Numerical modelling of linear and nonlinear diffusion equations by compact finite difference method

Applied Mathematics and Computation

Volumn 216, Issue 8, 2010, Pages 2472-2478

  Gürarslan, Gürhan ^a  

^a PAMUKKALE UNIVERSITY   (Turkey)

Author keywords

Compact schemes; Finite difference method; Nonlinear diffusion equation; Runge Kutta method

Indexed keywords

ALTERNATIVE METHODS; APPROXIMATE SOLUTION; COMPACT FINITE DIFFERENCES; COMPACT SCHEMES; DIFFUSION EQUATIONS; EXACT SOLUTION; LOW-STORAGE; NONLINEAR DIFFUSION EQUATIONS; NUMERICAL MODELLING; RUNGE-KUTTA; THIRD-ORDER; TOTAL VARIATION DIMINISHING;

DIFFERENCE EQUATIONS; DIFFUSION; FINITE DIFFERENCE METHOD; PARTIAL DIFFERENTIAL EQUATIONS; POWDERS; RUNGE KUTTA METHODS;

NONLINEAR EQUATIONS;

EID: 77953135506     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2010.03.093     Document Type: Article

References (41)

1. Saied E.A. The non-classical solution of the inhomogeneous non-linear diffusion equation. Appl. Math. Comput. 98 (1999) 103-108
2. Saied E.A., and Hussein M.M. New classes of similarity solutions of the inhomogeneous nonlinear diffusion equations. J. Phys. A 27 (1994) 4867-4874
3. Dresner L. Similarity Solutions of Nonlinear Partial Differential Equations (1983), Pitman, New York
4. Changzheng Q. Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method. IMA J. Appl. Math. 62 (1999) 283-302
5. Bluman G.W., and Kumei S. Symmetries and Differential Equations (1989), Springer, New York
6. King J.R. Exact results for the nonlinear diffusion equations. J. Phys. A 24 (1991) 5721-5745
7. Ames W.F. Nonlinear Partial Differential Equations (1972), Academic Press, New York
8. Peletier L.A. In: Amann H., Bazley N., and Kirchgassner K. (Eds). 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 (2001) 109-122
10. Kaptsov O.V. Determining equations in diffusion problems. Russ. J. Numer. Math. Modell. 15 2 (2000) 163-166
11. Kaptsov O.V. Linear determining equations for differential constraints. Sbornik: Math. 189 12 (1998) 1839-1854
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 (2003) 525-540
13. Wazwaz A.M. The variational iteration method: a powerful scheme for handling linear and nonlinear diffusion equations. Comput. Mat. Appl. 54 (2007) 933-939
14. Gürarslan G., and Sari M. Numerical solutions of linear and nonlinear diffusion equations by a differential quadrature method (DQM). Commun. Numer. Meth. En. (2009) 10.1002/cnm.1292
15. Meral G., and Tezer-Sezgin M. Differential quadrature solution of nonlinear reaction-diffusion equation with relaxation-type time integration. Int. J. Comput. Math. 86 3 (2009) 451-463
16. Meral G., and Tezer-Sezgin M. The differential quadrature solution of nonlinear reaction-diffusion and wave equations using several time-integration schemes. Commun. Numer. Meth. En. (2009) 10.1002/cnm.1305
17. Wazwaz A.M. Exact solutions to nonlinear diffusion equations obtained by the decomposition method. Appl. Math. Comput. 123 (2001) 109-122
18. Changzheng Q. Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method. IMA J. Appl. Math. 62 (1999) 283-302
19. Saied E.A. The non-classical solution of the inhomogeneous non-linear diffusion equation. Appl. Math. Comput. 98 (1999) 103-108
20. Hirsh R.S. Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique. J. Comput. Phys. 19 (1975) 90-109
21. Ciment M., Leventhal S.H., and Weinberg B.C. The operator compact implicit method for parabolic equations. J. Comput. Phys. 28 (1978) 135-166
22. Rigal A. High order difference schemes for unsteady one-dimensional diffusion-convection problems. J. Comput. Phys. 114 (1994) 59-76
23. Chu P.C., and Fan C. A three-point combined compact difference scheme. J. Comput. Phys. 140 (1998) 370-399
24. Singer I., and Turkel E. High-order finite difference methods for the Helmholtz equation. Comput. Method. Appl. M. 163 (1998) 343-358
25. Kolesnikov A., and Baker A.J. Efficient implementation of high order methods for the advection-diffusion equation. Comput. Method. Appl. M. 189 (2000) 701-722
26. Li M., and Tang T. A compact fourth-order finite difference scheme for unsteady viscous incompressible flows. J. Sci. Comput. 16 (2001) 29-45
27. Spotz W.F., and Carey G.F. Extension of high order compact schemes to time dependent problems. Numer. Meth. Part. D.E. 17 (2001) 657-672
28. Nihei T., and Ishii K. A fast solver of the shallow water equations on a sphere using a combined compact difference scheme. J. Comput. Phys. 187 (2003) 639-659
29. Karaa S., and Zhang J. High order ADI method for solving unsteady convection-diffusion problems. J. Comput. Phys. 198 (2004) 1-9
30. Uzun A., Blaisdell G.A., and Lyrintzis A.S. Application of compact schemes to large eddy simulation of turbulent jets. J. Sci. Comput. 21 (2004) 283-319
31. Li J. High-order finite difference schemes for differential equations containing higher derivatives. Appl. Math. Comput. 171 (2005) 1157-1176
32. Li J., and Visbal M.R. High-order compact schemes for nonlinear dispersive waves. J. Sci. Comput. 26 (2006) 1-23
33. Li J., Chen Y., and Liu G. High-order compact ADI Methods for parabolic equations. Comput. Math. Appl. 52 (2006) 1343-1356
34. Sari M., and Gürarslan G. A sixth-order compact finite difference scheme to the numerical solutions of Burger's equation. Appl. Math. Comput. 208 (2009) 475-483
35. Sari M. Solution of the porous media equation by a compact finite difference method. Math. Probl. Eng. 2009 (2009) Article ID 912541
36. Sari M., and Gürarslan G. A sixth-order compact finite difference method for the one-dimensional sine-Gordon equation. Commun. Numer. Meth. En. (2009) 10.1002/cnm.1349
37. Sari M., Gürarslan G., and Daǧ I. A compact finite difference method for the solution of the generalized Burgers-Fisher equation. Numer. Meth. Part. DE. 26 (2010) 125-134
38. Lele S.K. Compact finite difference schemes with spectral-like resolution. J. Comp. Phys. 103 (1992) 16-42
39. D.V. Gaitonde, M.R. Visbal, High-order schemes for Navier-Stokes equations: algorithms and implementation into FDL3DI, Technical Report AFRL-VA-WP-TR-1998-3060, Air Force Research Laboratory, Wright-Patterson AFB, Ohio, 1998.
40. Gottlieb S., and Shu C.-W. Total variation diminishing Runge-Kutta schemes. Math. Comput. 67 (1998)
41. Polyanin A.D., and Zaitsev V.F. Handbook of Nonlinear Partial Differential Equations (2004), Chapman & Hall/CRC, USA