Open Access
Issue
ESAIM: M2AN
Volume 57, Number 3, May-June 2023
Page(s) 1619 - 1655
DOI https://doi.org/10.1051/m2an/2023029
Published online 26 May 2023
  1. S. Gottlieb, D.I. Ketcheson and C.-W. Shu, Strong stability preserving Runge-Kutta and multistep time discretizations, World Scientific (2011). [Google Scholar]
  2. J.F.B.M. Kraaijevanger, Contractivity of Runge-Kutta methods. BIT Numer. Math. 31 (1991) 482–528. [Google Scholar]
  3. L. Bonaventura and A. Della Rocca, Unconditionally strong stability preserving extensions of the TR-BDF2 method. J. Sci. Comput. 70 (2017) 859–895. [Google Scholar]
  4. 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. [Google Scholar]
  5. C.-W. Shu, Total-variation-diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9 (1988) 1073–1084. [Google Scholar]
  6. X. Zhang and C.-W. Shu, Positivity-preserving high order finite difference WENO schemes for compressible Euler equations. J. Comput. Phys. 231 (2012) 2245–2258. [Google Scholar]
  7. X. Zhang and C.-W. Shu, On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229 (2010) 3091–3120. [Google Scholar]
  8. 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. [Google Scholar]
  9. W. Hundsdorfer, S.J. Ruuth and R.J. Spiteri, Monotonicity-preserving linear multistep methods. SIAM J. Numer. Anal. 41 (2003) 605–623. [Google Scholar]
  10. L. Ferracina and M.N. Spijker, Stepsize restrictions for the total-variation-diminishing property in general Runge-Kutta methods. SIAM J. Numer. Anal. 42 (2004) 1073–1093. [Google Scholar]
  11. L. Ferracina and M. Spijker, An extension and analysis of the Shu-Osher representation of Runge-Kutta methods. Math. Comput. 74 (2005) 201–219. [Google Scholar]
  12. I. Higueras, On strong stability preserving time discretization methods. J. Sci. Comput. 21 (2004) 193–223. [Google Scholar]
  13. I. Higueras, Representations of Runge-Kutta methods and strong stability preserving methods. SIAM J. Numer. Anal. 43 (2005) 924–948. [Google Scholar]
  14. S.J. Ruuth and R.J. Spiteri, Two barriers on strong-stability-preserving time discretization methods. J. Sci. Comput. 17 (2002) 211–220. [Google Scholar]
  15. H.W. Lenferink, Contractivity preserving explicit linear multistep methods. Numer. Math. 55 (1989) 213–223. [Google Scholar]
  16. J. Sand, Circle contractive linear multistep methods. BIT Numer. Math. 26 (1986) 114–122. [Google Scholar]
  17. R.J. Spiteri and S.J. Ruuth, A new class of optimal high-order strong-stability-preserving time discretization methods. SIAM J. Numer. Anal. 40 (2002) 469–491. [Google Scholar]
  18. D.I. Ketcheson, C.B. Macdonald and S. Gottlieb, Optimal implicit strong stability preserving Runge-Kutta methods. Appl. Numer. Math. 59 (2009) 373–392. [Google Scholar]
  19. G. Izzo and Z. Jackiewicz, Strong stability preserving general linear methods. J. Sci. Comput. 65 (2015) 271–298. [Google Scholar]
  20. M. Spijker, Contractivity in the numerical solution of initial value problems. Numer. Math. 42 (1983) 271–290. [Google Scholar]
  21. A. Bellen and L. Torelli, Unconditional contractivity in the maximum norm of diagonally split Runge-Kutta methods. SIAM J. Numer. Anal. 34 (1997) 528–543. [Google Scholar]
  22. K.J. in’t Hout, A note on unconditional maximum norm contractivity of diagonally split Runge-Kutta methods. SIAM J. Numer. Anal. 33 (1996) 1125–1134. [Google Scholar]
  23. C.B. Macdonald, S. Gottlieb and S.J. Ruuth, A numerical study of diagonally split Runge-Kutta methods for PDEs with discontinuities. J. Sci. Comput. 36 (2008) 89–112. [Google Scholar]
  24. D.I. Ketcheson, Step sizes for strong stability preservation with downwind-biased operators. SIAM J. Numer. Anal. 49 (2011) 1649–1660. [Google Scholar]
  25. S. Gottlieb, Z.J. Grant, J. Hu and R. Shu, High order unconditionally strong stability preserving multi-derivative implicit and IMEX Runge-Kutta methods with asymptotic preserving properties. Preprint arXiv:2102.11939 (2021). [Google Scholar]
  26. S. Blanes, A. Iserles and S. Macnamara, Positivity-preserving methods for ordinary differential equations. ESAIM: Math. Model. Numer. Anal. 56 (2022) 1843–1870. [Google Scholar]
  27. S. Ortleb and W. Hundsdorfer, Patankar-type Runge-Kutta schemes for linear PDEs, in AIP Conference Proceedings, Vol. 1863, AIP Publishing LLC (2017) 320008. [Google Scholar]
  28. S. Kopecz and A. Meister, Unconditionally positive and conservative third order modified Patankar–Runge–Kutta discretizations of production–destruction systems. BIT Numer. Math. 58 (2018) 691–728. [Google Scholar]
  29. J. Huang, T. Izgin, S. Kopecz, A. Meister and C.-W. Shu, On the stability of strong-stability-preserving modified Patankar Runge-Kutta schemes. Preprint arXiv:2205.01488 (2022). [Google Scholar]
  30. L. Ju, X. Li, Z. Qiao and J. Yang, Maximum bound principle preserving integrating factor Runge-Kutta methods for semilinear parabolic equations. J. Comput. Phys. 439 (2021) 110405. [Google Scholar]
  31. T. Tang and J. Yang, Implicit–explicit scheme for the Allen-Cahn equation preserves the maximum principle. J. Comput. Math. 34 (2016) 471–481. [Google Scholar]
  32. J. Shen, T. Tang and J. Yang, On the maximum principle preserving schemes for the generalized Allen-Cahn equation. Commun. Math. Sci. 14 (2016) 1517–1534. [Google Scholar]
  33. 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. [Google Scholar]
  34. L. Isherwood, Z.J. Grant and S. Gottlieb, Strong stability preserving integrating factor Runge-Kutta methods. SIAM J. Numer. Anal. 56 (2018) 3276–3307. [Google Scholar]
  35. L. Isherwood, Z.J. Grant and S. Gottlieb, Downwinding for preserving strong stability in explicit integrating factor Runge-Kutta methods. Pure Appl. Math. Q. 14 (2018) 3–25. [Google Scholar]
  36. L. Isherwood, Z.J. Grant and S. Gottlieb, Strong stability preserving integrating factor two-step Runge-Kutta methods. J. Sci. Comput. 81 (2019) 1446–1471. [Google Scholar]
  37. H. Zhang, J. Yan, X. Qian and S. Song, Numerical analysis and applications of explicit high order maximum principle preserving integrating factor Runge-Kutta schemes for Allen-Cahn equation. Appl. Numer. Math. 161 (2021) 372–390. [Google Scholar]
  38. 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. [Google Scholar]
  39. 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. [Google Scholar]
  40. J. Li, X. Li, L. Ju and X. Feng, Stabilized integrating factor Runge-Kutta method and unconditional preservation of maximum bound principle. SIAM J. Sci. Comput. 43 (2021) A1780–A1802. [Google Scholar]
  41. 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. [Google Scholar]
  42. H. Zhang, J. Yan, X. Qian and S. Song, Temporal high-order, unconditionally maximum-principle-preserving integrating factor multi-step methods for Allen–Cahn-type parabolic equations. Appl. Numer. Math. 186 (2023) 18–40. [Google Scholar]
  43. 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. [Google Scholar]
  44. 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) 177–22. [Google Scholar]
  45. J. Du, C. Wang, C. Qian and Y. Yang, High-order bound-preserving discontinuous Galerkin methods for stiff multispecies detonation. SIAM J. Sci. Comput. 41 (2019) B250–B273. [Google Scholar]
  46. J. Du and Y. Yang, Third-order conservative sign-preserving and steady-state-preserving time integrations and applications in stiff multispecies and multireaction detonations. J. Comput. Phys. 395 (2019) 489–510. [Google Scholar]
  47. J. Du, E. Chung and Y. Yang, Maximum-principle-preserving local discontinuous Galerkin methods for Allen-Cahn equations. Commun. Appl. Math. Comput. 4 (2022) 353–379. [Google Scholar]
  48. R. Yang, Y. Yang and Y. Xing, High order sign-preserving and well-balanced exponential Runge-Kutta discontinuous Galerkin methods for the shallow water equations with friction. J. Comput. Phys. 444 (2021) 110543. [Google Scholar]
  49. J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Contin. Dyn. Syst. 28 (2010) 1669–1691. [Google Scholar]
  50. J. Douglas Jr. and T. Dupont, Alternating-direction Galerkin methods on rectangles, in Numerical Solution of Partial Differential Equations–II, Elsevier (1971) 133–214. [Google Scholar]
  51. D.J. Eyre, An unconditionally stable one-step scheme for gradient systems. Unpublished article (1998). [Google Scholar]
  52. P. Smereka, Semi-implicit level set methods for curvature and surface diffusion motion. J. Sci. Comput. 19 (2003) 439–456. [Google Scholar]
  53. 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]
  54. Y. He, Y. Liu and T. Tang, On large time-stepping methods for the Cahn-Hilliard equation. Appl. Numer. Math. 57 (2007) 616–628. [Google Scholar]
  55. C.B. Macdonald and S.J. Ruuth, The implicit closest point method for the numerical solution of partial differential equations on surfaces. SIAM J. Sci. Comput. 31 (2010) 4330–4350. [Google Scholar]
  56. C.-B. Schönlieb and A. Bertozzi, Unconditionally stable schemes for higher order inpainting. Commun. Math. Sci. 9 (2011) 413–457. [Google Scholar]
  57. K. Chow and S.J. Ruuth, Linearly stabilized schemes for the time integration of stiff nonlinear PDEs. J. Sci. Comput. 87 (2021) 1–29. [Google Scholar]
  58. M. Hochbruck, A. Ostermann and J. Schweitzer, Exponential Rosenbrock-type methods. SIAM J. Numer. Anal. 47 (2009) 786–803. [Google Scholar]
  59. W. Hundsdorfer, A note on stability of the Douglas splitting method. Math. Comput. 67 (1998) 183–190. [Google Scholar]
  60. L. Duchemin and J. Eggers, The explicit–implicit-null method: Removing the numerical instability of PDEs. J. Comput. Phys. 263 (2014) 37–52. [Google Scholar]
  61. H. Wang, Q. Zhang, S. Wang and C.-W. Shu, Local discontinuous Galerkin methods with explicit–implicit-null time discretizations for solving nonlinear diffusion problems. Sci. China Math. 63 (2020) 183–204. [Google Scholar]
  62. L. Ju, J. Zhang, L. Zhu and Q. Du, Fast explicit integration factor methods for semilinear parabolic equations. J. Sci. Comput. 62 (2015) 431–455. [Google Scholar]
  63. 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]
  64. S. Krogstad, Generalized integrating factor methods for stiff PDEs. J. Comput. Phys. 203 (2005) 72–88. [Google Scholar]
  65. A. Ostermann, M. Thalhammer and W. Wright, A class of explicit exponential general linear methods. BIT Numer. Math. 46 (2006) 409–431. [Google Scholar]
  66. M. Bassenne, L. Fu and A. Mani, Time-accurate and highly-stable explicit operators for stiff differential equations. J. Comput. Phys. 424 (2021) 109847. [Google Scholar]
  67. M. Calvo, J.I. Montijano and L. Rández, A note on the stability of time–accurate and highly–stable explicit operators for stiff differential equations. J. Comput. Phys. 436 (2021) 110316. [Google Scholar]
  68. C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77 (1988) 439–471. [Google Scholar]
  69. S. Conde, S. Gottlieb, Z.J. Grant and J.N. Shadid, Implicit and implicit–explicit strong stability preserving Runge-Kutta methods with high linear order. J. Sci. Comput. 73 (2017) 667–690. [Google Scholar]
  70. J.C. Butcher, Trees and numerical methods for ordinary differential equations. Numer. Algorithms 53 (2010) 153–170. [Google Scholar]
  71. J.D. Lawson, Generalized Runge-Kutta processes for stable systems with large Lipschitz constants. SIAM J. Numer. Anal. 4 (1967) 372–380. [Google Scholar]
  72. Z. Jackiewicz, General Linear Methods for Ordinary Differential Equations, John Wiley & Sons (2009). [Google Scholar]
  73. H. Zhang, Repository to verify the order conditions of pRK. https://github.com/auseraccount/pRK (accessed 17 June 2022). [Google Scholar]
  74. S.M. Cox and P.C. Matthews, Exponential time differencing for stiff systems. J. Comput. Phys. 176 (2002) 430–455. [Google Scholar]
  75. S. Gottlieb and C.-W. Shu, Total variation diminishing Runge-Kutta schemes. Math. Comput. 67 (1998) 73–85. [Google Scholar]
  76. 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]
  77. A. Ostermann and M. Van Daele, Positivity of exponential Runge-Kutta methods. BIT Numer. Math. 47 (2007) 419–426. [Google Scholar]
  78. A. Ostermann and M. Thalhammer, Positivity of exponential multistep methods, in Numerical Mathematics and Advanced Applications, Springer (2006) 564–571. [Google Scholar]
  79. M. Hochbruck and A. Ostermann, Explicit exponential Runge-Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 43 (2005) 1069–1090. [Google Scholar]
  80. W. Hundsdorfer and J.G. Verwer, Numerical Solution of Time-Dependent Advection–Diffusion–Reaction Equations, Vol. 33, Springer Science & Business Media (2013). [Google Scholar]
  81. S. González-Pinto and D. Hernández-Abreu, Strong A-acceptability for rational functions. BIT Numer. Math. 43 (2003) 555–561. [Google Scholar]
  82. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Springer, Berlin, Heidelberg (1996). [Google Scholar]
  83. 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]
  84. Q. Du, L. Ju and J. Lu, Analysis of fully discrete approximations for dissipative systems and application to time-dependent nonlocal diffusion problems. J. Sci. Comput. 78 (2019) 1438–1466. [Google Scholar]
  85. Y. Li, H.G. Lee, D. Jeong and J. Kim, An unconditionally stable hybrid numerical method for solving the Allen-Cahn equation. Comput. Math. Appl. 60 (2010) 1591–1606. [Google Scholar]
  86. T. Ström, On logarithmic norms. SIAM J. Numer. Anal. 12 (1975) 741–753. [Google Scholar]

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