Communications on Applied Mathematics and Computation >

2022 , Vol. 4 >Issue 1: 353 - 379

DOI: https://doi.org/10.1007/s42967-020-00118-x

Maximum-Principle-Preserving Local Discontinuous Galerkin Methods for Allen-Cahn Equations

Expand
  • 1 Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China;
    2 Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China;
    3 Department of Mathematics, The Chinese University of Hong Kong, Hong Kong SAR, China;
    4 Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA

Received date: 2020-07-04

  Revised date: 2020-11-24

  Online published: 2022-03-01

Supported by

Jie Du is supported by the National Natural Science Foundation of China under Grant Number NSFC 11801302 and Tsinghua University Initiative Scientifc Research Program. Eric Chung is supported by Hong Kong RGC General Research Fund (Projects 14304217 and 14302018). The third author is supported by the NSF grant DMS-1818467.

Fold

Abstract

In this paper, we study the classical Allen-Cahn equations and investigate the maximumprinciple-preserving (MPP) techniques. The Allen-Cahn equation has been widely used in mathematical models for problems in materials science and fuid dynamics. It enjoys the energy stability and the maximum-principle. Moreover, it is well known that the AllenCahn equation may yield thin interface layer, and nonuniform meshes might be useful in the numerical solutions. Therefore, we apply the local discontinuous Galerkin (LDG) method due to its fexibility on h-p adaptivity and complex geometry. However, the MPP LDG methods require slope limiters, then the energy stability may not be easy to obtain. In this paper, we only discuss the MPP technique and use numerical experiments to demonstrate the energy decay property. Moreover, due to the stif source given in the equation, we use the conservative modifed exponential Runge-Kutta methods and thus can use relatively large time step sizes. Thanks to the conservative time integration, the bounds of the unknown function will not decay. Numerical experiments will be given to demonstrate the good performance of the MPP LDG scheme.

Key words： Maximum-principle-preserving; Local discontinuous Galerkin methods; Allen-Cahn equation; Conservative exponential integrations

Cite this article

Jie Du, Eric Chung, Yang Yang . Maximum-Principle-Preserving Local Discontinuous Galerkin Methods for Allen-Cahn Equations[J]. Communications on Applied Mathematics and Computation, 2022 , 4(1) : 353 -379 . DOI: 10.1007/s42967-020-00118-x

References

1. Arnold, D.:An interior penalty fnite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742-760 (1982)
2. Bassi, F., Rebay, S.:A high-order accurate discontinuous fnite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131, 267-279 (1997)
3. Cahn, J.W., Hilliard, J.E.:Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258-267 (1958)
4. Cahn, J.W., Hilliard, J.E.:Free energy of a nonuniform system. Ⅲ. Nucleation in a two-component incompressible fuid. J. Chem. Phys. 31, 688-699 (1959)
5. Chen, Z., Huang, H., Yan, J.:Third order maximum-principle-satisfying direct discontinuous Galerkin method for time dependent convection difusion equations on unstructured triangular meshes. J. Comput. Phys. 308, 198-217 (2016)
6. Cheng, Y., Shu, C.-W.:A discontinuous Galerkin fnite element method for time dependent partial differential equations with higher order derivatives. Math. Comput. 77, 699-730 (2008)
7. Chuenjarern, N., Xu, Z., Yang, Y.:High-order bound-preserving discontinuous Galerkin methods for compressible miscible displacements in porous media on triangular meshes. J. Comput. Phys. 378, 110-128 (2019)
8. Chung, E., Lee, C.S.:A staggered discontinuous Galerkin method for convection-difusion equations. J. Numer. Math. 20, 1-31 (2012)
9. Cockburn, B., Hou, S., Shu, C.-W.:The Runge-Kutta local projection discontinuous Galerkin fnite element method for conservation laws. IV:the multidimensional case. Math. Comput. 54, 545-581 (1990)
10. Cockburn, B., Lin, S.Y., Shu, C.-W.:TVB Runge-Kutta local projection discontinuous Galerkin fnite element method for conservation laws. Ⅲ:one-dimensional systems. J. Comput. Phys. 84, 90-113 (1989)
11. Cockburn, B., Shu, C.-W.:TVB Runge-Kutta local projection discontinuous Galerkin fnite element method for conservation laws. Ⅱ:general framework. Math. Comput. 52, 411-435 (1989)
12. Cockburn, B., Shu, C.-W.:The Runge-Kutta discontinuous Galerkin method for conservation laws. V:Multidimensional systems. J. Comput. Phys. 141, 199-224 (1998)
13. Cockburn, B., Shu, C.-W.:The local discontinuous Galerkin method for time-dependent convectiondifusion systems. SIAM J. Numer. Anal. 35, 2440-2463 (1998)
14. Du, J., Chung, E.:An adaptive staggered discontinuous Galerkin method for the steady state convection-difusion equation. J. Sci. Comput. 77, 1490-1518 (2018)
15. Du, J., Wang, C., Qian, C., Yang, Y.:High-order bound-preserving discontinuous Galerkin methods for stif multispecies detonation. SIAM J. Sci. Comput. 41, B250-B273 (2019)
16. Du, J., Yang, Y.:Maximum-principle-preserving third-order local discontinuous Galerkin methods on overlapping meshes. J. Comput. Phys. 377, 117-141 (2019)
17. Du, J., Yang, Y.:Third-order conservative sign-preserving and steady-state-preserving time integrations and applications in stif multispecies and multireaction detonations. J. Comput. Phys. 395, 489- 510 (2019)
18. Du, J., Yang, Y., Chung, E.:Stability analysis and error estimates of local discontinuous Galerkin method for convection-difusion equations on overlapping meshes. BIT Numer. Math. 59, 853-876 (2019)
19. Du, Q., Ju, L., Li, X., Qiao, Z.:Maximum principle preserving exponential time diferencing schemes for the nonlocal Allen-Cahn equation. SIAM J. Numer. Anal. 57, 875-898 (2018)
20. Gottlieb, S., Shu, C.-W., Tadmor, E.:Strong stability-preserving high-order time discretization method. SIAM Rev. 43, 89-112 (2001)
21. Guo, L., Yang, Y.:Positivity-preserving high-order local discontinuous Galerkin method for parabolic equations with blow-up solutions. J. Comput. Phys. 289, 181-195 (2015)
22. Guo, H., Yang, Y.:Bound-preserving discontinuous Galerkin method for compressible miscible displacement problem in porous media. SIAM J. Sci. Comput. 39, A1969-A1990 (2017)
23. Hou, T.L., Tang, T., Yang, J.:Numerical analysis of fully discretized Crank-Nicolson scheme for fractional-in-space Allen-Cahn equations. J. Sci. Comput. 72, 1214-1231 (2017)
24. Huang, J., Shu, C.-W.:Bound-preserving modifed exponential Runge-Kutta discontinuous Galerkin methods for scalar hyperbolic equations with stif source terms. J. Comput. Phys. 361, 111-135 (2018)
25. Li, X., Shu, C.-W., Yang, Y.:Local discontinuous Galerkin method for the Keller-Segel chemotaxis model. J. Sci. Comput. 73, 943-967 (2017)
26. Liu, H., Yan, J.:The direct discontinuous Galerkin methods for difusion problems. SIAM J. Numer. Anal. 47, 675-698 (2009)
27. Liu, Y., Shu, C.-W., Tadmor, E., Zhang, M.:Central local discontinuous Galerkin method on overlapping cells for difusion equations. ESAIM Math. Model. Numer. Anal. 45, 1009-1032 (2011)
28. Reed, W.H., Hill, T.R.:Triangular mesh method for the neutron transport equation. Los Alamos Scientifc Laboratory Report LA-UR-73-479, Los Alamos (1973)
29. Riviere, B., Wheeler, M.F., Girault, V.:Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I. Comput. Geosci. 8, 337-360 (1999)
30. Shen, J., Tang, T., Yang, J.:On the maximum principle preserving schemes for the generalized AllenCahn equation. Commun. Math. Sci. 14, 1517-1534 (2016)
31. Shen, J., Xu, J., Yang, J.:A new class of efcient and robust energy stable schemes for gradient fows. SIAM Rev. 61, 474-506 (2019)
32. Shen, J., Xu, J., Yang, J.:The scalar auxiliary variable (SAV) approach for gradient fows. J. Comput. Phys. 353, 407-416 (2018)
33. Shen, J., Yang, X.:Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Contin. Dyn. Syst. 28, 1669-1691 (2010)
34. Srinivasan, S., Poggie, J., Zhang, X.:A positivity-preserving high order discontinuous Galerkin scheme for convection-difusion equations. J. Comput. Phys. 366, 120-143 (2018)
35. Tang, T., Yang, J.:Implicit-explicit scheme for the Allen-Cahn equation preserves the maximum principle. J. Comput. Math. 34, 471-481 (2016)
36. Wang, C., Wise, S., Lowengrub, J.:An energy-stable and convergent fnite-diference scheme for the phase feld crystal equation. SIAM J. Numer. Anal. 47, 2269-2288 (2009)
37. Wheeler, M.:An elliptic collocation fnite element method with interior penalties. SIAM J. Numer. Anal. 15, 152-161 (1978)
38. Xiong, T., Qiu, J.-M., Xu, Z.:High order maximum-principle-preserving discontinuous Galerkin method for convection-difusion equations. SIAM J. Sci. Comput. 37, A583-A608 (2015)
39. Xu, C., Tang, T.:Stability analysis of large time-stepping methods for epitaxial growth models. SIAM J. Numer. Anal. 44, 1759-1779 (2006)
40. Xu, Z., Yang, Y., Guo, H.:High-order bound-preserving discontinuous Galerkin methods for wormhole propagation on triangular meshes. J. Comput. Phys. 390, 323-341 (2019)
41. Yang, X.:Linear, frst and second-order, unconditionally energy stable numerical schemes for the phase feld model of homopolymer blends. J. Comput. Phys. 327, 294-316 (2016)
42. Yang, X., Han, D.:Linearly frst- and second-order, unconditionally energy stable schemes for the phase feld crystal model. J. Comput. Phys. 330, 1116-1134 (2017)
43. Zhang, X., Shu, C.-W.:On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229, 3091-3120 (2010)
44. Zhang, Y., Zhang, X., Shu, C.-W.:Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection-difusion equations on triangular meshes. J. Comput. Phys. 234, 295-316 (2013)
Options
Outlines

/

〈 〉

Copyright © Editorial By Communications on Applied Mathematics and Computation
沪ICP备09014157