Abstract
A few novel radial basis function (RBF) discretization schemes for partial differential equations are developed in this study. For boundary-type methods, we derive the indirect and direct symmetric boundary knot methods. Based on the multiple reciprocity principle, the boundary particle method is introduced for general inhomogeneous problems without using inner nodes. For domain-type schemes, by using the Green integral we develop a novel Hermite RBF scheme called the modified Kansa method, which significantly reduces calculation errors at close-to-boundary nodes. To avoid Gibbs phenomenon, we present the least square RBF collocation scheme. Finally, five types of the kernel RBF are also briefly presented.
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References
Buhmann, M. D. (2000) Radial basis functions. Acta Numerica. 1–38.
Chen, W., Tanaka, M. (2000) New Insights into Boundary-only and Domaintype RBF Methods. Int. J. Nonlinear Sci. & Numer. Simulation. 1(2.3), 145–151.
Chen, W., Tanaka, M. (2000) Relationship between boundary integral equation and radial basis function. The 52th Symposium of JSCME on BEM (Tanaka, M. ed.), Tokyo.
Kansa, E. J. (1990) Multiquadrics: A scattered data approximation scheme with applications to computational fluid-dynamics, Comput. Math. Appl. 19, 147–161.
Fasshauer, G. E. (1996) Solving partial differential equations by collocation with radial basis functions. Proc. of Chamonix. (Mehaute, A., Rabut, C, Schu-maker, L. ed.), 1–8.
Wu, Z. (1998) Solving PDE with radial basis functions and the error estimation. Adv. Comput. Math. Lectures in Pure & Applied Mathematics, 202 (Chen, Z. Li, Y., Micchelli, C. A., Xu, Y., Dekkon M. ed.).
Golberg, M.A., Chen, C.S. (1998) The method of fundamental solutions for potential, Helmholtz and diffusion problems. Boundary Integral Methods – Numerical and Mathematical Aspects (Golberg M.A. ed.), 103–176, Comput. Mech. Publ. UK.
Chen, W. (2001) Direct linearization method for nonlinear PDE’s and the related kernel RBFs. http://xxx.Ianl.gov/abs/math.NA/0110005. CoRR preprint.
Chen, W. (2001) Several new domain-type and boundary-type numerical discretization schemes with radial basis function. CoRR preprint, http://xxx.lanl.gov/abs/cs.CC/0104018/abs/cs.CC/0104018.
Chen, W. (2001) RBF-based meshless boundary knot method and boundary particle method. Proc. of China Congress on Computational Mechanics ’2001, Guangzhou, China.
Partridge, P. W., Brebbia, C. A., Wrobel, L. W. (1992) The Dual Reciprocity Boundary Element Method. Comput. Mech. Publ. UK.
Chen, W. (2001) Boundary knot method for Laplace and biharmonic problems, Proc. of the 14th Nordic Seminar on Comput. Mech. 117–120, Lund, Sweden.
Nowak, A. J., Neves, A. C. (ed.) (1994) The Multiple Reciprocity Boundary Element Method. Comput. Mech. Publ., U.K..
Schaback, R., Hon, Y. C. (2001) On unsymmetric collocation by radial basis functions. J. Appl. Math. Comp. 119, 177–186.
Fedoseyev, A.L., Friedman, M.J., Kansa, E.J. (2001) Improved multiquadratic method for elliptic partial differential equations via PDE collocation on the boundary. Comput. Math. Appl. (in press).
Zhang, X., Song, K.Z., Lu, M.W., Liu, X. (2000) Meshless methods based on collocation with radial basis functions. Comput. Mech.26, 333–343.
Westphal Jr. T., de Barchellos, C. S. (1996) On general fundamental solutions of some linear elliptic differential operators. Engng. Anal. Boundary Elements, 17, 279–285.
Liu, W. K., Jun, S. (1998) Multiple-scale reproducing kernel particle methods for large deformation problems. Int. J. Numer. Methd. Engrg. 41, 1339–1362.
Chen, W. (20001) New RBF collocation schemes and their applications. Int. Workshop for Meshfree Methods for PDE’s, Bonn, Germany. CoRR preprint, http://xxx.lanl.gov/abs/math.NA/0111220.
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Chen, W. (2003). New RBF Collocation Methods and Kernel RBF with Applications. In: Griebel, M., Schweitzer, M.A. (eds) Meshfree Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol 26. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56103-0_6
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DOI: https://doi.org/10.1007/978-3-642-56103-0_6
Publisher Name: Springer, Berlin, Heidelberg
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