×

zbMATH — the first resource for mathematics

A comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations. (English) Zbl 0999.65135
Summary: We present a thorough numerical comparison between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of boundary value problems for partial differential equations. A series of test examples was solved with these two schemes, different problems with different type of governing equations, and boundary conditions. Particular emphasis was paid to the ability of these schemes to solve the steady-state convection-diffusion equation at high values of the Péclet number.
From the examples tested in this work, it was observed that the system of algebraic equations obtained with the symmetric method was in general simpler to solve than the one obtained with the unsymmetric method and that the resulting algorithm performs better. However, the unsymmetric method has the advantage of being simpler to implement.
Two main features about the results obtained in this work are worthy of special attention: first, with the symmetric method it was possible to solve convection-diffusion problems at a very high Péclet number without the need of any artificial damping term, and second, with these two approaches, symmetric and unsymmetric, it is possible to impose free boundary conditions for problems in unbounded domains.

MSC:
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Golberg, M.A.; Chen, C.S., Discrete projection methods for integral equations, (1997), Computation Mechanics Publications Southampton, UK · Zbl 0903.76065
[2] Micchelli, C.A., Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. approx., 2, 11-22, (1986) · Zbl 0625.41005
[3] Schaback, R., Multivariate interpolation and approximation by translates of a basis function, (), 1-8
[4] Franke, R., Scattered data interpolation: tests of some methods, Math. comp., 38, 181-200, (1982) · Zbl 0476.65005
[5] Stead, S., Estimation of gradients from scattered data, Rocky mount. J. math., 14, 265-279, (1984) · Zbl 0558.65009
[6] Duchon, J., Spline minimizing rotation—invariant seminorms in Sobolev spaces, (), 85-100
[7] Powell, M.J.D., The uniform convergence of thin plate spline interpolation in two dimensions, Numerische Mathematik, 68, 1, 107-128, (1994) · Zbl 0812.41005
[8] Madych, W.R.; Nelson, S.A., Multivariable interpolation and conditionally positive definite functions—II, Math. comput., 54, 211-230, (1990) · Zbl 0859.41004
[9] Tarwater, A.E., A parameter study of Hardy’s multiquadric method for scattered data interpolation, Technical report UCRL-563670, (1985), Lawrence Livermore National Laboratory
[10] Franke, R., A critical comparison of some methods for interpolation of scattered data, Technical report NPS-53-79-003, (1975), Naval Postgraduate School
[11] Carlson, R.E.; Foley, T.A., The parameter R2 in multiquadric interpolation, Computers math. applic., 21, 9, 29-42, (1991) · Zbl 0725.65009
[12] Kansa, E.J., Multiquadrics—A scattered data approximation scheme with applications to computation fluid-dynamics—I. surface approximations and partial derivatives estimates, Computers math. applic., 19, 8/9, 127-145, (1990) · Zbl 0692.76003
[13] Kansa, E.J., Multiquadrics—A scattered data approximation scheme with applications to computation fluid-dynamics—II. solution to parabolic, hyperbolic and elliptic partial differential equations, Computers math. applic., 19, 8/9, 147-161, (1990) · Zbl 0850.76048
[14] Fasshauer, G.E., Solving partial differential equations by collocation with radial basis functions, (), 1-8
[15] Wu, Z., Hermite-Birkhoff interpolation of scattered data by radial basis functions, Approx. theory, 8, 2, 1-11, (1992) · Zbl 0757.41009
[16] Wu, Z., Solving PDE with radial basis function and the error estimation, ()
[17] Schaback, R.; Franke, C., Covergence order estimates of meshless collocation methods using radial basis functions, Advances computational mathematics, 8, 4, 381-399, (1998) · Zbl 0909.65088
[18] Dubal, M.R., Domain decomposition and local refinement for multiquadric approximations. I: second-order equations in one-dimension, Journal of applied science, 1, 1, 146-171, (1994)
[19] Coleman, C.J., On the use of radial basis functions in the solution of elliptic boundary value problems, Computational mechanics, 17, 418-422, (1996) · Zbl 0851.76056
[20] Sharan, M.; Kansa, E.J.; Gupta, S., Application to the multiquadric method for numerical solution of elliptic partial differential equations, Appl. math. and comp., 84, 275-302, (1997) · Zbl 0883.65083
[21] Hon, Y.C.; Mao, X.Z., An efficient numerical scheme for Burgers’ equations, Appl. math. and comp., 95, 37-50, (1998) · Zbl 0943.65101
[22] Hon, Y.C.; Cheung, K.F.; Mao, X.Z., A multiquadric solution for the shallow water equations, ASCEJ, hydraulic engineering, 125, 5, 524-533, (1999)
[23] Dubal, M.R.; Olivera, S.R.; Matzner, R.A., ()
[24] Wendland, H., Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. comput. math., 4, 389-396, (1995) · Zbl 0838.41014
[25] R. Schaback and H. Wendland, Using compactly supported radial basis functions to solve partial differential equations, (preprint).
[26] A.I. Fedoseyev, M.J. Friedman and E.J. Kansa, Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary, (preprint). · Zbl 0999.65137
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.