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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 763630, 10 pages
General Formulation of Second-Order Semi-Lagrangian Methods for Convection-Diffusion Problems
1Department of Mathematics and Information, Ludong University, Yantai 264025, China
2Department of Mathematics and Information Science, Yantai University, Yantai 264005, China
Received 18 October 2012; Accepted 7 December 2012
Academic Editor: Xinguang Zhang
Copyright © 2013 Xiaohan Long and Chuanjun Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- R. Courant, E. Isaacson, and M. Rees, “On the solution of nonlinear hyperbolic differential equations by finite differences,” Communications on Pure and Applied Mathematics, vol. 5, pp. 243–255, 1952.
- A. Robert, “A stable numerical integration scheme for the primitive meteorological equations,” Atmosphere-Ocean, pp. 35–46, 1981.
- A. Staniforth and J. Côté, “Semi-Lagrangeian integration schemes for atmospheric models-A review,” Monthly Weather Review, vol. 119, pp. 2206–2223, 1991.
- P. K. Smolarkiewicz and J. A. Pudykiewicz, “A class of semi-Lagrangian approximations for fluids,” Journal of the Atmospheric Sciences, vol. 49, no. 22, pp. 2082–2096, 1992.
- J. Douglas Jr. and T. F. Russell, “Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures,” SIAM Journal on Numerical Analysis, vol. 19, pp. 871–885, 1982.
- C. N. Dawson, C. J. Van Duijn, and M. F. Wheeler, “Characteristic-galerkin methods for contaminant transport with nonequilibrium adsorption kinetics,” SIAM Journal on Numerical Analysis, vol. 31, no. 4, pp. 982–999, 1994.
- J. Douglas Jr., R. E. Ewing, and M. F. Wheeler, “The approximation of the pressure by a mixed method in the simulation of immiscible displacement,” RAIRO Journal on Numerical Analysis, vol. 17, pp. 17–33, 1983.
- T. F. Russell, “Time stepping along characteristics with incomplete iteration for a Galerkin approximation of immiscible displacement in porous media,” SIAM Journal on Numerical Analysis, vol. 22, no. 2, pp. 970–1013, 1985.
- R. Bermejo, “On the equivalence of semi-Lagrangian schemes and particle-in-cell finite element methods,” Monthly Weather Review, vol. 118, no. 4, pp. 979–987, 1990.
- E. Süli, “Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations,” Numerische Mathematik, vol. 53, no. 4, pp. 459–483, 1988.
- Y. Yuan, “The characteristic-mixed finite element method for enhanced oil recovery simulation and optimal order L2 error estimate,” Chinese Science Bulletin, vol. 38, pp. 1066–1070, 1993.
- R. E. Ewing and H. Wang, “A summary of numerical methods for time-dependent advection-dominated partial differential equations,” Journal of Computational and Applied Mathematics, vol. 128, no. 1-2, pp. 423–445, 2001.
- G. Strang, “On the construction and comparison of different splitting schemes,” SIAM Journal on Numerical Analysis, vol. 53, pp. 506–517, 1968.
- R. E. Ewing and T. F. Russell, “Multistep Galerkin methods along characteristics for convection-diffusion problems,” in Advances in Computer Methods for Partial Differential Equations IV, R. Vichnevetsky and R. S. Stepleman, Eds., pp. 28–36, IMACS Rutgers University, New Brunswich, NJ, USA, 1981.
- H. Rui and M. Tabata, “A second order characteristic finite element scheme for convection-diffusion problems,” Numerische Mathematik, vol. 92, no. 1, pp. 161–177, 2002.
- H. Notsu and M. Tabata, “A single-step characteristic-curve finite element scheme of second order in time for the incompressible Navier-Stokes equations,” Journal of Scientific Computing, vol. 38, no. 1, pp. 1–14, 2009.
- K. Boukir, Y. Maday, and B. Métivet, “A high order characteristics method for the incompressible Navier-Stokes equations,” Computer Methods in Applied Mechanics and Engineering, vol. 116, no. 1–4, pp. 211–218, 1994.
- K. Boukir, Y. Maday, B. Métivet, and E. Razafindrakoto, “A high-order characteristics/finite element method for the incompressible Navier-Stokes equations,” International Journal for Numerical Methods in Fluids, vol. 25, no. 12, pp. 1421–1454, 1997.
- A. Bermúdez, M. R. Nogueiras, and C. Vázquez, “Numerical analysis of convection-diffusion-reaction problems with higher order characteristics/finite elements. Part I: time discretization,” SIAM Journal on Numerical Analysis, vol. 44, no. 5, pp. 1829–1853, 2006.
- A. Bermúdez, M. R. Nogueirast, and C. Vázquez, “Numerical analysis of convection-diffusion-reaction problems with higher order characteristics/finite elements. Part II: fully discretized scheme and quadrature formulas,” SIAM Journal on Numerical Analysis, vol. 44, no. 5, pp. 1854–1876, 2006.
- G. Fourestey and S. Piperno, “A second-order time-accurate ALE Lagrange-Galerkin method applied to wind engineering and control of bridge profiles,” Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 39–41, pp. 4117–4137, 2004.
- D. Xiu and G. E. Karniadakis, “A semi-Lagrangian high-order method for Navier-Stokes equations,” Journal of Computational Physics, vol. 172, no. 2, pp. 658–684, 2001.
- D. Xiu, S. J. Sherwin, S. Dong, and G. E. Karniadakis, “Strong and auxiliary forms of the semi-Lagrangian method for incompressible flows,” Journal of Scientific Computing, vol. 25, no. 1, pp. 323–346, 2005.
- M. Al-Lawatia, R. C. Sharpley, and H. Wang, “Second-order characteristic methods for advection-diffusion equations and comparison to other schemes,” Advances in Water Resources, vol. 22, no. 7, pp. 741–768, 1999.
- M. Falcone and R. Ferretti, “Convergence analysis for a class of high-order semi-lagrangian advection schemes,” SIAM Journal on Numerical Analysis, vol. 35, no. 3, pp. 909–940, 1998.
- R. Bermejo and J. Conde, “A conservative quasi-monotone semi-Lagrangian scheme,” Monthly Weather Review, vol. 130, no. 2, pp. 423–430, 2002.
- F. Xiao and T. Yabe, “Completely conservative and oscillationless semi-Lagrangian schemes for advection transportation,” Journal of Computational Physics, vol. 170, no. 2, pp. 498–522, 2001.
- K. Toda, Y. Ogata, and T. Yabe, “Multi-dimensional conservative semi-Lagrangian method of characteristics CIP for the shallow water equations,” Journal of Computational Physics, vol. 228, no. 13, pp. 4917–4944, 2009.
- E. Hairer and H. Wanner, Solving Ordinary Differential Equations I-Nonstiff Problems, Springer, 1987.
- J. D. Lambert, Computational Methods in Ordinary Differential Equations, John Wiley & Sons, London, UK, 1973.
- M. Zlamal, “Finite element methods for nonliear parabolic equations,” RAIRO Journal of Numerical Analysis, vol. 11, no. 1, pp. 93–107, 1977.
- X. Long and Y. Yuan, “Multistep characteristic method for incompressible flow in porous media,” Applied Mathematics and Computation, vol. 214, no. 1, pp. 259–270, 2009.