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A fourth-order accurate finite volume method for ideal MHD via upwind constrained transport. (English) Zbl 1416.76147
Summary: We present a fourth-order accurate finite volume method for the solution of ideal magnetohydrodynamics (MHD). The numerical method combines high-order quadrature rules in the solution of semi-discrete formulations of hyperbolic conservation laws with the upwind constrained transport (UCT) framework to ensure that the divergence-free constraint of the magnetic field is satisfied. A novel implementation of UCT that uses the piecewise parabolic method (PPM) for the reconstruction of magnetic fields at cell corners in 2D is introduced. The resulting scheme can be expressed as the extension of the second-order accurate constrained transport (CT) Godunov-type scheme that is currently used in the Athena astrophysics code. After validating the base algorithm on a series of hydrodynamics test problems, we present the results of multidimensional MHD test problems which demonstrate formal fourth-order convergence for smooth problems, robustness for discontinuous problems, and improved accuracy relative to the second-order scheme.

##### MSC:
 76M12 Finite volume methods applied to problems in fluid mechanics 76W05 Magnetohydrodynamics and electrohydrodynamics 65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
##### Software:
ECHO; Athena ; HLLE; RIEMANN
Full Text:
##### References:
 [1] Amano, Takanobu, Divergence-free approximate Riemann solver for the quasi-neutral two-fluid plasma model, J. Comput. Phys., 299, 863-886, (2015) · Zbl 1351.76207 [2] Balsara, Dinshaw S., Multidimensional HLLE Riemann solver: application to Euler and magnetohydrodynamic flows, J. Comput. Phys., 229, 6, 1970-1993, (2010) · Zbl 1303.76140 [3] Balsara, Dinshaw S., A two-dimensional HLLC Riemann solver for conservation laws: application to Euler and magnetohydrodynamic flows, J. Comput. Phys., 231, 22, 7476-7503, (2012) · Zbl 1284.76261 [4] Balsara, Dinshaw 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 [5] Balsara, Dinshaw S.; Kim, Jongsoo, A comparison between divergence-cleaning and staggered-mesh formulations for numerical magnetohydrodynamics, Astrophys. J., 602, 2, 1079-1090, (2004) [6] Balsara, Dinshaw S.; Shu, Chi-Wang, Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. Comput. Phys., 160, 2, 405-452, (2000) · Zbl 0961.65078 [7] Balsara, Dinshaw S.; Spicer, Daniel S., A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations, J. Comput. Phys., 149, 2, 270-292, (1999) · Zbl 0936.76051 [8] Barad, Michael; Colella, Phillip, A fourth-order accurate local refinement method for Poisson’s equation, J. Comput. Phys., 209, 1, 1-18, (2005) · Zbl 1073.65126 [9] Brackbill, J. U.; Barnes, D. C., The effect of nonzero $$\operatorname{\nabla} \cdot B$$ on the numerical solution of the magnetohydrodynamic equations, J. Comput. Phys., 35, 3, 426-430, (1980) · Zbl 0429.76079 [10] Brio, M.; Wu, C. C., An upwind differencing scheme for the equations of ideal magnetohydrodynamics, J. Comput. Phys., 75, 2, 400-422, (1988) · Zbl 0637.76125 [11] Cargo, P.; Gallice, G., Roe matrices for ideal MHD and systematic construction of Roe matrices for systems of conservation laws, J. Comput. Phys., 136, 2, 446-466, (1997) · Zbl 0919.76053 [12] Colella, P.; Dorr, M.; Hittinger, J.; McCorquodale, P.; Martin, D. F., High-order finite-volume methods on locally-structured grids, (Numerical Modeling of Space Plasma Flows: ASTRONUM-2008, vol. 406, (2009)), 1-9 [13] Colella, P.; Dorr, M. R.; Hittinger, J. A.F.; Martin, D. F., High-order, finite-volume methods in mapped coordinates, J. Comput. Phys., 230, 8, 2952-2976, (2011) · Zbl 1218.65119 [14] Colella, Phillip; Sekora, Michael D., A limiter for PPM that preserves accuracy at smooth extrema, J. Comput. Phys., 227, 15, 7069-7076, (2008) · Zbl 1152.65090 [15] Colella, Phillip; Woodward, Paul R., The Piecewise Parabolic Method (PPM) for gas-dynamical simulations, J. Comput. Phys., 54, 1, 174-201, (1984) · Zbl 0531.76082 [16] Crockett, Robert K.; Colella, Phillip; Fisher, Robert T.; Klein, Richard I.; McKee, Christopher F., An unsplit, cell-centered Godunov method for ideal MHD, J. Comput. Phys., 203, 2, 422-448, (2005) · Zbl 1143.76599 [17] Dai, Wenlong; Woodward, Paul R., On the divergence-free condition and conservation laws in numerical simulations for supersonic magnetohydrodynamical flows, Astrophys. J., 494, 1, 317-335, (1998) [18] Davis, S. F., Simplified second-order Godunov-type methods, SIAM J. Sci. Stat. Comput., 9, 3, 445-473, (1988) · Zbl 0645.65050 [19] Núñez de la Rosa, Jonatan; Munz, Claus-Dieter, xtroem-fv: a new code for computational astrophysics based on very high order finite-volume methods - I. magnetohydrodynamics, Mon. Not. R. Astron. Soc., 455, 4, 3458-3479, (2016) [20] Dedner, A.; Kemm, F.; Kröner, D.; Munz, C.-D.; Schnitzer, T.; Wesenberg, M., Hyperbolic divergence cleaning for the MHD equations, J. Comput. Phys., 175, 2, 645-673, (2002) · Zbl 1059.76040 [21] Dumbser, Michael; Zanotti, Olindo; Hidalgo, Arturo; Balsara, Dinshaw S., ADER-WENO finite volume schemes with space-time adaptive mesh refinement, J. Comput. Phys., 248, 257-286, (2013) · Zbl 1349.76325 [22] Einfeldt, B.; Munz, C. D.; Roe, P. L.; Sjögreen, B., On Godunov-type methods near low densities, J. Comput. Phys., 92, 2, 273-295, (1991) · Zbl 0709.76102 [23] Einfeldt, Bernd, On Godunov-type methods for gas dynamics, SIAM J. Numer. Anal., 25, 2, 294-318, (1988) · Zbl 0642.76088 [24] Evans, Charles R.; Hawley, John F., Simulation of magnetohydrodynamic flows - a constrained transport method, Astrophys. J., 332, 2, 659, (1988) [25] Falle, S. A.E. G., Self-similar jets, Mon. Not. R. Astron. Soc., 250, 3, 581-596, (1991) [26] Fromang, S.; Hennebelle, P.; Teyssier, R., A high order Godunov scheme with constrained transport and adaptive mesh refinement for astrophysical magnetohydrodynamics, Astron. Astrophys., 457, 2, 371-384, (2006) [27] Gardiner, Thomas A.; Stone, James M., An unsplit Godunov method for ideal MHD via constrained transport, J. Comput. Phys., 205, 2, 509-539, (2005) · Zbl 1087.76536 [28] Gardiner, Thomas A.; Stone, James M., An unsplit Godunov method for ideal MHD via constrained transport in three dimensions, J. Comput. Phys., 227, 8, 4123-4141, (2008) · Zbl 1317.76057 [29] Gottlieb, Sigal; Shu, Chi-Wang, Total variation diminishing Runge-Kutta schemes, Math. Comput., 67, 221, 73-85, (1998) · Zbl 0897.65058 [30] Gottlieb, Sigal; Shu, Chi-Wang; Tadmor, Eitan, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43, 1, 89-112, (2001) · Zbl 0967.65098 [31] Gottlieb, Sigal; Ketcheson, David I.; Shu, Chi-Wang, High order strong stability preserving time discretizations, J. Sci. Comput., 38, 3, 251-289, (2009) · Zbl 1203.65135 [32] Guzik, Stephen M.; Gao, Xinfeng; Owen, Landon D.; McCorquodale, Peter; Colella, Phillip, A freestream-preserving fourth-order finite-volume method in mapped coordinates with adaptive-mesh refinement, Comput. Fluids, 123, 202-217, (2015) · Zbl 1390.65091 [33] Harten, Amiram; Lax, Peter D.; van Leer, Bram, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25, 1, 35-61, (1983) · Zbl 0565.65051 [34] Jiang, Guang-Shan; Shu, Chi-Wang, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 1, 202-228, (1996) · Zbl 0877.65065 [35] Ketcheson, David I., Highly efficient strong stability-preserving Runge-Kutta methods with low-storage implementations, SIAM J. Sci. Comput., 30, 4, 2113-2136, (2008) · Zbl 1168.65382 [36] Ketcheson, David I., Runge-Kutta methods with minimum storage implementations, J. Comput. Phys., 229, 5, 1763-1773, (2010) · Zbl 1183.65093 [37] Li, Fengyan; Shu, Chi-Wang, Locally divergence-free discontinuous Galerkin methods for MHD equations, J. Sci. Comput., 22-23, June, 413-442, (2005) · Zbl 1123.76341 [38] Loffeld, J.; Hittinger, J. A.F., On the arithmetic intensity of high-order finite-volume discretizations for hyperbolic systems of conservation laws, Int. J. High Perform. Comput. Appl., (2017) [39] Londrillo, P.; Del Zanna, L., High-order upwind schemes for multidimensional magnetohydrodynamics, Astrophys. J., 530, 1, 508-524, (2000) [40] Londrillo, P.; Del Zanna, L., On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: the upwind constrained transport method, J. Comput. Phys., 195, 1, 17-48, (2004) · Zbl 1087.76074 [41] Luo, Hong; Baum, Joseph D.; Löhner, Rainald, A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids, J. Comput. Phys., 227, 20, 8875-8893, (2008) · Zbl 1391.76350 [42] Matsumoto, Yosuke; Asahina, Yuta; Kudoh, Yuki; Kawashima, Tomohisa; Matsumoto, Jin; Takahashi, Hiroyuki R.; Minoshima, Takashi; Zenitani, Seiji; Miyoshi, Takahiro; Matsumoto, Ryoji, Magnetohydrodynamic simulation code CANS+: assessments and applications, (2016) [43] McCorquodale, Peter; Colella, Phillip, A high-order finite-volume method for conservation laws on locally refined grids, Commun. Appl. Math. Comput. Sci., 6, 1, 1-25, (2011) · Zbl 1252.65163 [44] Mignone, Andrea, High-order conservative reconstruction schemes for finite volume methods in cylindrical and spherical coordinates, J. Comput. Phys., 270, 784-814, (2014) · Zbl 1349.76364 [45] Mignone, Andrea; Tzeferacos, Petros; Bodo, Gianluigi, High-order conservative finite difference GLM-MHD schemes for cell-centered MHD, J. Comput. Phys., 229, 17, 5896-5920, (2010) · Zbl 1425.76305 [46] Miyoshi, Takahiro; Kusano, Kanya, A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics, J. Comput. Phys., 208, 1, 315-344, (2005) · Zbl 1114.76378 [47] Mocz, Philip; Vogelsberger, Mark; Sijacki, Debora; Pakmor, Rüdiger; Hernquist, Lars, A discontinuous Galerkin method for solving the fluid and magnetohydrodynamic equations in astrophysical simulations, Mon. Not. R. Astron. Soc., 437, 1, 397-414, (2014) [48] Olschanowsky, Catherine; Mills Strout, Michelle; Guzik, Stephen; Loffeld, John; Hittinger, Jeffrey, A study on balancing parallelism, data locality, and recomputation in existing PDE solvers, (SC14: International Conference for High Performance Computing, Networking, Storage and Analysis, (2014), IEEE) [49] Orszag, Steven A.; Tang, Cha-Mei, Small-scale structure of two-dimensional magnetohydrodynamic turbulence, J. Fluid Mech., 90, 01, 129, (1979) [50] Peterson, J. L.; Hammett, G. W., Positivity preservation and advection algorithms with applications to edge plasma turbulence, SIAM J. Sci. Comput., 35, 3, B576-B605, (2013) · Zbl 1273.76450 [51] Powell, Kenneth G.; Roe, Philip L.; Linde, Timur J.; Gombosi, Tamas I.; De Zeeuw, Darren L., A solution-adaptive upwind scheme for ideal magnetohydrodynamics, J. Comput. Phys., 154, 2, 284-309, (1999) · Zbl 0952.76045 [52] Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 2, 357-372, (1981) · Zbl 0474.65066 [53] Ryu, Dongsu; Jones, T. W., Numerical magnetohydrodynamics in astrophysics: algorithm and tests for one-dimensional flow, Astrophys. J., 442, 228, (1995) [54] Ryu, Dongsu; Miniati, Francesco; Jones, T. W.; Frank, Adam, A divergence-free upwind code for multidimensional magnetohydrodynamic flows, Astrophys. J., 509, 1, 244-255, (1998) [55] Shu, Chi-Wang; Osher, Stanley, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77, 2, 439-471, (1988) · Zbl 0653.65072 [56] Shu, Chi-Wang; Osher, Stanley, Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, J. Comput. Phys., 83, 1, 32-78, (1989) · Zbl 0674.65061 [57] Sod, Gary A., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 27, 1, 1-31, (1978) · Zbl 0387.76063 [58] Spiteri, Raymond J.; Ruuth, Steven J., A new class of optimal high-order strong-stability-preserving time discretization methods, SIAM J. Numer. Anal., 40, 2, 469-491, (2002) · Zbl 1020.65064 [59] Stone, James M.; Gardiner, Thomas, A simple unsplit Godunov method for multidimensional MHD, New Astron., 14, 2, 139-148, (2009) [60] Stone, James M.; Gardiner, Thomas A.; Teuben, Peter; Hawley, J. F.; Simon, J. B., Athena: a new code for astrophysical MHD, Astrophys. J. Suppl. Ser., 178, 137-177, (2008) [61] Susanto, A.; Ivan, L.; De Sterck, H.; Groth, C. P.T., High-order central ENO finite-volume scheme for ideal MHD, J. Comput. Phys., 250, 141-164, (2013) · Zbl 1349.65583 [62] Toro, E. F.; Spruce, M.; Speares, W., Restoration of the contact surface in the HLL-Riemann solver, Shock Waves, 4, 1, 25-34, (1994) · Zbl 0811.76053 [63] Tóth, Gábor, The $$\operatorname{\nabla} \cdot B = 0$$ constraint in shock-capturing magnetohydrodynamics codes, J. Comput. Phys., 161, 2, 605-652, (2000) · Zbl 0980.76051 [64] Del Zanna, L.; Zanotti, O.; Bucciantini, N.; Londrillo, P., ECHO: a Eulerian conservative high-order scheme for general relativistic magnetohydrodynamics and magnetodynamics, Astron. Astrophys., 473, 1, 11-30, (2007)
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