On a bivariate spectral relaxation method for unsteady magnetohydrodynamic flow in porous media
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Abstract
The paper presents a significant improvement to the implementation of the spectral relaxation method (SRM) for solving nonlinear partial differential equations that arise in the modelling of fluid flow problems. Previously the SRM utilized the spectral method to discretize derivatives in space and finite differences to discretize in time. In this work we seek to improve the performance of the SRM by applying the spectral method to discretize derivatives in both space and time variables. The new approach combines the relaxation scheme of the SRM, bivariate Lagrange interpolation as well as the Chebyshev spectral collocation method. The technique is tested on a system of four nonlinear partial differential equations that model unsteady threedimensional magnetohydrodynamic flow and mass transfer in a porous medium. Computed solutions are compared with previously published results obtained using the SRM, the spectral quasilinearization method and the Kellerbox method. There is clear evidence that the new approach produces results that as good as, if not better than published results determined using the other methods. The main advantage of the new approach is that it offers better accuracy on coarser grids which significantly improves the computational speed of the method. The technique also leads to faster convergence to the required solution.
Keywords
Bivariate Spectral relaxation method Magnetohydrodynamic flow Porous mediaMathematics Subject Classification
Primary 65N35 76W99Introduction
This work describes a new approach to the solution of a system of four partial differential equations that model the flow of unsteady threedimensional magnetohydrodynamic flow and mass transfer in porous media. As reported in Hayat et al. (2010), such equations arise in many applications including the aerodynamic extrusion of plastic sheets, the cooling of metallic sheets in a cooling bath and the manufacture of artificial film and fibers. Due to these important applications, many researchers have dedicated time and effort in studying these kind of problems and finding their solutions. The particular model equations considered in this work have been solved in Hayat et al. (2010) using the homotopy analysis method (HAM) and more recently, by Motsa et al. (2014a) using the spectral relaxation method (SRM) and the spectral quasilinearization method (SQLM). The HAM has been used extensively by researchers working on such problems Abbas et al. (2008), Ahmad et al. (2008), Ali and Mehmood (2008), Mehmood et al. (2008), AlizadehPahlavan and Sadeghy (2009), Fan et al. (2010), Xu et al. (2007), You et al. (2010). It is an analytic method for approximating solutions of differential equations developed by Liao (2012). The homotopy analysis method is an analytic method where accuracy and convergence are achieved by increasing the number of terms of the solution series. In some cases, such as when a large embedded physical parameter multiplies the nonlinear terms, far too many terms may be required to give accurate results. Retaining too many terms in the solution series is cumbersome, even with the use of symbolic computing software. The use of the HAM further depends on other arbitrarily introduced parameters such as the convergence controlling parameter which must be carefully selected through a separate procedure.
A popular numerical method used by many researchers to solve unsteady boundary layer flow problems is the Kellerbox method Ali et al. (2010a, b), Lok et al. (2010), Nazar et al. (2004a, b). The Kellerbox method is a finite difference based implicit numerical scheme which was developed by Cebeci and Bradshaw (1984). Recently, Motsa et al. (2014a, b) used spectral based relaxation and quasilinearization schemes to solve unsteady boundary layer problems. These schemes are accurate, easy to implement and are computationally efficient. As observed in Motsa et al. (2014a), the limitation of the spectral quasilinearization method is that the coupled highorder system of differential equations may often lead to very large systems of algebraic equations that may require significant computing resources. In addition, the actual process of developing the solution algorithm is timeconsuming in comparison to SRM. This is because with SQLM, the process begins with the quasilinearization step whereas with SRM the iteration scheme is obtained directly by requiring some terms to be evaluated at the current iteration and others at the previous iteration. The SRM works much like the familiar GaussSeidel iteration by decoupling a system of nonlinear PDEs into a system of linear PDEs which are then solved in succession. Consequently, the SRM is easy to implement and computationally efficient.
Both the original SRM and SQLM used in Motsa et al. (2014a) use finite differences to discretize derivatives in time. This is a disadvantage because finite difference schemes are known to converge slower than spectral methods. The use of finite differences effectively nullifies the benefits of fast convergence when spectral collocation is used to discretize in space. Furthermore, finite differences require fine grids with very small step sizes to guarantee accuracy, hence there is a huge computation time overhead each time the grid is refined. This paper provides a different approach to the implementation of the spectral relaxation method introduced in Motsa et al. (2014a). The innovation is that the spectral collocation method is used to discretize derivatives in both space and time. As a result, there are uniform convergence benefits in both directions. The scheme uses fewer grid points in space and time and thus it converges very fast. We refer to the improved SRM as the bivariate interpolation spectral relaxation method (BISRM). To test the viability of this innovation as a solution method, we have solved the coupled system of third and second order partial differential equations that describe a boundarylayer system. A careful comparison of the new results is made with the earlier SRM, SQLM and the Kellerbox results reported in Motsa et al. (2014a). We particularly compare the computational times for the different methods to reach the same level of accuracy.
Model equations
Bivariate interpolated spectral relaxation method (BISRM)
Results and discussion
In this section we present the numerical solutions of the three dimensional unsteady three dimensional magnetohydrodynamic flow and mass transfer in a porous media obtained using the BISQLM algorithm. In our computations the \(\eta \) domain was truncated to \(\eta _{\infty }=20\). This value gave accurate results for all the quantities of physical interest. To get accurate solutions, \(N_x = 60\) collocation points were used to discretize the space variable \(\eta \) and only \(N_t = 10\) collocation points were enough for the time variable \(\xi \).
Values of \(f''(0,\xi )\), \(g''(0,\xi )\), \(\theta '(0,\xi )\) and \(\phi '(0,\xi )\) when \(\lambda = 0.5, M = 2, c = 0.5, Sc = \gamma = 1, Pr = 1.5\)
\(\xi \)  BISRM  SRM  SQLM  Kellerbox 

(\(N_t = 10\))  (\(N_t = 2000\))  (\(N_t = 2000\))  (\(N_t = 2000\))  
\( f''(0,\xi )\)  
0.1  −0.851257  −0.851257  −0.851257  −0.851257 
0.3  −1.316705  −1.316705  −1.316705  −1.316705 
0.5  −1.685306  −1.685306  −1.685306  −1.685306 
0.7  −1.992608  −1.992608  −1.992608  −1.992608 
0.9  −2.259335  −2.259335  −2.259335  −2.259335 
\( g''(0,\xi )\)  
0.1  −0.417150  −0.417150  −0.417150  −0.417150 
0.3  −0.639602  −0.639602  −0.639602  −0.639602 
0.5  −0.817649  −0.817649  −0.817649  −0.817649 
0.7  −0.966603  −0.966603  −0.966603  −0.966603 
0.9  −1.095983  −1.095983  −1.095983  −1.095983 
\( \theta '(0,\xi )\)  
0.1  −0.710882  −0.710882  −0.710882  −0.710882 
0.3  −0.742842  −0.742842  −0.742842  −0.742842 
0.5  −0.765244  −0.765244  −0.765244  −0.765244 
0.7  −0.777270  −0.777270  −0.777270  −0.777270 
0.9  −0.770807  −0.770807  −0.770807  −0.770807 
\( \phi '(0,\xi )\)  
0.1  −0.634443  −0.634443  −0.634443  −0.634443 
0.3  −0.766867  −0.766867  −0.766867  −0.766867 
0.5  −0.891207  −0.891207  −0.891207  −0.891207 
0.7  −1.010045  −1.010045  −1.010045  −1.010045 
0.9  −1.125549  −1.125549  −1.125549  −1.125549 
CPU time  0.47  18.90  83.24  900.30 
Values of \(f''(0,\xi )\), \(g''(0,\xi )\), \(\theta '(0,\xi )\) and \(\phi '(0,\xi )\) when \(\lambda = 0.5, M = 2, c = 0.5, Sc = \gamma = 1, Pr = 1.5\)
\(\xi \backslash N_t\)  5  10  15  20 

\( f''(0,\xi )\)  
0.1  −0.85118289  −0.85125725  −0.85125723  −0.85125724 
0.3  −1.31678885  −1.31670509  −1.31670508  −1.31670508 
0.5  −1.68525477  −1.68530619  −1.68530619  −1.68530619 
0.7  −1.99262557  −1.99260827  −1.99260827  −1.99260827 
0.9  −2.25932899  −2.25933501  −2.25933501  −2.25933501 
\( g''(0,\xi )\)  
0.1  −0.41712280  −0.41715041  −0.41715040  −0.41715040 
0.3  −0.63962554  −0.63960199  −0.63960200  −0.63960200 
0.5  −0.81764133  −0.81764898  −0.81764898  −0.81764898 
0.7  −0.96660357  −0.96660340  −0.96660340  −0.96660340 
0.9  −1.09598187  −1.09598304  −1.09598304  −1.09598304 
\( \theta '(0,\xi )\)  
0.1  −0.71037577  −0.71087263  −0.71088151  −0.71088162 
0.3  −0.74357007  −0.74283215  −0.74284211  −0.74284190 
0.5  −0.76493472  −0.76524664  −0.76524370  −0.76524351 
0.7  −0.77722565  −0.77727704  −0.77727014  −0.77727005 
0.9  −0.77086219  −0.77080729  −0.77080662  −0.77080662 
\( \phi '(0,\xi )\)  
0.1  −0.63444437  −0.63444336  −0.63444326  −0.63444326 
0.3  −0.76685596  −0.76686699  −0.76686689  −0.76686689 
0.5  −0.89121805  −0.89120664  −0.89120666  −0.89120666 
0.7  −1.01004174  −1.01004487  −1.01004494  −1.01004494 
0.9  −1.12555031  −1.12554912  −1.12554913  −1.12554913 
CPU time  0.13  0.47  1.25  2.47 
The accuracy of the solutions for energy and mass transfer equations are not dependent on successive relaxation or iterations of the momentum equations since the convergence of the solutions doesn’t improve at all with an increase in the number of iterations. Hence, the results for \(\theta (\eta )\) are not dependent on successive approximations for \(f(\eta )\) and \(g(\eta )\). It was observed that it is enough to use one iteration of \(f(\eta )\) to give accurate result for \(\theta (\eta )\).
In Table 1, we present the results we obtained using the algorithm. The skin friction values, Nusselt number and Sherwood number for various values of \(\xi \) are displayed in Table 1. The results obtained using the method match those obtained using other methods. Computing time for the BISRM is much smaller than the other methods it is compared with. The method gives valid results when used with few collocation points. In particular, the BISRM requires only ten grid points to achieve the same valid results. The table validates the results obtained in this study. BISRM is computationally fast in generating valid results when compared with the SRM, SQLM and KellerBox. We can infer that the BISRM is better than finite differences coupled SRM in terms of computational speed and accuracy and the better accuracy could be the result of applying spectral collocation with uniform accuracy level in both \(\eta \) and \(\xi \) directions.
The residual errors and convergence rates of f when \(\lambda = 0.5, M = 2, c = 0.5, Sc = \gamma = 1, Pr = 1.5\)
Iter.  \(\Vert Res({\mathbf{f}} )\Vert _{\infty }\)  Convergence rates  

\(\xi = 0.25\)  \(\xi = 0.75\)  \(\xi = 1.00\)  \(\xi = 0.25\)  \(\xi = 0.75\)  \(\xi = 1.00\)  
1  \(2.14\times 10^{2}\)  \(2.36\times 10^{1}\)  \(3.72\times 10^{1}\)  1.14  1.00  0.98 
2  \(6.03\times 10^{4}\)  \(1.43\times 10^{2}\)  \(2.24\times 10^{2}\)  0.50  1.01  1.01 
3  \(1.05\times 10^{5}\)  \(8.58\times 10^{4}\)  \(1.43\times 10^{3}\)  1.53  1.00  1.00 
4  \(1.36\times 10^{6}\)  \(5.04\times 10^{5}\)  \(8.85\times 10^{5}\)  1.06  1.01  1.00 
5  \(5.95\times 10^{8}\)  \(2.98\times 10^{6}\)  \(5.52\times 10^{6}\)  0.97  1.02  1.00 
6  \(2.18\times 10^{9}\)  \(1.70\times 10^{7}\)  \(3.40\times 10^{7}\)  0.99  1.00  1.00 
7  \(8.84\times 10^{11}\)  \(9.06\times 10^{9}\)  \(2.07\times 10^{8}\)  0.95  1.00  1.00 
8  \(3.75\times 10^{12}\)  \(4.85\times 10^{10}\)  \(1.27\times 10^{9}\)  0.85  0.99  1.00 
The residual errors and convergence rates of g when \(\lambda = 0.5, M = 2, c = 0.5, Sc = \gamma = 1, Pr = 1.5\)
Iter.  \(\Vert Res({\mathbf{g}} )\Vert _{\infty }\)  Convergence Rates  

\(\xi = 0.25\)  \(\xi = 0.75\)  \(\xi = 1.00\)  \(\xi = 0.25\)  \(\xi = 0.75\)  \(\xi = 1.00\)  
1  \(4.17\times 10^{3}\)  \(4.93\times 10^{2}\)  \(7.83\times 10^{2}\)  0.94  1.01  0.99 
2  \(4.96\times 10^{5}\)  \(1.40\times 10^{3}\)  \(2.21\times 10^{3}\)  0.75  1.03  1.04 
3  \(7.66\times 10^{7}\)  \(3.81\times 10^{5}\)  \(6.39\times 10^{5}\)  1.26  1.13  1.18 
4  \(3.38\times 10^{8}\)  \(9.21\times 10^{7}\)  \(1.58\times 10^{6}\)  1.07  0.76  0.56 
5  \(6.56\times 10^{10}\)  \(1.38\times 10^{8}\)  \(2.01\times 10^{8}\)  0.83  0.68  0.92 
6  \(9.49\times 10^{12}\)  \(5.75\times 10^{10}\)  \(1.78\times 10^{9}\)  1.00  1.27  1.18 
7  \(2.82\times 10^{13}\)  \(6.58\times 10^{11}\)  \(1.93\times 10^{10}\)  0.89  1.08  1.04 
8  \(8.39\times 10^{15}\)  \(4.19\times 10^{12}\)  \(1.39\times 10^{11}\)  0.95  1.00  1.00 
Conclusion

The method gives accurate results in the whole space and time domains \(\xi \in [0,1]\) and \(\tau \in [0,\infty )\) with residual errors rapidly approaching zero.

The application of spectral collocation to both time and space derivatives ensures that the method performs significantly better than the SRM and the KellerBox method in terms of computational time.

The algorithm involves the usage of known formulas for discretization using Chebyshev spectral collocation.
Notes
Authors' contributions
All authors participated in the preparation of the manuscript. All authors read and approved the final manuscript.
Acknowledgements
This work was supported in part by the University of KwaZuluNatal.
Competing interests
The authors declare that they have no competing interests.
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