Numerical solution of Burgers' equation by cubic B-spline quasi-interpolation

Applied Mathematics and Computation

Volumn 208, Issue 1, 2009, Pages 260-272

  Zhu, Chun Gang ^a   Wang, Ren Hong ^a  

^a DALIAN UNIVERSITY OF TECHNOLOGY   (China)

Author keywords

B spline; Burgers' equation; Numerical solution; Quasi interpolation; Second order hyperbolic equation

Indexed keywords

INTERPOLATION; NUMERICAL ANALYSIS;

B-SPLINE; BURGERS' EQUATION; NUMERICAL SOLUTION; QUASI-INTERPOLATION; SECOND-ORDER HYPERBOLIC EQUATION;

SPLINES;

EID: 58549091278     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2008.11.045     Document Type: Article

References (29)

1. Bateman H. Some recent researches on the motion of fluids. Monthly Weather Rev. 43 (1915) 163-170
2. Inc M. On numerical solution of Burgers, equation by homotopy analysis method. Phys. Lett. A 372 (2008) 356-360
3. Burgers J.M. A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. 1 (1948) 171-199
4. xx = 0. Commun. Pure Appl. Math. 3 (1950) 201-230
5. Cole J.D. On a quasi-linear parabolic equation occurring in aerodynamics. Quart. Appl. Math. 9 (1951) 225-236
6. Christie I., Griffiths D.F., Mitchell A.R., and Sanz-Serna J.M. Product approximation for nonlinear problems in the finite element method. IMA J. Numer. Anal. 1 (1981) 253-266
7. Herbst B.M., Schoombie S.W., and Mitchell A.R. A moving Petrov-Galerkin method for transport equations. Int. J. Numer. Methods Eng. 18 (1982) 1321-1336
8. E.L. Miller, Predictor-Corrector Studies of Burgers' Model of Turbulent Flow, M.S. Thesis, University of Delaware, Newark, DE, 1966.
9. Christie I., and Mitchell A.R. Upwinding of high order Galerkin methods in conduction-convection problems. Int. J. Numer. Methods Eng. 12 (1978) 1764-1771
10. Ali A.H.A., Gardner L.R.T., and Gardner G.A. A collocation method for Burgers' equation using cubic B-splines. Comput. Methods Appl. Mech. Eng. 100 (1992) 325-337
11. Daǧ I., Irk D., and Saka B. A numerical solution of the Burgers' equation using cubic B-splines. Appl. Math. Comput. 163 (2005) 199-211
12. Daǧ I., Saka B., and Boz A. B-spline Galerkin methods for numerical solutions of the Burgers' equation. Appl. Math. Comput. 166 (2005) 506-522
13. Ramadan M.A., El-Danaf T.S., and Alaal F. A numerical solution of the Burgers' equation using septic B-splines. Chaos, Solitons Fract. 26 (2005) 795-804
14. Saka B., and Daǧ I. Quartic B-spline collocation methods to the numerical solutions of the Burgers' equation. Chaos, Solitons Fract. 32 (2007) 1125-1137
15. Saka B., and Daǧ I. A numerical study of the Burgers' equation. J. Franklin Inst. 345 (2008) 328C348
16. Zaki S.I. A quintic B-spline finite elements scheme for the KdVB equation. Comput. Methods Appl. Eng. 188 (2000) 121-134
17. Hardy R.L. Theory and applications of the multiquadric-biharmonic method, 20 years of discovery 1968-1988. Comput. Math. Appl. 19 (1990) 163-208
18. Kansa E.J. Multiquadricsa scattered data approximation scheme with applications to computational fluid dynamics I. Comput. Math. Appl. 19 (1990) 127-145
19. Kansa E.J. Multiquadricsa scattered data approximation scheme with applications to computational fluid dynamics II. Comput. Math. Appl. 19 (1990) 147-161
20. Hon Y.C., and Wu Z. A quasi-interpolation method for solving ordinary differential equations. Int. J. Numer. Methods Eng. 48 (2000) 1187-1197
21. Wu Z. Dynamically knots setting in meshless method for solving time dependent propagations equation. Comput. Methods Appl. Mech. Eng. 193 (2004) 1221-1229
22. Wu Z. Dynamical knot and shape parameter setting for simulating shock wave by using multi-quadric quasi-interpolation. Eng. Anal. Bound. Elem. 29 (2005) 354-358
23. Chen R., and Wu Z. Solving partial differential equation by using multiquadric quasi-interpolation. Appl. Math. Comput. 186 (2007) 1502-1510
24. Hon Y.C., and Mao X.Z. An efficient numerical scheme for Burgers' equation. Appl. Math. Comput. 95 (1998) 37-50
25. Chen R., and Wu Z. Applying multiquadric quasi-interpolation to solve Burgers' equation. Appl. Math. Comput. 172 (2006) 472-484
26. Farin G. Curves and Surfaces for CAGD. fifth ed. (2001), Morgan Kaufmann, San Francisco
27. P. Sablonnière, Quasi-interpolants splines sobre particiones uniforms, in: First Meeting in Approximation Theory of the University of Jaén, Ubeda, June 29-July 2, 2000. Prépublication IRMAR 00-38, Rennes, June 2000.
28. Sablonnière P. Univariate spline quasi-interpolants and applications to numerical analysis. Rend. Sem. Mat. Univ. Pol. Torino 63 (2005) 211-222
29. Nguyen H., and Reynen J. A space time finite element approach to Burgers's equation. In: Hinton E., et al. (Ed). Numerical Methods for Non-linear Problems vol. 3 (1987), Pineridge Press 718-728