Stochastic Runge-Kutta Methods for Preserving Maximum Bound Principle of Semilinear Parabolic Equations. Part II: Sinc Quadrature Rule

Abstract

The maximum bound principle (MBP) is an important property for a large class of semilinear parabolic equations. To propose MBP-preserving schemes with high spatial accuracy, in the first part of this series, we developed a class of time semidiscrete stochastic Runge-Kutta (SRK) methods for semilinear parabolic equations, and constructed the first- and second-order fully discrete MBP-preserving SRK schemes. In this paper, to develop higher order fully discrete MBP-preserving SRK schemes with spectral accuracy in space, we use the Sinc quadrature rule to approximate the conditional expectations in the time semidiscrete SRK methods and propose a class of fully discrete MBP-preserving SRK schemes with up to fourth-order accuracy in time for semilinear equations. Based on the property of the Sinc quadrature rule, we theoretically prove that the proposed fully discrete SRK schemes preserve the MBP and can achieve an exponential order convergence rate in space. In addition, we reveal that the conditional expectation with respect to the Bronwian motion in the time semidiscrete SRK method is essentially equivalent to the exponential Laplacian operator under the periodic boundary condition. Ample numerical experiments are also performed to demonstrate our theoretical results and to show the exponential order convergence rate in space of the proposed schemes.

Keywords

Semilinear parabolic equation / stochastic Runge-Kutta scheme / Sinc quadrature rule / MBP-preserving / exponential convergence rate

References

[1] P. Briand and R. Carmona, BSDEs with polynomial growth generators, J. Appl. Math. Stochastic Anal., 13:207-238, 2020.

[2] E. Burman and A. Ern, Stabilized Galerkin approximation of convection-diffusion-reaction equations: Discrete maximum principle and convergence, Math. Comp., 74:1637-1652, 2005.

[3] P. Chatzipantelidis, Z. Horvath, and V. Thomee, On preservation of positivity in some finite element methods for the heat equation, Comput. Methods Appl. Math., 15:417-437, 2015.

[4] W. Chen, C. Wang, X. Wang, and S. Wise, Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential, J. Comput. Phys. X, 3:100031, 2019.

[5] P. Ciarlet, Discrete maximum principle for finite-difference operators, Aequationes Math., 4:338352, 1970.

[6] P. Ciarlet and P. Raviart, Maximum principle and uniform convergence for the finite element method, Comput. Methods Appl. Mech. Engrg., 2:17-31, 1973.

[7] J. Cui, D. Hou, and Z. Qiao, Energy regularized models for logarithmic SPDEs and their numerical approximations, arXiv:2303.05003, 2023.

[8] 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., 57:875-898, 2019.

[9] 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., 63:317-359, 2021.

[10] Z. Fu, T. Tang, and J. Yang, Energy plus maximum bound preserving Runge-Kutta methods for the Allen-Cahn equation, J. Sci. Comput., 92:97, 2022.

[11] Z. Fu, T. Tang, and J. Yang, Energy diminishing implicit-explicit Runge-Kutta methods for gradient flows, Math. Comput., 2024. doi: 10.1090/mcom/3950.

[12] Z. Fu and J. Yang, Energy-decreasing exponential time differencing Runge-Kutta methods for phasefield models, J. Comput. Phys., 454:110943, 2022.

[13] D. Hou, L. Ju, and Z. Qiao, A linear second-order maximum bound principle-preserving BDF scheme for the Allen-Cahn equation with a general mobility, Math. Comput., 92:2515-2542, 2023.

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

[15] T. Hou, T. Tang, and J. Yang, Numerical analysis of fully discretized Crank-Nicolson scheme for fractional-in-space Allen-Cahn equations, J. Sci. Comput., 72:1214-1231, 2017.

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

[17] J. Huang, L. Ju, and Y. Xu, Efficient exponential integrator finite element method for semilinear parabolic equations, SIAM J. Sci. Comput., 45:A1545-A1570, 2023.

[18] L. Ju, X. Li, and Z. Qiao, Stabilized exponential-SAV schemes preserving energy dissipation law and maximum bound principle for the Allen-Cahn type equations, J. Sci. Comput., 92:66, 2022.

[19] L. Ju, X. Li, and Z. Qiao, Generalized SAV-exponential integrator schemes for Allen-Cahn type gradient flows, SIAM J. Numer. Anal., 60:1905-1931, 2022.

[20] L. Ju, X. Li, Z. Qiao, and J. Yang, Maximum bound principle preserving integrating factor RungeKutta methods for semilinear parabolic equations, J. Comput. Phys., 439:110405, 2021.

[21] N. Karoui, S. Peng, and M. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7:1-71, 1997.

[22] 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., 42:3957-3978, 2020.

[23] J. Li, R. Lan, Y. Cai, L. Ju, and X. Wang, Second-order semi-Lagrangian exponential time differencing method with enhanced error estimate for the convective Allen-Cahn equation, J. Sci. Comput., 97:7, 2023.

[24] 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:A1780-A1802, 2021.

[25] J. Lund and K. Bowers, Sinc Methods for Quadrature and Differential Equations, SIAM, 1992.

[26] L. Ma and Z. Qiao, An energy stable and maximum bound principle preserving scheme for the dynamic Ginzburg-Landau equations under the temporal gauge, SIAM J. Numer. Anal., 61:26952717, 2023.

[27] C. Nan and H. Song, The high-order maximum-principle-preserving integrating factor RungeKutta methods for nonlocal Allen-Cahn equation, J. Comput. Phys., 456:111028, 2022.

[28] É. Pardoux, BSDEs, weak convergence and homogenization of semilinear PDEs, in:Nonlinear Analysis, Differential Equations and Control. Nato Science Series C, Vol. 528, Springer, 503549, 1999.

[29] É. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14:55-61, 1990.

[30] G. Peng, Z. Gao, and X. Feng, A stabilized extremum-preserving scheme for nonlinear parabolic equation on polygonal meshes, Internat. J. Numer. Methods Fluids, 90:340-356, 2019.

[31] G. Peng, Z. Gao, W. Yan, and X. Feng, A positivity-preserving nonlinear finite volume scheme for radionuclide transport calculations in geological radioactive waste repository, Int. J. Numer. Methods Heat Fluid Flow, 30:516-534, 2019.

[32] S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Stochastics Rep., 37:61-74, 1991.

[33] J. Shen, T. Tang, and J. Yang, On the maximum principle preserving schemes for the generalized Allen-Cahn equation, Commun. Math. Sci., 14:1517-1534, 2016.

[34] F. Stenger, Handbook of Sinc Numerical Methods, CRC Press, 2016.

[35] Y. Sun and W. Zhao, Stochastic Runge-Kutta methods for preserving maximum bound principle of semilinear parabolic equations. Part I: Gaussian quadrature rule, CSIAM Trans. Appl. Math., 5:390-420, 2024.

[36] T. Tang and J. Yang, Implicit-explicit scheme for the Allen-Cahn equation preserves the maximum principle, J. Comput. Math., 34:471-481, 2016.

[37] R. Varga, On a discrete maximum principle, SIAM J. Numer. Anal., 3:355-359, 1966.

[38] X. Wang, W. Zhao, and T. Zhou, Sinc-$\theta $ schemes for backward stochastic differential equations, SIAM J. Numer. Anal., 60:1799-1823, 2022.

[39] X. Xiao, Z. Dai, and X. Feng, A positivity preserving characteristic finite element method for solving the transport and convection-diffusion-reaction equations on general surfaces, Comput. Phys. Commun., 247:106941, 2020.

[40] X. Xiao, X. Feng, and Y. He, Numerical simulations for the chemotaxis models on surfaces via a novel characteristic finite element method, Comput. Math. Appl., 78:20-34, 2019.

[41] J. Yang, Z. Yuan, and Z. Zhou, Arbitrarily high-order maximum bound preserving schemes with cut-off postprocessing for Allen-Cahn equations, J. Sci. Comput., 90:76, 2022.

[42] X. Yang, Error analysis of stabilized semi-implicit method of Allen-Cahn equation, Discrete Contin. Dyn. Syst. Ser. B, 11:1057-1070, 2009.

[43] W. Zhao, W. Zhang, and L. Ju, A numerical method and its error estimates for the decoupled forward-backward stochastic differential equations, Commun. Comput. Phys., 15:618-646, 2014.