Temporal error analysis of Euler semi-implicit scheme for the magnetohydrodynamics equations with variable density. (English) Zbl 1465.76054
Summary: In this paper, we consider the magnetohydrodynamics (MHD) equations with variable density, which are a coupled system by the incompressible Navier-Stokes equations with variable density and the Maxwell equations. Such MHD system describes the motions of several conducting incompressible immiscible fluids without surface tension in presence of a magnetic field. A first-order Euler semi-implicit time discrete scheme is proposed to approximate the MHD system such that we only need to solve the linearized subproblems at the discrete level. Moreover, it is unconditionally stable which is a key issue for problems of multiphysical fields. A rigorous error analysis is presented and the first-order temporal convergence rate \(O(\tau)\) is derived for small \(\tau\) by using the discrete \(L^p\)-regularity technique, where \(\tau\) is the time step. The numerical results are shown to confirm the unconditional stability and the convergence rate.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
Keywords:
magnetohydrodynamic Navier-Stokes equations; Euler semi-implicit scheme; finite element spatial discretization; unconditional stability; convergence analysis; error estimateReferences:
| [1] | Abidi, H.; Paicu, M., Global existence for the magnetohydrodynamic system in critical spaces, Proc. R. Soc. Edinb., Sect. A, Math., 138, 447-476 (2008) · Zbl 1148.35066 |
| [2] | Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0314.46030 |
| [3] | An, R., Error analysis of a new fractional-step method for the incompressible Navier-Stokes equations with variable density, J. Sci. Comput., 84, Article 3 pp. (2020) · Zbl 1450.65115 |
| [4] | An, R., Iteration penalty method for the incompressible Navier-Stokes equations with variable density based on the artificial compressible method, Adv. Comput. Math., 46, Article 5 pp. (2020), 29 pages · Zbl 1436.65169 |
| [5] | An, R., Error analysis of a time-splitting method for incompressible flows with variable density, Appl. Numer. Math., 150, 384-395 (2020) · Zbl 1448.76056 |
| [6] | An, R.; Li, Y., Error analysis of first-order projection method for time-dependent magnetohydrodynamics equations, Appl. Numer. Math., 112, 167-181 (2017) · Zbl 06657058 |
| [7] | An, R.; Zhou, C., Error analysis of a fractional-step method for magnetohydrodynamics equations, J. Comput. Appl. Math., 313, 168-184 (2017) · Zbl 1388.76121 |
| [8] | Bell, J. B.; Marcus, D. L., A second-order projection method for variable-density flows, J. Comput. Phys., 101, 334-348 (1992) · Zbl 0759.76045 |
| [9] | Bie, Q. Y.; Wang, Q. R.; Yao, Z. A., Global well-posedness of the 3D incompressible MHD equations with variable density, Nonlinear Anal., Real World Appl., 47, 85-105 (2019) · Zbl 1411.35013 |
| [10] | Cabannes, H., Theoretical Magnetofluiddynamics (1970), Academic Press: Academic Press New York |
| [11] | Cai, W. T.; Li, B. Y.; Li, Y., Error analysis of a fully discrete finite element method for variable density incompressible flows in two dimensions, ESAIM: M2AN (2020) |
| [12] | Chen, Q.; Tan, Z.; Wang, Y. J., Strong solutions to the incompressible magnetohydrodynamic equations, Math. Methods Appl. Sci., 34, 94-107 (2011) · Zbl 1254.35187 |
| [13] | Chorin, A., Numerical solution of the Navier-Stokes equations, Math. Comput., 22, 745-762 (1968) · Zbl 0198.50103 |
| [14] | Desjardins, B.; Le Bris, C., Remarks on a nonhomogeneous model of magnetohydrodynamics, Differ. Integral Equ., 11, 377-394 (1998) · Zbl 1067.76097 |
| [15] | Galdi, G. P., An Introduction to the Mathematical Theory of the Navier-Stokes Equations (1994), Springer-Verlag: Springer-Verlag New York · Zbl 0949.35004 |
| [16] | Gao, H. D.; Qiu, W. F., A semi-implicit energy conserving finite element method for the dynamical incompressible magnetohydrodynamics equations, Comput. Methods Appl. Mech. Eng., 346, 982-1001 (2019) · Zbl 1440.76061 |
| [17] | Gerbeau, J.; Le Bris, C., Existence of solution for a density-dependent magnetohydrodynamic equation, Adv. Differ. Equ., 2, 427-452 (1997) · Zbl 1023.35524 |
| [18] | Gerbeau, J.; Le Bris, C.; Lelièvre, T., Mathematical Methods for the Magnetohydrodynamics of Liquid Metals (2006), Oxford University Press: Oxford University Press Oxford · Zbl 1107.76001 |
| [19] | Guermond, J. L.; Quartapelle, L., A projection FEM for variable density incompressible flows, J. Comput. Phys., 165, 167-188 (2000) · Zbl 0994.76051 |
| [20] | Guermond, J. L.; Salgado, A., A splitting method for incompressible flows with variable density based on a pressure Poisson equation, J. Comput. Phys., 228, 2834-2846 (2009) · Zbl 1159.76028 |
| [21] | Guermond, J. L.; Salgado, A., Error analysis of a fractional time-stepping technique for incompressible flows with variable density, SIAM J. Numer. Anal., 49, 917-944 (2011) · Zbl 1241.76318 |
| [22] | Gui, G. L., Global well-posedness of the two-dimensional incompressible magnetohydrodynamics system with variable density and electrical conductivity, J. Funct. Anal., 267, 1488-1539 (2014) · Zbl 1294.35064 |
| [23] | He, Y. N., Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations, IMA J. Numer. Anal., 35, 767-801 (2015) · Zbl 1312.76061 |
| [24] | 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., 27, 353-384 (1990) · Zbl 0694.76014 |
| [25] | Huang, X. D.; Wang, Y., Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differ. Equ., 254, 511-527 (2013) · Zbl 1253.35121 |
| [26] | Hughes, W.; Young, F., The Electromagnetics of Fluids (1966), Wiley: Wiley New York |
| [27] | Kovács, B.; Li, B. Y.; Lubich, C., A-stable time discretizations preserve maximal parabolic regularity, SIAM J. Numer. Anal., 54, 3600-3624 (2016) · Zbl 1355.65122 |
| [28] | Kulikovskiy, A. G.; Lyubimov, G. A., Magnetohydrodynamics (1965), Addison-Wesley: Addison-Wesley Reading, MA |
| [29] | Li, B. Y.; Gao, H. D.; Sun, W. W., Unconditionally optimal error estimates of a Crank-Nicolson Galerkin method for the nonlinear thermistor equations, SIAM J. Numer. Anal., 52, 933-954 (2014) · Zbl 1298.65160 |
| [30] | Li, B. Y.; Sun, W. W., Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations, Int. J. Numer. Anal. Model., 10, 622-633 (2013) · Zbl 1281.65122 |
| [31] | Li, B. Y.; Sun, W. W., Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media, SIAM J. Numer. Anal., 51, 1959-1977 (2013) · Zbl 1311.76067 |
| [32] | Li, B. Y.; Wang, J. L.; Xu, L. W., A convergent linearized Lagrange finite element method for the magnetohydrodynamic equations in two-dimensional nonsmooth and nonconvex domains, SIAM J. Numer. Anal., 58, 430-459 (2020) · Zbl 1432.76162 |
| [33] | Li, X. L.; Wang, D. H., Global strong solution to the three-dimensional density-dependent incompressible magnetohydrodynamic flows, J. Differ. Equ., 251, 1580-1615 (2011) · Zbl 1296.76176 |
| [34] | Li, Y.; Mei, L. Q.; Ge, J. T.; Shi, F., A new fractional time-stepping method for variable density incompressible flows, J. Comput. Phys., 242, 124-137 (2013) · Zbl 1311.76068 |
| [35] | Li, Y.; Ma, Y. J.; An, R., Decoupled, semi-implicit scheme for a coupled system arising in magnetohydrodynamics problem, Appl. Numer. Math., 127, 142-163 (2018) · Zbl 1425.76303 |
| [36] | Li, Y.; Luo, X. L., Second-order semi-implicit Crank-Nicolson scheme for a coupled magnetohydrodynamics system, Appl. Numer. Math., 145, 48-68 (2019) · Zbl 1448.76194 |
| [37] | Moreau, R., Magnetohydrodynamics (1990), Kluwer Academic Publishers · Zbl 0714.76003 |
| [38] | Prohl, A., Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamic system, ESAIM: M2AN, 42, 1065-1087 (2008) · Zbl 1149.76029 |
| [39] | Pyo, J. H.; Shen, J., Gauge-Uzawa methods for incompressible flows with variable density, J. Comput. Phys., 221, 181-197 (2007) · Zbl 1109.76037 |
| [40] | Temam, R., Sur l’approximation de la solution des equations de Navier-Stokes par la méthode des pas fractionnaires II, Arch. Ration. Mech. Anal., 33, 377-385 (1969) · Zbl 0207.16904 |
| [41] | Temam, R., Navier-Stokes Equations (1977), North-Holland Publishing Company: North-Holland Publishing Company Amsterdam · Zbl 0335.35077 |
| [42] | Yang, X. F.; Zhang, G. D.; He, X. M., Convergence analysis of an Unconditionally energy stable projection scheme for magnetohydrodynamic equations, Appl. Numer. Math., 136, 235-256 (2019) · Zbl 1405.76026 |
| [43] | Zhang, G. D.; He, X. M.; Yang, X. F., A decoupled, linear and unconditionally energy stable scheme with finite element discretizations for magnetohydrodynamic equations, J. Sci. Comput., 81, 1678-1711 (2019) · Zbl 1432.76172 |
| [44] | Zhang, G. D.; He, X. M.; Yang, X. F., Fully decoupled, linear and unconditionally energy stable time discretization for solving the magnetohydrodynamic equations, J. Comput. Appl. Math., 369, Article #112636 pp. (2020) · Zbl 1447.65039 |
| [45] | Zhang, G. D.; He, Y. N., Decoupled schemes for unsteady MHD equations II: finite element spatial discretization and numerical implementation, Comput. Math. Appl., 69, 1390-1406 (2015) · Zbl 1443.65232 |
| [46] | Zhang, Y. H.; Hou, Y. R.; Shan, L., Numerical analysis of the Crank-Nicolson extrapolation time discrete scheme for magnetohydrodynamics flows, Numer. Methods Partial Differ. Equ., 31, 2169-2208 (2015) · Zbl 1331.76075 |
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.
