Open Access
Issue
ESAIM: M2AN
Volume 58, Number 1, January-February 2024
Page(s) 191 - 221
DOI https://doi.org/10.1051/m2an/2023101
Published online 16 February 2024
  1. A. Arrarás, K.J. in’t Hout, W. Hundsdorfer and L. Portero, Modified Douglas splitting methods for reaction–diffusion equations. BIT Numer. Math. 57 (2017) 261–285. [CrossRef] [Google Scholar]
  2. J. Butcher, Runge–Kutta methods for ordinary differential equations. Numerical Analysis and Optimization: NAO-III, Muscat, Oman, January 2014 (2015) 37–58. [CrossRef] [Google Scholar]
  3. M. Cheng and J.A. Warren, Controlling the accuracy of unconditionally stable algorithms in the Cahn–Hilliard equation. Phys. Rev. E 75 (2007) 017702. [CrossRef] [PubMed] [Google Scholar]
  4. W. Chen, S. Conde, C. Wang, X. Wang and S.M. Wise, A linear energy stable scheme for a thin film model without slope selection. J. Sci. Comput. 52 (2012) 546–562. [Google Scholar]
  5. A. Christlieb, K. Promislow and Z. Xu, On the unconditionally gradient stable scheme for the Cahn–Hilliard equation and its implementation with Fourier method. Commun. Math. Sci. 11 (2013) 345–360. [CrossRef] [MathSciNet] [Google Scholar]
  6. J.M. Church, Z. Guo, P.K. Jimack, A. Madzvamuse, K. Promislow, B. Wetton, S.M. Wise and F. Yang, High accuracy benchmark problems for Allen–Cahn and Cahn–Hilliard dynamics. Commun. Comput. Phys. 26 (2019). [Google Scholar]
  7. S.M. Cox and P.C. Matthews, Exponential time differencing for stiff systems. J. Comput. Phys. 176 (2002) 430–455. [Google Scholar]
  8. J. Douglas, Alternating direction methods for three space variables. Numer. Math. 4 (1962) 41–63. [CrossRef] [MathSciNet] [Google Scholar]
  9. J. Douglas and H.H. Rachford, On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82 (1956) 421–439. [CrossRef] [Google Scholar]
  10. Q. Du and W.-x. Zhu, Stability analysis and application of the exponential time differencing schemes. J. Comput. Math. (2004) 200–209. [Google Scholar]
  11. Q. Du, L. Ju, X. Li and Z. Qiao, Maximum principle preserving exponential time differencing schemes for the nonlocal Allen–Cahn equation. SIAM J. Numer. Anal. 57 (2019) 875–898. [CrossRef] [MathSciNet] [Google Scholar]
  12. Q. Du, L. Ju, X. Li and Z. Qiao, Maximum bound principles for a class of semilinear parabolic equations and exponential time-differencing schemes. SIAM Rev. 63 (2021) 317–359. [CrossRef] [MathSciNet] [Google Scholar]
  13. D.J. Eyre, An unconditionally stable one-step scheme for gradient systems. Unpublished Article (1998). [Google Scholar]
  14. Z. Fu and J. Yang, Energy-decreasing exponential time differencing Runge–Kutta methods for phase-field models. J. Comput. Phys. 454 (2022) 110943. [CrossRef] [Google Scholar]
  15. H. Gomez and T.J. Hughes, Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models. J. Comput. Phys. 230 (2011) 5310–5327. [CrossRef] [MathSciNet] [Google Scholar]
  16. Y. Gong, Q. Wang, Y. Wang and J. Cai, A conservative Fourier pseudo-spectral method for the nonlinear Schrödinger equation. J. Comput. Phys. 328 (2017) 354–370. [CrossRef] [MathSciNet] [Google Scholar]
  17. S. Gottlieb, D.I. Ketcheson and C.-W. Shu, Strong Stability Preserving Runge–Kutta and Multistep Time Discretizations. World Scientific (2011). [CrossRef] [Google Scholar]
  18. Y. He, Y. Liu and T. Tang, On large time-stepping methods for the Cahn–Hilliard equation. Appl. Numer. Math. 57 (2007) 616–628. [CrossRef] [MathSciNet] [Google Scholar]
  19. M. Hochbruck and A. Ostermann, Explicit exponential Runge–Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 43 (2005) 1069–1090. [CrossRef] [MathSciNet] [Google Scholar]
  20. M. Hochbruck, A. Ostermann and J. Schweitzer, Exponential Rosenbrock-type methods. SIAM J. Numer. Anal. 47 (2009) 786–803. [CrossRef] [Google Scholar]
  21. J. Huang and C.-W. Shu, Bound-preserving modified exponential Runge–Kutta discontinuous Galerkin methods for scalar hyperbolic equations with stiff source terms. J. Comput. Phys. 361 (2018) 111–135. [CrossRef] [MathSciNet] [Google Scholar]
  22. W. Hundsdorfer, A note on stability of the Douglas splitting method. Math. Comput. 67 (1998) 183–190. [CrossRef] [Google Scholar]
  23. L. Ju, X. Li and Z. Qiao, Generalized SAV-exponential integrator schemes for Allen–Cahn type gradient flows. SIAM J. Numer. Anal. 60 (2022) 1905–1931. [CrossRef] [MathSciNet] [Google Scholar]
  24. S. Lee and J. Kim, Effective time step analysis of a nonlinear convex splitting scheme for the Cahn–Hilliard equation. Comun. Sci. 25 (2019) 448–460. [Google Scholar]
  25. S. Lee, C. Lee, H.G. Lee and J. Kim, Comparison of different numerical schemes for the Cahn–Hilliard equation. J. Korean Soc. Ind. Appl. Math. 17 (2013) 197–207. [MathSciNet] [Google Scholar]
  26. D. Li, Effective maximum principles for spectral methods. Ann. Appl. Math. 37 (2021) 131–290. [CrossRef] [MathSciNet] [Google Scholar]
  27. D. Li, Z. Qiao and T. Tang, Characterizing the stabilization size for semi-implicit Fourier-spectral method to phase field equations. SIAM J. Numer. Anal. 54 (2016) 1653–1681. [Google Scholar]
  28. B. Li, J. Yang and Z. Zhou, Arbitrarily high-order exponential cut-off methods for preserving maximum principle of parabolic equations. SIAM J. Sci. Comput. 42 (2020) A3957–A3978. [CrossRef] [Google Scholar]
  29. H.-L. Liao, B. Ji and L. Zhang, An adaptive BDF2 implicit time-stepping method for the phase field crystal model. IMA J. Numer. Anal. 42 (2022) 649–679. [CrossRef] [MathSciNet] [Google Scholar]
  30. V.T. Luan, A. Ostermann, Exponential Rosenbrock methods of order five–construction, analysis and numerical comparisons. J. Comput. Appl. Math. 255 (2014) 417–431. [CrossRef] [MathSciNet] [Google Scholar]
  31. A. Ostermann and M. Van Daele, Positivity of exponential Runge–Kutta methods. BIT Numer. Math. 47 (2007) 419–426. [CrossRef] [Google Scholar]
  32. A. Prothero and A. Robinson, On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math. Comput. 28 (1974) 145–162. [CrossRef] [Google Scholar]
  33. C.-B. Schönlieb, A. Bertozzi, M. Burge and L. He, Image inpainting using a fourth-order total variation flow. In: SAMPTA’09 (2009). [Google Scholar]
  34. J. Shen and X. Yang, Numerical approximations of Allen–Cahn and Cahn–Hilliard equations. Discrete Contin. Dyn. Syst. 28 (2010) 1669–1691. [CrossRef] [MathSciNet] [Google Scholar]
  35. J. Shen and X. Zhang, Discrete maximum principle of a high order finite difference scheme for a generalized Allen–Cahn equation. Commun. Math. Sci. 20 (2022) 1409–1436. [CrossRef] [MathSciNet] [Google Scholar]
  36. J. Shen, T. Tang and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Vol. 41. Springer Science & Business Media (2011). [CrossRef] [Google Scholar]
  37. J. Shen, J. Xu and J. Yang, A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61 (2019) 474–506. [CrossRef] [MathSciNet] [Google Scholar]
  38. P. Smereka, Semi-implicit level set methods for curvature and surface diffusion motion. J. Sci. Comput. 19 (2003) 439–456. [CrossRef] [MathSciNet] [Google Scholar]
  39. T. Ström, On logarithmic norms. SIAM J. Numer. Anal. 12 (1975) 741–753. [CrossRef] [MathSciNet] [Google Scholar]
  40. Sum of the reciprocal of sine squared, https://math.stackexchange.com/questions/544228, Accessed Feb. 19 (2022). [Google Scholar]
  41. T. Tang and J. Yang, Implicit-explicit scheme for the Allen–Cahn equation preserves the maximum principle. J. Comput. Math. 34 (2016) 471–481. [MathSciNet] [Google Scholar]
  42. H. Wang, C.-W. Shu and Q. Zhang, Stability and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for advection-diffusion problems. SIAM J. Numer. Anal. 53 (2015) 206–227. [Google Scholar]
  43. C. Xu and T. Tang, Stability analysis of large time-stepping methods for epitaxial growth models. SIAM J. Numer. Anal. 44 (2006) 1759–1779. [Google Scholar]
  44. J. Xu and X. Xu, Lack of robustness and accuracy of many numerical schemes for phase-field simulations. Math. Models Methods Appl. Sci. (2023) 1–26. [Google Scholar]
  45. J. Xu, Y. Li, S. Wu and A. Bousquet, On the stability and accuracy of partially and fully implicit schemes for phase field modeling. Comput. Methods Appl. Mech. Eng. 345 (2019) 826–853. [Google Scholar]
  46. V.S. Yadav, V. Maurya, M.K. Rajpoot and J. Jaglan, Spatiotemporal pattern formations in stiff reaction-diffusion systems by new time marching methods. Appl. Math. Comput. 431 (2022) 127299. [Google Scholar]
  47. V.S. Yadav, N. Ganta, B. Mahato, M.K. Rajpoot and Y.G. Bhumkar, New time-marching methods for compressible Navier-Stokes equations with applications to aeroacoustics problems. Appl. Math. Comput. 419 (2022) 126863. [Google Scholar]
  48. X. Yang, Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys. 327 (2016) 294–316. [CrossRef] [MathSciNet] [Google Scholar]
  49. J. Yang, Q. Du and W. Zhang, Uniform lp-bound of the Allen–Cahn equation and its numerical discretization. Int. J. Numer. Anal. Model. 15 (2018) 213–227. [MathSciNet] [Google Scholar]
  50. H. Zhang, J. Yan, X. Qian, X. Chen and S. Song, Explicit third-order unconditionally structure-preserving schemes for conservative Allen–Cahn equations. J. Sci. Comput. 90 (2022) 1–29. [CrossRef] [Google Scholar]
  51. H. Zhang, J. Yan, X. Qian and S. Song, Up to fourth-order unconditionally structure-preserving parametric single-step methods for semilinear parabolic equations. Comput. Methods Appl. Mech. Eng. 393 (2022) 114817. [CrossRef] [Google Scholar]
  52. H. Zhang, X. Qian and S. Song, Third-order accurate, large time-stepping and maximum-principle-preserving schemes for the Allen-Cahn equation. Numer. Algorithms (2023) 1–38. [Google Scholar]
  53. H. Zhang, X. Qian, J. Xia and S. Song, Efficient inequality-preserving integrators for differential equations satisfying forward Euler conditions. ESAIM: Math. Model. Numer. Anal. 57 (2023) 1619–1655. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  54. H. Zhang, X. Qian, J. Xia and S. Song, Unconditionally maximum-principle-preserving parametric integrating factor two-step Runge–Kutta schemes for parabolic sine-Gordon equations. CSIAM Trans. Appl. Math. 4 (2023) 177–224. [CrossRef] [MathSciNet] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you