High-Order Central ENO Finite-Volume Scheme for MHD on Three-Dimensional Cubed-Sphere Grids

A high-order central essentially non-oscillatory (CENO) nitevolume scheme is developed for the compressible ideal magnetohydrodynamics (MHD) equations and applied to space-physics ows on three-dimensional cubed-sphere grids. The CENO scheme is based on a hybrid solution reconstruction procedure that provides high-order accuracy in smooth regions, even for smooth extrema, and non-oscillatory transitions at discontinuities. The scheme is applied in combination with the divergence correction technique proposed by Dedner et al. (J. Comput. Phys. 175 (2002) 645-673) to enforce the solenoidal condition for the magnetic eld. The cubed-sphere simulation framework represents a exible design based on a genuine multiblock implementation, leading to high-order accuracy, ux calculation, adaptivity and parallelism that are fully transparent to the boundaries between the six sectors of the cubed-sphere grid. Numerical results to demonstrate the accuracy, robustness and capability of the proposed high-order framework are discussed.

[1]  Phillip Colella,et al.  High-order finite-volume methods on locally-structured grids , 2009 .

[2]  Shian-Jiann Lin,et al.  Finite-volume transport on various cubed-sphere grids , 2007, J. Comput. Phys..

[3]  Clinton P. T. Groth,et al.  A parallel solution - adaptive method for three-dimensional turbulent non-premixed combusting flows , 2010, J. Comput. Phys..

[4]  Carla Jacobs,et al.  Magnetohydrodynamic modelling of the solar wind and coronal mass ejection. , 2007 .

[5]  Clinton P. T. Groth,et al.  High-Order CENO Finite-Volume Schemes for Multi-Block Unstructured Mesh , 2011 .

[6]  Patrick J. Roache,et al.  Verification and Validation in Computational Science and Engineering , 1998 .

[7]  Feng Xiao,et al.  Shallow water model on cubed-sphere by multi-moment finite volume method , 2008, J. Comput. Phys..

[8]  D. D. Zeeuw,et al.  Global three‐dimensional MHD simulation of a space weather event: CME formation, interplanetary propagation, and interaction with the magnetosphere , 2000 .

[9]  Lucian Ivan,et al.  Three-Dimensional MHD on Cubed-Sphere Grids: Parallel Solution-Adaptive Simulation Framework , 2011 .

[10]  V. Venkatakrishnan.  On the accuracy of limiters and convergence to steady state solutions , 1993 .

[11]  Xingliang Li,et al.  A multi-moment transport model on cubed-sphere grid , 2011 .

[12]  I. Sokolov,et al.  Three-dimensional MHD Simulation of the 2003 October 28 Coronal Mass Ejection: Comparison with LASCO Coronagraph Observations , 2008, 0805.3707.

[13]  John M. Dennis,et al.  A Comparison of Two Shallow-Water Models with Nonconforming Adaptive Grids , 2008 .

[14]  J. Brackbill,et al.  The Effect of Nonzero ∇ · B on the numerical solution of the magnetohydrodynamic equations☆ , 1980 .

[15]  P. Roe,et al.  A Solution-Adaptive Upwind Scheme for Ideal Magnetohydrodynamics , 1999 .

[16]  Dinshaw S. Balsara.  Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics , 2009, J. Comput. Phys..

[17]  Dinshaw Balsara,et al.  Second-Order-accurate Schemes for Magnetohydrodynamics with Divergence-free Reconstruction , 2003, astro-ph/0308249.

[18]  R. Sadourny.  Conservative Finite-Difference Approximations of the Primitive Equations on Quasi-Uniform Spherical Grids , 1972 .

[19]  Carlos A. Felippa,et al.  A compendium of FEM integration formulas for symbolic work , 2004 .

[20]  Dinshaw Balsara,et al.  A Comparison between Divergence-Cleaning and Staggered-Mesh Formulations for Numerical Magnetohydrodynamics , 2003 .

[21]  Long Chen.  FINITE ELEMENT METHOD , 2013 .

[22]  Chris Hill,et al.  Implementation of an Atmosphere-Ocean General Circulation Model on the Expanded Spherical Cube , 2004 .

[23]  C. Munz,et al.  Hyperbolic divergence cleaning for the MHD equations , 2002 .

[24]  Enrico Camporeale,et al.  Numerical modeling of space plasma flows , 2009 .

[25]  A. Chorin.  A Numerical Method for Solving Incompressible Viscous Flow Problems , 1997 .

[26]  Hans De Sterck,et al.  High-order central ENO finite-volume scheme for ideal MHD , 2013 .

[27]  Xinfeng Gao,et al.  A Parallel Solution-Adaptive Method for Turbulent Non-Premixed Combusting Flows , 2008 .

[28]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[29]  Henry M. Tufo,et al.  High-order Galerkin methods for scalable global atmospheric models , 2007, Comput. Geosci..

[30]  S. Cable,et al.  Intermediate shocks in three‐dimensional magnetohydrodynamic bow‐shock flows with multiple interacting shock fronts , 2007 .

[31]  Clinton P. T. Groth,et al.  High-order solution-adaptive central essentially non-oscillatory (CENO) method for viscous flows , 2011, J. Comput. Phys..

[32]  Eric Ronald Priest,et al.  Solar magneto-hydrodynamics , 1982 .

[33]  Hans d,et al.  Multi-dimensional upwind constrained transport on unstructured grids for 'shallow water' magnetohydrodynamics , 2001 .

[34]  H. Tufo,et al.  A spectral finite volume transport scheme on the cubed-sphere , 2007 .

[35]  Hans De Sterck,et al.  Hyperbolic Conservation Laws on Three-Dimensional Cubed-Sphere Grids : A Parallel Solution-Adaptive Simulation Framework , 2012 .

[36]  J. Sachdev,et al.  A parallel solution-adaptive scheme for multi-phase core flows in solid propellant rocket motors , 2005 .

[37]  George Em Karniadakis,et al.  A Discontinuous Galerkin Method for the Viscous MHD Equations , 1999 .

[38]  Lucian Ivan,et al.  Development of High-order CENO Finite-volume Schemes with Block-based Adaptive Mesh Refinement (AMR) , 2011 .

[39]  Paul A. Ullrich,et al.  Atmospheric Modeling with High-Order Finite-Volume Methods , 2011 .

[40]  Jay D. Salmonson,et al.  APPLICATION OF THE CUBED-SPHERE GRID TO TILTED BLACK HOLE ACCRETION DISKS , 2009 .

[41]  Stephen J. Thomas,et al.  A Discontinuous Galerkin Transport Scheme on the Cubed Sphere , 2005 .

[42]  A. V. Koldoba,et al.  Three-dimensional Magnetohydrodynamic Simulations of Accretion to an Inclined Rotator: The “Cubed Sphere” Method , 2002, astro-ph/0209598.

[43]  Bram van Leer,et al.  Design of Optimally Smoothing Multi-Stage Schemes for the Euler Equations , 1989 .

[44]  Clinton P. T. Groth,et al.  Parallel High-Order Anisotropic Block-Based Adaptive Mesh Refinement Finite-Volume Scheme , 2011 .

[45]  Dimitri J. Mavriplis,et al.  Revisiting the Least-squares Procedure for Gradient Reconstruction on Unstructured Meshes , 2003 .

[46]  Clinton P. T. Groth,et al.  High-Order Central ENO Finite-Volume Scheme with Adaptive Mesh Refinement , 2007 .

[47]  Kenneth G. Powell,et al.  AN APPROXIMATE RIEMANN SOLVER FOR MAGNETOHYDRODYNAMICS (That Works in More than One Dimension) , 1994 .

[48]  W. Gander,et al.  Adaptive Quadrature—Revisited , 2000 .

[49]  Clinton P. T. Groth,et al.  International Journal of Computational Fluid Dynamics a Parallel Adaptive Mesh Refinement Algorithm for Predicting Turbulent Non-premixed Combusting Flows a Parallel Adaptive Mesh Refinement Algorithm for Predicting Turbulent Non-premixed Combusting Flows , 2022 .

[50]  Joe F. Thompson,et al.  Numerical grid generation: Foundations and applications , 1985 .

[51]  Quentin F. Stout,et al.  Solution-adaptive magnetohydrodynamics for space plasmas: Sun-to-Earth simulations , 2004, Computing in Science & Engineering.

[52]  Phillip Colella,et al.  High-order, finite-volume methods in mapped coordinates , 2010, J. Comput. Phys..

[53]  Andrea Lani,et al.  A finite volume implicit time integration method for solving the equations of ideal magnetohydrodynamics for the hyperbolic divergence cleaning approach , 2011, J. Comput. Phys..

[54]  C. P. T. Groth,et al.  High-Order CENO Finite-Volume Scheme for Low-Speed Viscous Flows on Three-Dimensional Unstructured Mesh , 2011 .

[55]  Timothy J. Barth,et al.  Recent developments in high order K-exact reconstruction on unstructured meshes , 1993 .

[56]  J. Hawley,et al.  Simulation of magnetohydrodynamic flows: A Constrained transport method , 1988 .

[57]  Charles L. Lawson,et al.  Solving least squares problems , 1976, Classics in applied mathematics.

[58]  Bram van Leer,et al.  High-order finite-volume methods for the shallow-water equations on the sphere , 2010, J. Comput. Phys..

[59]  G. Tóth.  The ∇·B=0 Constraint in Shock-Capturing Magnetohydrodynamics Codes , 2000 .

[60]  William H. Press,et al.  Numerical recipes: the art of scientific computing, 3rd Edition , 2007 .