An unsplit staggered mesh scheme for multidimensional magnetohydrodynamics

A new unsplit staggered mesh algorithm (USM) that solves multidimensional magnetohydrodynamics (MHD) on a staggered mesh is introduced and studied. Proper treatments of multidimensional flow problems are required for MHD simulations to avoid unphysical results that can even introduce numerical instability. The research work in this dissertation, which is based on an approach that combines the high-order Godunov method and the constrained transport (CT) scheme, uses such multidimensional consideration in a spatial reconstruction-evolution step. The core problem of MHD simulation is the nonlinear evolution of solutions using well-designed algorithms that maintain the divergence-free constraint of the magnetic field components. The USM algorithm proposed in this dissertation ensures the solenoidal constraint by using Stokes' Theorem as applied to a set of induction equations. In CT-type of MHD schemes, one solves the discrete induction equations to proceed temporal evolutions of the staggered magnetic fields using electric fields. The accuracy of the computed electric fields therefore directly influence the solution quality of the magnetic fields. To meet this end, an accurate and improved electric field construction (IEC) scheme has been introduced as one of the essential parts of the current dissertation work. Another important feature in this work is a development of a new algorithm that solves the induction equations with an added capability that controls numerical (anti)dissipations of the magnetic fields. This staggered dissipation-control differencing algorithm (SDDA) makes use of extra dissipation terms, for which their derivations are established from modified equations of the induction equations. A series of comparison studies in a suite of numerical results of the USM-IEC-SDDA scheme will show a great deal of qualitative improvements in many stringent multidimensional MHD test problems.

[1]  Arnold Neumaier,et al.  Introduction to Numerical Analysis , 2001 .

[2]  P. Roe,et al.  Divergence- and curl-preserving prolongation and restriction formulas , 2002 .

[3]  P. Teuben,et al.  Athena: A New Code for Astrophysical MHD , 2008, 0804.0402.

[4]  M. Berger,et al.  Adaptive mesh refinement for hyperbolic partial differential equations , 1982 .

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

[6]  Philip L. Roe,et al.  An upwind scheme for magnetohydrodynamics , 1995 .

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

[8]  S. Orszag,et al.  Small-scale structure of two-dimensional magnetohydrodynamic turbulence , 1979, Journal of Fluid Mechanics.

[9]  Rho-Shin Myong,et al.  On Godunov-Type Schemes for Magnetohydrodynamics , 1998 .

[10]  P. Londrillo,et al.  On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: the upwind constrained transport method , 2004 .

[11]  P. Colella.  Multidimensional upwind methods for hyperbolic conservation laws , 1990 .

[12]  K. Murawski,et al.  Modern Numerical Schemes for Solving Magnetohydrodynamic Equations , 1997 .

[13]  J. Stone,et al.  An unsplit Godunov method for ideal MHD via constrained transport , 2005, astro-ph/0501557.

[14]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[15]  M. J. Rycroft,et al.  Plasma physics: An introduction to the theory of astrophysical, geophysical & laboratory plasmas: Sturrock P. A., 1994, 335 pp. Cambridge University Press, £17.95 pb, $24.95, ISBN 0-521-44810-7 , 1995 .

[16]  G. Parks,et al.  Physics Of Space Plasmas: An Introduction , 1991 .

[17]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[18]  Richard I. Klein,et al.  An unsplit, cell-centered Godunov method for ideal MHD - eScholarship , 2003 .

[19]  M. Norman,et al.  ZEUS-2D : a radiation magnetohydrodynamics code for astrophysical flows in two space dimensions. II : The magnetohydrodynamic algorithms and tests , 1992 .

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

[21]  T. Linde,et al.  A three-dimensional adaptive multifluid MHD model of the heliosphere. , 1998 .

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

[23]  John Lyon,et al.  A time dependent three‐dimensional simulation of the Earth's magnetosphere: Reconnection events , 1982 .

[24]  M. Goossens,et al.  An Introduction to Plasma Astrophysics and Magnetohydrodynamics , 2003 .

[25]  D. A. Roberts,et al.  North-south flows observed in the outer heliosphere at solar minimum: Vortex streets or corotating interaction regions? , 2001 .

[26]  John Lyon,et al.  A simulation study of east-west IMF effects on the magnetosphere , 1981 .

[27]  Francesco Miniati,et al.  A Divergence-free Upwind Code for Multidimensional Magnetohydrodynamic Flows , 1998 .

[28]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..

[29]  Dongsu Ryu,et al.  Numerical magnetohydrodynamics in astrophysics: Algorithm and tests for multidimensional flow , 1995 .

[30]  Ian Hutchinson,et al.  Principles of Magnetohydrodynamics , 2005 .

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

[32]  Shengtai Li,et al.  High order central scheme on overlapping cells for magneto-hydrodynamic flows with and without constrained transport method , 2008, J. Comput. Phys..

[33]  D. A. Roberts,et al.  Numerical simulation of Alfvénic turbulence in the solar wind , 1999 .

[34]  Manuel Torrilhon,et al.  High order WENO schemes: investigations on non-uniform converges for MHD Riemann problems , 2004 .

[35]  J. Huba.  NRL: Plasma Formulary , 2004 .

[36]  Eitan Tadmor,et al.  Non-Oscillatory Hierarchical Reconstruction for Central and Finite Volume Schemes , 2006 .

[37]  Dongwook Lee,et al.  A Parallel Unsplit Staggered Mesh Algorithm for Magnetohydrodynamics , 2006 .

[38]  D. Aaron Roberts,et al.  Quasi-Two-Dimensional MHD Turbulence in Three-Dimensional Flows , 1999 .

[39]  Gabor Toth,et al.  Conservative and orthogonal discretization for the Lorentz force , 2002 .

[40]  K. Kusano,et al.  A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics , 2005 .

[41]  P. Colella,et al.  A Finite Difference Domain Decomposition Method Using Local Corrections for the Solution of Poisson's Equation , 1999 .

[42]  Joachim Birn,et al.  Magnetic Reconnection in Plasmas , 2002 .

[43]  John B. Bell,et al.  An Adaptive Mesh Projection Method for Viscous Incompressible Flow , 1997, SIAM J. Sci. Comput..

[44]  Phillip Colella,et al.  A Higher-Order Godunov Method for Multidimensional Ideal Magnetohydrodynamics , 1994, SIAM J. Sci. Comput..

[45]  A. Gautschy,et al.  Computational methods for astrophysical fluid flow , 1998 .

[46]  C. Hines.  Electromagnetic Waves , 2021, Nature.

[47]  P. Londrillo,et al.  High-Order Upwind Schemes for Multidimensional Magnetohydrodynamics , 1999, astro-ph/9910086.

[48]  Roberts,et al.  Magnetohydrodynamic simulation of the radial evolution and stream structure of solar-wind turbulence. , 1991, Physical review letters.

[49]  Dongsu Ryu,et al.  Numerical magetohydrodynamics in astronphysics: Algorithm and tests for one-dimensional flow` , 1995 .

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

[51]  D. Birchall,et al.  Computational Fluid Dynamics , 2020, Radial Flow Turbocompressors.

[52]  P. Woodward,et al.  The Piecewise Parabolic Method (PPM) for Gas Dynamical Simulations , 1984 .

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

[54]  James M. Stone,et al.  An unsplit Godunov method for ideal MHD via constrained transport in three dimensions , 2007, J. Comput. Phys..

[55]  V. G. Weirs,et al.  On Validating an Astrophysical Simulation Code , 2002, astro-ph/0206251.

[56]  Julian F. Scott,et al.  An Introduction to Turbulent Flow , 2000 .

[57]  Dinshaw S. Balsara,et al.  Notes on the Eigensystem of Magnetohydrodynamics , 1996, SIAM J. Appl. Math..

[58]  Frank H. Shu,et al.  The Physics of Astrophysics, Volume II: Gas Dynamics , 2009 .

[59]  Ue-Li Pen,et al.  A Free , Fast , Simple and Efficient TVD MHD Code , 2003 .

[60]  D. Balsara,et al.  A Staggered Mesh Algorithm Using High Order Godunov Fluxes to Ensure Solenoidal Magnetic Fields in Magnetohydrodynamic Simulations , 1999 .

[61]  Dinshaw S. Balsara,et al.  Linearized Formulation of the Riemann Problem for Adiabatic and Isothermal Magnetohydrodynamics , 1998 .

[62]  A. Ecer,et al.  Parallel Computational Fluid Dynamics, '91 , 1992 .

[63]  S.,et al.  Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media , 1966 .

[64]  Gérard Gallice,et al.  Roe Matrices for Ideal MHD and Systematic Construction of Roe Matrices for Systems of Conservation Laws , 1997 .

[65]  Kenneth G. Powell,et al.  Three‐dimensional multiscale MHD model of cometary plasma environments , 1996 .

[66]  Dinshaw Balsara,et al.  Divergence-free adaptive mesh refinement for Magnetohydrodynamics , 2001 .

[67]  A. Harten.  On the symmetric form of systems of conservation laws with entropy , 1983 .

[68]  Michael Wiltberger,et al.  Nonequilibrium, large-amplitude MHD fluctuations in the solar wind , 1995 .

[69]  M. Brio,et al.  An upwind differencing scheme for the equations of ideal magnetohydrodynamics , 1988 .

[70]  Ue-Li Pen,et al.  A FREE, FAST, SIMPLE, AND EFFICIENT TOTAL VARIATION DIMINISHING MAGNETOHYDRODYNAMIC CODE , 2003 .

[71]  Paul R. Woodward,et al.  A Simple Finite Difference Scheme for Multidimensional Magnetohydrodynamical Equations , 1998 .

[72]  Paul R. Woodward,et al.  An approximate Riemann solver for ideal magnetohydrodynamics , 1994 .

[73]  Phillip Colella,et al.  A higher-order Godunov method for the equations of ideal magnetohydrodynamics , 1992 .

[74]  C. Richard DeVore,et al.  Flux-corrected transport techniques for multidimensional compressible magnetohydrodynamics , 1989 .

[75]  D. Aaron Roberts,et al.  Velocity shear generation of solar wind turbulence , 1992 .

[76]  Lisa L. Lowe,et al.  Multigrid elliptic equation solver with adaptive mesh refinement , 2005 .

[77]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[78]  Shengtai Li,et al.  Note on Upwinding Constrained Transport Method for Ideal Magnetohydrodynamics 12 , 2022 .

[79]  P. Janhunen,et al.  A Positive Conservative Method for Magnetohydrodynamics Based on HLL and Roe Methods , 2000 .

[80]  J. Anderson,et al.  Fundamentals of Aerodynamics , 1984 .

[81]  D. A. Roberts,et al.  Alfvén wave phase mixing driven by velocity shear in two‐dimensional open magnetic configurations , 1999 .

[82]  P. Colella,et al.  Local adaptive mesh refinement for shock hydrodynamics , 1989 .

[83]  J. Saltzman,et al.  An unsplit 3D upwind method for hyperbolic conservation laws , 1994 .

[84]  Dinshaw S. Balsara,et al.  Total Variation Diminishing Scheme for Adiabatic and Isothermal Magnetohydrodynamics , 1998 .