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

Authors:
J. Wertz;E. J. Kansa;L. Ling
Affiliations:
Box 14154 Evansville, IN 47728, U.S.A.;Department of Mechanical and Aeronautical Engineering University of California, Davis, CA 95616, U.S.A.;University of Tokyo, Department of Mathematical Sciences 3-8-1 Komaba Meguro Tokyo 153, Japan
Venue:
Computers & Mathematics with Applications
Year:
2006

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Abstract

This study examines the generalized multiquadrics (MQ), @f"j(x) = [(x-x"j)^2+c"j^2]^@b in the numerical solutions of elliptic two-dimensional partial differential equations (PDEs) with Dirichlet boundary conditions. The exponent @b as well as c"j^2 can be classified as shape parameters since these affect the shape of the MQ basis function. We examined variations of @b as well as c"j^2 where c"j^2 can be different over the interior and on the boundary. The results show that increasing ,@b has the most important effect on convergence, followed next by distinct sets of (c"j^2)@W@?@W @? (c"j^2)@?@W. Additional convergence accelerations were obtained by permitting both (c"j^2)@W@?@W and (c"j^2)@?@W 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.