Accurate solutions of the porous media equation that usually occurs in nonlinear problems of heat and mass transfer and in biological systems are obtained using a compact finite difference method in space and a low-storage total variation diminishing third-order Runge-Kutta scheme in time. In the calculation of the numerical derivatives, only a tridiagonal band matrix algorithm is encountered. Therefore, this scheme causes to less accumulation of numerical errors and less use of storage space. The computed results obtained by this way have been compared with the exact solutions to show the accuracy of the method. The approximate solutions to the equation have been computed without transforming the equation and without using linearization. Comparisons indicate that there is a very good agreement between the numerical solutions and the exact solutions in terms of accuracy. This method is seen to be a very good alternative method to some existing techniques for such realistic problems.

Most physical phenomena and processes encountered in various fields of
science are governed by partial differential equations. The nonlinear heat equation describing various physical phenomena,
for instance see references [

Computing the solutions that have a physical or biological interpretation for the
nonlinear equations of the form (

Equation (

For the motion of thin viscous films,
(

In this work, it is aimed to effectively employ a
combination of a sixth-order compact finite difference (CFD6) scheme in space
[

The compact finite difference schemes can be dealt within two essential categories: explicit compact and implicit compact approaches. Whilst the first category computes the numerical derivatives directly at each grid by using large stencils, the second one obtains all the numerical derivatives along a grid line using smaller stencils and solving a linear system of equations. Because of the reasons given previously, the present work uses the second approach. To attain the solutions of the equation, discretizations are needed in both space and time.

Spatial
derivatives are evaluated by the compact finite difference scheme [

In the current work, the equations are integrated in time with
consideration of the low-storage TVD-RK3 scheme [

The CFD6 scheme will be applied to
physical models to illustrate the strength of the present method. Each spatial
derivative on the right-hand side of (

In order to see numerically whether the present methodology leads to accurate solutions, the CFD6 solutions are evaluated for some examples of the porous media equations given above.

Now, numerical solutions of the
porous media equation are obtained to
validate the current numerical scheme. To verify the efficiency, measure its
accuracy and the versatility of the present scheme for our problem in
comparison with the exact solution, the absolute errors are reported which are
defined by

Consider the porous media equation
in the form (

The absolute
errors for various values of

Absolute error | ||
---|---|---|

0.10 | 0.01 | 1.07E-15 |

0.10 | 1.08E-14 | |

1.00 | 6.51E-14 | |

0.50 | 0.01 | 1.11E-15 |

0.10 | 1.10E-14 | |

1.00 | 1.32E-13 | |

0.90 | 0.01 | 1.33E-15 |

0.10 | 3.11E-15 | |

1.00 | 3.51E-14 |

The absolute
errors for various values of

Absolute error | ||
---|---|---|

0.10 | 0.01 | 1.09E-14 |

0.10 | 1.20E-14 | |

1.00 | 3.70E-14 | |

0.50 | 0.01 | 1.28E-15 |

0.10 | 2.32E-15 | |

1.00 | 4.80E-15 | |

0.90 | 0.01 | 1.35E-14 |

0.10 | 1.74E-14 | |

1.00 | 4.86E-14 |

The absolute
errors for various values of

Absolute error | ||
---|---|---|

0.10 | 0.01 | 7.05E-13 |

0.10 | 1.01E-12 | |

1.00 | 2.91E-11 | |

0.50 | 0.01 | 1.07E-13 |

0.10 | 2.09E-13 | |

1.00 | 3.72E-12 | |

0.90 | 0.01 | 1.26E-12 |

0.10 | 1.73E-12 | |

1.00 | 3.68E-11 |

The absolute
errors for various values of

Absolute error | ||
---|---|---|

0.10 | 0.01 | 8.10E-07 |

0.10 | 1.22E-07 | |

1.00 | 1.92E-10 | |

0.50 | 0.01 | 6.49E-08 |

0.10 | 4.75E-07 | |

1.00 | 1.54E-11 | |

0.90 | 0.01 | 1.01E-06 |

0.10 | 1.00E-06 | |

1.00 | 1.85E-10 |

Comparison
of the CFD6 with the FD4 for various values of

Exact | Numerical | Absolute error | ||||
---|---|---|---|---|---|---|

FD4 | CFD6 | FD4 | CFD6 | |||

1.1 | 0.10 | 1.110366 | 1.110365 | 1.110367 | 1.38E-06 | 1.31E-06 |

1.00 | 6.717324 | 6.717289 | 6.717331 | 3.45E-05 | 7.52E-06 | |

5.00 | 20024.059813 | 20023.956641 | 20024.082234 | 1.03E-01 | 2.24E-02 | |

1.5 | 0.10 | 0.814269 | 0.814235 | 0.814268 | 3.33E-05 | 7.68E-07 |

1.00 | 4.926037 | 4.925791 | 4.926030 | 2.47E-04 | 6.95E-06 | |

5.00 | 14684.310530 | 14683.574752 | 14684.289675 | 7.36E-01 | 2.09E-02 | |

1.9 | 0.10 | 0.642844 | 0.642830 | 0.642842 | 1.34E-05 | 1.70E-06 |

1.00 | 3.888977 | 3.888881 | 3.888964 | 9.61E-05 | 1.25E-05 | |

5.00 | 11592.876734 | 11592.589375 | 11592.839334 | 2.87E-01 | 3.74E-02 |

Convergence rate (CR) of the present scheme:
Comparisons on various values of

Average absolute error | CR | Average absolute error | CR | Average absolute error | CR | |||

5.84E-09 | 6.82E-10 | 8.95E-10 | ||||||

6.86E-10 | 5.28 | 3.84E-10 | 5.45 | 5.06E-10 | 5.42 | |||

1.48E-10 | 5.34 | 2.27E-10 | 5.54 | 2.99E-10 | 5.51 | |||

4.36E-11 | 5.46 | 1.39E-10 | 5.62 | 1.84E-10 | 5.60 | |||

1.58E-11 | 5.57 | 8.80E-11 | 5.71 | 1.17E-10 | 5.68 | |||

6.57E-12 | 5.69 | 5.72E-11 | 5.80 | 7.60E-11 | 5.77 | |||

3.03E-12 | 5.80 | 3.81E-11 | 5.89 | 5.07E-11 | 5.86 | |||

1.51E-12 | 5.91 | 2.59E-11 | 5.98 | 3.45E-11 | 5.95 | |||

8.02E-13 | 6.02 | 1.79E-11 | 6.07 | 2.39E-11 | 6.05 | |||

4.47E-13 | 6.14 | 1.26E-11 | 6.16 | 1.68E-11 | 6.15 |

The absolute errors for
various values of

Absolute error | ||
---|---|---|

0.1 | 11 | 4.77E-14 |

21 | 3.04E-14 | |

51 | 2.97E-14 | |

81 | 3.00E-14 | |

101 | 3.00E-14 | |

0.5 | 11 | 8.17E-14 |

21 | 7.66E-14 | |

51 | 7.64E-14 | |

81 | 7.68E-14 | |

101 | 7.68E-14 | |

0.9 | 11 | 1.15E-13 |

21 | 2.28E-14 | |

51 | 2.25E-14 | |

81 | 2.33E-14 | |

101 | 2.31E-14 |

The absolute errors for
various values of

Absolute error | CPU (s) | Absolute error | CPU (s) | ||
---|---|---|---|---|---|

0.1 | 1.0E-04 | 1.11E-14 | 0.0 | 1.74E-04 | 0.0 |

1.0E-05 | 7.09E-14 | 2.0 | 3.04E-15 | 3.0 | |

1.0E-06 | 4.77E-14 | 16.0 | 3.04E-14 | 30.0 | |

1.0E-07 | 2.49E-13 | 162.0 | 3.03E-13 | 295.0 | |

1.0E-08 | 3.25E-12 | 1640.0 | 3.27E-12 | 2964.0 | |

0.5 | 1.0E-04 | 2.26E-15 | 0.0 | 8.19E-04 | 0.0 |

1.0E-05 | 1.19E-14 | 2.0 | 7.61E-15 | 3.0 | |

1.0E-06 | 8.17E-14 | 16.0 | 7.66E-14 | 30.0 | |

1.0E-07 | 8.26E-13 | 162.0 | 7.84E-13 | 295.0 | |

1.0E-08 | 8.74E-12 | 1640.0 | 8.62E-12 | 2964.0 | |

0.9 | 1.0E-04 | 1.65E-14 | 0.00 | 7.02E-04 | 0.0 |

1.0E-05 | 8.23E-14 | 2.00 | 2.50E-15 | 3.0 | |

1.0E-06 | 1.15E-13 | 16.00 | 2.28E-14 | 30.0 | |

1.0E-07 | 3.66E-13 | 162.00 | 2.12E-13 | 295.0 | |

1.0E-08 | 3.07E-12 | 1640.00 | 2.76E-12 | 2964.0 |

As various
problems of science were modeled by nonlinear partial differential equations,
and since therefore the porous media equation is of high importance, the following examples [

Let us take

In Table

For

The absolute
errors have been shown for various values of

When

The absolute errors have been shown
for various values of

when

When

In this paper,
it is shown that the
approximate solutions of the porous media equation
are very close to the exact solutions. The absolute errors have been
calculated for

The method was also applied to realistic problem sizes (see Table

In this paper, numerical simulations of the porous media equation were dealt with using a combination of the CFD6 scheme in space and a low-storage explicit TVD-RK3 scheme in time. The method successfully worked to give very reliable and accurate solutions to the equation. The method gives convergent approximations and handles nonlinear problems. In this method, there is no need for linearization of nonlinear terms. Nonlinear scientific models arise frequently in scientific problems for expressing nonlinear phenomena. For nonlinear problems, the present method is seen to be a very good choice to achieve a high degree of accuracy while dealing with the problems. The computed results justify the advantage of this method. The present method needs less use of storage space.

The author would like to thank anonymous referees of the journal of MPE for their valuable comments and suggestions to improve most of this paper. The author is very grateful to G. Gürarslan (Department of Civil Engineering, Faculty of Engineering, Pamukkale University, Denizli, Turkey) for his comments and careful reading the paper.