Abstract. Many numerical solution alternatives based on Navier Stokes equations or Reynolds equations with single or double phase closure turbulence model were proposed. These have the drawback of : i) High computational cost, ii) The detailed complexity of the flow are partially accounted for, iii) Parameters estimation is still an open problem. In this paper, we proceeded by depth averaging o...

In this paper, we are interested in some convergent formulations for the unsymmetric collocation method or the so-called Kansa’s method. We review some newly developed theories on solvability and convergence. The rates of convergence of these variations of Kansa’s method are examined and verified in arbitrary–precision computations. Numerical examples confirm with the theories that the modified...

Based on the radial basis function (RBF), non-singular general solution and dual reciprocity method (DRM), this paper presents an inherently meshless, exponential convergence, integration-free, boundary-only collocation techniques for numerical solution of general partial differential equation systems. The basic ideas behind this methodology are very mathematically simple and generally effectiv...

The boundary knot method (BKM) is a recent meshfree boundary-type radial basis function (RBF) collocation technique. Compared with the method of fundamental solution, the BKM uses the nonsingular general solution instead of the singular fundamental solution to evaluate the homogeneous solution, while as such the dual reciprocity method (DRM) is still employed to approximate the particular solut...

During the last decade, three main variations have been proposed for solving elliptic PDEs by means of collocation with radial basis functions (RBFs). In this study, we have implemented them for infinitely smooth RBFs, and then compared them across the full range of values for the shape parameter of the RBFs. This was made possible by a recently discovered numerical procedure that bypasses the ...

Numerical solution of multi-dimensional PDEs is a challenging problem with respect to computational cost and memory requirements, as well as regarding representation of realistic geometries and adaption to solution features. Meshfree methods such as global radial basis function approximation have been successfully applied to several types of problems. However, due to the dense linear systems th...

Abstract. Radial basis function (RBF) interpolation can be very effective for scattered data in any number of dimensions. As one of their many applications, RBFs can provide highly accurate collocation-type numerical solutions to several classes of PDEs. To better understand the accuracy that can be obtained, we survey here derivative approximations based on RBFs using a similar Fourier analysi...

The total variation (TV) denoising method is a PDE-based technique that preserves the edges well but has undesirable staircase effect in some cases, namely, the translation of smooth regions (ramps) into piecewise constant regions (stairs). This paper introduces a novel mesh-free approach using TV (ROFmodel) regularization and radial basis function (RBF) for the numerical approximation of TV-ba...

In this article, we consider the discretization of the time-dependent Schrödinger equation using radial basis functions (RBFs). We formulate the discretized problem over an unbounded domain without imposing explicit boundary conditions. Since we can show that time stability of the discretization is not guaranteed for an RBF-collocation method, we propose to employ a Galerkin ansatz instead. For...

This paper deals with the numerical solution of a class of reaction diffusion equations arises from ecological phenomena. When two species are introduced into unoccupied habitat, they can spread across the environment as two travelling waves with the wave of the faster reproducer moving ahead of the slower.The mathematical modelling of invasions of species in more complex settings that include ...