A novel radial basis function method for 3D linear and nonlinear advection diffusion reaction equations with variable co
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ORIGINAL ARTICLE
A novel radial basis function method for 3D linear and nonlinear advection diffusion reaction equations with variable coefficients Xia Tian1 · Ji Lin1,2,3 Received: 17 June 2020 / Accepted: 28 August 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract The radial basis function method for 3D advection diffusion reaction equations with variable coefficients is presented. The proposed method implements the linear combination of radial basis functions which impose boundary conditions in advance, and thus such a combination with weighted parameters can be used to construct the final approximation. Furthermore, the weighted parameters are solved by substituting the approximation into governing equations. This method leads to crucial improvements in the feasibility and accuracy which can now be easily applied to general 3D nonlinear problems through linearized techniques. Finally, accuracy and efficiency of the proposed method are verified by several examples. Keywords Radial basis function · Meshless method · Quasilinearization technique · Advection diffusion reaction
1 Introduction Advection diffusion reaction equation (ADRE) is a fundamental equation which models a wide range of phenomena in engineering and science, such as the simulation of the underground water [1], the heat conduction problem [2], and the diffusion of pollutants in stream [3]. Analytical solutions are rare for several idealized problems. Therefore, numerical methods have attracted much attentions, such as the finite difference method [4–7], the finite element method [8, 9], the finite volume method [10]. During the last several decades, much effort has been devoted to developing meshless methods to overcome the mesh used in traditional mesh-based methods [11–20]. Gharib [21] investigated the meshless generalized reproducing kernel method for time-dependent ADRE with variable coefficients. Li provided the meshless local Petrov Galerkin method for multi-dimensional convection diffusion reaction * Ji Lin [email protected] 1
College of Mechanics and Materials, Hohai University, Nanjing 211100, China
2
State Key Laboratory of Acoustic, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China
3
State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
(CDRE) based on collocation method [22]. Oruç provided a meshless method based on Pascal polynomials to solve Berger equations [23] and elliptic problems [24]. Zhang proposed the element free Galerkin method for CDRE with small diffusion [25]. Dehghan [26] considered the stochastic advection-diffusion equations based on the meshless method. A local radial basis function (RBF) method was proposed for ADRE on complexly shaped domains [27]. Zhang provided a fast and stabilized meshless method for the solution of convection diffusion problems [28]. A semianalytical meshless method was proposed for 2D nonlinear ADRE in [29]. Recently, Yue [
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