Solution to a Convection-Diffusion Problem Using a New Variable Inverse-Multiquadric Parameter in a Collocation Meshfree Scheme

Sayan Kaennakham ,Nissaya Chuathong

doi:10.21152/1750-9548.11.4.359

Abstract

The lack of reliable judgment on the choice of shape parameter is acknowledged as a drawback in the methodology of collocation meshfree by radial basis function (RBF). Some attempts have been proposed and tested only with specific classes of PDEs. Moreover, while most of this focusses on multiquadric (MQ) RBF, there has not been work done on invers-multiquadric (IMQ) RBF despite its’ increasing popularity. As a consequence, our main tasks in this work are, firstly, to numerically investigate the quality of each adaptive/variable RBF-shape parameter approaches presented in literature by applying them to the same type of problem, convection-diffusion class. Secondly, we proposed a new form of shape-parameter scheme to be used with the inverse-multiquadric (IMQ) type of RBF. The Kansa meshless method is implemented and it is interestingly found that the proposed form produces good results quality in terms of both matrix condition number, and the accuracy.

Highlights

Convection-diffusion equation describes phenomena including both convection and diffusion effects, and appears in various fields of natural sciences, e.g., heat transfer, weather prediction and atmospheric radioactivity propagation
In 2013, Sun and Li [3] proposed a method which combines a 6th-order compact difference scheme and 2nd-order Crank-Nicolson scheme called the alternating direction implicit method (ADI). Their numerical examples have proven that ADI preserves the higher order accuracy for convection-dominated problem
One of our previous works, Chanthawara et al [4], done under the context of the boundary element method has confirmed the difficulty encountered and this is the main reason for focusing on this challenging problem in this investigation

Summary

Introduction

Convection-diffusion equation describes phenomena including both convection and diffusion effects, and appears in various fields of natural sciences, e.g., heat transfer, weather prediction and atmospheric radioactivity propagation It may be treated as a simplified model of the system of the Navier-Stokes equations, which are representative equations in fluid dynamics. Amongst several attempts and remedies; the enlargement of the local node adaptive, upwind scheme, the biased domain, and the nodal refinement and the adaptive analysis, for this challenge have been proposed, revisited, and nicely numerically demonstrated by Gu and Liu [1] In this work, they concluded that with extra efforts included, the meshfree method has some attractive advantages over the traditional schemes such as finite element and finite volume. One of our previous works, Chanthawara et al [4], done under the context of the boundary element method has confirmed the difficulty encountered and this is the main reason for focusing on this challenging problem in this investigation

Objectives

Results

Conclusion