Haar wavelet collocation method for the numerical solution of boundary layer fluid flow problems

• Published in: International journal of thermal sciences 2011-05, Vol.50 (5), p.686-697
• Main Authors: Siraj-ul-Islam, Šarler, Božidar, Aziz, Imran, Fazal-i-Haq
• Format: Article
• Language: English
• Subjects: Boundary layer flow; Boundary-value problems (BVPs); Collocation method; Collocation methods; Computational efficiency; Computational methods in fluid dynamics; Computing time; Convection and heat transfer; Exact sciences and technology; Fluid dynamics; Fundamental areas of phenomenology (including applications); Haar wavelets; Laminar boundary layers; Laminar flows; Mathematical analysis; Mathematical models; Navier Stokes equations; Numerical analysis; Numerical method; Physics; Similarity transformations; System of coupled second-order ordinary differential equations; Turbulent flows, convection, and heat transfer; Wavelet
• ISSN: 1290-0729; 1778-4166
• DOI: 10.1016/j.ijthermalsci.2010.11.017

Based on Haar wavelets an efficient numerical method is proposed for the numerical solution of system of coupled Ordinary Differential Equations (ODEs) related to the natural convection boundary layer fluid flow problems with high Prandtl number (Pr). The numerical study of these flow models is necessary as the existing literature is more focused on the flow problems with small values of Pr. In this work, the problem of natural convection which consists of coupled nonlinear ODEs is solved simultaneously. The ODEs are obtained from the Navier Stokes equations through the similarity transformations. The effects of variation of Pr on heat transfer are investigated. Performance of the Haar Wavelets Collocation Method (HWCM) is compared with the finite difference method (FDM), Runge–Kutta Method (RKM), homotopy analysis method (HAM) and exact solution for the last problem. More accurate solutions are obtained by wavelets decomposition in the form of a multi-resolution analysis of the function which represents solution of the given problems. Through this analysis the solution is found on the coarse grid points and then refined towards higher accuracy by increasing the level of the Haar wavelets. Neumann’s boundary conditions which are problematic for most of the numerical methods are automatically coped with. A distinctive feature of the proposed method is its simple applicability for a variety of boundary conditions. Efficiency analysis of HWCM versus RKM is performed using Timing command in Mathematica software. A brief convergence analysis of the proposed method is given. Numerical tests are performed to test the applicability, efficiency and accuracy of the method.