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A fully-implicit finite-volume method for multi-fluid reactive and collisional magnetized plasmas on unstructured meshes. (English) Zbl 1349.76299
Summary: We present a Finite Volume scheme for solving Maxwell’s equations coupled to magnetized multi-fluid plasma equations for reactive and collisional partially ionized flows on unstructured meshes. The inclusion of the displacement current allows for studying electromagnetic wave propagation in a plasma as well as charge separation effects beyond the standard magnetohydrodynamics (MHD) description, however, it leads to a very stiff system with characteristic velocities ranging from the speed of sound of the fluids up to the speed of light. In order to control the fulfillment of the elliptical constraints of the Maxwell’s equations, we use the hyperbolic divergence cleaning method. In this paper, we extend the latter method applying the CIR scheme with scaled numerical diffusion in order to balance those terms with the Maxwell flux vectors. For the fluids, we generalize the \(AUSM^{+}\)-up to multiple fluids of different species within the plasma. The fully implicit second-order method is first verified on the Hartmann flow (including comparison with its analytical solution), two ideal MHD cases with strong shocks, namely, Orszag-Tang and the MHD rotor, then validated on a much more challenging case, representing a two-fluid magnetic reconnection under solar chromospheric conditions. For the latter case, a comparison with pioneering results available in literature is provided.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
82D10 Statistical mechanics of plasmas
Software:
PETSc; AUSM; COOLFluiD
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References:
[1] National Research Council, (2008), The National Academies Press Washington, DC, Severe space weather events - understanding societal and economic impacts: a workshop report
[2] Braams, C. M.; Stott, P. E., Nuclear fusion: half a century of magnetic confinement research, Plasma Phys. Control. Fusion, 44, 8, 1767, (2002)
[3] Giordano, D., Hypersonic-flow governing equations with electromagnetic fields, (2002), AIAA Paper 2002-2165
[4] Yamada, M.; Yoo, J.; Jara-Almonte, J.; Ji, H.; Kulsrud, R. M.; Myers, C. E., Conversion of magnetic energy in the magnetic reconnection layer of a laboratory plasma, Nat. Commun., 5, 4774, (September 2014)
[5] Tóth, G., The div.B=0 constraint in shock-capturing magnetohydrodynamics codes, J. Comput. Phys., 161, 605-652, (July 2000)
[6] Yalim, M. S.; vanden Abeele, D.; Lani, A.; Quintino, T.; Deconinck, H., A finite volume implicit time integration method for solving the equations of ideal magnetohydrodynamics for the hyperbolic divergence cleaning approach, J. Comput. Phys., 230, 6136-6154, (July 2011)
[7] Kissmann, R.; Pomoell, J., A semidiscrete finite volume constrained transport method on orthogonal curvilinear grids, SIAM J. Sci. Comput., 34, 2, 763-791, (March 2012)
[8] Munz, C.-D.; Ommes, P.; Schneider, R., A three-dimensional finite-volume solver for the Maxwell equations with divergence cleaning on unstructured meshes, Comput. Phys. Commun., 130, 83-117, (July 2000)
[9] MacCormack, R., (2008), Numerical simulation of aerodynamic flow including induced magnetic and electric fields
[10] MacCormack, R., (2009), Flow simulations within induced magnetic and electric fields
[11] D’Ambrosio, D.; Giordano, D., Electromagnetic fluid dynamics for aerospace applications, J. Thermophys. Heat Transf., 21, 2, 284-302, (2007)
[12] Thompson, R. J.; Wilson, A.; Moeller, T.; Merkle, C. L., A strong conservative Riemann solver for the solution of the coupled Maxwell and Navier-Stokes equations, J. Comput. Phys., 258, 431-450, (2014) · Zbl 1349.76538
[13] Shumlak, U.; Loverich, J., Approximate Riemann solver for the two-fluid plasma model, J. Comput. Phys., 187, 2, 620-638, (May 2003)
[14] Hakim, A.; Loverich, J.; Shumlak, U., A high resolution wave propagation scheme for ideal two-fluid plasma equations, J. Comput. Phys., 219, 1, 418-442, (November 2006)
[15] Loverich, J.; Shumlak, U., A discontinuous Galerkin method for the full two-fluid plasma model, Comput. Phys. Commun., 169, 1, 251-255, (July 2005)
[16] Srinivasan, B.; Shumlak, U., Analytical and computational study of the ideal full two-fluid plasma model and asymptotic approximations for Hall-magnetohydrodynamics, Phys. Plasmas, 18, 9, (September 2011)
[17] Ofman, L.; Davila, J. M., Three-fluid 2.5-dimensional magnetohydrodynamic model of the effective temperature in coronal holes, Astrophys. J., 553, 935-940, (June 2001)
[18] Ofman, L.; Abbo, L.; Giordano, S., Multi-fluid model of a streamer at solar minimum and comparison with observations, Astrophys. J., 734, 30, (June 2011)
[19] Meier, E. T.; Shumlak, U., A general nonlinear fluid model for reacting plasma-neutral mixtures, Phys. Plasmas, 19, 7, (July 2012)
[20] Leake, J. E.; Lukin, V. S.; Linton, M. G.; Meier, E. T., Multi-fluid simulations of chromospheric magnetic reconnection in a weakly ionized reacting plasma, Astrophys. J., 760, 109, (December 2012)
[21] Courant, R.; Isaacson, E.; Rees, M., On the solution of nonlinear hyperbolic differential equations by finite differences, Commun. Pure Appl. Math., 5, 243-255, (1952) · Zbl 0047.11704
[22] Liou, M., A sequel to AUSM: AUSM+, J. Comput. Phys., 129, 364-382, (December 1996)
[23] Liou, M.-S., A sequel to AUSM, part II: AUSM^{+}-up for all speeds, J. Comput. Phys., 214, 137-170, (May 2006)
[24] Lani, A.; Quintino, T.; Kimpe, D.; Deconinck, H.; Vandewalle, S.; Poedts, S., The coolfluid framework: design solutions for high-performance object oriented scientific computing software, (Sloot, P. M.A.; Sunderan, V. S.; van Albada, G. D.; Dongarra, J. J., Computational Science ICCS 2005, vol. 1, LNCS, vol. 3514, (May 2005), Emory University, Springer Atlanta, GA, USA), 281-286 · Zbl 1129.65328
[25] Lani, A.; Quintino, T.; Kimpe, D.; Deconinck, H.; Vandewalle, S.; Poedts, S., Reusable object-oriented solutions for numerical simulation of PDEs in a high performance environment, Special Edition on POOSC 2005, Sci. Program, 14, 2, 111-139, (2006)
[26] Kimpe, D.; Lani, A.; Quintino, T.; Poedts, S.; Vandewalle, S., The coolfluid parallel architecture, (Kranzlmüller, D.; Di Martino, B.; Dongarra, J. J., Proc. 12th European Parallel Virtual Machine and Message Passing Interface Conference, Sorento, (Oct. 2005), Springer), 520-527
[27] Lani, A., An object oriented and high performance platform for aerothermodynamics simulation, (2008), Université Libre de Bruxelles, PhD thesis
[28] Quintino, T., A component environment for high-performance scientific computing. design and implementation, (2008), Katholieke Universiteit Leuven, PhD thesis
[29] Lani, A.; Villedieu, N.; Bensassi, K.; Kapa, L.; Panesi, M.; Yalim, M. S., Coolfluid: an open computational platform for multi-physics simulation, (21st AIAA CFD Conference, San Diego, WA, (June 2013))
[30] Degrez, G.; Lani, A.; Panesi, M.; Chazot, O.; Deconinck, H., Modelling of high-enthalpy, high-Mach number flows, J. Phys. D, Appl. Phys., 41, (2009)
[31] Lani, A.; Mena, J. Garicano; Deconinck, H., A residual distribution method for symmetrized systems in thermochemical nonequilibrium, (20th AIAA CFD Conference, Honolulu, Hawaii, (Jun 2011)), AIAA-2011-3546
[32] Knight, D.; Longo, J.; Drikakis, D.; Gaitonde, D.; Lani, A., Assessment of CFD capability for prediction of hypersonic shock interactions, Prog. Aerosp. Sci., 48-49, 8-26, (2012)
[33] Lani, A.; Panesi, M.; Deconinck, H., Conservative residual distribution method for viscous double cone flows in thermochemical nonequilibrium, Commun. Comput. Phys., 13, 479-501, (2013)
[34] Panesi, M.; Lani, A., Collisional radiative coarse-grain model for ionization in air, Phys. Fluids, 25, (2013)
[35] Munafo, A.; Lani, A.; Bultel, A.; Panesi, M., Modeling of non-equilibrium phenomena in expanding flows by means of a collisional-radiative model, Phys. Plasmas, 20, 7, (2013)
[36] Yalim, M. S.; Abeele, D. V.; Lani, A., Simulation of field-aligned ideal MHD flows around perfectly conducting cylinders using an artificial compressibility approach, (Proc. of the 11th International Conference on Hyperbolic Problems, July 17-21, Lyon, France, (2006), Ecole Normale Supérieure, Springer-Verlag), 1085-1092 · Zbl 1372.76114
[37] Lani, A.; Yalim, M. S.; Poedts, S., A GPU-enabled finite volume solver for global magnetospheric simulations on unstructured grids, Comput. Phys. Commun., 185, 10, 2538-2557, (2014) · Zbl 1360.76357
[38] Vranjes, J.; Poedts, S., A new paradigm for solar coronal heating, Europhys. Lett., 86, 39001, (May 2009)
[39] Vranjes, J.; Poedts, S., The universally growing mode in the solar atmosphere: coronal heating by drift waves, Mon. Not. R. Astron. Soc., 398, 918-930, (September 2009)
[40] Lukin, V., Home page of hifi framework
[41] Voronov, G. S., A practical fit formula for ionization rate coefficients of atoms and ions by electron impact: \(\operatorname{Z} = 1 - 28\), At. Data Nucl. Data Tables, 65, 1, (1997)
[42] Smirnov, B. M., Physics of atoms, (2003), Springer New York
[43] Braginskii, S. I., Transport processes in a plasma, Rev. Plasma Phys., 1, 205, (1965)
[44] Yu, Sheng-Tao, (Jan. 1997), Treatment of stiff source terms in conservation laws by the method of space-time conservation element and solution element
[45] Barth, T., (1994), Aspects of unstructured grids and finite-volume solvers for the Euler and Navier-Stokes equations
[46] Venkatakrishnan, V., (1993), On the accuracy of limiters and convergence to steady state solutions
[47] Liou, M.-S.; Steffen, C. J., A new flux splitting scheme, J. Comput. Phys., 107, 23-39, (July 1993)
[48] Liou, M.-S., A further development of the AUSM^{+} towards robust and accurate solutions for all speeds, (June 2003), AIAA paper 4116
[49] Kim, S. E.; Makarov, B.; Caraeni, D., Multi-dimensional linear reconstruction scheme for arbitrary unstructured mesh, (16th AIAA CFD Conference, Orlando, Florida, (Jun 2003)), AIAA-2003-3990
[51] Dedner, A.; Kemm, F.; Kröner, D.; Munz, C.-D.; Schnitzer, T.; Wesenberg, M., Hyperbolic divergence cleaning for the MHD equations, J. Comput. Phys., 175, 645-673, (January 2002)
[52] Brackbill, J. U.; Barnes, D. C., The effect of nonzero product of magnetic gradient and B on the numerical solution of the magnetohydrodynamic equations, J. Comput. Phys., 35, 426-430, (May 1980)
[53] Balsara, D. S., Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction, Astrophys. J. Suppl. Ser., 151, 149-184, (March 2004)
[54] Balsara, D. S., Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics, J. Comput. Phys., 228, 5040-5056, (August 2009)
[55] Balsara, D. S.; Rumpf, T.; Dumbser, M.; Munz, C.-D., Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics, J. Comput. Phys., 228, 2480-2516, (April 2009)
[56] Shen, Y.; Zha, G.; Huerta, M. A., E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO scheme, J. Comput. Phys., 231, 6233-6247, (August 2012)
[57] Mignone, A.; Tzeferacos, P.; Bodo, G., High-order conservative finite difference GLM-MHD schemes for cell-centered MHD, J. Comput. Phys., 229, 5896-5920, (August 2010)
[59] Mitchner, M.; Kruger, C. H., Partially ionized gases, (1973), Department of Mechanical Engineering, Stanford Engineering, Wiley-Interscience Publication
[60] Orszag, S. A.; Tang, C.-M., Small-scale structure of two-dimensional magnetohydrodynamic turbulence, J. Fluid Mech., 90, 129-143, (January 1979)
[61] Christlieb, Andrew J.; Rossmanith, James A.; Tang, Qi, Finite difference weighted essentially non-oscillatory schemes with constrained transport for ideal magnetohydrodynamics, J. Comput. Phys., 268, 302-325, (2014) · Zbl 1349.76442
[62] Balsara, D. S.; Spicer, D. S., A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations, J. Comput. Phys., 149, 270-292, (March 1999)
[63] Biskamp, D., Magnetic reconnection in plasmas, (2005), Cambridge University Press · Zbl 0891.76094
[64] Loverich, J.; Hakim, A.; Shumlak, U., A discontinuous Galerkin method for ideal two-fluid plasma equations, Commun. Comput. Phys., 9, 240, (2011) · Zbl 1364.35278
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