A GENERALIZED SCALAR AUXILIARY VARIABLE METHOD FOR THE TIME-DEPENDENT GINZBURG-LANDAU EQUATIONS

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doi:10.1007/s10473-024-0215-y

Abstract: This paper develops a generalized scalar auxiliary variable (SAV) method for the time-dependent Ginzburg-Landau equations. The backward Euler method is used for discretizing the temporal derivative of the time-dependent Ginzburg-Landau equations. In this method, the system is decoupled and linearized to avoid solving the non-linear equation at each step. The theoretical analysis proves that the generalized SAV method can preserve the maximum bound principle and energy stability, and this is confirmed by the numerical result, and also shows that the numerical algorithm is stable.

Key words: time-dependent Ginzburg-Landau equation, generalized scalar auxiliary variable algorithm, maximum bound principle, energy stability

CLC Number:

65M12

Zhiyong SI. A GENERALIZED SCALAR AUXILIARY VARIABLE METHOD FOR THE TIME-DEPENDENT GINZBURG-LANDAU EQUATIONS[J].Acta mathematica scientia,Series B, 2024, 44(2): 650-670.

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[1] Chen Z, Dai S.Adaptive Galerkin methods with error control for a dynamical Ginzburg-Landau model in superconductivity. SIAM J Numer Anal, 2001, 38: 1961-1985
[2] Chen Z, Hoffmann K. Numerical studies of a non-stationary Ginzburg-Landau model for superconductivity. Adv Math Sci Appl, 1995, 5: 363-389
[3] Chen Z, Hoffmann K, Liang J. On a non-stationary Ginzburg-Landau superconductivity model. Math Method Appl Sci, 1993, 16: 855-875
[4] Du Q. Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau model for superconductivity. Appl Anal, 1994, 53: 1-17
[5] Du Q. Finite element methods for the time-dependent Ginzburg-Landau model of superconductivity. Comput Math Appl, 1994, 27: 119-133
[6] Ganesh M, Thompson T. A spectrally accurate algorithm and analysis for a Ginzburg-Landau model on superconducting surfaces. Multiscale Model Sim, 2018, 16: 78-105
[7] Gao H, Ju L, Xie W. A stabilized semi-implicit Euler gauge-invariant method for the time-dependent Ginzburg-Landau equations. J Sci Comput, 2019, 80: 1083-1115
[8] Gao H, Li B, Sun W. Optimal error estimates of linearized Crank-Nicolson Galerkin FEMs for the time-dependent Ginzburg-Landau equations in superconductivity. SIAM J Numer Anal, 2014, 52: 1183-1202
[9] Gao H, Sun W. An efficient fully linearized semi-implicit Galerkin-mixed FEM for the dynamical Ginzburg-Landau equations of superconductivity. J Comput Phys, 2015, 294: 329-345
[10] Gao H, Sun W. A new mixed formulation and efficient numerical solution of Ginzburg-Landau equations under the temporal gauge. SIAM J Sci Comput, 2016, 38: A1339-A1357
[11] Gao H, Sun W. Analysis of linearized Galerkin-mixed FEMs for the time-dependent Ginzburg-Landau equations of superconductivity. Adv Comput Math, 2018, 44: 923-949
[12] Gor'kov L, Éliashberg G. Generalization of the Ginburg-Landau equations for non-stationary problems in the case of alloys with paramagnetic impurities. Sov J Exp Theo Phys, 1968, 27: 328-334
[13] Huang F, Shen J. A new class of implicit-explicit BDF

k

$k$ SAV schemes for general dissipative systems and their error analysis. Comput Meth Appl Mech Engrg, 2022, 392: 114718
[14] Huang F, Shen J, Yang Z. A highly efficient and accurate new scalar auxiliary variable approach for gradient flows. SIAM J Sci Comput, 2020, 42: A2514-A2536
[15] Hecht F. New development in FreeFem++. J Numer Math, 2012, 20: 251-266
[16] Heywood J, Rannacher R. Finite-element approximation of the nonstationary Navier-Stokes problem. part IV: Error analysis for second-order time discretization. SIAM J Numer Anal, 1990, 27: 353-384
[17] Jiang M, Zhang Z, Zhao J. Improving the accuracy and consistency of the scalar auxiliary variable (SAV) method with relaxation. J Comput Phys, 2022, 456: 110954
[18] Ju L, Li X, Qiao Z.Generalized SAV-exponential integrator Schemes for Allen-Cahn type gradient flows. SIAM J Numer Anal, 2022, 60(4): 1905-1931
[19] Ju L, Li X, Qiao Z. Stabilized exponential-SAV schemes preserving energy dissipation law and maximum bound principle for the Allen-Cahn type equations. J Sci Comput, 2022, 92: Art 66
[20] Li B, Wang K, Zhang Z. A Hodge decomposition method for dynamic Ginzburg-Landau equations in nonsmooth domains-a second approach. Commun Comput Phys, 2020, 28: 768-802
[21] Li B, Zhang Z. Mathematical and numerical analysis of the time-dependent Ginzburg-Landau equations in nonconvex polygons based on Hodge decomposition. Math Comput, 2017, 86: 1579-1608
[22] Li B, Zhang Z. A new approach for numerical simulation of the time-dependent Ginzburg-Landau equations. J Comput Phys, 2015, 303: 238-250
[23] Liu F, Mondello M, Goldenfeld N. Kinetics of the superconducting transition. Phys Rev Lett, 1991, 66: 3071-3074
[24] Mu M, Huang Y. An alternating Crank-Nicolson method for decoupling the Ginzburg-Landau equations. SIAM J Numer Anal, 1998, 35: 1740-1761
[25] Nochetto R, Pyo J. Optimal relaxation parameter for the Uzawa method. Numer Math, 2004, 98: 695-702
[26] Shen J, Xu J, Yang J. The scalar auxiliary variable (SAV) approach for gradient flows. J Comput Phys, 2018, 353: 407-416
[27] Shen J, Xu J, Yang J. A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev, 2019, 61: 474-506
[28] Shen J, Zhang X X. Discrete maximum principle of a high order finite difference scheme for a generalized Allen-Cahn equation. Commun Math Sci, 2022, 20: 1409-1436
[29] Tan Z, Tang H. A general class of linear unconditionally energy stable schemes for the gradient flows. J Comput Phys, 2022, 464: 111372
[30] Tang T, Qiao Z.Efficient numerical methods for phase-field equations
(in Chinese). Sci Sin Math, 2020, 50: 1-20
[31] Wang Y, Si Z. A convex splitting method for the time-dependent Ginzburg-Landau equation. Numer Algor, 2023. https://doi.org/10.1007/s11075-023-01672-0
[32] Wu C, Sun W. Analysis of Galerkin FEMs for mixed formulation of time-dependent Ginzburg-Landau equations under temporal gauge. SIAM J Numer Anal, 2018, 56: 1291-1312
[33] Zhang Y, Shen J. A generalized SAV approach with relaxation for dissipative systems. J Comput Phys, 2022, 464: 111311

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