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Divergence-Free WENO Reconstruction-Based Finite Volume Scheme for Solving Ideal MHD Equations on Triangular Meshes

Part of: Astronomy and astrophysics (General) Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  12 April 2016

Zhiliang Xu ,
Dinshaw S. Balsara  and
Huijing Du
Zhiliang Xu*
Affiliation:
Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA
Dinshaw S. Balsara*
Affiliation:
Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA
Huijing Du
Affiliation:
Department of Mathematics, University of California, Irvine, Irvine, CA 92697, USA
*
*Corresponding author. Email addresses:[email protected] (Z. Xu), Dinshaw.S. [email protected] (D. S. Balsara), [email protected] (H. Du)
*Corresponding author. Email addresses:[email protected] (Z. Xu), Dinshaw.S. [email protected] (D. S. Balsara), [email protected] (H. Du)
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Abstract

In this paper, we introduce a high-order accurate constrained transport type finite volume method to solve ideal magnetohydrodynamic equations on two-dimensional triangular meshes. A new divergence-free WENO-based reconstruction method is developed to maintain exactly divergence-free evolution of the numerical magnetic field. In this formulation, the normal component of the magnetic field at each face of a triangle is reconstructed uniquely and with the desired order of accuracy. Additionally, a new weighted flux interpolation approach is also developed to compute the z-component of the electric field at vertices of grid cells. We also present numerical examples to demonstrate the accuracy and robustness of the proposed scheme.

Keywords

Finite volume methodMHD equationsglobally divergence-free

MSC classification

Secondary: 65M08: Finite volume methods 65Z05: Applications to physics 85-08: Computational methods
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 4 , April 2016 , pp. 841 - 880
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Abgrall, R.. On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. J. Comput. Phys., 144:45–58, 1994.CrossRefGoogle Scholar
[2]Balbás, J. and Tadmor, E.. Non-oscillatory central scheme for one- and two-dimensional MHD equations. SIAM J. Sci. Comput., 28(2):533–560, 2006.CrossRefGoogle Scholar
[3]Balbás, J., Tadmor, E. and Wu, C.-C.. Non-oscillatory central scheme for one- and two-dimensional MHD equations: I. J. Comput. Phys., 201:261–285, 2004.CrossRefGoogle Scholar
[4]Balsara, D.S. and Spicer, D.. Maintaining pressure positivity in magnetohydrodynamic simulations. J. Comput. Phys., 148:133–148, 1999.CrossRefGoogle Scholar
[5]Balsara, D.S. and Spicer, D.. A Staggered mesh Algorithm Using High Order Godunov Fluxes to Ensure Solenoidal Magnetic Fields in Magnetohydrodynamic Simulations. J. Comput. Phys., 149:270–292, 1999.CrossRefGoogle Scholar
[6]Balsara, D.S.. Divergence-free adaptive mesh refinement for magnetohydrodynamics. J. Comput. Phys., 174:614–648, 2001.CrossRefGoogle Scholar
[7]Balsara, D.S.. Second-Order-Accurate Schemes for Magnetohydrodynamics with Divergence-Free Reconstruction. The Astrophysical Journal Supplement Series, 151:149–184, 2004.CrossRefGoogle Scholar
[8]Balsara, D.S. and Kim, J.-S.. A Comparison between Divergence-Cleaning and Staggered-Mesh Formulations for Numerical Magnetohydrodynamics. Astrophysical Journal, 602(2):1079–1090, 2004.CrossRefGoogle Scholar
[9]Balsara, D.S., Rumpf, T., Dumbser, M. and Munz, C.D.. Efficient, High Accuracy ADER-WENO Schemes for Hydrodynamics and Divergence-Free MHD. J. Comput. Phys., 228:2480–2516, 2009.CrossRefGoogle Scholar
[10]Balsara, D.S.. Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics. J. Comput. Phys., 228(14):5040–5056, 2009.CrossRefGoogle Scholar
[11]Balsara, D.S.. Multidimensional HLLE Riemann solver: Application to Euler and magnetohydrodynamic flows. J. Comput. Phys., 229:1970–1993, 2010.CrossRefGoogle Scholar
[12]Balsara, D.S., Dumbser, M. and Abgrall, R.. Multidimensional HLLC Riemann solver for unstructured meshes With application to Euler and MHD flows. J. Comput. Phys., 231(22):7476–7503, 2014.CrossRefGoogle Scholar
[13]Balsara, D.S.. A two-dimensional HLLC Riemann solver for conservation laws: Application to Euler and magnetohydrodynamic flows. J. Comput. Phys., 261:172–208, 2014.CrossRefGoogle Scholar
[14]Balsara, D.S., Meyer, C., Dumbser, M., Du, H. and Xu, Z.-L.. Efficient Implementation of ADER Schemes for Euler and Magnetohydrodynamical Flows on Structured Meshes C Comparison with Runge-Kutta Methods. J. Comput. Phys., 235(15):934, 2013.CrossRefGoogle Scholar
[15]Brackbill, J.U. and Barnes, D.C.. The effect of nonzero ▽.B on the numerical solution of the magnetohydrodynamic equations. J. Comput. Phys., 35:426–430, 1980.CrossRefGoogle Scholar
[16]Brecht, S.H., Lyon, J.G., Fedder, J.A. and Hain, K.. A simulation study of east-west IMF effets on the magnetosphere. Geophysical Research Letters., 8:397–400, 1981.CrossRefGoogle Scholar
[17]Cargo, P. and Gallice, G.. Roe Matrices for Ideal MHD and Systematic Construction of Roe Matrices for Systems of Conservation Laws. J. Comput. Phys., 136:446–466, 1997.CrossRefGoogle Scholar
[18]Cockburn, B., Li, F. and Shu, C.-W.. Locally divergence-free discontinuous Galerkin methods for the Maxwell equations. J. Comput. Phys., 22-23:413–442, 2005.Google Scholar
[19]Dai, W. and Woodward, P.R.. On the divergence-free condition and conservation laws in numerical simulations for supersonic magnetohydrodynamic flows. Astrophysical Journal, 494:317–335, 1998.CrossRefGoogle Scholar
[20]Dedner, A., Kemm, F., Kroner, D., Munz, C.D., Schnitzer, T. and Wesenberg, M.. Hyperbolic divergence-cleaning for the MHD equations. J. Comput. Phys., 175:645, 2002.CrossRefGoogle Scholar
[21]Dumbser, M. and Käser, M.. Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. J. Comput. Phys., 221:693–723, 2007.CrossRefGoogle Scholar
[22]Friedrich, O.. Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids. J. Comput. Phys., 144:194–212, 1998.CrossRefGoogle Scholar
[23]Gardiner, T. and Stone, J.M.. An unsplit Godunov method for ideal MHD via constrained transport. J. Comput. Phys., 205(2):509–539, 2005.CrossRefGoogle Scholar
[24]Harten, A. and Chakravarthy, S.. Multi-dimensional ENO schemes for general geometries. Technical Report 91-76, ICASE, 1991.Google Scholar
[25]Hu, C. and Shu, C.-W.. Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys., 150:97–127, 1999.CrossRefGoogle Scholar
[26]Käser, M. and Iske, A.. ADER schemes on adaptive triangular meshes for scalar conservation laws. J. Comput. Phys., 205(2):486–508, 2005.CrossRefGoogle Scholar
[27]Levy, D., Puppo, G. and Russo, G.. Central WENO schemes for hyperbolic systems of conservation laws. ESAIM: Math. Modell. Numer. Anal., 33:547–571, 1999.CrossRefGoogle Scholar
[28]Li, F., Xu, L. and Yakovlev, S.. Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field. J. Comput. Phys., 230(12):4828–4847, 2011.CrossRefGoogle Scholar
[29]Li, F. and Shu, C.-W.. Locally divergence-free discontinuous Galerkin methods for MHD equations. Journal of Scientific Computing, 22-23:413–442, 2005.CrossRefGoogle Scholar
[30]Li, S.. High order central scheme on overlapping cells for magnetohydrodynamic flows with and without constrained transport method, J. Comput. Phys., 227:7368–7393, 2008.CrossRefGoogle Scholar
[31]Londrillo, P. and DelZanna, L.. On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: the upwind constrained transport method, J. Comput. Phys., 195:17–48, 2004.CrossRefGoogle Scholar
[32]Miyoshi, T. and Kusano, K.. A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics. J. Comput. Phys., 208:315–344, 2005.CrossRefGoogle Scholar
[33]Orszag, S.A. and Tang, C.M.. Small-scale structure of two-dimensional magnetohydrodynamic turbulence. J. Fluid Mech., 90:129, 1979.CrossRefGoogle Scholar
[34]Powell, K.G.. An Approximate Riemann Solver for Magnetohydrodynamics. Technical Report ICASE Report, 94-24, ICASE, NASA Langley, 1994.Google Scholar
[35]Roe, P. L. and Balsara, D. S.. Notes on the eigensystem of magnetohydrodynamics. SIAM Journal of applied Mathematics, 56(1):57–67, 1996.CrossRefGoogle Scholar
[36]Ryu, D. and Jones, T.W.. Numerical Magnetohydrodynamics in Astrophysics: Algorithm and Tests for One-Dimensional Flow. Astrophysical J., 442:228–258, 1995.CrossRefGoogle Scholar
[37]Ryu, D., Miniati, F., Jones, T.W. and Frank, A.. A divergence-free upwind code for multidimensional magnetohydrodynamic flows. Astrophys. J., 509:244–255, 1998.CrossRefGoogle Scholar
[38]Stone, J.M. and Norman, M.L.. ZEUS-2D: A radiation magnetohydrodynamics code for astrophysical flows in two space dimensions. II The magnetohydrodynamic algorithms and tests. Astrophysical Journal Supplement Series., 80:791C818, 1992.Google Scholar
[39]Shu, C.-W. and Osher, S.. Efficient implementation of essentially non-scillatory capturing schemes. J. Comput. Phys., 77:439–471, 1988.CrossRefGoogle Scholar
[40]Shu, C.-W.. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Cockburn, B., Johnson, C., Shu, C.-W. and Tadmor, E. (Editor: Quarteroni, A.), Lecture Notes in Mathematics, Berlin. Springer, 1697, 1998.Google Scholar
[41]Sonar, T.. On the construction of essentially non-oscillatory finite volume approximations to hyperbolic conservation laws on general triangulations: polynomial recovery, accuracy and stencil selection. Comput. Methods Appl. Mech. Engrg., 140:157–181, 1997.CrossRefGoogle Scholar
[42]Tóth, G.. The ▽.B = 0 constraint in shock-capturing magnetohydrodynamics codes. J. Comput. Phys., 161:605–652, 2000.CrossRefGoogle Scholar
[43]Xu, Z.-L. and Liu, Y.-J. and Shu, C.-W.. Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO type linear reconstruction and partial neighboring cells, J. Comput. Phys., 228:2194–2212, 2009.CrossRefGoogle Scholar
[44]Yee, K.S.. Numerical solution of initial boundary value problems involving Maxwell's equatons in isotropic media. IEEE Transactions on Antenna Propagation, AP-14:302–307, 1966.Google Scholar
[45]Zachary, A.L., Malagoli, A. and Colella, P.. A Higher-Order Godunov Method for Multidimensional Ideal Magnetohydrodynamics. SIAM Journal on Scientific Computing, 15(2):263–284, 1994.CrossRefGoogle Scholar
[46]Balsara, D.S. and Dumbser, M.Multidimensional Riemann problem with self-similar internal structure. Part II Application to hyperbolic conservation laws on unstructured meshes Original. J. Comput. Phys., 287:269–292, 2015.CrossRefGoogle Scholar
[47]Balsara, D.S.. Self-adjusting, positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics. J. Comput. Phys., 231(22):7504–7517, 2012.CrossRefGoogle Scholar
[48]Cheng, Y., Li, F., Qiu, J.and Xu, L.. Positivity-preserving DG and central DG methods for ideal MHD equations. J. Comput. Phys., 238:255–280, 2013.CrossRefGoogle Scholar