A Computational Meshfree Technique for the Numerical Solution of the Two-Dimensional Coupled Burgers' Equations

In this paper we employ a meshfree technique for the numerical solution of the two-dimensional coupled Burgers' equations. Combination of the collocation method using the radial basis functions (RBFs) with first-order accurate forward difference approximation is employed for obtaining a meshfree solution of the problem. Three different types of RBFs are used for this purpose. Performance of the proposed method is successfully tested in terms of various error norms. In the case of non-availability of exact solution, performance of the new method is compared with the existing methods. The elementary stability analysis is established theoretically and is also supported by numerical results.

[1]  Zongmin Wu,et al.  Convergence error estimate in solving free boundary diffusion problem by radial basis functions method , 2003 .

[2]  E. Kansa.  Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates , 1990 .

[3]  J. Cole.  On a quasi-linear parabolic equation occurring in aerodynamics , 1951 .

[4]  Clive A. J. Fletcher,et al.  Generating exact solutions of the two‐dimensional Burgers' equations , 1983 .

[5]  Paul Arminjon,et al.  Numerical solution of burgers' equations in two space dimensions☆ , 1979 .

[6]  Robert Schaback,et al.  On unsymmetric collocation by radial basis functions , 2001, Appl. Math. Comput..

[7]  Benny Y. C. Hon,et al.  An efficient numerical scheme for Burgers' equation , 1998, Appl. Math. Comput..

[8]  Abdul-Majid Wazwaz,et al.  Multiple-front solutions for the Burgers equation and the coupled Burgers equations , 2007, Appl. Math. Comput..

[9]  Clive A. J. Fletcher,et al.  A comparison of finite element and finite difference solutions of the one- and two-dimensional Burgers' equations , 1983 .

[10]  Carsten Franke,et al.  Solving partial differential equations by collocation using radial basis functions , 1998, Appl. Math. Comput..

[11]  Samir F. Radwan.  On the Fourth-Order Accurate Compact ADI Scheme for Solving the Unsteady Nonlinear Coupled Burgers’ Equations , 1999 .

[12]  A. Refik Bahadir,et al.  A fully implicit finite-difference scheme for two-dimensional Burgers' equations , 2003, Appl. Math. Comput..

[13]  J. Burgers.  A mathematical model illustrating the theory of turbulence , 1948 .

[14]  R. E. Carlson,et al.  The parameter R2 in multiquadric interpolation , 1991 .

[15]  Ching-Shyang Chen,et al.  A numerical method for heat transfer problems using collocation and radial basis functions , 1998 .

[16]  Kwok Fai Cheung,et al.  Multiquadric Solution for Shallow Water Equations , 1999 .

[17]  Xiong Zhang,et al.  Meshless methods based on collocation with radial basis functions , 2000 .

[18]  Jichun Li,et al.  Radial basis function method for 1-D and 2-D groundwater contaminant transport modeling , 2003 .

[19]  S. G. Rubin,et al.  A cubic spline approximation for problems in fluid mechanics , 1975 .

[20]  Joseph D. Ward,et al.  Collocation discretizations of the transport equation with radial basis functions , 2003, Appl. Math. Comput..

[21]  Siraj-ul-Islam,et al.  A comparative study of numerical solutions of a class of KdV equation , 2008, Appl. Math. Comput..

[22]  Mehdi Dehghan,et al.  The solution of coupled Burgers' equations using Adomian-Pade technique , 2007, Appl. Math. Comput..

[23]  Siraj-ul-Islam,et al.  A meshfree method for the numerical solution of the RLW equation , 2009 .

[24]  Carsten Franke,et al.  Convergence order estimates of meshless collocation methods using radial basis functions , 1998, Adv. Comput. Math..

[25]  E. Kansa,et al.  Exponential convergence and H‐c multiquadric collocation method for partial differential equations , 2003 .

[26]  Jinqiao Duan,et al.  Limit set of trajectories of the coupled viscous Burgers' equations , 1996, Applied Mathematics Letters.