An unsplit Godunov method for ideal MHD via constrained transport

By Thomas A. Gardiner, James M. Stone. 2005

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We describe a single step, second-order accurate Godunov scheme for ideal MHD based on combining the piecewise parabolic method (PPM) for performing spatial reconstruction, the corner transport upwind (CTU) method of Colella for multidimensional integration, and the constrained transport (CT) algorithm for preserving the divergence-free constraint on the magnetic field. We adopt the most compact form of CT, which requires the field be represented by area-averages at cell faces. We demonstrate that the fluxes of the area-averaged field used by CT can be made consistent with the fluxes of the volume-averaged field returned by a Riemann solver if they obey certain simple relationships. We use these relationships to derive new algorithms for constructing the CT fluxes at grid cell corners which reduce exactly to the equivalent one-dimensional solver for plane-parallel, grid-aligned flow. We show that the PPM reconstruction algorithm must include multidimensional terms for MHD, and we describe a number of important extensions that must be made to CTU in order for it to be used for MHD with CT. We present the results of a variety of test problems to demonstrate the method is accurate and robust.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/b59649429c07b7d30bce63862e18936d908e9b6d

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In Applied Mathematics

In Computational Mechanics

Citations in the corpus

A characteristic scale for cold gas (2016, methodology): “In all of our simulations, we integrate the usual equations of ideal hydrodynamics using the conservative code Athena (Stone et al., 2008; Gardiner & Stone, 2008).”
MAXIMALLY STAR-FORMING GALACTIC DISKS. I. STARBURST REGULATION VIA FEEDBACK-DRIVEN TURBULENCE (2011, methodology): “We integrate the dynamical equations using a version of the Athena code (Stone et al. 2008), which employs a single-step, directionally-unsplit Godunov scheme (Gardiner & Stone 2005, 2008) to obtain conservative, second-order accurate solutions.”
EFFECT OF AMBIPOLAR DIFFUSION ON THE NONLINEAR EVOLUTION OF MAGNETOROTATIONAL INSTABILITY IN WEAKLY IONIZED DISKS (2011, methodology): “Numerical Method We use Athena, a higher-order Godunov MHD code with constrained transport technique to enforce the divergence-free constraint on the magnetic field (Gardiner & Stone 2005, 2008; Stone et al. 2008) for all calculations presented in this paper.”
Simulating Radiating and Magnetized Flows in Multiple Dimensions with ZEUS-MP (2005, methodology): “Indeed, the past decade has seen the emergence of several newMHDcodes based onGodunov methods (Ryu et al. 1998; Londrillo & Del Zanna 2000, 2004; Balsara 2004; Gardiner & Stone 2005).”
Linear and non-linear evolution of the vertical shear instability in accretion discs (2012, methodology): “NIRVANA utilizes the same finitedifference scheme as the ZEUS code, whereas NIRVANA-III applies a finite-volume Godunov scheme very similar to the ones used in, e.g., RAMSES (Teyssier 2002) or ATHENA (Gardiner & Stone 2008).”
PLUTO: A Numerical Code for Computational Astrophysics (2007, methodology): “The CTU method has been recently extended to magnetohydrodynamics in the work of Crockett et al. (2005), Gardiner & Stone (2005), and Fromang et al. (2006), and to relativistic hydrodynamics by Mignone et al. (2005b).”
Global Simulations of the Inner Regions of Protoplanetary Disks with Comprehensive Disk Microphysics (2017, methodology): “We use the newly developed grid-based higher-order Godunov MHD code Athena++ (J. M. Stone et al. 2017, in preparation), which is the successor of the widely used Athena MHD code (Gardiner & Stone 2005, 2008; Stone et al. 2008), to carry out global simulations of PPDs in this work.”
Athena: A New Code for Astrophysical MHD (2008, methodology): “…gravitational source terms for both a static gravitational potential and self-gravity (T. A. Gardiner & J. M. Stone, in preparation), the shearing box (Gardiner & Stone 2005b), anisotropic heat conduction (Parrish & Stone 2005, 2007), and transfer of ionizing radiation (Krumholz et al. 2007).”
The impact of star formation feedback on the circumgalactic medium (2016, methodology): “We ran 3-dimensional hydrodynamic simulations with an ideal gas equation of state using the athena code (Stone et al. 2008; Gardiner & Stone 2008), which integrates the standard fluid equations.”
The Athena++ Adaptive Mesh Refinement Framework: Design and Magnetohydrodynamic Solvers (2020, methodology): “Following Gardiner & Stone (2005), the magnetic fields are initialized with a vector potential of the form Az = B0 exp [ − (z − z0) 2 σ2 ] ×max ( R− √ (x− x0)2 + (y − y0)2, 0 ) , (29) where (x0, y0, z0) = (− √ 2/2, 0, √ 2/2) is the initial center of the loop, B0 the magnetic field strength, R = 0.5…”
THE MASS AND SIZE DISTRIBUTION OF PLANETESIMALS FORMED BY THE STREAMING INSTABILITY. I. THE ROLE OF SELF-GRAVITY (2015, methodology): “A detailed description of the base Athena algorithm and the results of various test problems are given in Gardiner & Stone (2005), Gardiner & Stone (2008), and Stone et al. (2008).”
Formation of precessing jets by tilted black hole discs in 3D general relativistic MHD simulations (2017, methodology): “H-AMR features a staggered grid for constrained transport of magnetic field (Gardiner & Stone 2005), adaptive mesh-refinement (AMR, not used adaptively here), and a locally adaptive time-step (details to be described in future work).”
IMPLEMENTATION OF THE SHEARING BOX APPROXIMATION IN ATHENA (2010, methodology): “This test has been useful for developing the CT algorithm implemented in Athena (Gardiner & Stone 2005a; 2008; Stone et al. 2008).”
An HLLC Riemann solver for relativistic flows – II. Magnetohydrodynamics (2006, methodology): “Computation of time-centered left and right zone edge values proceeds using the CTU of Colella (1990), recently extended to relativistic hydrodynamics by Mignone, Plewa & Bodo (2005) and to classical MHD by Gardiner & Stone (2005).”
SUSTAINED MAGNETOROTATIONAL TURBULENCE IN LOCAL SIMULATIONS OF STRATIFIED DISKS WITH ZERO NET MAGNETIC FLUX (2009, methodology): “All simulations make use of Athena’s orbital advection scheme (Stone & Gardiner 2010), allowing us to consider domains with large radial extent.”
Smoothed particle hydrodynamics and magnetohydrodynamics (2010, methodology): “e very dicult for Eulerian schemes are almost trivial in a Lagrangian scheme such as SPMHD. The problem involves the advection of a loop of current across the computational domain, was introduced by Gardiner and Stone (2005) to test their Athena MHD code and presents a challenging problem for grid-b”