Radial Basis Function Methods for the Solution of

Radial Basis Function Methods for the Solution
of Convective Diffusive Problems
Pedro González-Casanova Henrı́quez,1
1
Unidad de Investigación en Computo Aplicado, DGSCA-UNAM
Gustavo Rodrı́guez Gómez,2
José Antonio Muñoz Gómez3
2,3
Coordinación de Ciencias Computacionales, INAOE
The numerical solution of PDE by radial basis function methods is still in
its infancy. Since Kansa’s works [1] and [2], on unsymmetrical RBF collocation method for the solution of PDE, a substantial amount of research has
been done in this filed. Still, many numerical and theoretical problems remain to be solved. In this talk we shall review some of the most significant
techniques in this field as well as their most successful applications for evolutionary problems. In the first part of this talk, we will discuss the solution of
parabolic convective diffusive problems emphasizing the use of multiquadric
radial basis function kernels. An h-c algorithm -which is a radial basis function equivalent of the h-p finite element method- will be presented and the
role of the shape parameter of the multiquadric kernel on behavior of the numerical solution will be discussed. Numerical results showing the exponential
rate of convergence of this technique will be presented. In the second part of
this talk we address the problem of approximating the solution of non-linear
hyperbolic conservation laws in one dimension. A consistent, conservative,
TVD, meshfree RBF approach to these problems will be reviewed and its
possible extension to multidimensional problems will be discussed.
1.- E. J. Kansa, ”Multiquadrics- A scattered data approximation scheme
with applications to computational fluid dynamics: I. Surface approximations and partial derivative estimates”, Comput. Math. Appl., 19, (6-8):
127-145 (1990).
2.- E. J. Kansa, ”Multiquadrics- A scattered data approximation scheme with
applications to computational fluid dynamics: II. Solutions to parabolic, hyperbolic, and elliptic partial differential equations”, Comput. Math. Appl.,
19, (6-8): 147-161 (1990).
1