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Second order unconditionally convergent and energy stable linearized scheme for MHD equations

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Abstract

In this paper, we propose an efficient numerical scheme for magnetohydrodynamics (MHD) equations. This scheme is based on a second order backward difference formula for time derivative terms, extrapolated treatments in linearization for nonlinear terms. Meanwhile, the mixed finite element method is used for spatial discretization. We present that the scheme is unconditionally convergent and energy stable with second order accuracy with respect to time step. The optimal L 2 and H 1 fully discrete error estimates for velocity, magnetic variable and pressure are also demonstrated. A series of numerical tests are carried out to confirm our theoretical results. In addition, the numerical experiments also show the proposed scheme outperforms the other classic second order schemes, such as Crank-Nicolson/Adams-Bashforth scheme, linearized Crank-Nicolson’s scheme and extrapolated Gear’s scheme, in solving high physical parameters MHD problems.

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References

  1. Abboud, H., Girault, V., Sayah, T.: A second order accuracy for a full discretized time-dependent Navier-Stokes equations by a two-grid scheme. Numer. Math. 114, 189–231 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Akbas, M., Kaya, S., Mohebujjaman, M., Rebholz, L.G.: Numerical analysis and testing of a fully discrete, decoupled penalty-projection algorithm for MHD in elsässer variable. Int. J. Numer. Anal. Model. 13, 90–113 (2016)

    MathSciNet  MATH  Google Scholar 

  3. Akbas, M., Kaya, S., Rebholz, L.G.: On the stability at all times of linearly extrapolated BDF2 timestepping for multiphysics incompressible flow problems. Numer. Methods PDEs (2016)

  4. Akrivis, G.: Implicit-explicit multistep methods for nonlinear parabolic equations. Math. Comput. 82, 45–68 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ascher, U.M., Ruuth, S.J., Wetton, B.T.R.: Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32, 797–823 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Badia, S., Codina, R.: Analysis of a stabilized finite element approximation of the transient convection-diffusion equation using an Ale framework. SIAM J. Numer. Anal. 44, 2159–2197 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bermejo, R., Galán Del Sastre, P., Saavedra, L.: A second order in time modified lagrange-galerkin finite element method for the incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 50, 3084–3109 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Borzì, A., Andrade, S.G.: Second-order approximation and fast multigrid solution of parabolic bilinear optimization problems. Adv. Comput. Math. 41, 457–488 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cao, C., Wu, J.: Two regularity criteria for the 3D MHD equations. J. Differ. Equ. 248, 2263–2274 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Case, M.A., Labovsky, A., Rebholz, L.G., Wilson, N.E.: A high physical accuracy method for incompressible magnetohydrodynamics. Int. J. Numer. Anal. Model. Ser. B 1, 217–236 (2010)

    MathSciNet  MATH  Google Scholar 

  11. Chen, W., Gunzburger, M., Sun, D., Wang, X.: Efficient and long-time accurate second-order methods for Stokes-Darcy System. SIAM J. Numer. Anal. 51, 2563–2584 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Del Sastre, P.G., Bermejo, R.: Error analysis for hp-FEM semi-Lagrangian second order BDF method for convection-dominated diffusion problems. J. Sci. Comput. 49, 211–237 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. De Los Reyes, J.C., Andrade, S.G.: Numerical simulation of thermally convective viscoplastic fluids by semismooth second order type methods. J. Non-Newtonian Fluid Mech. 193, 43–48 (2013)

    Article  Google Scholar 

  14. Dolejší, V., Vlasák, M.: Analysis of a BDF-DGFE scheme for nonlinear convection-diffusion problems. Numer. Math. 110, 405–447 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Emmrich, E.: Error of the two-step BDF for the incompressible Navier-Stokes problem. ESAIM: M2AN 38, 757–764 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Emmrich, E.: Convergence of a time discretization for a class of non-Newtonian fluid flow. Commun. Math. Sci. 6, 827–843 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Emmrich, E.: Convergence of the variable two-step BDF time discretisation of nonlinear evolution problems governed by a monotone potential operator. BIT Numer. Math. 49, 297–323 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Georgescu, V.: Some boundary value problems for differenttial forms on compact Riemannian manifolds. Ann. Mat. Pura Appl. 4, 159–198 (1979)

    Article  MATH  Google Scholar 

  19. Gerbeau, J.F.: Problèmes mathématiques et numériques posés par la modélisation de l’électrolyse de l’aluminium. Ph.D. Thesis, École Nationale des Ponts et Chaussées (1998)

  20. Gerbeau, J.F.: A stabilized finite element method for the incompressible magnetohydrodynamic equations. Numer. Math. 87, 83–111 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gerbeau, J.F., Le Bris, C., Lelièvre, T.: Mathematical Method for the Magnetohydrodynamics of Liquid Metals. Oxford University Press, Oxford (2006)

    Book  MATH  Google Scholar 

  22. Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equtions: Theory and Algorithms. Springer, New York (1986)

    Book  MATH  Google Scholar 

  23. Greif, C., Li, D., Schötzau, D., Wei, X.: A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics. Comput. Methods Appl. Mech. Engrg. 199, 2840–2855 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gunzburger, M.D., Meir, A.J., Peterson, J.P.: On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics. Math. Comp. 56, 523–563 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  25. He, C., Wang, Y.: On the regularity criteria for weak solutions to the magnetohydrodynamic equations. J. Differ. Equ. 238, 1–17 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. He, Y.N.: Unconditional convergence of the Euler semi-implicit scheme for the 3D incompressible MHD equations. IMA J. Numer. Anal. 35, 767–801 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Heister, T., Mohebujjaman, M., Rebholz, L.G.: Decoupled, Unconditionally stable, higher order discretizations for MHD flow simulation. J. Sci. Comput. 71, 21–43 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Heister, T., Olshanskii, M.A., Rebholz, L.G.: Unconditional long-time stability of a velocity-vorticity method for the 2D Navier-Stokes equations. Numer. Math. 135, 143–167 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Heywood, J.G., Rannacher, R.: Finite-element approximations of the nonstationary Navier-Stokes problem I: regularity of solutions and second-order spatial discretization. SIAM J. Numer. Anal. 19, 275–311 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hill, A.T., Söli, E.: Approximation of the global attractor for the incompressible Navier-Stokes equations. IMA J. Numer. Anal. 20, 633–667 (2000)

    Article  MathSciNet  Google Scholar 

  31. Huang, C., Vandewalle, S.: Unconditionally stable difference methods for delay partial differential equations. Numer. Math. 122, 579–601 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kim, S., Lee, E.B., Choi, W.: Newton’s algorithm for magnetohydrodynamic equations with the initial guess from Stokes-like problem. J. Comput. Appl. Math. 309, 1–10 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Layton, W., Mays, N., Neda, M., Trenchea, C.: Numerical analysis of modular regularization methods for the BDF2 time discretization of the Navier-Stokes equations. ESAIM: M2AN. 48, 765–793 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Layton, W., Tran, H., Trenchea, C.: Numerical analysis of two partitioned methods for uncoupling evolutionary MHD flows. Numer. Method Partial Differ. Equs. 30, 1083–1102 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lifschitz, A.E.: Magnetohydrodynamics and spectral theory. Kluwer, Dordrecht (1989)

    Book  MATH  Google Scholar 

  36. Lin, F., Zhang, P.: Global small solutions to an MHD-type system: the three-dimensional case. Comm. Pure Appl. Math. 67, 531–580 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Lin, F., Xu, L., Zhang, P.: Global small solutions to 2-D incompressible MHD system. J. Differ. Equ. 259, 5440–5485 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Meir, A.J., Schmidt, P.G.: Analysis and numerical approximation of a stationary MHD flow problem with nonideal boundary. SIAM J. Numer. Anal. 36, 1304–1332 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  39. Moreau, R.: Magnetohydrodynamics. Kluwer, Dordrecht (1990)

    Book  MATH  Google Scholar 

  40. Phillips, E.G., Elman, H.C., Cyr, E.C., Shadid, J.N., Pawlowski, R.P.: A block preconditioner for an exact penalty formulation for stationary MHD. SIAM J. Sci. Comput. 36(6), B930–B951 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. Phillips, E.G., Shadid, J.N., Cyr, E.C., Elman, H.C., Pawlowski, R.P.: Block preconditioners for stable mixed nodal and edge finite element representations of incompressible resistive MHD. SIAM J. Sci. Comput. 38(6), B1009–B1031 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Priest, E.R., Hood, A.W.: Advances in Solar System Magnetohydrodynamics. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  43. Prohl, A.: Convergent finite element discretizations of the nonstationary incompressible Magnetohydrodynamics system, M2AN. Math. Model. Numer. Anal. 42, 1065–1087 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  44. Rong, Y., Hou, Y., Zhang, Y.: Numerical analysis of a second order algorithm for simplified magnetohydrodynamic flows. Adv. Comput. Math. http://doi.org/10.1007/s10444-016-9508-6 (2016)

  45. Schonbek, M.E., Schonbek, T.P., Söli, E.: Large-time behaviour of solutions to the magnetohydrodynamics equations. Math. Ann. 304, 717–756 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  46. Schötzau, D.: Mixed finite element methods for stationary incompressible magnetohydrodynamics. Numer. Math. 96, 771–800 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  47. Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Comm. Pure Appl. Math. 36, 635–664 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  48. Temam, R.: Navier-Stokes Equations, Theory and Numerical Analysis (3rd Ed.), North-Holland, Amsterdam (1983)

  49. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Spinger, New York (1988)

    Book  MATH  Google Scholar 

  50. Tone, F.: On the long-time H 2-stability of the implicit Euler scheme for the 2D magnetohydrodynamics equations. J. Sci. Comput. 38, 331–348 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  51. Trenchea, C.: Unconditional stability of a partitioned IMEX method for magnetohydrodynamic flows. Appl. Math. Lett. 27(2014), 97–100 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  52. Varah, J.M.: Stability restrictions on second-order, three level finite difference schemes for parabolic equations. SIAM J. Numer. Anal. 17, 300–309 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  53. Wacker, B., Arndt, D., Lube, G.: Nodal-based finite element methods with local projection stabilization for linearized incompressible magnetohydrodynamics. Comput. Methods Appl. Mech. Engrg. 302, 170–192 (2016)

    Article  MathSciNet  Google Scholar 

  54. Wang, X.: An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier-Stokes equations. Numer. Math. 121, 753–779 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  55. Wiedmer, M.: Finite element approximation for equations of magnetohydrodynamics. Math. Comput. 69, 83–101 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  56. Yuksel, G., Ingram, R.: Numerical analysis of a finite element, Crank-Nicolson discretization for MHD flow at small magnetic Reynolds number. Int. J. Numer. Anal. Model. 10, 74–98 (2013)

    MathSciNet  MATH  Google Scholar 

  57. Yuksel, G., Isik, O.R.: Numerical analysis of Backward-Euler discretization for simplified magnetohydrodynamic flows. Appl. Math. Model. 39, 1889–1898 (2015)

    Article  MathSciNet  Google Scholar 

  58. Zhang, G.D., He, Y.N., Zhang, Y.: Streamline diffusion finite element method for stationary incompressible magnetohydrodynamics. Numer. Method PDEs. 30, 1877–1901 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  59. Zhang, G.D., He, Y.N.: Unconditional convergence of the Euler semi-implicit scheme for the 3D incompressible MHD equations: numerical implementation. Internat. J. Numer. Methods Heat Fluid Flow. 25, 1912–1923 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  60. Zhang, G.D., He, Y.N.: Decoupled schemes for unsteady MHD equations II: finite element discretization and numerical implementation. Comput. Math. Appl. 69, 1390–1406 (2015)

    Article  MathSciNet  Google Scholar 

  61. Zhang, K., Wong, J.C.F., Zhang, R.: Second-order implicit-explicit scheme for the Gray-Scott model. J. Comput. Appl. Math. 213, 559–581 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  62. Zhang, Y., Hou, Y., Shan, L.: Numerical analysis of the Crank-Nicolson extrapolation time discrete scheme for magnetohydrodynamics flows. Numer. Method. for PDEs. 31, 2169–2208 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

We gratefully acknowledge the anonymous referees for their pertinent and perceptive comments which have significantly improved our paper.

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Authors and Affiliations

  1. School of Mathematics and Information Sciences, Yantai University, Yantai, 264005, China

    Guo-Dong Zhang & Chunjia Bi

  2. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China

    Jinjin Yang

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  1. Guo-Dong Zhang
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Correspondence to Guo-Dong Zhang.

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Communicated by: Long Chen

Guodong Zhang is supported by National Science Foundation of China (11601468); Chunjia Bi is supported by National Science Foundation of China (11571297) and Shandong Province Natural Science Foundation (ZR2014AM003).

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Zhang, GD., Yang, J. & Bi, C. Second order unconditionally convergent and energy stable linearized scheme for MHD equations. Adv Comput Math 44, 505–540 (2018). https://doi.org/10.1007/s10444-017-9552-x

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