4th Dolomites Workshop on Constructive Approximation and Applications
Alba di Canazei (Italy), September 8-13, 2016
IMPLEMENTATION OF MULTIPLE BOUNDARY CONDITIONS IN RBF COLLOCATION METHOD
ALI SAFDARI-VAIGHANI1∗, ELISABETH LARSSON2, ALFA HERYUDONO3
1 Department of Mathematics, Allameh Tabataba’i University, Tehran, Iran [email protected]
2 Department of Information Technology, Uppsala University, Uppsala, Sweden [email protected]
3 Department of Mathematics, University of Massachusetts Dartmouth, Dartmouth, Massachusetts, USA
Numerical solutions of the PDEs are routinely computed by re- searchers in many different areas. Collocation methods based on RBFs have become important when trying to obtain the numerical solution of various ordinary differential equations (ODEs) and partial differ- ential equations (PDEs). Even for the one-dimensional case, how to implement multiple boundary conditions for a time-dependent global collocation problem is not obvious. In this case, we need to enforce two boundary conditions at each end point resulting in a total of four boundary conditions at the two boundary points. Fictitious or ghost point methods have been commonly used as a way to enforce multiple boundary conditions in finite difference methods. The implementation for global collocation methods such as pseudospectral methods is due to Fornberg [1].
The aim of this talk is to show that the Rosenau equation, as an initial-boundary value problem with multiple boundary conditions, can be implemented using RBF approximation methods [2]. For this aim, the fictitious point method and the resampling method are studied in combination with an RBF collocation method. The numerical experi- ments show that both methods perform well.
Keywords: collocation method, radial basis function, multiple bound- ary conditions
Classification: MSC 65M70, MSC 35G31
∗ Speaker.
1
2 A. SAFDARI, E. LARSSON, A. HERYUDONO
References
1. Fornberg, B.: A pseudospectral fictitious point method for high order initial- boundary value problems. SIAM J. Sci. Comput. 28(5), 1716-1729 (electronic) (2006).
2. Safdari-Vaighani, A., Larsson, E., Heryudono, A.: Fictitious point and resam- pling radial basis function methods for solving the Rosenau equation (2016, in preparation).
