A fourth-order accurate finite volume method for ideal MHD via upwind constrained transport

Felker, Kyle Gerard; Stone, James M.

doi:10.1016/j.jcp.2018.08.025

A fourth-order accurate finite volume method for ideal MHD via upwind constrained transport

Journal Article · Wed Aug 22 04:00:00 EDT 2018 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2018.08.025· OSTI ID:1610340

^[1]; Stone, James M. ^[2]

Princeton University, NJ (United States); DOE/OSTI
Princeton University, NJ (United States)

Here 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.

View Accepted Manuscript (DOE)

Research Organization:: Krell Institute, Ames, IA (United States); Princeton University, NJ (United States)

Sponsoring Organization:: USDOE Office of Science (SC); National Science Foundation (NSF)

Grant/Contract Number:: FG02-97ER25308

OSTI ID:: 1610340

Alternate ID(s):: OSTI ID: 23081832

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: C Vol. 375; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (60)

A high-order finite-volume method for conservation laws on locally refined grids McCorquodale, Peter; Colella, Phillip Communications in Applied Mathematics and Computational Science, Vol. 6, Issue 1 https://doi.org/10.2140/camcos.2011.6.1	journal	January 2011
Efficient Implementation of Weighted ENO Schemes Jiang, Guang-Shan; Shu, Chi-Wang Journal of Computational Physics, Vol. 126, Issue 1 https://doi.org/10.1006/jcph.1996.0130	journal	June 1996
Roe Matrices for Ideal MHD and Systematic Construction of Roe Matrices for Systems of Conservation Laws Cargo, Patricia; Gallice, Gérard Journal of Computational Physics, Vol. 136, Issue 2 https://doi.org/10.1006/jcph.1997.5773	journal	September 1997
A Staggered Mesh Algorithm Using High Order Godunov Fluxes to Ensure Solenoidal Magnetic Fields in Magnetohydrodynamic Simulations Balsara, Dinshaw S.; Spicer, Daniel S. Journal of Computational Physics, Vol. 149, Issue 2 https://doi.org/10.1006/jcph.1998.6153	journal	March 1999
A Solution-Adaptive Upwind Scheme for Ideal Magnetohydrodynamics Powell, Kenneth G.; Roe, Philip L.; Linde, Timur J. Journal of Computational Physics, Vol. 154, Issue 2 https://doi.org/10.1006/jcph.1999.6299	journal	September 1999
Monotonicity Preserving Weighted Essentially Non-oscillatory Schemes with Increasingly High Order of Accuracy Balsara, Dinshaw S.; Shu, Chi-Wang Journal of Computational Physics, Vol. 160, Issue 2 https://doi.org/10.1006/jcph.2000.6443	journal	May 2000
The ∇·B=0 Constraint in Shock-Capturing Magnetohydrodynamics Codes Tóth, Gábor Journal of Computational Physics, Vol. 161, Issue 2 https://doi.org/10.1006/jcph.2000.6519	journal	July 2000
Hyperbolic Divergence Cleaning for the MHD Equations Dedner, A.; Kemm, F.; Kröner, D. Journal of Computational Physics, Vol. 175, Issue 2 https://doi.org/10.1006/jcph.2001.6961	journal	January 2002
Restoration of the contact surface in the HLL-Riemann solver Toro, E. F.; Spruce, M.; Speares, W. Shock Waves, Vol. 4, Issue 1 https://doi.org/10.1007/BF01414629	journal	July 1994
Locally Divergence-Free Discontinuous Galerkin Methods for MHD Equations Li, Fengyan; Shu, Chi-Wang Journal of Scientific Computing, Vol. 22-23, Issue 1-3 https://doi.org/10.1007/s10915-004-4146-4	journal	June 2005
High Order Strong Stability Preserving Time Discretizations Gottlieb, Sigal; Ketcheson, David I.; Shu, Chi-Wang Journal of Scientific Computing, Vol. 38, Issue 3 https://doi.org/10.1007/s10915-008-9239-z	journal	September 2008
A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws Sod, Gary A. Journal of Computational Physics, Vol. 27, Issue 1 https://doi.org/10.1016/0021-9991(78)90023-2	journal	April 1978
The Effect of Nonzero ∇ · B on the numerical solution of the magnetohydrodynamic equations Brackbill, J. U.; Barnes, D. C. Journal of Computational Physics, Vol. 35, Issue 3 https://doi.org/10.1016/0021-9991(80)90079-0	journal	May 1980
Approximate Riemann solvers, parameter vectors, and difference schemes Roe, P. L. Journal of Computational Physics, Vol. 43, Issue 2 https://doi.org/10.1016/0021-9991(81)90128-5	journal	October 1981
The Piecewise Parabolic Method (PPM) for gas-dynamical simulations Colella, Phillip; Woodward, Paul R. Journal of Computational Physics, Vol. 54, Issue 1 https://doi.org/10.1016/0021-9991(84)90143-8	journal	April 1984
An upwind differencing scheme for the equations of ideal magnetohydrodynamics Brio, M.; Wu, C. C. Journal of Computational Physics, Vol. 75, Issue 2 https://doi.org/10.1016/0021-9991(88)90120-9	journal	April 1988
Efficient implementation of essentially non-oscillatory shock-capturing schemes Shu, Chi-Wang; Osher, Stanley Journal of Computational Physics, Vol. 77, Issue 2 https://doi.org/10.1016/0021-9991(88)90177-5	journal	August 1988
Efficient implementation of essentially non-oscillatory shock-capturing schemes, II Shu, Chi-Wang; Osher, Stanley Journal of Computational Physics, Vol. 83, Issue 1 https://doi.org/10.1016/0021-9991(89)90222-2	journal	July 1989
On Godunov-type methods near low densities Einfeldt, B.; Munz, C. D.; Roe, P. L. Journal of Computational Physics, Vol. 92, Issue 2 https://doi.org/10.1016/0021-9991(91)90211-3	journal	February 1991
A freestream-preserving fourth-order finite-volume method in mapped coordinates with adaptive-mesh refinement Guzik, Stephen M.; Gao, Xinfeng; Owen, Landon D. Computers & Fluids, Vol. 123 https://doi.org/10.1016/j.compfluid.2015.10.001	journal	December 2015
On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: the upwind constrained transport method Londrillo, P.; Del Zanna, L. Journal of Computational Physics, Vol. 195, Issue 1 https://doi.org/10.1016/j.jcp.2003.09.016	journal	March 2004
An unsplit, cell-centered Godunov method for ideal MHD Crockett, Robert K.; Colella, Phillip; Fisher, Robert T. Journal of Computational Physics, Vol. 203, Issue 2 https://doi.org/10.1016/j.jcp.2004.08.021	journal	March 2005
An unsplit Godunov method for ideal MHD via constrained transport Gardiner, Thomas A.; Stone, James M. Journal of Computational Physics, Vol. 205, Issue 2 https://doi.org/10.1016/j.jcp.2004.11.016	journal	May 2005
A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics Miyoshi, Takahiro; Kusano, Kanya Journal of Computational Physics, Vol. 208, Issue 1 https://doi.org/10.1016/j.jcp.2005.02.017	journal	September 2005
A fourth-order accurate local refinement method for Poisson’s equation Barad, Michael; Colella, Phillip Journal of Computational Physics, Vol. 209, Issue 1 https://doi.org/10.1016/j.jcp.2005.02.027	journal	October 2005
An unsplit Godunov method for ideal MHD via constrained transport in three dimensions Gardiner, Thomas A.; Stone, James M. Journal of Computational Physics, Vol. 227, Issue 8 https://doi.org/10.1016/j.jcp.2007.12.017	journal	April 2008
A limiter for PPM that preserves accuracy at smooth extrema Colella, Phillip; Sekora, Michael D. Journal of Computational Physics, Vol. 227, Issue 15 https://doi.org/10.1016/j.jcp.2008.03.034	journal	July 2008
A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids Luo, Hong; Baum, Joseph D.; Löhner, Rainald Journal of Computational Physics, Vol. 227, Issue 20 https://doi.org/10.1016/j.jcp.2008.06.035	journal	October 2008
Runge–Kutta methods with minimum storage implementations Ketcheson, David I. Journal of Computational Physics, Vol. 229, Issue 5 https://doi.org/10.1016/j.jcp.2009.11.006	journal	March 2010
Multidimensional HLLE Riemann solver: Application to Euler and magnetohydrodynamic flows Balsara, Dinshaw S. Journal of Computational Physics, Vol. 229, Issue 6 https://doi.org/10.1016/j.jcp.2009.11.018	journal	March 2010
High-order conservative finite difference GLM–MHD schemes for cell-centered MHD Mignone, Andrea; Tzeferacos, Petros; Bodo, Gianluigi Journal of Computational Physics, Vol. 229, Issue 17 https://doi.org/10.1016/j.jcp.2010.04.013	journal	August 2010
High-order, finite-volume methods in mapped coordinates Colella, P.; Dorr, M. R.; Hittinger, J. A. F. Journal of Computational Physics, Vol. 230, Issue 8 https://doi.org/10.1016/j.jcp.2010.12.044	journal	April 2011
A two-dimensional HLLC Riemann solver for conservation laws: Application to Euler and magnetohydrodynamic flows Balsara, Dinshaw S. Journal of Computational Physics, Vol. 231, Issue 22 https://doi.org/10.1016/j.jcp.2011.12.025	journal	September 2012
ADER-WENO finite volume schemes with space–time adaptive mesh refinement Dumbser, Michael; Zanotti, Olindo; Hidalgo, Arturo Journal of Computational Physics, Vol. 248 https://doi.org/10.1016/j.jcp.2013.04.017	journal	September 2013
High-order central ENO finite-volume scheme for ideal MHD Susanto, A.; Ivan, L.; De Sterck, H. Journal of Computational Physics, Vol. 250 https://doi.org/10.1016/j.jcp.2013.04.040	journal	October 2013
High-order conservative reconstruction schemes for finite volume methods in cylindrical and spherical coordinates Mignone, A. Journal of Computational Physics, Vol. 270 https://doi.org/10.1016/j.jcp.2014.04.001	journal	August 2014
Multidimensional Riemann problem with self-similar internal structure. Part I – Application to hyperbolic conservation laws on structured meshes Balsara, Dinshaw S. Journal of Computational Physics, Vol. 277 https://doi.org/10.1016/j.jcp.2014.07.053	journal	November 2014
Divergence-free approximate Riemann solver for the quasi-neutral two-fluid plasma model Amano, Takanobu Journal of Computational Physics, Vol. 299 https://doi.org/10.1016/j.jcp.2015.07.035	journal	October 2015
A simple unsplit Godunov method for multidimensional MHD Stone, James M.; Gardiner, Thomas New Astronomy, Vol. 14, Issue 2 https://doi.org/10.1016/j.newast.2008.06.003	journal	February 2009
Small-scale structure of two-dimensional magnetohydrodynamic turbulence Orszag, Steven A.; Tang, Cha-Mei Journal of Fluid Mechanics, Vol. 90, Issue 1 https://doi.org/10.1017/S002211207900210X	journal	January 1979
A high order Godunov scheme with constrained transport and adaptive mesh refinement for astrophysical magnetohydrodynamics Fromang, S.; Hennebelle, P.; Teyssier, R. Astronomy & Astrophysics, Vol. 457, Issue 2 https://doi.org/10.1051/0004-6361:20065371	journal	September 2006
ECHO: a Eulerian conservative high-order scheme for general relativistic magnetohydrodynamics and magnetodynamics Del Zanna, L.; Zanotti, O.; Bucciantini, N. Astronomy & Astrophysics, Vol. 473, Issue 1 https://doi.org/10.1051/0004-6361:20077093	journal	July 2007
Simulation of magnetohydrodynamic flows - A constrained transport method Evans, Charles R.; Hawley, John F. The Astrophysical Journal, Vol. 332 https://doi.org/10.1086/166684	journal	September 1988
Numerical magetohydrodynamics in astrophysics: Algorithm and tests for one-dimensional flow` Ryu, Dongsu; Jones, T. W. The Astrophysical Journal, Vol. 442 https://doi.org/10.1086/175437	journal	March 1995
On the Divergence‐free Condition and Conservation Laws in Numerical Simulations for Supersonic Magnetohydrodynamical Flows Dai, Wenlong; Woodward, Paul R. The Astrophysical Journal, Vol. 494, Issue 1 https://doi.org/10.1086/305176	journal	February 1998
A Divergence‐free Upwind Code for Multidimensional Magnetohydrodynamic Flows Ryu, Dongsu; Miniati, Francesco; Jones, T. W. The Astrophysical Journal, Vol. 509, Issue 1 https://doi.org/10.1086/306481	journal	December 1998
High‐Order Upwind Schemes for Multidimensional Magnetohydrodynamics Londrillo, P.; Del Zanna, L. The Astrophysical Journal, Vol. 530, Issue 1 https://doi.org/10.1086/308344	journal	February 2000
A Comparison between Divergence‐Cleaning and Staggered‐Mesh Formulations for Numerical Magnetohydrodynamics Balsara, Dinshaw S.; Kim, Jongsoo The Astrophysical Journal, Vol. 602, Issue 2 https://doi.org/10.1086/381051	journal	February 2004
Athena: A New Code for Astrophysical MHD Stone, James M.; Gardiner, Thomas A.; Teuben, Peter The Astrophysical Journal Supplement Series, Vol. 178, Issue 1 https://doi.org/10.1086/588755	journal	September 2008
Total variation diminishing Runge-Kutta schemes Gottlieb, Sigal; Shu, Chi-Wang Mathematics of Computation of the American Mathematical Society, Vol. 67, Issue 221 https://doi.org/10.1090/S0025-5718-98-00913-2	journal	January 1998
Self-similar jets Falle, S. A. E. G. Monthly Notices of the Royal Astronomical Society, Vol. 250, Issue 3 https://doi.org/10.1093/mnras/250.3.581	journal	June 1991
A discontinuous Galerkin method for solving the fluid and magnetohydrodynamic equations in astrophysical simulations Mocz, Philip; Vogelsberger, Mark; Sijacki, Debora Monthly Notices of the Royal Astronomical Society, Vol. 437, Issue 1 https://doi.org/10.1093/mnras/stt1890	journal	October 2013
xtroem-fv: a new code for computational astrophysics based on very high order finite-volume methods – I. Magnetohydrodynamics Núñez-de la Rosa, Jonatan; Munz, Claus-Dieter Monthly Notices of the Royal Astronomical Society, Vol. 455, Issue 4 https://doi.org/10.1093/mnras/stv2531	journal	November 2015
Highly Efficient Strong Stability-Preserving Runge–Kutta Methods with Low-Storage Implementations Ketcheson, David I. SIAM Journal on Scientific Computing, Vol. 30, Issue 4 https://doi.org/10.1137/07070485X	journal	January 2008
On Godunov-Type Methods for Gas Dynamics Einfeldt, Bernd SIAM Journal on Numerical Analysis, Vol. 25, Issue 2 https://doi.org/10.1137/0725021	journal	April 1988
Simplified Second-Order Godunov-Type Methods Davis, S. F. SIAM Journal on Scientific and Statistical Computing, Vol. 9, Issue 3 https://doi.org/10.1137/0909030	journal	May 1988
On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws Harten, Amiram; Lax, Peter D.; Leer, Bram van SIAM Review, Vol. 25, Issue 1 https://doi.org/10.1137/1025002	journal	January 1983
Positivity Preservation and Advection Algorithms with Applications to Edge Plasma Turbulence Peterson, J. L.; Hammett, G. W. SIAM Journal on Scientific Computing, Vol. 35, Issue 3 https://doi.org/10.1137/120888053	journal	January 2013
A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods Spiteri, Raymond J.; Ruuth, Steven J. SIAM Journal on Numerical Analysis, Vol. 40, Issue 2 https://doi.org/10.1137/S0036142901389025	journal	January 2002
Strong Stability-Preserving High-Order Time Discretization Methods Gottlieb, Sigal; Shu, Chi-Wang; Tadmor, Eitan SIAM Review, Vol. 43, Issue 1 https://doi.org/10.1137/S003614450036757X	journal	January 2001

Cited By (3)

An arbitrary high-order Spectral Difference method for the induction equation Han Veiga, Maria; Velasco-Romero, David A.; Wenger, Quentin Journal of Computational Physics, Vol. 438 https://doi.org/10.1016/j.jcp.2021.110327	journal	August 2021
Numerical Methods for Simulating Star Formation Teyssier, Romain; Commerçon, Benoît Frontiers in Astronomy and Space Sciences, Vol. 6 https://doi.org/10.3389/fspas.2019.00051	journal	July 2019
Digital Infrastructure in Astrophysics timmes, frank; Townsend, Rich; Bildsten, Lars Vol. 52, Issue 2, Vol. 52, Issue 2 https://doi.org/10.3847/25c2cfeb.13af6a03	journal	January 2020

Similar Records

Constrained-Transport Magnetohydrodynamics with Adaptive-Mesh-Refinement in CHARM

Journal Article · Tue May 24 00:00:00 EDT 2011 · The Astrophysical Journal Supplement · OSTI ID:1050649

CONSTRAINED-TRANSPORT MAGNETOHYDRODYNAMICS WITH ADAPTIVE MESH REFINEMENT IN CHARM

Journal Article · Fri Jul 01 00:00:00 EDT 2011 · Astrophysical Journal, Supplement Series · OSTI ID:21560357

Constrained-transport magnetohydrodynamics with adaptive mesh refinement in CHARM

Journal Article · Thu Jun 23 00:00:00 EDT 2011 · The Astrophysical Journal. Supplement Series · OSTI ID:1511331

Related Subjects

97 MATHEMATICS AND COMPUTING
constrained transport
high-order finite volume method
magnetohydrodynamics
numerical methods

A fourth-order accurate finite volume method for ideal MHD via upwind constrained transport

Citation Formats

References (60)

Cited By (3)

Similar Records

Related Subjects