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

In this work, accurate solutions to linear and nonlinear diffusion equations were 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 have been used for treatment of these equations. The computed results with the use of this technique have been compared with the exact solution to show the accuracy of it. Here, the approximate solution to the diffusion equations has been 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.

[1]  G. Meral,et al.  Differential quadrature solution of nonlinear reaction–diffusion equation with relaxation-type time integration , 2009, Int. J. Comput. Math..

[2]  G. Bluman,et al.  Symmetries and differential equations , 1989 .

[3]  Murat Sari,et al.  A sixth-order compact finite difference scheme to the numerical solutions of Burgers' equation , 2009, Appl. Math. Comput..

[4]  Graham F. Carey,et al.  Extension of high‐order compact schemes to time‐dependent problems , 2001 .

[5]  Lawrence Dresner,et al.  Similarity solutions of nonlinear partial differential equations , 1983 .

[6]  Miguel R. Visbal,et al.  High-order Compact Schemes for Nonlinear Dispersive Waves , 2006, J. Sci. Comput..

[7]  Effat A. Saied,et al.  New classes of similarity solutions of the inhomogeneous nonlinear diffusion equations , 1994 .

[8]  John R. King,et al.  Exact results for the nonlinear diffusion equations delta u/ delta t= delta / delta x(u-4/3 delta u/ delta x) and delta u/ delta t- delta / delta x(u-2/3 delta u/ delta x) , 1991 .

[9]  Effat A. Saied,et al.  The non-classical solution of the inhomogeneous non-linear diffusion equation , 1999, Appl. Math. Comput..

[10]  Tomonori Nihei,et al.  A fast solver of the shallow water equations on a sphere using a combined compact difference scheme , 2003 .

[11]  Anastasios S. Lyrintzis,et al.  Application of Compact Schemes to Large Eddy Simulation of Turbulent Jets , 2004, J. Sci. Comput..

[12]  Abdul-Majid Wazwaz.  Exact solutions to nonlinear diffusion equations obtained by the decomposition method , 2001, Appl. Math. Comput..

[13]  Tao Tang,et al.  A Compact Fourth-Order Finite Difference Scheme for Unsteady Viscous Incompressible Flows , 2001, J. Sci. Comput..

[14]  P. Chu,et al.  A Three-Point Combined Compact Difference Scheme , 1998 .

[15]  O. Kaptsov.  LINEAR DETERMINING EQUATIONS FOR DIFFERENTIAL CONSTRAINTS , 1998, Glasgow Mathematical Journal.

[16]  Alain Rigal.  High order difference schemes for unsteady one-dimensional diffusion-convection problems , 1994 .

[17]  Guoqing Liu,et al.  High-Order Compact ADI Methods for Parabolic Equations , 2006, Comput. Math. Appl..

[18]  Gurhan Gurarslan,et al.  Numerical solutions of linear and nonlinear diffusion equations by a differential quadrature method (DQM) , 2011 .

[20]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..

[21]  Jichun Li,et al.  High-order finite difference schemes for differential equations containing higher derivatives , 2005, Appl. Math. Comput..

[22]  R. Hirsh,et al.  Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique , 1975 .

[23]  A. J. Baker,et al.  Efficient implementation of high order methods for the advection–diffusion equation , 2000 .

[24]  M. Ciment,et al.  Review. The Operator Compact Implicit Method for Parabolic Equations , 1978 .

[25]  A. Polyanin,et al.  Handbook of Nonlinear Partial Differential Equations , 2003 .

[26]  Murat Sari,et al.  Solution of the Porous Media Equation by a Compact Finite Difference Method , 2009 .

[27]  Yoshikazu Giga,et al.  Nonlinear Partial Differential Equations , 2004 .

[28]  I. Singer,et al.  High-order finite difference methods for the Helmholtz equation , 1998 .

[29]  Miguel R. Visbal,et al.  High-Order Schemes for Navier-Stokes Equations: Algorithm and Implementation Into FDL3DI , 1998 .

[30]  Abdul-Majid Wazwaz,et al.  The variational iteration method: A powerful scheme for handling linear and nonlinear diffusion equations , 2007, Comput. Math. Appl..

[31]  S. Lele.  Compact finite difference schemes with spectral-like resolution , 1992 .

[32]  Gurhan Gurarslan,et al.  A sixth‐order compact finite difference method for the one‐dimensional sine‐Gordon equation , 2011 .

[33]  Qu Changzheng,et al.  Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method , 1999 .

[34]  Klaus Kirchgässner,et al.  Applications of nonlinear analysis in the physical sciences : invited papers presented at a workshop at Bielefeld, Federal Republic of Germany, 1-10 October 1979 , 1981 .

[35]  Jun Zhang,et al.  High order ADI method for solving unsteady convection-diffusion problems , 2004 .

[36]  Murat Sari,et al.  A compact finite difference method for the solution of the generalized Burgers–Fisher equation , 2010 .

[37]  O. Kaptsov.  Determining equations in diffusion problems , 2000 .

[38]  Abdul-Majid Wazwaz,et al.  Several new exact solutions for a fast diffusion equation by the differential constraints of the linear determining equations , 2003, Appl. Math. Comput..