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 three-dimensional magneto-hydrodynamic 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 Keller-box 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.
Highlights
This work describes a new approach to the solution of a system of four partial differential equations that model the flow of unsteady three-dimensional magneto-hydrodynamic flow and mass transfer in porous media
Bivariate interpolated spectral relaxation method (BI‐SRM) we introduce the Bivariate Interpolated Spectral Relaxation Method (BISRM) for solving the system of nonlinear partial differential equations (1)–(3)
It can be observed from the table that the Keller-box method takes a significant amount of computational time than the SRM and spectral quasilinearization method (SQLM)
Summary
This work describes a new approach to the solution of a system of four partial differential equations that model the flow of unsteady three-dimensional magneto-hydrodynamic 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 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), Alizadeh-Pahlavan 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. 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
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