A comparison between weak form meshless local Petrov–Galerkin method (MLPG) and strong form meshless diffuse approximate method (DAM) is performed for the diffusion equation in two dimensions. The shape functions are in both methods obtained by moving least squares (MLS) approximation with the polynomial weight function of the fourth order on the local support domain with 13 closest nodes. The weak form test functions are similar to the MLS weight functions but defined over the square integration domain. Implicit time-stepping discretization is used. The methods are tested and compared in terms of average and maximum error norms on uniform and non-uniform node arrangements on a square without and with a hole for a Dirichlet jump problem and involvement of Dirichlet and Neumann boundary conditions. The results are compared also to the results of the finite difference and finite element method. It has been found out that both meshless methods provide a similar accuracy and the same convergence rate. The advantage of DAM is in simpler numerical implementation and lower computational cost.
COBISS.SI-ID: 2024699
We have recently published a comparative study of global versus local meshless methods for two-dimensional parabolic partial differential equations (PDEs). This paper extends this study to important three-dimensional situations. The performance of the methods in assessed in terms of accuracy and efficiency. In both the global and local methods discussed, the time discretization is performed in explicit and implicit way, and the multiquadric (MQ) radial basis functions (RBFs) are used in spatial discretization of the PDE. The collocation approach is used for calculating the coefficients of the RBFs. A uniform and non-uniform node arrangements are used to test the validity of the numerical results and the ability of the methods to treat problems with Dirichlet and Neumann boundary conditions. Our tests show superiority of local radial basis function collocation method, including explicit and implicit time discretization, especially for the problems with Dirichlet boundary conditions. Local explicit method is very accurate but also is sensitive to the node distribution. Performance of the local implicit method is comparatively better than others when larger number of nodes and mixed boundary conditions are used. Global methods are efficient and accurate when small amount of data is involved in computations.
COBISS.SI-ID: 2405371
This paper examines the numerical solution of the transient nonlinear coupled Burgers’ equations by a Local Radial Basis Functions Collocation Method (LRBFCM) for large values of Reynolds number (Re) up to 10**3. The LRBFCM belongs to a class of truly meshless methods which do not need any underlying mesh but works on a set of uniform or random nodes without any a priori node to node connectivity. The time discretization is performed in an explicit way and collocation with the multiquadric radial basis functions (RBFs) are used to interpolate diffusion–convection variable and its spatial derivatives on five noded subdomains. Adaptive upwind technique is used for stabilization of the method for large Re in the case of mixed boundary conditions. Accuracy of the method is assessed as a function of time and space discretizations. The method is tested on two benchmark problems having Dirichlet and mixed boundary conditions. The numerical solution obtained from the LRBFCM for different value of Re is compared with analytical solution as well as other numerical methods. It is shown that the proposed method is efficient, accurate and stable for flow with reasonably high Reynolds numbers.
COBISS.SI-ID: 24965415