Monotone finite difference schemes for nonlinear system with mixed quasi-montonicity. (English) Zbl 0997.65098
The authors propose some monotone finite difference schemes for nonlinear systems with mixed quasi-monotonicity. Two monotone iteration processes for the corresponding discrete problems are investigated. They converge to the quasi-solutions of the discrete problems, which are the exact solutions under appropriate conditions.
A monotone finite difference scheme on a uniform mesh with fourth-order accuracy is constructed. Numerical results are also given.
A monotone finite difference scheme on a uniform mesh with fourth-order accuracy is constructed. Numerical results are also given.
Reviewer: Laura-Iulia Aniţa (Iaşi)
MSC:
65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
65L12 | Finite difference and finite volume methods for ordinary differential equations |
34B15 | Nonlinear boundary value problems for ordinary differential equations |
Keywords:
numerical results; monotone finite difference schemes; nonlinear systems; mixed quasi-monotonicity; monotone iteration; fourth-order accuracyReferences:
[1] | Pao, C. V., Nonlinear Parabolic and Elliptic Equations (1992), Plenum Press: Plenum Press New York · Zbl 0780.35044 |
[2] | Guo, Ben-yu; Miller, J. J.H., Iterative and Petrov-Galerkin methods for solving a system of one-dimensional nonlinear elliptic equations, Math. Comput., 58, 531-547 (1992) · Zbl 0812.65077 |
[3] | Wang, Yuan-ming; Guo, Ben-yu, Petrov-Galerkin methods for nonlinear systems without monotonicity, Appl. Numer. Math., 36, 57-78 (2001) · Zbl 0969.65071 |
[4] | Ishihara, K., Monotone explicit iterations of the finite element approximations for the nonlinear boundary value problem, Numer. Math., 43, 419-437 (1984) · Zbl 0531.65061 |
[5] | Ishihara, K., Explicit iterations with monotonicity for the finite element approximations applied to a system of nonlinear elliptic equations, J. Approx. Theory, 44, 241-252 (1985) · Zbl 0599.65074 |
[6] | Guo, Ben-yu; Miller, J. J.H., Iterative and finite difference methods for solving a system of nonlinear elliptic equations, J. Appl. Sci., 10, 1-24 (1992) · Zbl 0812.65077 |
[7] | Greenspan, D.; Parter, S. V., Mildly nonlinear elliptic partial differential equations and their numerical solution, II, Numer. Math., 45, 419-437 (1965) · Zbl 0135.38302 |
[8] | Pao, C. V., Monotone iterative methods for finite difference system of reaction-diffusion equations, Numer. Math., 46, 571-586 (1985) · Zbl 0589.65072 |
[9] | Pao, C. V., Numerical solutions for some coupled systems of nonlinear boundary value problems, Numer. Math., 51, 381-394 (1987) · Zbl 0632.65111 |
[10] | Pao, C. V., Block monotone iterative methods for numerical solutions of nonlinear elliptic equations, Numer. Math., 72, 239-262 (1995) · Zbl 0838.65104 |
[11] | Lazer, A.; Leung, A.; Murio, D., Monotone scheme for finite difference equations concerning steady-state prey-predator interaction, J. Comput. Appl. Math., 8, 243-251 (1982) · Zbl 0494.65052 |
[12] | Leung, A., Monotone schemes for semilinear elliptic systems related to ecology, Math. Methods Appl. Sci., 4, 272-285 (1982) · Zbl 0493.35044 |
[13] | Pao, C. V., Numerical methods for semilinear parabolic equations, SIAM J. Numer. Anal., 24, 24-35 (1987) · Zbl 0623.65100 |
[14] | Pao, C. V., Numerical analysis of coupled systems of nonlinear parabolic equations, SIAM J. Numer. Anal., 36, 393-416 (1999) · Zbl 0921.65061 |
[15] | Hoff, D., Stability and convergence of finite difference methods for systems of nonlinear reaction-diffusion equations, SIAM J. Numer. Anal., 15, 1161-1177 (1978) · Zbl 0411.76062 |
[16] | Pao, C. V., Numerical methods for systems of nonlinear parabolic equations with time delays, J. Math. Anal. Appl., 240, 249-279 (1999) · Zbl 0941.65083 |
[17] | Wang, J.; Pao, C. V., Finite difference reaction-diffusion equations with nonlinear diffusion coefficients, Numer. Math., 85, 485-502 (2000) · Zbl 0962.65073 |
[18] | Guo, Ben-yu; Wang, Yuan-ming, Almost monotone approximation to nonlinear two-points problem, Adv. Comput. Math., 8, 65-96 (1998) · Zbl 0892.65054 |
[19] | Collatz, L., The Numerical Treatment of Differential Equations (1960), Springer-Verlag: Springer-Verlag Berlin · Zbl 0086.32601 |
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.