Numerical solution of the initial boundary value problem with vacuum boundary conditions for the magnetic field induction equation in a ball

I. V. Bychin   ; A. V. Gorelikov   ; A. V. Ryakhovskii

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 64 (2020), pp. 15-30

Résumé

The aim of this work is to develop and test an algorithm for numerical solution of the initialboundary value problem with vacuum boundary conditions for the equation of magnetic field induction in a ball based on a modification of finite-difference methods of computational fluid dynamics and electrodynamics for use in orthogonal curvilinear coordinates. The problem of the numerical solution of the magnetic field induction equation in the ball $\mathrm{G}$ is considered in this paper using the magnetohydrodynamics (MHD) model. It is assumed that outside the ball the magnetic field induction $\mathbf{B} = \mathrm{grad}\, \psi$, and the potential $\psi$ is a harmonic function regular at infinity. At the boundary $\partial G$, the vacuum boundary conditions are set. They include the requirement that the magnetic field is continuous and the normal component of the current density is equal to zero. The discretization of the induction equation is carried out using the control-volume method and the FDTD (Finite Difference Time Domain) method modified for orthogonal curvilinear coordinates. When integrating over time, a completely implicit scheme is used. The total (convective and diffusion) flows are approximated according to the power-law scheme. The proposed algorithm consists in sequentially executing the following steps at each time layer: $\underline{\text{Step 1}}$: The approximation of the radial component of the magnetic field at $\partial G$ is determined from the discretized continuity equation for the boundary control volumes. $\underline{\text{Step 2}}$: The external Neumann problem for the Laplace equation is solved using the Kelvin transformation. The approximation for the tangential components of the magnetic field at the $\partial G$ is calculated using the found potential $\psi$. $\underline{\text{Step 3}}$: The found components of $\mathbf{B}$ are used as boundary conditions for finding a solution to the induction equation. The steps 1 to 3 are repeated until convergence is achieved at a given moment in time. The algorithm was tested on the problem of the magnetic field diffusion in a conducting ball, which has an analytical solution. In this paper, the discretization of the magnetic field induction equation the solution of which satisfies the solenoidality condition is obtained in arbitrary orthogonal curvilinear coordinates. A new algorithm for numerical solution of the initial-boundary value problem with vacuum boundary conditions for the equation of magnetic field induction in a ball is developed. The comparison between the numerical solution and the analytical solution demonstrates the convergence of the constructed finite-difference scheme.

Détail

Keywords: induction equation, vacuum boundary conditions, numerical solution.

Bibliographie

[1] P. L. Olson, G. A. Glatzmaier, R. S. Coe, “Complex polarity reversals in a geodynamo model”, Earth Planet. Sci. Lett., 304 (2011), 168–179 | DOI

[2] P. Olson, P. Driscoll, H. Amit, “Dipole collapse and reversal precursors in a numerical dynamo”, Phys. Earth. Planet. Inter., 173 (2009), 121–140 | DOI

[3] L. Jouve, A. S. Brun, R. Arlt, A. Brandenburg, M. Dikpati, A. Bonanno, P. J. Käpylä, D. Moss, M. Rempel, P. Gilman, M. J. Korpi, A. G. Kosovichev, “A solar mean field dynamo benchmark”, Astron. Astrophys., 483 (2008), 949–960 | DOI

[4] G. A. Glatzmaier, “Computer simulations of Jupiter's deep internal dynamics help interpret what Juno sees”, Proc. Nat. Acad. Sci., 115:27 (2018), 6896–6904 | DOI

[5] C. Jones, “A dynamo model of Jupiter's magnetic field”, Icarus, 241 (2014), 148–159 | DOI

[6] C. A. Jones, “Convection-driven geodynamo models”, Phil. Trans. R. Soc. Lond. A, 358 (2000), 873–897 | DOI | MR | Zbl

[7] A. A. Jackson et al, “A spherical shell numerical dynamo benchmark with pseudo-vacuum magnetic boundary conditions”, Geophysical Journal International, 196:2 (2014), 712–723 | DOI

[8] V. Bandaru, T. Boeck, D. Krasnov, J. Schumacher, “A hybrid finite difference/boundary element procedure for the simulation of turbulent MHD duct flow at finite magnetic Reynolds number”, Journal of Computational Physics, 304 (2016), 320–339 | DOI | MR | Zbl

[9] G. A. Glatzmaier, “Numerical simulations of stellar convective dynamos I. The model and method”, Journal of Computational Physics, 55 (1984), 461–484 | DOI

[10] M. Yu. Reshetnyak, Simulation in Geodynamo, Lambert, Saarbrucken, 2013, 180 pp.

[11] A. Kageyama, T. Miyagoshi, T. Sato, “Formation of current coils in geodynamo simulations”, Nature, 454 (2008), 1106–1109 | DOI

[12] M. P. Galanin, Yu. P. Popov, Quasistatonary electromagnetic fields in inhomogeneous media: Mathematical simulation, Nauka; Fizmatlit, M., 1995

[13] P. Hejda, M. Reshetnyak, “Control volume method for the dynamo problem in the sphere with the free rotating inner core”, Studia geoph. et geod., 47 (2003), 147–159 | DOI

[14] A. G. Kulikovskij, N. V. Pogorelov, A. Yu. Semenov, Mathematical problems of numerical solution of hyperbolic systems of equations, Fizmatlit, M., 2001 | MR

[15] V. B. Betelin, V. A. Galkin, A. V. Gorelikov, “Predictor-corrector type algorithm for the numerical solution of the induction equation in problem of magnetic hydrodynamics of viscous incompressible fluid”, Doklady Akademii nauk, 464:5 (2015), 525–528 | DOI | MR | Zbl

[16] V. V. Kolmychkov, O. S. Mazhorova, E. E. Fedoseev, “Numerical Method for MHD Equations”, Preprinty IPM im. M.V. Keldysha, 2009, 030, 28 pp.

[17] G. Toth, “The $\nabla\cdot\mathbf{B=0}$ constraint in shock-capturing magnetohydrodynamics codes”, Journal of Computational Physics, 161 (2000), 605–652 | DOI | MR | Zbl

[18] M. V. Popov, T. G. Elizarova, “Simulation of 3D MHD flows within the framework of magneto-quasi-gasdynamics”, Preprinty IPM im. M.V. Keldysha, 2013, 23, 32 pp.

[19] K. G. Powell, P. L. Roe, T. J. Linde, T. I. Gombosi, D. L. De Zeeuw, “A Solution-Adaptive Upwind Scheme for Ideal Magnetohydrodynamics”, Journal of Computational Physics, 154 (1999), 284–309 | DOI | MR | Zbl

[20] R. Teyssier, B. Commercon, “Numerical Methods for Simulating Star Formation”, Frontiers in Astronomy and Space Sciences, 6 (2019), 51 | DOI

[21] L. D. Landau, E. M. Lifshitz, A Course of Theoretical Physics, v. 8, Electrodynamics of Continuous Media, Pergamon Press, Oxford, 1984 | MR

[22] A. G. Kulikovskiy, G. A. Lyubimov, Magnetohydrodynamics, Logos, M., 2011

[23] T. G. Cowling, Magnetohydrodynamics, Adam Hilger, Bristol, 1976, 136 pp. | MR | Zbl

[24] S. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing, Washington DC, 1980, 197 pp. | Zbl

[25] H. K. Versteeg, W. Malalasekera, An Introduction to Computational Fluid Dynamics, Pearson Education Limited, Harlow, 2007, 503 pp.

[26] J. H. Ferziger, M. Peric, Computational Methods for Fluid Dynamics, Springer, Berlin, 2002, 423 pp. | MR | Zbl

[27] A. Taflove, S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd edition, Artech House, Boston, 2000, 866 pp. | MR | Zbl

[28] A. D. Grigorev, Methods of computational electrodynamics, Fizmatlit, M., 2012

[29] A. Ates, O. Altun, A. Kilicman, “On A Comparison of Numerical Solution Methods for General Transport Equation on Cylindrical Coordinates”, Applied Mathematics Information Sciences, 11:2 (2017), 433–439 | DOI

[30] S. K. Dewangan, S. L. Sinha, “Comparison of various numerical differencing schemes in predicting non-newtonian transition flow through an eccentric annulus with inner cylinder in rotation”, International Journal of Mechanical and Industrial Engineering, 3:1 (2013), 119–129

[31] K. Arshad, M. Rafique, A. Majid, S. Jabeen, “Analyses of MHD Pressure Drop in a Curved Bend for Different Liquid Metals”, Journal of Applied Sciences, 7:1 (2007), 72–78 | DOI | MR

[32] A. N. Tihonov, A. A. Samarskij, Equations of mathematical physics, Nauka, M., 1972 | MR

[33] A. V. Gorelikov, “An axisymmetric solution of the problem about demagnetization of a conducting ball”, Vestnik kibernetiki, 2018, no. 4 (32), 9–15