## Numerical solutions of the Burgers’ equation by the least-squares quadratic B-spline finite element method.(English)Zbl 1052.65094

Summary: A least-squares quadratic B-spline finite element method for calculating the numerical solutions of the one-dimensional Burgers-like equations with appropriate boundary and initial conditions is presented. Three test problems have been studied to demonstrate the accuracy of the present method. Results obtained by the method have been compared with the exact solution of each problem and are found to be in good agreement with each other. A Fourier stability analysis of the method is also investigated.

### MSC:

 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 35Q53 KdV equations (Korteweg-de Vries equations) 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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### References:

 [1] Abd-el-Malek, M.B.; El-Mansi, S.M.A., Group theoretic methods applied to burgers’ equation, J. comput. appl. math., 115, 1-12, (2000) · Zbl 0942.35157 [2] Ali, A.H.A.; Gardner, G.A.; Gardner, L.R.T., A collocation solution for burgers’ equation using cubic B-spline finite elements, Comput. methods appl. mech. engrg., 100, 325-337, (1992) · Zbl 0762.65072 [3] Bateman, H., Some recent researches on the motion of fluids, Monthly weather rev., 43, 163-170, (1915) [4] Benton, E.; Platzman, G.W., A table of solutions of the one-dimensional Burgers equations, Quart. appl. math., 30, 195-212, (1972) · Zbl 0255.76059 [5] Burgers, J.M., A mathematical model illustrating the theory of turbulence, Adv. appl. mech., 1, 171-199, (1948) [6] Caldwell, J.; Smith, P., Solution of Burgers equation with a large Reynolds number, Appl. math. modelling, 6, 381-385, (1982) · Zbl 0496.76029 [7] Cole, J.D., On a quasi-linear parabolic equations occurring in aerodynamics, Quart. appl. math., 9, 225-236, (1951) · Zbl 0043.09902 [8] Evans, D.J.; Abdullah, A.R., The group explicit method for the solution of Burger’s equation, Computing, 32, 239-253, (1984) · Zbl 0523.65071 [9] Gardner, L.R.T.; Gardner, G.A.; Dogan, A., A petrov – galerkin finite element scheme for burgers’ equation, Arab. J. sci. engrg., 22, 99-109, (1997) · Zbl 0908.65089 [10] L.R.T. Gardner, G.A. Gardner, A. Dogan, A least-squares finite element scheme for Burgers’ equation, University of Wales, Bangor, Mathematics, Preprint 96.01, 1996. · Zbl 0867.76040 [11] Hopf, E., The partial differential equation ut + uux = μuxx, Comm. pure appl. math., 3, 201-230, (1950) [12] Kakuda, K.; Tosaka, N., The generalized boundary element approach to burgers’ equation, Internat. J. numer. methods engrg., 29, 245-261, (1990) · Zbl 0712.76070 [13] Kutluay, S.; Bahadir, A.R.; Özdes, A., Numerical solution of one-dimensional Burgers equationexplicit and exact-explicit finite difference methods, J. comput. appl. math., 103, 251-261, (1999) · Zbl 0942.65094 [14] E.L. Miller, Predictor-corrector studies of Burgers’ model of turbulent flow, M.S. Thesis, University of Delaware, Newark, DE, 1966. [15] Mittal, R.C.; Singhal, P., Numerical solution of Burger’s equation, Comm. numer. methods engrg., 9, 397-406, (1993) · Zbl 0782.65147 [16] Mittal, R.C.; Singhal, P., Numerical solution of periodic burger equation, Indian J. pure appl. math., 27, 7, 689-700, (1996) · Zbl 0859.76053 [17] H. Nguyen, J. Reynen, A space-time finite element approach to Burgers’ equation, in: C. Taylor, E. Hinton, D.R.J. Owen, E. Onate (Eds.), Numerical Methods for Non-Linear Problems, Vol. 2, Pineridge Publisher, Swansea, 1982, pp. 718-728. [18] Nguyen, H.; Reynen, J., A space-time least square finite element scheme for advection – diffusion equations, Comput. methods appl. mech. engrg., 42, 331-342, (1984) · Zbl 0517.76089 [19] Özis, T.; Özdes, A., A direct variational methods applied to burgers’ equation, J. comput. appl. math., 71, 163-175, (1996) · Zbl 0856.65114 [20] Prenter, P.M., Splines and variational methods, (1975), Wiley New York · Zbl 0344.65044 [21] Smith, G.D., Numerical solution of partial differential equations: finite difference methods, (1987), Clarendon Press Oxford [22] Varoglu, E.; Finn, W.D.L., Space time finite elements incorporating characteristics for the burgers’ equations, Internat. J. numer. methods engrg., 16, 171-184, (1980) · Zbl 0449.76076
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