Applications of Mathematics, Vol. 64, No. 5, pp. 531-556, 2019
A higher order pressure segregation scheme for the time-dependent magnetohydrodynamics equations
Yun-Bo Yang, Yao-Lin Jiang, Qiong-Xiang Kong
Received March 10, 2017. Published online September 1, 2019.
Abstract: A higher order pressure segregation scheme for the time-dependent incompressible magnetohydrodynamics (MHD) equations is presented. This scheme allows us to decouple the MHD system into two sub-problems at each time step. First, a coupled linear elliptic system is solved for the velocity and the magnetic field. And then, a Poisson-Neumann problem is treated for the pressure. The stability is analyzed and the error analysis is accomplished by interpreting this segregated scheme as a higher order time discretization of a perturbed system which approximates the MHD system. The main results are that the convergence for the velocity and the magnetic field are strongly second-order in time while that for the pressure is strongly first-order in time. Some numerical tests are performed to illustrate the theoretical predictions and demonstrate the efficiency of the proposed scheme.
References: [1] R. A. Adams, J. J. F. Fournier: Sobolev Spaces. Pure and Applied Mathematics 140, Academic press, New York (2003). DOI 10.1016/S0079-8169(03)80012-0 | MR 2424078 | Zbl 1098.46001
[2] R. An, Y. Li: Error analysis of first-order projection method for time-dependent magnetohydrodynamics equations. Appl. Numer. Math. 112 (2017), 167-181. DOI 10.1016/j.apnum.2016.10.010 | MR 3574248 | Zbl 06657058
[3] F. Armero, J. C. Simo: Long-term dissipativity of time-stepping algorithms for an abstract evolution equation with applications to the incompressible MHD and Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 131 (1996), 41-90. DOI 10.1016/0045-7825(95)00931-0 | MR 1393572 | Zbl 0888.76042
[4] S. Badia, R. Codina, R. Planas: On an unconditionally convergent stabilized finite element approximation of resistive magnetohydrodynamics. J. Comput. Phys. 234 (2013), 399-416. DOI 10.1016/j.jcp.2012.09.031 | MR 2999784 | Zbl 1284.76248
[5] S. Badia, R. Planas, J. V. Gutiérrez-Santacreu: Unconditionally stable operator splitting algorithms for the incompressible magnetohydrodynamics system discretized by a stabilized finite element formulation based on projections. Int. J. Numer. Methods Eng. 93 (2013), 302-328. DOI 10.1002/nme.4392 | MR 3008267 | Zbl 1352.76122
[6] J. Blasco, R. Codina: Error estimates for an operator-splitting method for incompressible flows. Appl. Numer. Math. 51 (2004), 1-17. DOI 10.1016/j.apnum.2004.02.004 | MR 2083322 | Zbl 1126.76339
[7] J. Blasco, R. Codina, A. Huerta: A fractional-step method for the incompressible Navier-Stokes equations related to a predictor-multicorrector algorithm. Int. J. Numer. Methods Fluids 28 (1998), 1391-1419. DOI 10.1002/(SICI)1097-0363(19981230)28:10<1391::AID-FLD699>3.0.CO;2-5 | MR 1663560 | Zbl 0935.76041
[8] H. Choi, J. Shen: Efficient splitting schemes for magneto-hydrodynamic equations. Sci. China, Math. 59 (2016), 1495-1510. DOI 10.1007/s11425-016-0280-5 | MR 3528499 | Zbl 1388.76224
[9] A. J. Chorin: Numerical solution of the Navier-Stokes equations. Math. Comput. 22 (1968), 745-762. DOI 10.2307/2004575 | MR 0242392 | Zbl 0198.50103
[10] J.-F. Gerbeau: A stabilized finite element method for the incompressible magnetohydrodynamic equations. Numer. Math. 87 (2000), 83-111. DOI 10.1007/s002110000193 | MR 1800155 | Zbl 0988.76050
[11] J.-F. Gerbeau, C. Le Bris, T. Lelièvre: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford (2006). DOI 10.1093/acprof:oso/9780198566656.001.0001 | MR 2289481 | Zbl 1107.76001
[12] V. Girault, P.-A. Raviart: Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer Series in Computational Mathematics 5, Springer, Berlin (1986). DOI 10.1007/978-3-642-61623-5 | MR 0851383 | Zbl 0585.65077
[13] C. Greif, D. Li, D. Schötzau, X. Wei: A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 199 (2010), 2840-2855. DOI 10.1016/j.cma.2010.05.007 | MR 2740762 | Zbl 1231.76146
[14] J. L. Guermond, P. Minev, J. Shen: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195 (2006), 6011-6045. DOI 10.1016/j.cma.2005.10.010 | MR 2250931 | Zbl 1122.76072
[15] J. L. Guermond, J. Shen: A new class of truly consistent splitting schemes for incompressible flows. J. Comput. Phys. 192 (2003), 262-276. DOI 10.1016/j.jcp.2003.07.009 | MR 2045709 | Zbl 1032.76529
[16] J. L. Guermond, J. Shen: Velocity-correction projection methods for incompressible flows. SIAM J. Numer. Anal. 41 (2003), 112-134. DOI 10.1137/S0036142901395400 | MR 1974494 | Zbl 1130.76395
[17] J. L. Guermond, J. Shen: On the error estimates for the rotational pressure-correction projection methods. Math. Comput. 73 (2004), 1719-1737. DOI 10.1090/S0025-5718-03-01621-1 | MR 2059733 | Zbl 1093.76050
[18] M. D. Gunzburger, A. J. Meir, J. S. Peterson: On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics. Math. Comput. 56 (1991), 523-563. DOI 10.2307/2008394 | MR 1066834 | Zbl 0731.76094
[19] F. Hecht: New development in freefem++. J. Numer. Math. 20 (2012), 251-265. DOI 10.1515/jnum-2012-0013 | MR 3043640 | Zbl 1266.68090
[20] Y.-L. Jiang, Y.-B. Yang: Semi-discrete Galerkin finite element method for the diffusive Peterlin viscoelastic model. Comput. Methods Appl. Math. 18 (2018), 275-296. DOI 10.1515/cmam-2017-0021 | MR 3776046 | Zbl 1391.76337
[21] W. Layton, H. Tran, C. Trenchea: Numerical analysis of two partitioned methods for uncoupling evolutionary MHD flows. Numer. Methods Partial Differ. Equations 30 (2014), 1083-1102. DOI 10.1002/num.21857 | MR 3200267 | Zbl 1364.76088
[22] A. Linke, M. Neilan, L. G. Rebholz, N. E. Wilson: A connection between coupled and penalty projection timestepping schemes with FE spatial discretization for the Navier-Stokes equations. J. Numer. Math. 25 (2017), 229-248. DOI 10.1515/jnma-2016-1024 | MR 3767412 | Zbl 06857557
[23] A. Prohl: Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system. ESAIM, Math. Model. Numer. Anal. 42 (2008), 1065-1087. DOI 10.1051/m2an:2008034 | MR 2473320 | Zbl 1149.76029
[24] Y. Qian, T. Zhang: The second order projection method in time for the time-dependent natural convection problem. Appl. Math., Praha 61 (2016), 299-315. DOI 10.1007/s10492-016-0133-y | MR 3502113 | Zbl 06587854
[25] S. S. Ravindran: An extrapolated second order backward difference time-stepping scheme for the magnetohydrodynamics system. Numer. Funct. Anal. Optim. 37 (2016), 990-1020. DOI 10.1080/01630563.2016.1181651 | MR 3532388 | Zbl 1348.76189
[26] P. G. Schmidt: A Galerkin method for time-dependent MHD flow with nonideal boundaries. Commun. Appl. Anal. 3 (1999), 383-398. MR 1696344 | Zbl 0931.76099
[27] D. Schötzau: Mixed finite element methods for stationary incompressible magneto-hydrodynamics. Numer. Math. 96 (2004), 771-800. DOI 10.1007/s00211-003-0487-4 | MR 2036365 | Zbl 1098.76043
[28] M. Sermange, R. Temam: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36 (1983), 635-664. DOI 10.1002/cpa.3160360506 | MR 0716200 | Zbl 0524.76099
[29] J. Shen: On error estimates of projection methods for Navier-Stokes equations: First-order schemes. SIAM J. Numer. Anal. 29 (1992), 57-77. DOI 10.1137/0729004 | MR 1149084 | Zbl 0741.76051
[30] J. Shen: On error estimates of the projection methods for the Navier-Stokes equations: Second-order schemes. Math. Comput. 65 (1996), 1039-1065. DOI 10.1090/S0025-5718-96-00750-8 | MR 1348047 | Zbl 0855.76049
[31] J. Shen, X. Yang: Error estimates for finite element approximations of consistent splitting schemes for incompressible flows. Discrete Contin. Dyn. Syst., Ser. B. 8 (2007), 663-676. DOI 10.3934/dcdsb.2007.8.663 | MR 2328729 | Zbl 1220.76046
[32] J. C. Simo, F. Armero: Unconditional stability and long-term behavior of transient algorithms for the incompressible Navier-Stokes and Euler equations. Comput. Methods Appl. Mech. Eng. 111 (1994), 111-154. DOI 10.1016/0045-7825(94)90042-6 | MR 1259618 | Zbl 0846.76075
[33] R. Temam: Une méthode d'approximation de la solution des équations de Navier-Stokes. Bull. Soc. Math. Fr. 96 (1968), 115-152. (In French.) DOI 10.24033/bsmf.1662 | MR 0237972 | Zbl 0181.18903
[34] C. Trenchea: Unconditional stability of a partitioned IMEX method for magnetohydrodynamic flows. Appl. Math. Lett. 27 (2014), 97-100. DOI 10.1016/j.aml.2013.06.017 | MR 3111615 | Zbl 1311.76096
[35] Y.-B. Yang, Y.-L. Jiang: Numerical analysis and computation of a type of IMEX method for the time-dependent natural convection problem. Comput. Methods Appl. Math. 16 (2016), 321-344. DOI 10.1515/cmam-2016-0006 | MR 3483620 | Zbl 1336.65155
[36] Y.-B. Yang, Y.-L. Jiang: Analysis of two decoupled time-stepping finite element methods for incompressible fluids with microstructure. Int. J. Comput. Math. 95 (2018), 686-709. DOI 10.1080/00207160.2017.1294688 | MR 3760370 | Zbl 1387.65108
[37] Y.-B. Yang, Y.-L. Jiang: An explicitly uncoupled VMS stabilization finite element method for the time-dependent Darcy-Brinkman equations in double-diffusive convection. Numer. Algorithms 78 (2018), 569-597. DOI 10.1007/s11075-017-0389-7 | MR 3803360 | Zbl 1402.65139
[38] G. Yuksel, R. Ingram: Numerical analysis of a finite element, Crank-Nicolson discretization for MHD flows at small magnetic Reynolds numbers. Int. J. Numer. Anal. Model 10 (2013), 74-98. MR 3011862 | Zbl 1266.76066
[39] G. Yuksel, O. R. Isik: Numerical analysis of Backward-Euler discretization for simplified magnetohydrodynamic flows. Appl. Math. Modelling 39 (2015), 1889-1898. DOI 10.1016/j.apm.2014.10.007 \goodbreak | MR 3325585
Affiliations: Yun-Bo Yang, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, P. R. China; Department of Mathematics, Yunnan Normal University, Kunming, Yunnan, 650500, P. R. China, e-mail: ybyang13@126.com; Yao-Lin Jiang, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, P. R. China, e-mail: yljiang@mail.xjtu.edu.cn; Qiong-Xiang Kong (corresponding author), School of Human Settlements and Civil Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, P. R. China, e-mail: qxkong@mail.xjtu.edu.cn
