Tailored finite point method for solving one-dimensional Burgers' equation

References

• E.N. Aksan, Quadratic B-spline finite element method for numerical solution of the Burgers' equation, Appl. Math. Comput. 174 (2006), pp. 884–896. doi: 10.1016/j.amc.2005.05.020
   Web of Science ®Google Scholar
• A. Asaithambi, Numerical solution of the Burgers' equation by automatic differentiation, Appl. Math. Comput. 216 (2010), pp. 2700–2708. doi: 10.1016/j.amc.2010.03.115
   Web of Science ®Google Scholar
• A.R. Bahadir and M. Sağlam, A mixed finite difference and boundary element approach to one-dimensional Burgers' equation, Appl. Math. Comput. 160 (2005), pp. 663–673. doi: 10.1016/j.amc.2003.11.024
   Web of Science ®Google Scholar
• A.N. Brooks and T.J.R. Hughes, Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg. 32 (1982), pp. 199–259. doi: 10.1016/0045-7825(82)90071-8
   Web of Science ®Google Scholar
• J.D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9(3) (1951), pp. 225–236.
   Web of Science ®Google Scholar
• A. Dogan, A Galerkin finite element approach to Burgers' equation, Appl. Math. Comput. 157 (2004), pp. 331–346. doi: 10.1016/j.amc.2003.08.037
   Web of Science ®Google Scholar
• F. Gao and C. Chi, Numerical solution of nonlinear Burgers' equation using high accuracy multi-quadric quasi-interpolation, Appl. Math. Comput. 229 (2014), pp. 414–421. doi: 10.1016/j.amc.2013.12.035
   Web of Science ®Google Scholar
• R.J. Gelinas and S.K. Doss, and K. Miller, The moving finite element method: Applications to general partial differential equations with multiple large gradients, J. Comput. Phys. 40 (1981), pp. 202–249. doi: 10.1016/0021-9991(81)90207-2
   Web of Science ®Google Scholar
• H. Han, Z. Huang, and R.B. Kellogg, A tailored finite point method for a singular perturbation problem on an unbounded domain, J. Sci. Comput. 36 (2008), pp. 243–261. doi: 10.1007/s10915-008-9187-7
   Web of Science ®Google Scholar
• H. Han, Y.-T. Shih, and C.-C. Tsai, Tailored finite point method for numerical solutions of singular perturbed eigenvalue problems, Adv. Appl. Math. Mech. 6(3) (2014), pp. 376–402.
   Web of Science ®Google Scholar
• A. Hashemian and H.M. Shodja, A meshless approach for solution of Burgers' equation, J. Comput. Appl. Math. 220 (2008), pp. 226–239. doi: 10.1016/j.cam.2007.08.014
   Web of Science ®Google Scholar
• I.A. Hassanien, A.A. Salama, and H.A. Hosham, Fourth-order finite difference method for solving Burgers equation, Appl. Math. Comput. 170 (2005), pp. 781–800. doi: 10.1016/j.amc.2004.12.052
   Web of Science ®Google Scholar
• E. Hopf, The partial differential equation ut+uux=νuxx, Comm. Pure Appl. Math. 3 (1950), pp. 201–230. doi: 10.1002/cpa.3160030302
   Web of Science ®Google Scholar
• Z. Huang and X. Yang, Tailored finite point method for first order wave equation, J. Sci. Comput. 49 (2011), pp. 351–366. doi: 10.1007/s10915-011-9468-4
   Web of Science ®Google Scholar
• C. Johson and A. Szepessy, On the convergence of a finite element method for a nonlinear hyperbolic conservation law, Math. Comput. 49 (1987), pp. 427–444. doi: 10.1090/S0025-5718-1987-0906180-5
   Web of Science ®Google Scholar
• S. Kutluay, A.R. Bahadir, and A. Özdes, Numerical solution of the one-dimensional Burgers' equation: Explicit and exact-explicit finite difference methods, J. Comput. Appl. Math. 103 (1999), pp. 251–261. doi: 10.1016/S0377-0427(98)00261-1
   Web of Science ®Google Scholar
• S. Kutluay, A. Esen, and I. Dag, Numerical solutions of the Burgers' equation by the least-squares quadratic B-spline finite element method, J. Comput. Appl. Math. 167 (2004), pp. 21–33. doi: 10.1016/j.cam.2003.09.043
   Web of Science ®Google Scholar
• W. Liao and J. Zhu, Efficient and accurate finite difference schemes for solving one-dimensional Burgers' equation, Int. J. Comput. Math. 88 (2011), pp. 2575–2590. doi: 10.1080/00207160.2010.548519
   Web of Science ®Google Scholar
• K. Pandey, L. Verma, and A.K. Verma, On a finite difference scheme for Burgers' equation, Appl. Math. Comput. 215 (2009), pp. 2206–2214. doi: 10.1016/j.amc.2009.08.018
   Web of Science ®Google Scholar
• Y.-T. Shih, R.B. Kellogg, and P. Tsai, A tailored finite point method for convection–diffusion–reaction problems, J. Sci. Comput. 43(2) (2010), pp. 239–260. doi: 10.1007/s10915-010-9354-5
   Web of Science ®Google Scholar
• Y.-T. Shih, R.B. Kellogg, and Y. Chang, Characteristic tailored finite point method for convection dominated convection–diffusion–reaction problems, J. Sci. Comput. 47 (2011), pp. 198–215. doi: 10.1007/s10915-010-9433-7
   Web of Science ®Google Scholar
• W.L. Wood, An exact solution for Burger's equation, Comm. Numer. Methods. Engrg. 22(7) (2006), pp. 797–798. doi: 10.1002/cnm.850
   Web of Science ®Google Scholar
• P.-G. Zhang and J.-P. Wang, A predictor–corrector compact finite difference scheme for Burgers' equation, Appl. Math. Comput. 219 (2012), pp. 892–898. doi: 10.1016/j.amc.2012.06.064
   Web of Science ®Google Scholar