The Eulerian-Lagrangian method of fundamental solutions for two-dimensional unsteady Burgers’ equations. (English) Zbl 1244.76096

Summary: The Eulerian-Lagrangian method of fundamental solutions is proposed to solve the two-dimensional unsteady Burgers’ equations. Through the Eulerian-Lagrangian technique, the quasi-linear Burgers’ equations can be converted to the characteristic diffusion equations. The method of fundamental solutions is then adopted to solve the diffusion equation through the diffusion fundamental solution; in the meantime the convective term in the Burgers’ equations is retrieved by the back-tracking scheme along the characteristics. The proposed numerical scheme is free from mesh generation and numerical integration and is a truly meshless method. Two-dimensional Burgers’ equations of one and two unknown variables with and without considering the disturbance of noisy data are analyzed. The numerical results are compared very well with the analytical solutions as well as the results by other numerical schemes. By observing these comparisons, the proposed meshless numerical scheme is convinced to be an accurate, stable and simple method for the solutions of the Burgers’ equations with irregular domain even using very coarse collocating points.


76M25 Other numerical methods (fluid mechanics) (MSC2010)
76D99 Incompressible viscous fluids
65M80 Fundamental solutions, Green’s function methods, etc. for initial value and initial-boundary value problems involving PDEs
Full Text: DOI


[1] Bateman, H., Some recent researches on the motion of fluids, Mon weather rev, 43, 163-170, (1915)
[2] Burgers, J.M., A mathematical model illustrating the theory of turbulence, () · JFM 65.0988.03
[3] Fletcher, C.A.J., Burgers’ equation: a model for all reasons, () · Zbl 0496.76091
[4] Cole, J.D., On a quasi-linear parabolic equation occurring in aerodynamics, Q appl math, 19, 225-236, (1951) · Zbl 0043.09902
[5] Hopf, E., The partial differential equation \(u_t + u u_x = \mu_{\mathit{xx}}\), Commun pure appl math, 3, 201-230, (1950)
[6] Benton, E.R.; Platzman, G.W., A table of solutions of the one-dimensional Burgers equation, Q appl math, 30, 195-212, (1972) · Zbl 0255.76059
[7] Fletcher, C.A.J., Generating exact solutions of the two-dimensional burgers’ equation, Int J numer methods fluids, 3, 213-216, (1983) · Zbl 0563.76082
[8] Bahadir, A.R., A fully implicit finite-difference scheme for two-dimensional burgers’ equation, Appl math comput, 137, 131-137, (2003) · Zbl 1027.65111
[9] Kutluay, S.; Bahadir, A.R.; Ozdes, A., Numerical solution of one-dimensional Burgers equation: explicit and exact-explicit finite difference methods, J comput appl math, 103, 251-261, (1999) · Zbl 0942.65094
[10] Radwan, S.F., Comparison of high-order accurate schemes for solving the two-dimensional unsteady burgers’ equation, J comput appl math, 103, 383-397, (2005) · Zbl 1063.65085
[11] Froncioni, A.M.; Labbe, P.; Garon, A.; Camarero, R., Interpolation-free space – time remeshing for the Burgers equation, Commun numer methods eng, 13, 875-884, (1997) · Zbl 0902.76059
[12] Kutluay, S.; Esen, A.; Dag, I., Numerical solutions of the burgers’ equation by the least-squares quadratic B-spline finite element method, J comput appl math, 167, 21-33, (2004) · Zbl 1052.65094
[13] Chino, E.; Tosaka, N., Dual reciprocity boundary element analysis of time-independent burgers’ equation, Eng anal bound elem, 21, 261-270, (1998) · Zbl 0954.76058
[14] Kakuda, K.; Tosaka, N., The generalized boundary element approach to burgers’ equation, Int J numer methods eng, 29, 245-261, (1990) · Zbl 0712.76070
[15] Hon, Y.C.; Mao, X.Z., An efficient numerical scheme for burgers’ equation, Appl math comput, 95, 37-50, (1998) · Zbl 0943.65101
[16] Li, J.C.; Hon, Y.C.; Chen, C.S., Numerical comparisons of two meshless methods using radial basis functions, Eng anal bound elem, 26, 205-225, (2002) · Zbl 1003.65132
[17] Atluri, S.N., The meshless method (MLPG) for domain & bie discretizations, (2004), Tech Science Press Forsyth, GA, USA · Zbl 1105.65107
[18] Atluri, S.N.; Shen, S., The meshless local Petrov-Galerkin (MLPG) method: a simple & less-costly alternative to the finite element and boundary element methods, Cmes, 3, 11-51, (2002) · Zbl 0996.65116
[19] Atluri, S.N.; Zhu, T., A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput mech, 22, 117-127, (1998) · Zbl 0932.76067
[20] Lin, H.; Atluri, S.N., The meshless local Petrov-Galerkin (MLPG) method for solving incompressible navier – stokes equations, Cmes, 2, 117-142, (2001)
[21] Golberg, M.A., The method of fundamental solutions for Poisson’s equations, Eng anal bound elem, 16, 205-213, (1995)
[22] Hu, S.P.; Fan, C.M.; Chen, C.W.; Young, D.L., Method of fundamental solutions for stokes’ first and second problems, J mech, 21, 25-31, (2005)
[23] Karageorghis, A.; Fairweather, G., The method of fundamental solutions for the numerical solution of the biharmonic equation, J comput phys, 69, 434-459, (1987) · Zbl 0618.65108
[24] Kupradze, V.D.; Aleksidze, M.A., The method of functional equations for the approximate solution of certain boundary value problem, Comput math math phys, 4, 4, 82-126, (1964) · Zbl 0154.17604
[25] Young, D.L.; Fan, C.M.; Tsai, C.C.; Chen, C.W.; Murugesan, K., Eulerian – lagrangian method of fundamental solutions for multi-dimensional advection – diffusion equation, Int math forum, 1, 14, 687-706, (2006) · Zbl 1143.65384
[26] Young, D.L.; Ruan, J.W., Method of fundamental solutions for scattering problems of electromagnetic waves, Cmes, 7, 223-232, (2005) · Zbl 1106.78008
[27] Young, D.L.; Chen, C.W.; Fan, C.M.; Murugesan, K.; Tsai, C.C., The method of fundamental solutions for Stokes flow in a rectangular cavity with cylinders, Eur J mech B-fluids, 24, 703-716, (2005) · Zbl 1103.76319
[28] Young, D.L.; Tsai, C.C.; Murugesan, K.; Fan, C.M.; Chen, C.W., Time-dependent fundamental solutions for homogeneous diffusion problems, Eng anal bound elem, 29, 1463-1473, (2004) · Zbl 1098.76622
[29] Young, D.L.; Tsai, C.C.; Fan, C.M., Direct approach to solve nonhomogeneous diffusion problems using fundamental solutions and dual reciprocity methods, J chin inst eng, 27, 597-609, (2004)
[30] Young, D.L.; Wang, Y.F.; Eldho, T.I., Solution of the advection – diffusion equation using the eulerian – lagrangian boundary element method, Eng anal bound elem, 24, 449-457, (2000) · Zbl 0973.76065
[31] Young, D.L., An eulerian – lagrangian method of fundamental solutions for Burger’s equation, () · Zbl 1180.76049
[32] Young DL, Tsai CC, Fan CM, Chen CW. The method of fundamental solutions and condition number analysis for inverse problem of Laplace equation. Comput Math Appl, 2007, in press, doi:10.1016/j.camwa.2007.05.015.
[33] Marin, L.; Elliott, L.; Heggs, P.J.; Ingham, D.B.; Lesnic, D.; Wen, X., Comparison of regularization methods for solving the Cauchy problem associated with the Helmholtz equation, Int J numer methods eng, 60, 1933-1947, (2004) · Zbl 1062.78015
[34] Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P., Numerical recipes in Fortran 77: the art of scientific computing, (1999), Cambridge University Press Cambridge
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.