic equation with nonlocal boundary conditions arising from electrochemistry. Nonlinear Analysis, Theory, Method., Apli., 18(4):317–331, 1992. https://doi.org/10.1016/0362-546X(92)90148-8
R. Čiegis, O. Suboč and A. Bugajev. Parallel algorithms for threedimensional parabolic and pseudoparabolic problems with different boundary conditions. Nonlinear. Anal. Model. Control, 19(3):382–395, 2014. https://doi.org/10.15388/NA.2014.3.5
R. Čiegis, O. Suboč and Rem. Čiegis. Numerical simulation of nonlocal delayed contraller for simple bioreactors. Informatica, 29(2):233–249, 2018. https://doi.org/10.15388/Informatica.2018.165
R. Čiegis and N. Tumanova. Numerical solution of parabolic problems with nonlocal boundary conditions. Numer. Funct. Anal. Optim., 31(12):1318–1329, 2010. https://doi.org/10.1080/01630563.2010.526734
R. Čiegis, A. Štikonas, O. Štikonienė and O. Suboč. A monotonic finite-difference scheme for a parabolic problem with nonlocal condition. Differ. Equ., 38(7):1027–1037, 2002. https://doi.org/10.1023/A:1021167932414
W.A. Day. Extensions of a property of the heat equation to linear thermo-elasticity and order theories. Q. Appl. Math., 40:319–330, 1982. https://doi.org/10.1090/qam/678203
G. Ekolin. Finite-difference methods for a nonlocal boundary-value problem for heat equation. BIT, 31:245–261, 1991. https://doi.org/10.1007/BF01931285
G. Fairweather and J.C. Lopez-Marcos. Galerkin methods for a semilinear parabolic problem with nonlocal conditions. Adv. Comp. Math., 6:243–262, 1996. https://doi.org/10.1007/BF02127706
A.V. Gulin, N.I. Ionkin and V.A. Morozova. Stability of a nonlocal twodimensional finite-difference problem. Differ. Equ., 37(7):970–978, 2001. https://doi.org/10.1023/A:1011961822115
F. Ivanauskas, V. Laurinavičius, M. Sapagovas and A. Nečiporenko. Reactiondiffusion equation with nonlocal boundary condition subject to PIDcontroller bioreactor. Nonlinear. Anal. Model. Control, 22(2):261–272, 2017. https://doi.org/10.15388/NA.2017.2.8
F. Ivanauskas, T. Meškauskas and M. Sapagovas. Stability of difference schemes for two-dimensional parabolic equations with nonlocal conditions. Appl. Math. Comp., 215(7):2716–2732, 2009. https://doi.org/10.1016/j.amc.2009.09.012
J. Jachimavičienė, Ž. Jesevičiūtė and M. Sapagovas. The stability of finite-difference schemes for a pseudoparabolic equation with nonlocal conditions. Numer. Funct. Anal. Optim., 30(9):988–1001, 2009. https://doi.org/10.1080/01630560903405412
J. Jachimavičienė, M. Sapagovas, A. Štikonas and O. Štikonienė. On the stability of explicit finite difference schemes for a pseudoparabolic equation with nonlocal conditions. Nonlinear. Anal. Model. Control, 19(2):225–240, 2014. https://doi.org/10.15388/NA.2014.2.6
K. Jakubėlienė, R. Čiupaila and M. Sapagovas. Semi-implicit difference scheme for a two-dimensional parabolic equation with an integral boundary condition. Math. Model. Anal., 22(5):617–633, 2017. https://doi.org/10.3846/13926292.2017.1342709
L.I. Kamynin. A boundary value problem in the theory of the heat conduction with nonclassical boundary condition. Zh. Vychisl. Mat. Mat. Fiz., 4(6):1006– 1024 (in Russian), 1964. https://doi.org/10.1016/0041-5553(64)90080-1
T. Leonavičienė, A. Bugajev, G. Jankevičiūtė and R. Čiegis. On stability analysis of finite difference schemes for generalized Kuramoto-Tsuzuki equation with nonlocal boundary conditions. Math. Model. Anal., 21(5):630–643, 2016. https://doi.org/10.3846/13926292.2016.1198836
Y. Lin, S. Xu and H. M. Yin. Finite difference approximations for a class of non-local parabolic equations. Intern. J. Math. and Mat. Sci., 20(1):147–164, 1997. https://doi.org/10.1155/S0161171297000215
J. Martin-Vaquero, A. Queiruga-Dios and A. H. Encinas. Numerical algorithms for diffusion-reaction problems with non-classical conditions. Appl. Math. Compt., 218(9):5487–5492, 2012. https://doi.org/10.1016/j.amc.2011.11.037
A.M. Nakhushev. On certain approximate method for boundary-value problems for differential equations and its applications in ground waters dynamics. Differ. Uravn., 18(1):72–81 (in Russian), 1988.
C.V. Pao. Numerical solutions of reaction-diffusion equations with nonlocal boundary conditions. J. Comp. Appi. Math., 136:227–243, 2001. https://doi.org/10.1016/S0377-0427(00)00614-2
A.A. Samarskii. The Theory of Difference Schemes. Moscow, Nauka, 1977. https://doi.org/10.1201/9780203908518 (in Russian); Marcel Dekler, Inc., New York and Basel, 2001.
M. Sapagovas. On the stability of a finite-difference scheme for nonlocal parabolic boundary-value problems. Lith. Math. J., 48(3):339–356, 2008. https://doi.org/10.1007/s10986-008-9017-5
M. Sapagovas. On the spectral properties of three-layer difference schemes for parabolic equations with nonlocal conditions. Differ. Equ., 48(7):1018–1027, 2012. https://doi.org/10.1134/S0012266112070129
M. Sapagovas, V. Griškonienė and O. Štikonienė. Application of M-matrices for solution of a nonlinear elliptic equation with an integral condition. Nonlinear. Anal. Model. Control, 22(4):489–504, 2017. https://doi.org/10.15388/NA.2017.4.5
M. Sapagovas and K. Jakubėlienė. Alternating direction method for two-dimensional parabolic equation with nonlocal integral condition. Nonlinear. Anal. Model. Control, 17(1):91–98, 2012. https://doi.org/10.15388/NA.17.1.14080
M. Sapagovas, T. Meškauskas and F. Ivanauskas. Numerical spectral analysis of a difference operator with non-local boundary conditions. Appl. Math. Comput., 218(14):7515–7527, 2012. https://doi.org/10.1016/j.amc.2012.01.017
M. Sapagovas, T. Meškauskas and F. Ivanauskas. Influence of complex coefficients on the stability of difference scheme for parabolic equations with nonlocal conditions. Appl. Math. Comp., 332:228–240, 2018. https://doi.org/10.1016/j.amc.2018.03.072
M. Sapagovas, A. Štikonas and O. Štikonienė. Alternating direction method for the Poisson equation with weight coefficients in an integral condition. Differ. Equ., 47(8):1163–1174, 2011. https://doi.org/10.1134/S0012266111080118
M. Sapagovas, O. Štikonienė, K. Jakubėlienė and R. Čiupaila. Finite difference method for boundary value problem for nonlinear elliptic equation with nonlocal conditions. Bound. Value Probl., 2019(94):1–16, 2019. https://doi.org/10.1186/s13661-019-1202-4
M. Sapagovas, R. Čiupaila, Z. Jokšienė and T. Meškauskas. Computational experiment for stability analysis of difference schemes with nonlocal conditions. Informatica, 24(2):275–290, 2013.
M.P. Sapagovas and A.D. Štikonas. On the stucture of the spectrum of a differ-ˇ ential operator with a nonlocal condition. Differ. Equ., 41(7):1010–1018, 2005. https://doi.org/10.1007/s10625-005-0242-y
A. Štikonas. Investigation of characteristic curve for Sturm-Liouville problem with nonlocal boundary conditions on torus. Math. Model. Anal., 16(1):1–22, 2011. https://doi.org/10.3846/13926292.2011.552260
A. Štikonas. A survey on stationary problems, Green’s functions and spectrum of Sturm-Liouville problem with nonlocal boundary conditions. Nonlinear. Anal. Model. Control, 19(3):301–334, 2014. https://doi.org/10.15388/NA.2014.3.1
O. Štikonienė, M. Sapagovas and R. Čiupaila. On iterative methods for some elliptic equations with nonlocal conditions. Nonlinear. Anal. Model. Control, 19(3):517–535, 2014. https://doi.org/10.15388/NA.2014.3.14
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. https://doi.org/10.1137/0732025
J.R. Cannon. The solution of the heat equation subject to specification of energy. Q. Appl. Math., 21(2):155–160, 1963. https://doi.org/10.1090/qam/160437
J.R. Cannon, Y. Lin and A.L. Matheson. The solution of the diffusion equation in two-space variables subject to specification of mass. Appl. Anal., 50(1):1–15, 1993. https://doi.org/10.1080/00036819308840181
Y.S. Choi and K.Y. Chan. A parabolic equation with nonlocal boundary conditions arising from electrochemistry. Nonlinear Analysis, Theory, Method., Apli., 18(4):317–331, 1992. https://doi.org/10.1016/0362-546X(92)90148-8
R. Čiegis, O. Suboč and A. Bugajev. Parallel algorithms for threedimensional parabolic and pseudoparabolic problems with different boundary conditions. Nonlinear. Anal. Model. Control, 19(3):382–395, 2014. https://doi.org/10.15388/NA.2014.3.5
R. Čiegis, O. Suboč and Rem. Čiegis. Numerical simulation of nonlocal delayed contraller for simple bioreactors. Informatica, 29(2):233–249, 2018. https://doi.org/10.15388/Informatica.2018.165
R. Čiegis and N. Tumanova. Numerical solution of parabolic problems with nonlocal boundary conditions. Numer. Funct. Anal. Optim., 31(12):1318–1329, 2010. https://doi.org/10.1080/01630563.2010.526734
R. Čiegis, A. Štikonas, O. Štikonienė and O. Suboč. A monotonic finite-difference scheme for a parabolic problem with nonlocal condition. Differ. Equ., 38(7):1027–1037, 2002. https://doi.org/10.1023/A:1021167932414
W.A. Day. Extensions of a property of the heat equation to linear thermo-elasticity and order theories. Q. Appl. Math., 40:319–330, 1982. https://doi.org/10.1090/qam/678203
G. Ekolin. Finite-difference methods for a nonlocal boundary-value problem for heat equation. BIT, 31:245–261, 1991. https://doi.org/10.1007/BF01931285
G. Fairweather and J.C. Lopez-Marcos. Galerkin methods for a semilinear parabolic problem with nonlocal conditions. Adv. Comp. Math., 6:243–262, 1996. https://doi.org/10.1007/BF02127706
A.V. Gulin, N.I. Ionkin and V.A. Morozova. Stability of a nonlocal twodimensional finite-difference problem. Differ. Equ., 37(7):970–978, 2001. https://doi.org/10.1023/A:1011961822115
F. Ivanauskas, V. Laurinavičius, M. Sapagovas and A. Nečiporenko. Reactiondiffusion equation with nonlocal boundary condition subject to PIDcontroller bioreactor. Nonlinear. Anal. Model. Control, 22(2):261–272, 2017. https://doi.org/10.15388/NA.2017.2.8
F. Ivanauskas, T. Meškauskas and M. Sapagovas. Stability of difference schemes for two-dimensional parabolic equations with nonlocal conditions. Appl. Math. Comp., 215(7):2716–2732, 2009. https://doi.org/10.1016/j.amc.2009.09.012
J. Jachimavičienė, Ž. Jesevičiūtė and M. Sapagovas. The stability of finite-difference schemes for a pseudoparabolic equation with nonlocal conditions. Numer. Funct. Anal. Optim., 30(9):988–1001, 2009. https://doi.org/10.1080/01630560903405412
J. Jachimavičienė, M. Sapagovas, A. Štikonas and O. Štikonienė. On the stability of explicit finite difference schemes for a pseudoparabolic equation with nonlocal conditions. Nonlinear. Anal. Model. Control, 19(2):225–240, 2014. https://doi.org/10.15388/NA.2014.2.6
K. Jakubėlienė, R. Čiupaila and M. Sapagovas. Semi-implicit difference scheme for a two-dimensional parabolic equation with an integral boundary condition. Math. Model. Anal., 22(5):617–633, 2017. https://doi.org/10.3846/13926292.2017.1342709
L.I. Kamynin. A boundary value problem in the theory of the heat conduction with nonclassical boundary condition. Zh. Vychisl. Mat. Mat. Fiz., 4(6):1006– 1024 (in Russian), 1964. https://doi.org/10.1016/0041-5553(64)90080-1
T. Leonavičienė, A. Bugajev, G. Jankevičiūtė and R. Čiegis. On stability analysis of finite difference schemes for generalized Kuramoto-Tsuzuki equation with nonlocal boundary conditions. Math. Model. Anal., 21(5):630–643, 2016. https://doi.org/10.3846/13926292.2016.1198836
Y. Lin, S. Xu and H. M. Yin. Finite difference approximations for a class of non-local parabolic equations. Intern. J. Math. and Mat. Sci., 20(1):147–164, 1997. https://doi.org/10.1155/S0161171297000215
J. Martin-Vaquero, A. Queiruga-Dios and A. H. Encinas. Numerical algorithms for diffusion-reaction problems with non-classical conditions. Appl. Math. Compt., 218(9):5487–5492, 2012. https://doi.org/10.1016/j.amc.2011.11.037
A.M. Nakhushev. On certain approximate method for boundary-value problems for differential equations and its applications in ground waters dynamics. Differ. Uravn., 18(1):72–81 (in Russian), 1988.
C.V. Pao. Numerical solutions of reaction-diffusion equations with nonlocal boundary conditions. J. Comp. Appi. Math., 136:227–243, 2001. https://doi.org/10.1016/S0377-0427(00)00614-2
A.A. Samarskii. The Theory of Difference Schemes. Moscow, Nauka, 1977. https://doi.org/10.1201/9780203908518 (in Russian); Marcel Dekler, Inc., New York and Basel, 2001.
M. Sapagovas. On the stability of a finite-difference scheme for nonlocal parabolic boundary-value problems. Lith. Math. J., 48(3):339–356, 2008. https://doi.org/10.1007/s10986-008-9017-5
M. Sapagovas. On the spectral properties of three-layer difference schemes for parabolic equations with nonlocal conditions. Differ. Equ., 48(7):1018–1027, 2012. https://doi.org/10.1134/S0012266112070129
M. Sapagovas, V. Griškonienė and O. Štikonienė. Application of M-matrices for solution of a nonlinear elliptic equation with an integral condition. Nonlinear. Anal. Model. Control, 22(4):489–504, 2017. https://doi.org/10.15388/NA.2017.4.5
M. Sapagovas and K. Jakubėlienė. Alternating direction method for two-dimensional parabolic equation with nonlocal integral condition. Nonlinear. Anal. Model. Control, 17(1):91–98, 2012. https://doi.org/10.15388/NA.17.1.14080
M. Sapagovas, T. Meškauskas and F. Ivanauskas. Numerical spectral analysis of a difference operator with non-local boundary conditions. Appl. Math. Comput., 218(14):7515–7527, 2012. https://doi.org/10.1016/j.amc.2012.01.017
M. Sapagovas, T. Meškauskas and F. Ivanauskas. Influence of complex coefficients on the stability of difference scheme for parabolic equations with nonlocal conditions. Appl. Math. Comp., 332:228–240, 2018. https://doi.org/10.1016/j.amc.2018.03.072
M. Sapagovas, A. Štikonas and O. Štikonienė. Alternating direction method for the Poisson equation with weight coefficients in an integral condition. Differ. Equ., 47(8):1163–1174, 2011. https://doi.org/10.1134/S0012266111080118
M. Sapagovas, O. Štikonienė, K. Jakubėlienė and R. Čiupaila. Finite difference method for boundary value problem for nonlinear elliptic equation with nonlocal conditions. Bound. Value Probl., 2019(94):1–16, 2019. https://doi.org/10.1186/s13661-019-1202-4
M. Sapagovas, R. Čiupaila, Z. Jokšienė and T. Meškauskas. Computational experiment for stability analysis of difference schemes with nonlocal conditions. Informatica, 24(2):275–290, 2013.
M.P. Sapagovas and A.D. Štikonas. On the stucture of the spectrum of a differ-ˇ ential operator with a nonlocal condition. Differ. Equ., 41(7):1010–1018, 2005. https://doi.org/10.1007/s10625-005-0242-y
A. Štikonas. Investigation of characteristic curve for Sturm-Liouville problem with nonlocal boundary conditions on torus. Math. Model. Anal., 16(1):1–22, 2011. https://doi.org/10.3846/13926292.2011.552260
A. Štikonas. A survey on stationary problems, Green’s functions and spectrum of Sturm-Liouville problem with nonlocal boundary conditions. Nonlinear. Anal. Model. Control, 19(3):301–334, 2014. https://doi.org/10.15388/NA.2014.3.1
O. Štikonienė, M. Sapagovas and R. Čiupaila. On iterative methods for some elliptic equations with nonlocal conditions. Nonlinear. Anal. Model. Control, 19(3):517–535, 2014. https://doi.org/10.15388/NA.2014.3.14