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 |
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