Semi-analytical Time Differencing Methods for Stiff Problems

详细信息    查看全文

• 作者：Chang-Yeol Jung (1)
  Thien Binh Nguyen (1)
  
  1. Department of Mathematical Sciences ; School of Natural Science ; Ulsan National Institute of Science and Technology ; UNIST-gil 50 ; Ulsan ; 689-798 ; Republic of Korea
  
• 关键词：Semi ; analytical time differencing ; Stiff problems ; Singular perturbation analysis ; Transition layers ; Boundary layers ; Initial layers ; Nonlinear ordinary and partial differential equations ; 74S25 ; 65L04 ; 34E15 ; 80M35 ; 76R50 ; 35B40 ; 80M12
• 刊名：Journal of Scientific Computing
• 出版年：2015
• 出版时间：May 2015
• 年：2015
• 卷：63
• 期：2
• 页码：355-373
• 全文大小：576 KB
• 参考文献：1. Ascher, UM, Ruuth, SJ, Wetton, BTR (1995) Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32: pp. 797-823 CrossRef
  2. Ascher, UM, Ruuth, SJ, Spiteri, RJ (1997) Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25: pp. 151-167 CrossRef
  3. Ashi, H.: Numerical methods for stiff systems, Ph.D. thesis, the University of Nottingham (2008)
  4. Aziz, et al. Z.A.: Fourth-order time stepping for stiff PDEs via integrating factor. Adv. Sci. Lett. 19(1), 170鈥?73 (2013)
  5. Boyd, JP (2001) Chebyshev and Fourier Spectral Methods. Dover, Mineola, NY
  6. Cash, JR (2003) Efficient numerical methods for the solution of stiff initial-value problems and differential algebraic equations. Proc. R. Soc. Lond. A. 459: pp. 797-815 CrossRef
  7. Canuto, C, Hussaini, MY, Quarteroni, A, Zang, TA (1988) Spectral Methods in Fluid Dynamics. Springer Series in Computational Physics. Springer, Berlin CrossRef
  8. Cox, SM, Matthews, PC (2002) Exponential time differencing for stiff systems. J. Comput. Phys. 176: pp. 430-455 CrossRef
  9. Weinan, E, Engquist, Bjorn, Li, Xiantao, Ren, Weiqing, Vanden-Eijnden, Eric (2007) The heterogeneous multiscale method: a review. Commun. Comput. Phys. 2: pp. 367-450
  10. Engquist, B, Tsai, Y-H (2005) Heterogeneous multiscale methods for stiff ordinary differential equations. Math. Comput. 74: pp. 1707-1742 CrossRef
  11. Fornberg, B (1996) A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge, UK
  12. Holmes, MH (1995) Introduction to Perturbation Methods. Springer, New York CrossRef
  13. Han, H., Kellogg, R. B.: A method of enriched subspaces for the numerical solution of a parabolic singular perturbation problem. In: Computational and Asymptotic Methods for Boundary and Interior Layers, Boole Press Conf. Ser. 4, Dublin, pp. 46鈥?2 (1982)
  14. Hyman, JM, Nicolaenko, B (1986) The Kuramoto-Sivashinsky equation: a bridge between PDE鈥檚 and dynamical systems. Physica D 18: pp. 113-126 CrossRef
  15. Hoz, FDL, Vadillo, F (2008) An exponential time differencing method for the nonlinear Schr枚dinger equation. Comput. Phys. Commun. 179: pp. 449-456 CrossRef
  16. Hairer, E, Wanner, G (1996) Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems. Springer, Berlin CrossRef
  17. Johnson, RS (2005) Singular Perturbation Theory. Springer Science+Business Media Inc, New York
  18. Jung, C (2008) Finite elements scheme in enriched subspaces for singularly perturbed reaction-diffusion problems on a square domain. Asymptot. Anal. 57: pp. 41-69
  19. Jin, S, Levermore, CD (1996) Numerical schemes for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys. 126: pp. 449-467 CrossRef
  20. Jung, C, Nguyen, TB (2013) Semi-analytical numerical methods for convection-dominated problems with turning points. Int. J. Numer. Anal. Model. 10: pp. 314-332
  21. Jung, C., Nguyen, T.B.: New time differencing methods for spectral methods (submitted)
  22. Jung, C, Temam, R (2007) Asymptotic analysis for singularly perturbed convection-diffusion equations with a turning point. J. Math. Phys. 48: pp. 065301 CrossRef
  23. Jung, C, Temam, R (2009) Finite volume approximation of one-dimensional stiff convection-diffusion equations. J. Sci. Comput. 41: pp. 384-410 CrossRef
  24. Jin, S, Xin, Z (1995) The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math. 48: pp. 235-276 CrossRef
  25. Kevorkian, J, Cole, JD (1996) Multiple Scale and Singular Perturbation Methods. Springer, Berlin CrossRef
  26. Kellogg, RB, Stynes, M (2008) Layers and corner singularities in singularly perturbed elliptic problems. BIT 48: pp. 309-314 CrossRef
  27. Kevrekidis, IG, Samaey, G (2009) Equation-free multiscale computation: algorithms and applications. Annu. Rev. Phys. Chem. 60: pp. 321-344 CrossRef
  28. Kassam, A.-K., Trefethen, L.N.: Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26(4), 1214鈥?233.
  29. Krogstad, S (2005) Generalized integrating factors methods for stiff PDEs. J. Comput. Phys. 203: pp. 72-88 CrossRef
  30. Marion, M, Temam, R (1989) Nonlinear Galerkin methods. SIAM J. Numer. Anal. 26: pp. 1139-1157 CrossRef
  31. Mai-Duy, N, Pan, D, Phan-Thien, N, Khoo, BC (2013) Dissipative particle dynamics modeling of low Reynolds number incompressible flows. J. Rheol. 57: pp. 585 CrossRef
  32. Nicolaenko, B, Scheurer, B, Temam, R (1985) Some global dynamical properties of the Kuramoto-Sivashisky equations: nonlinear stability and attractors. Physica D 16: pp. 155-183 CrossRef
  33. O鈥橫alley, RE (2008) Singularly perturbed linear two-point boundary value problems. SIAM Rev. 50: pp. 459-482 CrossRef
  34. Shih, S, Kellogg, RB (1987) Asymptotic analysis of a singular perturbation problem. SIAM J. Math. Anal. 18: pp. 1467-1511 CrossRef
  35. Stynes, M (2005) Steady-state convection-diffusion problems. Acta Numer. 14: pp. 445-508 CrossRef
  36. Trefethen, LN (2000) Spectral methods in Matlab. Society for Industrial and Applied Mathematics, Philadelphia CrossRef
  37. Tao, M, Owhadi, H, Marsden, JE (2010) Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and Hamiltonian systems with hidden slow dynamics via flow averaging. Multiscale Model. Simul. 8: pp. 1269-1324 CrossRef
  38. Wasow, W (1985) Linear Turning Point Theory. Spinger, New York CrossRef
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algorithms
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Mathematical and Computational Physics
  
• 出版者：Springer Netherlands
• ISSN：1573-7691

文摘

A semi-analytical method is developed based on conventional integrating factor (IF) and exponential time differencing (ETD) schemes for stiff problems. The latter means that there exists a thin layer with a large variation in their solutions. The occurrence of this stiff layer is due to the multiplication of a very small parameter \(\varepsilon \) with the transient term of the equation. Via singular perturbation analysis, an analytic approximation of the stiff layer, which is called a corrector, is sought for and embedded into the IF and ETD methods. These new schemes are then used to approximate the non-stiff part of the solution. Since the stiff part is resolved analytically by the corrector, the new method outperforms the conventional ones in terms of accuracy. In this paper, we apply our new method for both problems of ordinary differential equations and some partial differential equations.