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.

COBISS.SI-ID: 1740027

This chapter represents a local meshless solution of natural convection phenomena. We systematicaly describe the method and local algorithm for pressure-velocity coupling. We present the numerical solution on three groups of tests. The first group is the standard de Vahl Davis test, which tackles natural convection of air in closed rectangular cavita. We solve five different scenarios of laminar flow. The second group is dedicated to natural convection in porous media, where we present the solution in different cavities at different conditions. The last group represents melting of pure substance, governed by natiral convection. All results are avaluated agains standard methods and compared with already published results. We obtain a good agreement with the results of outher authors in all numerical test cases.

COBISS.SI-ID: 2019067

This paper tackles an improved Localized Radial Basis Functions Collocation Method (LRBFCM) for the numerical solution of hyperbolic partial differential equations (PDEs). The LRBFCM is based on multiquadric (MQ) Radial Basis Functions (RBFs) and belongs to a class of truly meshless methods which do not need any underlying mesh. This method can be implemented on a set of uniform or random nodes, without any a priori knowledge of node to node connectivity. We have chosen uniform nodal arrangement due their suitability and better accuracy. Five nodded domains of influence are used in the local support for the calculation of the spatial partial derivatives. This approach results in a small interpolation matrix for each data center and hence the time integration is comparatively lower computational cost than the related global method. Different sizes of domain of influence i.e m = 5; 13 are considered. Shape parameter sensitivity of MQ is handled through scaling technique. The time derivative is approximated by forward difference formula. An adaptive upwind technique is used for stabilization of the method. Capabilities of the LRBFCM are tested by applying it to one and two dimensional benchmark problems with discontinuities, shock pattern and periodic initial conditions. Performance of the LRBFCM is compared with analytical solution, other numerical methods and the results reported earlier in the literature. We have also made comparison with implicit first order time discretization and first order upwind spatial descretization (FVM1) and implicit second order time descretization and first order upwind spatial descretization (FVM2) as well. Accuracy of the method is assessed as a function of time and space. Numerical convergence is also shown for both one and two dimensional test problems. It has been observed that the proposed method is more effcient in terms of less memory requirement and less computational efforts due to one time inversion of 5 x 5 (size of local domain of inuence) coeffcient matrix. The results obtained through LBRFCM are stable and comparable with the existing methods for a variety of problems with practical applications.

COBISS.SI-ID: 1998075