# zbMATH — the first resource for mathematics

Correspondence between constrained transport and vector potential methods for magnetohydrodynamics. (English) Zbl 1406.76090
Summary: We show that one can formulate second-order field- and flux-interpolated constrained transport/central difference (CT/CD) type methods as cell-centered magnetic vector potential schemes. We introduce four vector potential CTA/CDA schemes – three of which correspond to CT/CD methods of G. Tóth [ibied. 161, No. 2, 605–652 (2000; Zbl 0980.76051)] and one of which is a new simple flux-CT-like scheme – where the centroidal vector potential is the primal update variable. These algorithms conserve a discretization of the $$\nabla \cdot \mathbf{B} = 0$$ condition to machine precision and may be combined with shock-capturing Godunov type base schemes for magnetohydrodynamics. Recasting CT in terms of a centroidal vector potential allows for some simple generalizations of divergence-preserving methods to unstructured meshes, and potentially new directions to generalize CT schemes to higher-order.

##### MSC:
 76W05 Magnetohydrodynamics and electrohydrodynamics 76M20 Finite difference methods applied to problems in fluid mechanics
##### Software:
IllinoisGRMHD; HE-E1GODF; Pluto
Full Text:
##### References:
 [1] Tóth, G., The $$\operatorname{\nabla} \cdot \operatorname{B} = 0$$ constraint in shock-capturing magnetohydrodynamics codes, J. Comput. Phys., 161, 605-652, (2000) · Zbl 0980.76051 [2] 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, (1980) · Zbl 0429.76079 [3] Brackbill, J., Fluid modeling of magnetized plasmas, Space Sci. Rev., 42, 153-167, (1985) [4] Evans, C. R.; Hawley, J. F., Simulation of magnetohydrodynamic flows - a constrained transport method, Astrophys. J., 332, 659-677, (1988) [5] Yee, K., Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas Propag., 14, 302-307, (1966) · Zbl 1155.78304 [6] Fromang, S.; Hennebelle, P.; Teyssier, R., A high order Godunov scheme with constrained transport and adaptive mesh refinement for astrophysical magnetohydrodynamics, Astron. Astrophys., 457, 371-384, (2006) [7] Mignone, A.; Bodo, G.; Massaglia, S.; Matsakos, T.; Tesileanu, O.; Zanni, C.; Ferrari, A., PLUTO: a numerical code for computational astrophysics, Astrophys. J. Suppl. Ser., 170, 228-242, (2007) [8] Cunningham, A. J.; Frank, A.; Varnière, P.; Mitran, S.; Jones, T. W., Simulating magnetohydrodynamical flow with constrained transport and adaptive mesh refinement: algorithms and tests of the astrobear code, Astrophys. J. Suppl. Ser., 182, 519-542, (2009) [9] Christlieb, A. J.; Rossmanith, J. A.; Tang, Q., Finite difference weighted essentially non-oscillatory schemes with constrained transport for ideal magnetohydrodynamics, J. Comput. Phys., 268, 302-325, (2014) · Zbl 1349.76442 [10] White, C. J.; Stone, J. M.; Gammie, C. F., An extension of the athena++ code framework for GRMHD based on advanced Riemann solvers and staggered-mesh constrained transport, Astrophys. J. Suppl. Ser., 225, 22, (2016) [11] Del Zanna, L.; Bucciantini, N.; Londrillo, P., An efficient shock-capturing central-type scheme for multidimensional relativistic flows. II. magnetohydrodynamics, Astron. Astrophys., 400, 397-413, (2003) · Zbl 1222.76122 [12] Etienne, Z. B.; Paschalidis, V.; Haas, R.; Mösta, P.; Shapiro, S. L., Illinoisgrmhd: an open-source, user-friendly GRMHD code for dynamical spacetimes, Class. Quantum Gravity, 32, 17, 175009, (2015) · Zbl 1327.83153 [13] Mocz, P.; Vogelsberger, M.; Hernquist, L., A constrained transport scheme for MHD on unstructured static and moving meshes, Mon. Not. R. Astron. Soc., 442, 43-55, (2014) [14] Balsara, D. S.; Dumbser, M., Divergence-free MHD on unstructured meshes using high order finite volume schemes based on multidimensional Riemann solvers, J. Comput. Phys., 299, 687-715, (2015) · Zbl 1351.76092 [15] Balsara, D. S.; Dumbser, M., Multidimensional Riemann problem with self-similar internal structure. part II - application to hyperbolic conservation laws on unstructured meshes, J. Comput. Phys., 287, 269-292, (2015) · Zbl 1351.76091 [16] Dai, W.; Woodward, P. R., A simple finite difference scheme for multidimensional magnetohydrodynamical equations, J. Comput. Phys., 142, 331-369, (1998) · Zbl 0932.76048 [17] Dai, W.; Woodward, P. R., On the divergence-free condition and conservation laws in numerical simulations for supersonic magnetohydrodynamical flows, Astrophys. J., 494, 317-335, (1998) [18] 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, (1999) · Zbl 0936.76051 [19] Mocz, P.; Pakmor, R.; Springel, V.; Vogelsberger, M.; Marinacci, F.; Hernquist, L., A moving mesh unstaggered constrained transport scheme for magnetohydrodynamics, Mon. Not. R. Astron. Soc., 463, 477-488, (2016) [20] Helzel, C.; Rossmanith, J. A.; Taetz, B., An unstaggered constrained transport method for the 3D ideal magnetohydrodynamic equations, J. Comput. Phys., 230, 3803-3829, (2011) · Zbl 1369.76061 [21] Helzel, C.; Rossmanith, J. A.; Taetz, B., A high-order unstaggered constrained transport method for the 3D ideal magnetohydrodynamic equations based on the method of lines, arXiv e-prints · Zbl 1369.76062 [22] Ryu, D.; Miniati, F.; Jones, T. W.; Frank, A., A divergence-free upwind code for multidimensional magnetohydrodynamic flows, Astrophys. J., 509, 244-255, (1998) [23] Gardiner, T. A.; Stone, J. M., An unsplit Godunov method for ideal MHD via constrained transport, J. Comput. Phys., 205, 509-539, (2005) · Zbl 1087.76536 [24] Gardiner, T. A.; Stone, J. M., An unsplit Godunov method for ideal MHD via constrained transport in three dimensions, J. Comput. Phys., 227, 4123-4141, (2008) · Zbl 1317.76057 [25] Balsara, D. S., A two-dimensional HLLC Riemann solver for conservation laws: application to Euler and magnetohydrodynamic flows, J. Comput. Phys., 231, 7476-7503, (2012) · Zbl 1284.76261 [26] Balsara, D. S., Multidimensional Riemann problem with self-similar internal structure. part I - application to hyperbolic conservation laws on structured meshes, J. Comput. Phys., 277, 163-200, (2014) · Zbl 1349.76303 [27] Balsara, D. S., Three dimensional HLL Riemann solver for conservation laws on structured meshes; application to Euler and magnetohydrodynamic flows, J. Comput. Phys., 295, 1-23, (2015) · Zbl 1349.76584 [28] Balsara, D. S.; Dumbser, M.; Abgrall, R., Multidimensional HLLC Riemann solver for unstructured meshes - with application to Euler and MHD flows, J. Comput. Phys., 261, 172-208, (2014) · Zbl 1349.76426 [29] Balsara, D. S., Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction, Astrophys. J. Suppl. Ser., 151, 149-184, (2004) [30] Balsara, D. S., Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics, J. Comput. Phys., 228, 5040-5056, (2009) · Zbl 1280.76030 [31] Balsara, D. S.; Amano, T.; Garain, S.; Kim, J., A high-order relativistic two-fluid electrodynamic scheme with consistent reconstruction of electromagnetic fields and a multidimensional Riemann solver for electromagnetism, J. Comput. Phys., 318, 169-200, (2016) · Zbl 1349.76425 [32] Toro, E. F., Riemann solvers and numerical methods for fluid dynamics: A practical introduction, (2013), Springer Science & Business Media [33] Van Leer, B., Upwind and high-resolution methods for compressible flow: from donor cell to residual-distribution schemes, Commun. Comput. Phys., 1, 192-206, (2006), 138 · Zbl 1114.76049 [34] Miyoshi, T.; Kusano, K., A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics, J. Comput. Phys., 208, 315-344, (2005) · Zbl 1114.76378 [35] Orszag, S. A.; Tang, C.-M., Small-scale structure of two-dimensional magnetohydrodynamic turbulence, J. Fluid Mech., 90, 129-143, (1979) [36] Miniati, F.; Martin, D. F., Constrained-transport magnetohydrodynamics with adaptive mesh refinement in CHARM, Astrophys. J. Suppl. Ser., 195, 5, (2011)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.