The role of the multiquadric shape parameters in solving elliptic partial differential equations

✍ Scribed by J. Wertz; E.J. Kansa; L. Ling

• Publisher: Elsevier Science
• Year: 2006
• Tongue: English
• Weight: 880 KB
• Volume: 51
• Category: Article
• ISSN: 0898-1221
• DOI: 10.1016/j.camwa.2006.04.009

✦ Synopsis

This study examines the generalized multiquadrics (MQ), Cj (x) = [(x -xj)2 + c~]~ in the numerical solutions of elliptic two-dimensional partial differential equations (PDEs) with Dirichlet boundary conditions. The exponent /9 as well as c~ can be classified as shape parameters since these affect the shape of the MQ basis function. We examined variations of 3 as well as c~ where c~ can be different over the interior and on the boundary. The results show that increasing ¢3 has the most important effect on convergence, followed next by distinct sets of (c~)n\on << (c~)oa. Additional convergence accelerations were obtained by permitting both (c~)n\oa and (c~)oa to oscillate about its mean value with amplitude of approximately 1/2 for odd and even values of the indices. Our results show high orders of accuracy as the number of data centers increases with some simple heuristics.