The radial basis function differential quadrature method with ghost points

Abstract We propose a simple approach to improve the accuracy of the Radial Basis Function Differential Quadrature (RBF-DQ) method for the solution of elliptic boundary value problems. While the traditional RBF-DQ method places the centers exclusively inside the domain, the proposed method expands the region for the centers allowing them to lie both inside and outside the computational domain. Furthermore, we seek an improvement to determine the shape parameter for the radial basis function by using the modified Franke’s formula to find an initial search interval for the leave-one-out cross-validation method, which is a widely used method for the determination of the shape parameter. Both 2D and 3D numerical examples are presented to demonstrate the effectiveness of the proposed method.

[1]  R. Bellman,et al.  DIFFERENTIAL QUADRATURE: A TECHNIQUE FOR THE RAPID SOLUTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS , 1972 .

[2]  Y. L. Wu,et al.  Development of RBF-DQ method for derivative approximation and its application to simulate natural convection in concentric annuli , 2002 .

[3]  Gregory E. Fasshauer,et al.  On choosing “optimal” shape parameters for RBF approximation , 2007, Numerical Algorithms.

[4]  Alper Korkmaz,et al.  Solitary wave simulations of Complex Modified Korteweg-de Vries Equation using differential quadrature method , 2009, Comput. Phys. Commun..

[5]  T. Tran-Cong,et al.  Incompressible smoothed particle hydrodynamics-moving IRBFN method for viscous flow problems , 2015 .

[6]  Yinnian He,et al.  Stabilized multiscale finite element method for the stationary Navier–Stokes equations☆ , 2009 .

[7]  Fawang Liu,et al.  A novel finite volume method for the Riesz space distributed-order diffusion equation , 2017, Comput. Math. Appl..

[8]  Yong Duan,et al.  Multi-quadric quasi-interpolation method coupled with FDM for the Degasperis-Procesi equation , 2016, Appl. Math. Comput..

[9]  E. Kansa.  MULTIQUADRICS--A SCATTERED DATA APPROXIMATION SCHEME WITH APPLICATIONS TO COMPUTATIONAL FLUID-DYNAMICS-- II SOLUTIONS TO PARABOLIC, HYPERBOLIC AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS , 1990 .

[10]  Lei-Hsin Kuo,et al.  On the Selection of a Good Shape Parameter for RBF Approximation and Its Application for Solving PDEs , 2015 .

[11]  Chang Shu,et al.  Integrated radial basis functions‐based differential quadrature method and its performance , 2007 .

[12]  Xuejun Xu,et al.  A monotone finite volume method for time fractional Fokker-Planck equations , 2017, 1711.00598.

[13]  Shmuel Rippa,et al.  An algorithm for selecting a good value for the parameter c in radial basis function interpolation , 1999, Adv. Comput. Math..

[14]  Graeme Fairweather,et al.  The method of fundamental solutions for elliptic boundary value problems , 1998, Adv. Comput. Math..

[15]  B. Fornberg,et al.  A numerical study of some radial basis function based solution methods for elliptic PDEs , 2003 .

[16]  Yi Xu,et al.  Simulation of linear and nonlinear advection-diffusion-reaction problems by a novel localized scheme , 2020, Appl. Math. Lett..

[17]  R. Franke.  Scattered data interpolation: tests of some methods , 1982 .

[18]  Yunqing Huang,et al.  An efficient two grid method for miscible displacement problem approximated by mixed finite element methods , 2019, Comput. Math. Appl..

[19]  Chuanzeng Zhang,et al.  A local radial basis function collocation method for band structure computation of phononic crystals with scatterers of arbitrary geometry , 2018, Applied Mathematical Modelling.

[20]  D. Young,et al.  Evaluation of Multi-Order Derivatives by Local Radial Basis Function Differential Quadrature Method , 2013 .

[21]  Artur Krowiak,et al.  Hermite type radial basis function-based differential quadrature method for higher order equations , 2016 .

[22]  Daniel W. Watson,et al.  The radial basis function-differential quadrature method for elliptic problems in annular domains , 2020, J. Comput. Appl. Math..

[23]  C. Fan,et al.  The method of approximate particular solutions for solving certain partial differential equations , 2012 .