Exponential Convergence of Multiquadric Collocation Method: a Numerical Study

Abstract

RECENT NUMERICAL STUDIES HAVE PROVED THAT MULTIQUADRIC COLLOCATION METHODS CAN ACHIEVE EXPONENTIAL RATE OF CONVERGENCE FOR ELLIPTIC PROBLEMS. ALTHOUGH SOME INVESTIGATIONS HAS BEEN PERFORMED FOR TIME DEPENDENT PROBLEMS, THE INFLUENCE OF THE SHAPE PARAMETER OF THE MULTIQUADRIC KERNEL ON THE CONVERGENCE RATE OF THESE SCHEMES HAS NOT BEEN STUDIED. IN THIS ARTICLE, WE INVESTIGATE THIS ISSUE AND THE INFLUENCE OF THE PÉCLET NUMBER ON THE RATE OF CONVERGENCE FOR A CONVECTION DIFFUSION PROBLEM BY USING BOTH AN EXPLICIT AND IMPLICIT MULTIQUADRIC COLLOCATION TECHNIQUES. WE FOUND THAT FOR LOW TO MODERATE PÉCLET NUMBER AN EXPONENTIAL RATE OF CONVERGENCE CAN BE ATTAINED. IN ADDITION, WE FOUND THAT INCREASING THE VALUE OF THE PÉCLET NUMBER PRODUCES A VALUE REDUCTION OF THE COEFFICIENT THAT DETERMINES THE EXPONENTIAL RATE OF CONVERGENCE. MORE OVER, WE NUMERICALLY SHOWED THAT THE OPTIMAL VALUE OF THE SHAPE PARAMETER DECREASES MONOTONICALLY WHEN THE DIFFUSIVE COEFFICIENT IS REDUCED