INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS Int. J. Numer. Model. 2013; 26 Published online 15 January 2013 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/jnm.1882 High-order absorbing boundary conditions for the meshless radial point interpolation method in the frequency domain Thomas Kaufmann 1, ,  and Christian Engström 2 1 The School of Electrical & Electronic Engineering, University of Adelaide, Adelaide, SA, 5005, Australia 2 Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden SUMMARY The meshless radial point interpolation method (RPIM) in frequency domain for electromagnetic scattering prob- lems is presented. This method promises high accuracy in a simple collocation approach using radial basis functions. The treatment of high-order non-reflecting boundary conditions for open waveguides is discussed and implemented up to fourth-order. RPIM allows the direct calculation of high-order spatial derivatives without the introduction of auxiliary variables. High-order absorbing boundary conditions offer a choice of absorbing angles for each degree of spatial derivatives. For general applications, a set of these absorbing angles is calculated using global optimization. Numerical experiments show that at the same computational cost, the numerical reflections of the absorbing boundary conditions are much lower than conventional perfectly matched layers, especially at high angles of incidence. Copyright © 2013 John Wiley & Sons, Ltd. Received 24 May 2012; Revised 24 October 2012; Accepted 4 November 2012 KEY WORDS: absorbing boundary conditions; radial point interpolation; finite-difference methods; radial basis functions; meshless methods; frequency domain modeling 1. INTRODUCTION Methods based on radial basis functions (RBFs) interpolation are an emerging new field in many areas of science including statistical analysis [1], computer vision [2], and neural networks [3]. In this paper, a collocation method is implemented. This type of methods seek an approximate solution from a finite dimensional subspace by requiring that Maxwell’s equations are satisfied at a set of so called col- location points. This approach avoids an explicit mesh topology and all the computational overhead associated with its generation. Also, the movement and addition of nodes is greatly simplified and modeling of complex geometries is possible by conformal node placement. In the classical colloca- tion approach, low-order polynomial basis functions are used [4]. However, it was recently realized that RBFs have superior interpolation properties [5, 6]. Collocation methods based on non-polynomial basis functions or radially dependent high-order polynomials are frequently called meshless meth- ods. Many different types of meshless methods are known, such as the moving least squares methods [7], or smooth particle hydrodynamics [8] to name a few. Collocation methods based on RBFs have been introduced as the Kansa method [9, 10] or the radial point interpolation method (RPIM) [11, 12] where the basis functions are transformed into explicit form. Theoretical investigations have shown that exponential convergence rates can be expected when solving differential equations [13,14]. These are much higher rates than in commonly used numerical methods such as the finite-difference time- domain (FDTD) scheme or the finite element method (FEM) with h-refinement. Practically, this means that when increasing the discretization, the accuracy is dramatically improved in comparison with the  Correspondence to: Thomas Kaufmann, The School of Electrical & Electronic Engineering, University of Adelaide, Adelaide, SA, 5005, Australia.  Email: thomas.kaufmann@adelaide.edu.au Copyright © 2013 John Wiley & Sons, Ltd :478–492