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Numerical modelling of linear and nonlinear diffusion equations by compact finite difference method. (English) Zbl 1193.65156

Summary: Accurate solutions to linear and nonlinear diffusion equations are introduced. A combination of a sixth-order compact finite difference scheme in space and a low-storage third-order total variation diminishing Runge-Kutta scheme in time is used for a treatment of these equations. The computed results with the use of this technique are compared with the exact solution to show the accuracy of it. Here, the approximate solution to the diffusion equations is obtained easily and elegantly with neither transforming nor linearizing the equation. The present method is seen to be a very good alternative method to some existing techniques for realistic problems.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
35K55 Nonlinear parabolic equations

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[1] Saied, E. A., The non-classical solution of the inhomogeneous non-linear diffusion equation, Appl. Math. Comput., 98, 103-108 (1999) · Zbl 0929.35065
[2] Saied, E. A.; Hussein, M. M., New classes of similarity solutions of the inhomogeneous nonlinear diffusion equations, J. Phys. A, 27, 4867-4874 (1994) · Zbl 0842.35002
[3] Dresner, L., Similarity Solutions of Nonlinear Partial Differential Equations (1983), Pitman: Pitman New York · Zbl 0526.35002
[4] Changzheng, Q., Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method, IMA J. Appl. Math., 62, 283-302 (1999) · Zbl 0936.35039
[5] Bluman, G. W.; Kumei, S., Symmetries and Differential Equations (1989), Springer: Springer New York · Zbl 0718.35004
[6] King, J. R., Exact results for the nonlinear diffusion equations, J. Phys. A, 24, 5721-5745 (1991) · Zbl 0760.35038
[7] Ames, W. F., Nonlinear Partial Differential Equations (1972), Academic Press: Academic Press New York · Zbl 0255.35001
[8] Peletier, L. A., (Amann, H.; Bazley, N.; Kirchgassner, K., Applications of Nonlinear Analysis in the Physical Sciences (1981), Pitman)
[9] Wazwaz, A. M., Exact solutions to nonlinear diffusion equations by the decomposition method, Appl. Math. Comput., 123, 1, 109-122 (2001) · Zbl 1027.35019
[10] Kaptsov, O. V., Determining equations in diffusion problems, Russ. J. Numer. Math. Modell., 15, 2, 163-166 (2000) · Zbl 0972.35050
[11] Kaptsov, O. V., Linear determining equations for differential constraints, Sbornik: Math., 189, 12, 1839-1854 (1998) · Zbl 0935.35004
[12] Wazwaz, A. M., Several new exact solutions for a fast diffusion equation by the differential constraints of the linear determining equations, Appl. Math. Comput., 145, 525-540 (2003) · Zbl 1027.35059
[13] Wazwaz, A. M., The variational iteration method: a powerful scheme for handling linear and nonlinear diffusion equations, Comput. Mat. Appl., 54, 933-939 (2007) · Zbl 1141.65077
[14] Gürarslan, G.; Sari, M., Numerical solutions of linear and nonlinear diffusion equations by a differential quadrature method (DQM), Commun. Numer. Meth. En. (2009)
[15] Meral, G.; Tezer-Sezgin, M., Differential quadrature solution of nonlinear reaction-diffusion equation with relaxation-type time integration, Int. J. Comput. Math., 86, 3, 451-463 (2009) · Zbl 1160.65333
[16] Meral, G.; Tezer-Sezgin, M., The differential quadrature solution of nonlinear reaction-diffusion and wave equations using several time-integration schemes, Commun. Numer. Meth. En. (2009) · Zbl 1160.65333
[17] Wazwaz, A. M., Exact solutions to nonlinear diffusion equations obtained by the decomposition method, Appl. Math. Comput., 123, 109-122 (2001) · Zbl 1027.35019
[18] Changzheng, Q., Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method, IMA J. Appl. Math., 62, 283-302 (1999) · Zbl 0936.35039
[19] Saied, E. A., The non-classical solution of the inhomogeneous non-linear diffusion equation, Appl. Math. Comput., 98, 103-108 (1999) · Zbl 0929.35065
[20] Hirsh, R. S., Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique, J. Comput. Phys., 19, 90-109 (1975) · Zbl 0326.76024
[21] Ciment, M.; Leventhal, S. H.; Weinberg, B. C., The operator compact implicit method for parabolic equations, J. Comput. Phys., 28, 135-166 (1978) · Zbl 0393.65038
[22] Rigal, A., High order difference schemes for unsteady one-dimensional diffusion-convection problems, J. Comput. Phys., 114, 59-76 (1994) · Zbl 0807.65096
[23] Chu, P. C.; Fan, C., A three-point combined compact difference scheme, J. Comput. Phys., 140, 370-399 (1998) · Zbl 0923.65071
[24] Singer, I.; Turkel, E., High-order finite difference methods for the Helmholtz equation, Comput. Method. Appl. M., 163, 343-358 (1998) · Zbl 0940.65112
[25] Kolesnikov, A.; Baker, A. J., Efficient implementation of high order methods for the advection-diffusion equation, Comput. Method. Appl. M., 189, 701-722 (2000) · Zbl 0977.76046
[26] Li, M.; Tang, T., A compact fourth-order finite difference scheme for unsteady viscous incompressible flows, J. Sci. Comput., 16, 29-45 (2001) · Zbl 1172.76344
[27] Spotz, W. F.; Carey, G. F., Extension of high order compact schemes to time dependent problems, Numer. Meth. Part. D.E., 17, 657-672 (2001) · Zbl 0998.65101
[28] Nihei, T.; Ishii, K., A fast solver of the shallow water equations on a sphere using a combined compact difference scheme, J. Comput. Phys., 187, 639-659 (2003) · Zbl 1061.76513
[29] Karaa, S.; Zhang, J., High order ADI method for solving unsteady convection-diffusion problems, J. Comput. Phys., 198, 1-9 (2004) · Zbl 1053.65067
[30] Uzun, A.; Blaisdell, G. A.; Lyrintzis, A. S., Application of compact schemes to large eddy simulation of turbulent jets, J. Sci. Comput., 21, 283-319 (2004) · Zbl 1071.76028
[31] Li, J., High-order finite difference schemes for differential equations containing higher derivatives, Appl. Math. Comput., 171, 1157-1176 (2005) · Zbl 1090.65101
[32] Li, J.; Visbal, M. R., High-order compact schemes for nonlinear dispersive waves, J. Sci. Comput., 26, 1-23 (2006) · Zbl 1089.76043
[33] Li, J.; Chen, Y.; Liu, G., High-order compact ADI Methods for parabolic equations, Comput. Math. Appl., 52, 1343-1356 (2006) · Zbl 1121.65092
[34] Sari, M.; Gürarslan, G., A sixth-order compact finite difference scheme to the numerical solutions of Burger’s equation, Appl. Math. Comput., 208, 475-483 (2009) · Zbl 1159.65343
[35] Sari, M., Solution of the porous media equation by a compact finite difference method, Math. Probl. Eng., 2009 (2009), Article ID 912541 · Zbl 1181.80013
[36] Sari, M.; Gürarslan, G., A sixth-order compact finite difference method for the one-dimensional sine-Gordon equation, Commun. Numer. Meth. En. (2009)
[37] Sari, M.; Gürarslan, G.; Dağ, İ., A compact finite difference method for the solution of the generalized Burgers-Fisher equation, Numer. Meth. Part. DE., 26, 125-134 (2010) · Zbl 1183.65114
[38] Lele, S. K., Compact finite difference schemes with spectral-like resolution, J. Comp. Phys., 103, 16-42 (1992) · Zbl 0759.65006
[40] Gottlieb, S.; Shu, C.-W., Total variation diminishing Runge-Kutta schemes, Math. Comput., 67 (1998) · Zbl 0897.65058
[41] Polyanin, A. D.; Zaitsev, V. F., Handbook of Nonlinear Partial Differential Equations (2004), Chapman & Hall/CRC: Chapman & Hall/CRC USA · Zbl 1024.35001
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