In previous work, a new adaptive meshfree advection scheme for numerically solving linear transport equations has been proposed. The scheme, being a combination of an adaptive semi-Lagrangian method and local radial basis function interpolation, is essentially a method of backward characteristics. The adaptivity of the meshfree advection scheme relies on customized rules for the refinement and ...

In this paper, we analyse the iterated collocation method for Hammerstein equations with smooth and weakly singular kernels. The paper expands the study which began in [14] concerning the superconvergence of the iterated Galerkin method for Hammerstein equations. We obtain in this paper a similar superconvergence result for the iterated collocation method for Hammerstein equations. We also disc...

A Multistep collocation techniques is used in this paper to develop a 3-point explicit and implicit block methods, which are suitable for generating solutions of the general second-order ordinary differential equations of the form y′′ f x, y, y′ , y x0 a, y′ x0 b. The derivation of both explicit and implicit block schemes is given for the purpose of comparison of results. The Stability and Conv...

The development of computational acoustics allows simulation of sound generation and propagation in complex environment. In particular, meshfree methods are widely used to solve acoustics problems through arbitrarily distributed field points and approximation smoothness flexibility. As a Lagrangian meshfree method, smoothed particle hydrodynamics (SPH) method reduce the difficulty in solving pr...

In this paper, new numerical modeling of material flow in the thermo-mechanical friction stir welding process is presented. In this numerical model, the discretization in space is derived by the meshfree Galerkin method using a Lagrangian meshfree convex approximation. The discrete thermal and mechanical equations are weakly coupled as the time advances using a forward difference scheme. A mort...

This document discusses accuracy of the collocation technique we employ to calculate the velocity field in ambient inviscid fluid above an interface swollen by a denser source fluid.

Time domain boundary integral formulations of transient scattering problems involve retarded potential integral equations (RPIEs). We outline how Fourier and Laplace transforms can be used to obtain an O(h) convergence result for a “polar” piecewise–linear collocation approximation of a scalar RPIE on an infinite flat plate.