A Local-Adaptive Multiquadric Shape Parameter Applied with DRBEM to Convection-Diffusion Problem

Abstract

While receiving more and more attention from scientists and engineers, the Dual Reciprocity Boundary Element Method (DRBEM) is known to face many factors and one of which is the choice of the Radial Basis Functions (RBFs) used. Amongst the popular choices of RBFs, the Multiquadric form is known to yield reliable solutions and yet, finding the optimal value of what is called ‘shape parameter’ noted, is known not to be straightforward. Nevertheless, it is well-known that the choice of the fixed value of is difficult to pinpoint and highly depends on the problem at hand. In this work, therefore, we propose a shape parameter that is a variable which can locally adapt itself correspondingly to the local change of the physics of the problem under investigation. For this reason, the convection-diffusion type of PDEs is focused on when the shape parameter is linked to the local Peclet number via the proposed formula. The results produced in this work show that the proposed shape variable is promising in producing satisfactory numerical solutions, particularly when compared with the fixed ones.