An adaptive meshless method of line based on radial basis functions

Document Type : Research Article


Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran.


In this paper, an adaptive meshless method of line is applied to distribute the nodes in the spatial domain. In many cases in meshless methods, it is also necessary for the chosen nodes to have certain smoothness properties. The set of nodes is also required to satisfy certain constraints. In this paper, one of these constraints is investigated. The aim of this manuscript is the implementation of an algorithm for selection of the nodes satisfying a given constraint, in the meshless method of line. This algorithm is applied to some illustrative examples to show the efficiency of the algorithm and its ability to increase the accuracy.


1. Behrens, J and Iske, A. Grid-free adaptive semi-Lagrangian advection using radial basis functions, Computers & Mathematics with Applications, 43 (3{5) (2002) 319-327.
2. Belytschko, T., Krongauze, Y., Organ, D., Fleming, M. and Krysl, P Meshless methods: An overview and recent developments, Computer Methods in Applied Mechanics and Engineering, 139 (1996) 3-47 (special issue on Meshless Methods).
3. Bozzini, M., Lenarduzzi, L. and Schaback, R. Adaptive interpolation by scaled multiquadrics, Advances in Computational Mathematics, 16(4) (2002) 375-387.
4. Cao, W., Huang, W. and Russell, R. D. A study of monitor functions for two dimensional adaptive mesh generation, SIAM Journal on Scientific Computing, 20(6)(1999) 1978{1994.
5. Carey, G. F. and Dinh, H. T. Grading functions and mesh redistribution,
SIAM Journalon Numerical Analysis , 22(5)(1985) 1028-1040.
6. Fasshauer, G. E. Mesh free Approximation Methods with MATLAB. World Scientific Co. Pte. Ltd., Singapore, 2007.
7. Ferreira, A. J. M., Kansa, E. J., Fasshauer, G. E. and Leitao, V. M. A. Progress on Meshless Methods, Computational Methods in Applied Sciences, Springer 2009.
8. Hon, Y. C. Multiquadric collocation method with adaptive technique for problems with boundary layer, International Journal of Applied Science and Computations, 6(3) (1999) 173-184.
9. Hon, Y. C., Chen, C. S. and Schaback, R. Scientific Computing with Radial Basis Functions. Draft version 0.0, Cambridge, 2003.
10. Hon, Y. C, Schaback, R. and Zhou, X. An adaptive greedy algorithm for solving large RBF collocation problems, Numerical Algorithms, 32(1) (2003)13-25.
11. Kansa, E. J. Multiquadrics-A scattered data approximation scheme with applications to computational fuid-dynamics-I surface approximations and partial derivative estimates, Computer and Mathematics with Applications, 19 (1990) 127-145.
12. Kansa, E. J. Multiquadrics-A scattered data approximation scheme with applications to omputational fuid dynamics- II. Solution to parabolic, hyperbolic and elliptic partial differential equations, Computer and Mathematics with Appllications, 19 (1990) 147-161.
13. Kautsky, J. and Nichols, N. K. Equi-distributing meshes with constraints, SIAM Journal on Scientificc and Statistical Computing, 1(4) (1980) 449-511.
14. Quan, S. A meshless method of lines for the numerical solution of KdV equation using radial basis functions, Engineering Analysis with Boundary Elements, 33 (2009) 1171-1180.
15. Sanz-Serna, J. and Christie, I. A simple adaptive technique for nonlinear wave problems, Journal of Computational Physics, 67 (1986) 348-360.
16. Sarra, S. A. Adaptive radial basis function methods for time dependent partial differential equations, Applied Numerical Mathematics, 54 (1)(2005) 7994.
17. Schaback, R. and Wendland, H. Adaptive greedy techniques for approximate solution of large RBF systems, Numerical Algorithms, 24(3)(2000)239-254.
18. Schiesser, W. E. The numerical method of lines: integration of partial differential equations, San Diego, California: Academic Press; 1991.