An analysis of the linear advection –diffusion equation using mesh-free and mesh-dependent methods I. Boztosun a, * , A. Charafi b a Department of Physics, Erciyes University, 38039 Kayseri, Turkey b Computational Mathematics Group, University of Portsmouth, Mercantile House PO1 2EG, UK Received 5 November 2001; received in accepted form 31 May 2002; accepted 5 June 2002 Abstract The numerical solution of advection–diffusion equations has been a long standing problem and many numerical methods that attempt to find stable and accurate solutions have to resort to artificial methods to stabilize the solution. In this paper, we present a meshless method based on thin plate radial basis functions (RBF). The efficiency of the method in terms of computational processing time, accuracy and stability is discussed. The results are compared with the findings from the dual reciprocity/boundary element and finite difference methods as well as the analytical solution. Our analysis shows that the RBFs method, with its simple implementation, generates excellent results and speeds up the computational processing time, independent of the shape of the domain and irrespective of the dimension of the problem. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Meshless methods; Radial basis functions; Thin plate spline; Finite difference method; Boundary element method; Dual reciprocity method; Partial differential equation; Linear advection–diffusion problem 1. Introduction The solution of the advection–diffusion equation is a long standing problem and many numerical methods have been introduced to model accurately the interaction between advective and diffusive processes. This modeling is the most challenging task in the numerical approximation of the partial differential equations [1] and the available numerical solutions are very sophisticated in order to avoid two undesirable features: oscillatory behavior and numerical diffusion, which are mainly due to the advection term when it dominates (see Refs. [2–5] for a detailed discussion). The advection–diffusion equation is the basis of many physical and chemical phenomena, and its use has also spread into economics, financial forecasting and other fields [1]. In general, the numerical solution of advection– diffusion equations has been dominated by either finite difference, finite element or boundary element methods. These methods are derived from local interpolation schemes and require a mesh to support the application. It is well known that finite difference and finite element solutions of the advection– diffusion equation present numerical pro- blems of oscillations and damping. On the other hand, boundary element solutions seem to be relatively free from these problems, as shown by Brebbia and Skerget [3]. The numerical solution of this equation is a difficult task because of two reasons; Firstly, the nature of the governing equation, which includes first-order and second-order partial derivatives in space. According to the value of k (diffusion coefficient) and v (advection coefficient), the equation becomes parabolic for diffusion dominated processes or hyperbolic for advection dominated processes. Traditional finite difference methods (FDM) are generally accurate for solving the former but not the latter, in which case oscillations and smoothing of the wave front are introduced. This can be interpreted as the artificial diffusion intrinsic to these methods [3–8]. Secondly, since the above-mentioned numerical methods are all mesh-dependent, it is vital to construct an appropriate mesh to obtain a better approxi- mation to the problem. However, the construction of an appropriate mesh is not an easy task and sometimes the problems cannot be solved because of the lack of an appropriate mesh structure [1,9–13]. Because of the complexity of mesh-generation, con- siderable effort has been devoted in recent years to the development of mesh-free methods, also called meshless methods. These methods aim to eliminate the structure of the mesh and approximate the solution using a set of quasi- random points rather than points from a grid discretization. 0955-7997/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S0955-7997(02)00053-X Engineering Analysis with Boundary Elements 26 (2002) 889–895 www.elsevier.com/locate/enganabound * Corresponding author. E-mail addresses: boztosun@erciyes.edu.tr (I. Boztosun), abdel.charafi@port.ac.uk (A. Charafi).