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Low Regularity Integrators for the Conservative Allen–Cahn Equation with a Nonlocal Constraint

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Abstract

In contrast to the classical Allen–Cahn equation, the conservative Allen–Cahn equation with a nonlocal Lagrange multiplier not only satisfies the maximum bound principle (MBP) and energy dissipation law but also ensures mass conservation. Many existing schemes often fail to preserve all these properties at the discrete level or require high regularity in time on the exact solution for convergence analysis. In this paper, we construct a new class of low regularity integrators (LRIs) for time discretization of the conservative Allen–Cahn equation by repeatedly using Duhamel’s formula. The proposed first- and second-order LRI schemes are shown to conserve mass unconditionally and satisfy the MBP under some time step size constraints. Temporal error estimates for these schemes are derived under a low regularity assumption that the exact solution is only Lipschitz continuous in time, followed by a rigorous proof for energy stability of the corresponding time-discrete solutions. Various numerical experiments and comparisons in two and three dimensions are presented to verify the theoretical results and illustrate the performance of the LRI schemes, especially when the interfacial parameter approaches zero.

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References

  1. Aihara, S., Takaki, T., Takada, N.: Multi-phase-field modeling using a conservative Allen-Cahn equation for multiphase flow. Comput. Fluids 178, 141–151 (2019)

    MathSciNet  Google Scholar 

  2. Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27(6), 1085–1095 (1979)

    Google Scholar 

  3. Brassel, M., Bretin, E.: A modified phase field approximation for mean curvature flow with conservation of the volume. Math. Meth. Appl. Sci. 34(10), 1157–1180 (2011)

    MathSciNet  Google Scholar 

  4. Calvo, M.C., Schratz, K.: Uniformly accurate low regularity integrators for the Klein-Gordon equation from the classical to nonrelativistic limit regime. SIAM J. Numer. Anal. 60(2), 888–912 (2022)

    MathSciNet  Google Scholar 

  5. Chen, X., Hilhorst, D., Logak, E.: Mass conserving Allen-Cahn equation and volume preserving mean curvature flow. Interf. Free Bound. 12(4), 527–549 (2011)

    MathSciNet  Google Scholar 

  6. Cheng, Q., Wang, C.: Error estimate of a second order accurate scalar auxiliary variable (SAV) numerical method for the epitaxial thin film equation. Adv. Appl. Math. Mech. 13, 1318–1354 (2021)

    MathSciNet  Google Scholar 

  7. Choi, Y., Kim, J.: Maximum principle preserving and unconditionally stable scheme for a conservative Allen-Cahn equation. Eng. Anal. Bound. Elem. 150, 111–119 (2023)

    MathSciNet  Google Scholar 

  8. Choi, J.W., Lee, H.G., Jeong, D., Kim, J.: An unconditionally gradient stable numerical method for solving the Allen-Cahn equation. Phys. A 388(9), 1791–1803 (2009)

    MathSciNet  Google Scholar 

  9. Copetti, M.I.M., Elliott, C.M.: Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy. Numer. Math. 63(1), 39–65 (1992)

    MathSciNet  Google Scholar 

  10. Debussche, A., Dettori, L.: On the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal. Theory Methods Appl. 24(10), 1491–1514 (1995)

    MathSciNet  Google Scholar 

  11. Doan, C.-K., Hoang, T.-T.-P., Ju, L., Schratz, K.: Low regularity integrators for semilinear parabolic equations with maximum bound principles. BIT Numer. Math. 63(1), 2 (2023)

    MathSciNet  Google Scholar 

  12. Doan, C.-K., Hoang, T.-T.-P., Ju, L.: Fully discrete error analysis of first-order low regularity integrators for the Allen-Cahn equation. Numer. Methods Partial Differ. Equ. 39(5), 3594–3608 (2023)

    MathSciNet  Google Scholar 

  13. Du, Q., Ju, L., Li, X., Qiao, Z.: Maximum principle preserving exponential time differencing schemes for the nonlocal Allen-Cahn equation. SIAM J. Numer. Anal. 57(2), 875–898 (2019)

    MathSciNet  Google Scholar 

  14. Du, Q., Ju, L., Li, X., Qiao, Z.: Maximum bound principles for a class of semilinear parabolic equations and exponential time-differencing schemes. SIAM Rev. 63(2), 317–359 (2021)

    MathSciNet  Google Scholar 

  15. Feng, X., Prohl, A.: Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math. 94(1), 33–65 (2003)

    MathSciNet  Google Scholar 

  16. Feng, J., Zhou, Y., Hou, T.: A maximum-principle preserving and unconditionally energy-stable linear second-order finite difference scheme for Allen-Cahn equations. Appl. Math. Lett. 118, 107179 (2021)

    MathSciNet  Google Scholar 

  17. Geng, S., Li, T., Ye, Q., Yang, X.: A new conservative Allen-Cahn type Ohta-Kawaski phase-field model for diblock copolymers and its numerical approximations. Adv. Appl. Math. Mech. 14(1), 101–124 (2022)

    MathSciNet  Google Scholar 

  18. He, D., Pan, K., Hu, H.: A spatial fourth-order maximum principle preserving operator splitting scheme for the multi-dimensional fractional Allen-Cahn equation. Appl. Numer. Math. 151, 44–63 (2020)

    MathSciNet  Google Scholar 

  19. Hou, T., Leng, H.: Numerical analysis of a stabilized Crank-Nicolson/Adams-Bashforth finite difference scheme for Allen-Cahn equations. Appl. Math. Lett. 102, 106150 (2020)

    MathSciNet  Google Scholar 

  20. Hou, T., Tang, T., Yang, J.: Numerical analysis of fully discretized Crank-Nicolson scheme for fractional-in-space Allen-Cahn equations. J. Sci. Comput. 72(3), 1214–1231 (2017)

    MathSciNet  Google Scholar 

  21. Hou, T., Xiu, D., Jiang, W.: A new second-order maximum-principle preserving finite difference scheme for Allen-Cahn equations with periodic boundary conditions. Appl. Math. Lett. 104, 106265 (2020)

    MathSciNet  Google Scholar 

  22. Huang, Z., Lin, G., Ardekani, A.M.: Consistent and conservative scheme for incompressible two-phase flows using the conservative Allen-Cahn model. J. Comput. Phys. 420, 109718 (2020)

    MathSciNet  Google Scholar 

  23. Huang, Z., Lin, G., Ardekani, A.M.: A consistent and conservative volume distribution algorithm and its applications to multiphase flows using phase-field models. Int. J. Multiph. Flow 142, 103727 (2021)

    MathSciNet  Google Scholar 

  24. Jeong, D., Kim, J.: Conservative Allen-Cahn-Navier-Stokes system for incompressible two-phase fluid flows. Comput. Fluids 156, 239–246 (2017)

    MathSciNet  Google Scholar 

  25. Jiang, K., Ju, L., Li, J., Li, X.: Unconditionally stable exponential time differencing schemes for the mass-conserving Allen-Cahn equation with nonlocal and local effects. Numer. Methods Partial Differ. Equ. 38(6), 1636–1657 (2022)

    MathSciNet  Google Scholar 

  26. Ju, L., Zhang, J., Zhu, L., Du, Q.: Fast explicit integration factor methods for semilinear parabolic equations. J. Sci. Comput. 62(2), 431–455 (2015)

    MathSciNet  Google Scholar 

  27. Kim, J., Jeong, D., Yang, S.D., Choi, Y.: A finite difference method for a conservative Allen-Cahn equation on non-flat surfaces. J. Comput. Phys. 334, 170–181 (2017)

    MathSciNet  Google Scholar 

  28. Kim, J., Lee, S., Choi, Y.: A conservative Allen-Cahn equation with a space-time dependent Lagrange multiplier. Int. J. Eng. Sci. 84, 11–17 (2014)

    MathSciNet  Google Scholar 

  29. Lee, H.G.: High-order and mass conservative methods for the conservative Allen-Cahn equation. Comput. Math. Appl. 72(3), 620–631 (2016)

    MathSciNet  Google Scholar 

  30. Lee, D., Kim, J.: Comparison study of the conservative Allen-Cahn and the Cahn-Hilliard equations. Math. Comput. Simul. 119, 35–56 (2016)

    MathSciNet  Google Scholar 

  31. Lee, H.G., Shin, J., Lee, J.Y.: A high-order and unconditionally energy stable scheme for the conservative Allen-Cahn equation with a nonlocal Lagrange multiplier. J. Sci. Comput. 90, 1–12 (2022)

    MathSciNet  Google Scholar 

  32. Li, J., Ju, L., Cai, Y., Feng, X.: Unconditionally maximum bound principle preserving linear schemes for the conservative Allen-Cahn equation with nonlocal constraint. J. Sci. Comput. 87(3), 1–32 (2021)

    MathSciNet  Google Scholar 

  33. Li, B., Ma, S., Schratz, K.: A semi-implicit exponential low-regularity integrator for the Navier-Stokes equations. SIAM J. Numer. Anal. 60(4), 2273–2292 (2022)

    MathSciNet  Google Scholar 

  34. Liao, H.L., Tang, T., Zhou, T.: A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen-Cahn equations. J. Comput. Phys. 414, 109473 (2020)

    MathSciNet  Google Scholar 

  35. Ostermann, A., Rousset, F., Schratz, K.: Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity. Found. Comput. Math. 21(3), 725–765 (2021)

    MathSciNet  Google Scholar 

  36. Ostermann, A., Schratz, K.: Low regularity exponential-type integrators for semilinear Schrödinger equations. Found. Comput. Math. 18, 731–755 (2018)

    MathSciNet  Google Scholar 

  37. Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Springer, New York (1984)

    Google Scholar 

  38. Ren, F., Song, B., Sukop, M.C., Hu, H.: Improved lattice Boltzmann modeling of binary flow based on the conservative Allen-Cahn equation. Phys. Rev. E 94(2), 023311 (2016)

    MathSciNet  Google Scholar 

  39. Rousset, F., Schratz, K.: A general framework of low regularity integrators. SIAM J. Numer. Anal. 59(3), 1735–1768 (2021)

    MathSciNet  Google Scholar 

  40. Rubinstein, J., Sternberg, P.: Nonlocal reaction-diffusion equations and nucleation. IMA J. Appl. Math. 48(3), 249–264 (1992)

    MathSciNet  Google Scholar 

  41. Schratz, K., Wang, Y., Zhao, X.: Low-regularity integrators for nonlinear Dirac equations. Math. Comput. 90(327), 189–214 (2021)

    MathSciNet  Google Scholar 

  42. Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61, 474–506 (2019)

    MathSciNet  Google Scholar 

  43. Shen, J., Tang, T., Yang, J.: On the maximum principle preserving schemes for the generalized Allen-Cahn equation. Commun. Math. Sci. 14(6), 1517–1534 (2016)

    MathSciNet  Google Scholar 

  44. Shen, J., Yang, X.: Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Contin. Dyn. Syst 28(4), 1669–1691 (2010)

    MathSciNet  Google Scholar 

  45. Shin, J., Lee, H.G., Lee, J.Y.: Unconditionally stable methods for gradient flow using convex splitting Runge-Kutta scheme. J. Comput. Phys. 347, 367–381 (2017)

    MathSciNet  Google Scholar 

  46. Tan, Z., Zhang, C.: The discrete maximum principle and energy stability of a new second-order difference scheme for Allen-Cahn equations. Appl. Numer. Math. 166, 227–237 (2021)

    MathSciNet  Google Scholar 

  47. Tang, T., Yang, J.: Implicit-explicit scheme for the Allen-Cahn equation preserves the maximum principle. J. Comput. Math. 34(5), 471–481 (2016)

    MathSciNet  Google Scholar 

  48. Wang, C., Wise, S.M.: An energy stable and convergent finite-difference scheme for the modified phase field crystal equation. SIAM J. Numer. Anal. 49(3), 945–969 (2011)

    MathSciNet  Google Scholar 

  49. Wu, J., Yang, J., Tan, Z.: Unconditionally energy-stable time-marching methods for the multi-phase conservative Allen-Cahn fluid models based on a modified SAV approach. Comput. Meth. Appl. Mech. Eng. 398, 115291 (2022)

    MathSciNet  Google Scholar 

  50. Wu, Y., Zhao, X.: Optimal convergence of a second-order low-regularity integrator for the KdV equation. IMA J. Numer. Anal. 42(4), 3499–3528 (2022)

    MathSciNet  Google Scholar 

  51. Wu, Y., Zhao, X.: Embedded exponential-type low-regularity integrators for KdV equation under rough data. BIT Numer. Math. 62(3), 1049–1090 (2022)

    MathSciNet  Google Scholar 

  52. Xiao, X., He, R., Feng, X.: Unconditionally maximum principle preserving finite element schemes for the surface Allen-Cahn type equations. Numer. Methods Partial Differ. Equ. 36(2), 418–438 (2020)

    MathSciNet  Google Scholar 

  53. Xu, J., Li, Y., Wu, S., Bousquet, A.: On the stability and accuracy of partially and fully implicit schemes for phase field modeling. Comput. Methods Appl. Mech. Eng. 345, 826–853 (2019)

    MathSciNet  Google Scholar 

  54. Xu, Z., Yang, X., Zhang, H., Xie, Z.: Efficient and linear schemes for anisotropic Cahn-Hilliard model using the stabilized-invariant energy quadratization (S-IEQ) approach. Comput. Phys. Commun. 238, 36–49 (2019)

    MathSciNet  Google Scholar 

  55. Yang, J., Jeong, D., Kim, J.: A fast and practical adaptive finite difference method for the conservative Allen-Cahn model in two-phase flow system. Int. J. Multiph. Flow 137, 103561 (2021)

    MathSciNet  Google Scholar 

  56. Yang, J., Kim, J.: Numerical study of incompressible binary fluids on 3D curved surfaces based on the conservative Allen-Cahn-Navier-Stokes model. Comput. Fluids 228, 105094 (2021)

    MathSciNet  Google Scholar 

  57. Yang, X., Zhang, G.: Convergence analysis for the invariant energy quadratization (IEQ) schemes for solving the Cahn-Hilliard and Allen-Cahn equations with general nonlinear potential. J. Sci. Comput. 82(3), 55 (2020)

    MathSciNet  Google Scholar 

  58. Zhai, S., Weng, Z., Feng, X.: Investigations on several numerical methods for the non-local Allen-Cahn equation. Int. J. Heat Mass Transfer 87, 11118 (2015)

    Google Scholar 

  59. Zhai, S., Weng, Z., Feng, X.: Fast explicit operator splitting method and time-step adaptivity for fractional non-local Allen-Cahn model. Appl. Math. Model. 40(2), 1315–1324 (2016)

    MathSciNet  Google Scholar 

  60. Zhang, H., Yan, J., Qian, X., Chen, X., Song, S.: Explicit third-order unconditionally structure-preserving schemes for conservative Allen-Cahn equations. J. Sci. Comput. 90, 1–29 (2022)

    MathSciNet  Google Scholar 

  61. Zhang, H., Yan, J., Qian, X., Song, S.: Numerical analysis and applications of explicit high order maximum principle preserving integrating factor Runge-Kutta schemes for Allen-Cahn equation. Appl. Numer. Math. 161, 372–390 (2021)

    MathSciNet  Google Scholar 

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Funding

T.-T.-P. Hoang’s work is partially supported by U.S. National Science Foundation under Grant Number DMS-2041884. L. Ju’s work is partially supported by U.S. National Science Foundation under Grant Number DMS-2109633.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Auburn University, Auburn, AL, 36849, USA

    Cao-Kha Doan & Thi-Thao-Phuong Hoang

  2. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Lili Ju

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Doan, CK., Hoang, TTP. & Ju, L. Low Regularity Integrators for the Conservative Allen–Cahn Equation with a Nonlocal Constraint. J Sci Comput 101, 57 (2024). https://doi.org/10.1007/s10915-024-02703-1

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