Analysis and applications of the exponential time differencing schemes and their contour integration modifications

BIT Numerical Mathematics

Volumn 45, Issue 2, 2005, Pages 307-328

  Du, Qiang ^a   Zhu, Wenxiang ^a  

^a PENNSYLVANIA STATE UNIVERSITY   (United States)

Author keywords

Allen Cahn equations; Contour integrals; Energy estimates; Exponential time differencing; Fourier analysis; Fourier spectral methods; Maximum principle; Phase transitions; Stability; Time integration schemes

Indexed keywords

EID: 27144518174     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10543-005-7141-8     Document Type: Article

References (18)

1. G. Akrivis, M. Crouzeix, and C. Makridakis, Implicit-explicit multistep methods for quasilinear parabolic equations, Numer. Math., 82 (1999), pp. 521-541.
2. G. Beylkin, J. M. Keiser, and L. Vozovoi, A new class of time discretization schemes for the solution of nonlinear PDEs, J. Comput. Phys., 147 (1998), pp. 362-387.
3. G. Caginalp and E. Socolovsky, Efficient computation of a sharp interface by spreading via phase field methods, Applied Mathematics Letters, 2 (1989), pp. 117-120.
4. L.-Q. Chen, C. Wolverton, V. Vaithyanathan, and Z. Liu, Modeling solid-state phase transformations and microstructure evolution, MRS Bulletin, 26 (2001), pp. 197-292.
5. L.-Q. Chen and J. Shen, Applications of semi-implicit Fourier-spectral method to phase field equations, Comput. Physics Comm., 108 (1998), pp. 147-158.
6. S. M. Cox and P. C. Matthews, Exponential time differencing for stiff systems, J. Comp. Phys., 176 (2002), pp. 430-455.
7. Q. Du and W. Zhu, Stability analysis and applications of the exponential time differencing schemes, Journal of Computational Mathematics, 22(2) (2004), pp. 200-209.
8. E. Weinan, Analysis of the heterogeneous multiscale method for ordinary differential equations, Comm. Math. Sci., 3 (2003), pp. 423-436.
9. W. Gear and I. Kevrekidis, Projective methods for stiff differential equations: problems with gaps in their eigenvalue spectrum, SIAM Journal on Scientific Computing, 24 (2003), pp. 1091-1106.
10. W. Gear, I. Kevrekidis, J. Hyman, P. Kevrekidis, O. Runborg, and C. Theodoropoulos, Equation Free Multiscale Computation: enabling microscopic simulators to perform system-level tasks, to appear in Communications in the Mathematical Sciences, 2003.
11. W. George and J. Warren, A parallel 3d dendritic growth simulator using the phase-field method, Journal of Computational Physics, 177 (2002), pp. 264-283.
12. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems, Springer-Verlag, New York, 1999.
13. A. Kassam and L. N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAM Journal on Scientific Computing, 26 (2005), pp. 1214-1233.
14. A. Khachaturyan, Theory of Structural Transformations in Solids, John Wiley, New York, 1983.
15. J. Lambert, Numerical Methods for Ordinary Differential Systems The Initial Value Problem, John Wiley and Sons, New York, 1991.
16. D. R. Mott, E. S. Oran, and B. van Leer, A quasi-steady-state solver for the stiff ordinary differential equations of reaction kinetics, J. Comp. Phys., 164 (2002), pp. 407-468.
17. P. Yu and Q. Du, A Variational Construction of Anisotropic Mobility in Phase-Field Simulation, preprint (2003).
18. J. Zhu and L.Q. Chen, private communications, 2003.