A new Lagrange multiplier approach for constructing structure preserving schemes. I: Positivity preserving. (English) Zbl 1507.65195
Summary: We propose a new Lagrange multiplier approach to construct positivity preserving schemes for parabolic type equations. The new approach introduces a space-time Lagrange multiplier to enforce the positivity with the Karush-Kuhn-Tucker (KKT) conditions. We then use a predictor-corrector approach to construct a class of positivity schemes: with a generic semi-implicit or implicit scheme as the prediction step, and the correction step, which enforces the positivity, can be implemented with negligible cost. We also present a modification which allows us to construct schemes which, in addition to positivity preserving, is also mass conserving. This new approach is not restricted to any particular spatial discretization and can be combined with various time discretization schemes. We establish stability results for our first- and second-order schemes under a general setting, and present ample numerical results to validate the new approach.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65K15 | Numerical methods for variational inequalities and related problems |
| 65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |
References:
| [1] | Lu, Changna; Huang, Weizhang; Van Vleck, Erik S., The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and lubrication-type equations, J. Comput. Phys., 242, 24-36 (2013) · Zbl 1297.65097 |
| [2] | Li, Buyang; Yang, Jiang; Zhou, Zhi, Arbitrarily high-order exponential cut-off methods for preserving maximum principle of parabolic equations, SIAM J. Sci. Comput., 42, 6, A3957-A3978 (2020) · Zbl 1456.65117 |
| [3] | Du, Qiang; Ju, Lili; Li, Xiao; Qiao, Zhonghua, Maximum bound principles for a class of semilinear parabolic equations and exponential time differencing schemes (2020), arXiv preprint arXiv:2005.11465 · Zbl 1465.35081 |
| [4] | Li, Hao; Xie, Shusen; Zhang, Xiangxiong, A high order accurate bound-preserving compact finite difference scheme for scalar convection diffusion equations, SIAM J. Numer. Anal., 56, 6, 3308-3345 (2018) · Zbl 1405.65097 |
| [5] | Li, Hao; Zhang, Xiangxiong, On the monotonicity and discrete maximum principle of the finite difference implementation of \(C^0- Q^2\) finite element method, Numer. Math., 145, 2, 437-472 (2020) · Zbl 1451.65200 |
| [6] | Zhang, Xiangxiong; Shu, Chi-Wang, Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments, Proc. R. Soc. A, 467, 2134, 2752-2776 (2011) · Zbl 1222.65107 |
| [7] | Zhang, Xiangxiong; Shu, Chi-Wang, On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, J. Comput. Phys., 229, 23, 8918-8934 (2010) · Zbl 1282.76128 |
| [8] | Chen, Wenbin; Wang, Cheng; Wang, Xiaoming; Wise, Steven M., Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential, J. Comput. Phys.: X, 3, Article 100031 pp. (2019) · Zbl 07785514 |
| [9] | Liu, Jian-Guo; Wang, Li; Zhou, Zhennan, Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations, Math. Comp., 87, 311, 1165-1189 (2018) · Zbl 1383.65111 |
| [10] | Hu, Jingwei; Huang, Xiaodong, A fully discrete positivity-preserving and energy-dissipative finite difference scheme for Poisson-Nernst-Planck equations, Numer. Math., 1-39 (2020) · Zbl 1442.35459 |
| [11] | Huang, Fukeng; Shen, Jie, Bound/positivity preserving and energy stable SAV schemes for dissipative systems: applications to keller-segel and Poisson-Nernst-Planck equations, SIAM J. Sci. Comput., 43, 3, A1832-A1857 (2021) · Zbl 1512.65229 |
| [12] | van der Vegt, Jaap J. W.; Xia, Yinhua; Xu, Yan, Positivity preserving limiters for time-implicit higher order accurate discontinuous Galerkin discretizations, SIAM J. Sci. Comput., 41, 3, A2037-A2063 (2019) · Zbl 1503.65249 |
| [13] | Ito, Kazufumi; Kunisch, Karl, On a semi-smooth Newton method and its globalization, Math. Program., 118, 2, 347-370 (2009) · Zbl 1164.65018 |
| [14] | Ito, Kazufumi; Kunisch, Karl, Lagrange Multiplier Approach to Variational Problems and Applications (2008), SIAM · Zbl 1156.49002 |
| [15] | Harker, Patrick T.; Pang, Jong-Shi, Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Math. Program., 48, 1, 161-220 (1990) · Zbl 0734.90098 |
| [16] | Facchinei, Francisco; Pang, Jong-Shi, Finite-Dimensional Variational Inequalities and Complementarity Problems (2007), Springer Science & Business Media · Zbl 1062.90001 |
| [17] | Vázquez, Juan Luis, The Porous Medium Equation: Mathematical Theory (2007), Oxford University Press on Demand · Zbl 1107.35003 |
| [18] | Bergounioux, Maïtine; Ito, Kazufumi; Kunisch, Karl, Primal-dual strategy for constrained optimal control problems, SIAM J. Control Optim., 37, 4, 1176-1194 (1999) · Zbl 0937.49017 |
| [19] | Curtiss, Charles Francis; Hirschfelder, Joseph O., Integration of stiff equations, Proc. Natl. Acad. Sci. USA, 38, 3, 235 (1952) · Zbl 0046.13602 |
| [20] | Cheng, Qing; Shen, Jie, Global constraints preserving scalar auxiliary variable schemes for gradient flows, SIAM J. Sci. Comput., 42, 4, A2489-A2513 (2020) · Zbl 1456.65087 |
| [21] | Cheng, Qing; Shen, Jie, Multiple scalar auxiliary variable (MSAV) approach and its application to the phase-field vesicle membrane model, SIAM J. Sci. Comput., 40, 6, A3982-A4006 (2018) · Zbl 1403.65045 |
| [22] | Liu, Yuanyuan; Shu, Chi-Wang; Zhang, Mengping, High order finite difference WENO schemes for nonlinear degenerate parabolic equations, SIAM J. Sci. Comput., 33, 2, 939-965 (2011) · Zbl 1227.65074 |
| [23] | Zhornitskaya, Liya; Bertozzi, Andrea L., Positivity-preserving numerical schemes for lubrication-type equations, SIAM J. Numer. Anal., 37, 2, 523-555 (1999) · Zbl 0961.76060 |
| [24] | Elliott, C. M.; Stuart, A. M., The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal., 30, 6, 1622-1663 (1993) · Zbl 0792.65066 |
| [25] | Shen, Jie; Xu, Jie; Yang, Jiang, A new class of efficient and robust energy stable schemes for gradient flows, SIAM Rev., 61, 3, 474-506 (2019) · Zbl 1422.65080 |
| [26] | Cheng, Qing; Liu, Chun; Shen, Jie, Generalized SAV approaches for gradient systems, J. Comput. Appl. Math., 394, Article 113532 pp. (2021) · Zbl 1475.65066 |
| [27] | Butcher, John Charles, Numerical Methods for Ordinary Differential Equations (2016), John Wiley & Sons · Zbl 1354.65004 |
| [28] | Guermond, Jean-Luc; Minev, Peter; Shen, Jie, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Engrg., 195, 44-47, 6011-6045 (2006) · Zbl 1122.76072 |
| [29] | Shen, Jie; Tang, Tao; Wang, Li-Lian, Spectral Methods: Algorithms, Analysis and Applications, Vol. 41 (2011), Springer Science & Business Media · Zbl 1227.65117 |
| [30] | Allen, Samuel M.; Cahn, John W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27, 6, 1085-1095 (1979) |
| [31] | Temam, Roger, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Vol. 68 (2012), Springer Science & Business Media |
| [32] | He, Dongdong; Pan, Kejia; Yue, Xiaoqiang, A positivity preserving and free energy dissipative difference scheme for the Poisson-Nernst-Planck system, J. Sci. Comput., 81, 1, 436-458 (2019) · Zbl 1427.65165 |
| [33] | Bertozzi, Andrea L.; Pugh, Mary, The lubrication approximation for thin viscous films: Regularity and long-time behavior of weak solutions, Comm. Pure Appl. Math., 49, 2, 85-123 (1996) · Zbl 0863.76017 |
| [34] | Gómez, Héctor; Calo, Victor M.; Bazilevs, Yuri; Hughes, Thomas J. R., Isogeometric analysis of the Cahn-Hilliard phase-field model, Comput. Methods Appl. Mech. Engrg., 197, 49-50, 4333-4352 (2008) · Zbl 1194.74524 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
