An efficient numerical method based on uniform Haar wavelets is proposed for the numerical solution of second-order boundary-value problems (BVPs) arising in the mathematical modelling of deformation of beams and plates, deflection theory, deflection of a cantilever beam under a concentrated load, obstacle problems and many other engineering applications. The Haar wavelet basis permits to enlarge the class of functions used so far in the collocation framework. The performance of the Haar wavelets is compared with the Walsh wavelets, semi-orthogonal B-spline wavelets, spline functions, Adomian decomposition method (ADM), finite difference method, and Runge-Kutta method coupled with nonlinear shooting method. A more accurate solution can be obtained by wavelet decomposition in the form of a multi-resolution analysis of the function which represents the solution of a given problem. 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. The main advantage of the Haar wavelet based method is its efficiency and simple applicability for a variety of boundary conditions. The convergence analysis of the proposed method alongside numerical procedure for multi-point boundary-value problems are given to test wider applicability and accuracy of the method.
COBISS.SI-ID: 2203899
The purpose of the paper is to extend and explore the application of a novel meshless Local Radial Basis Function Collocation Method (LRBFCM) in solution of a steady, laminar, natural convection flow, influenced by magnetic field. The problem is defined by coupled mass, momentum, energy and induction equations that are solved in two dimensions by using local collocation with multiquadrics radial basis functions on an overlapping five nodded sub-domains and explicit time-stepping. The fractional step method is used to couple the pressure and velocity fields. The considered problem is calculated in a square cavity with two insulated horizontal and two differentially heated vertical walls with magnetic field applied in the horizontal direction. Numerical predictions are calculated for different Grashof numbers, ranging from 10**4 to 10**6, and Hartman numbers, ranging from 0 to 100, at Prandtl numbers 0.71 and 0.14. The results of the method are compared to predictions, obtained by other numerical methods, including FLUENT code. Good agreement has been achieved. The LRBFCM has been used in this kind of problems for the first time. The main advantage of the method is its simple numerical implementation and no need for polygonisation (mesh).
COBISS.SI-ID: 2827003
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 has comparatively low computational cost than the related global method. Different sizes of domain of influence i.e. m=5,13 are considered. Shape parameter sensitivity minimization of MQ is taken into account through scaling technique. The time derivative is approximated by first order 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 discretization (FVM1) and implicit second order time discretization and first order upwind spatial discretization (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 efficient in terms of memory requirement and computational efforts due to one-time inversion of 5×5 (size of local domain of influence) coefficient matrix. The results obtained through LBRFCM are stable, accurate and comparable with the existing methods for a variety of problems with practical applications.
COBISS.SI-ID: 1998075