A linearity preserving nodal variation limiting algorithm for continuous Galerkin discretization of ideal MHD equations

Mabuza, Sibusiso; Shadid, John N.; Cyr, Eric C.; Pawlowski, Roger P.; Kuzmin, Dmitri

doi:10.1016/j.jcp.2020.109390

Title: A linearity preserving nodal variation limiting algorithm for continuous Galerkin discretization of ideal MHD equations

Journal Article · Tue Mar 10 00:00:00 EDT 2020 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2020.109390· OSTI ID:1618098

^[1]; Shadid, John N. ^[2]; Cyr, Eric C. ^[3]; Pawlowski, Roger P. ^[3]; Kuzmin, Dmitri ^[4]

Clemson Univ., SC (United States)
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Univ. of New Mexico, Albuquerque, NM (United States)
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
TU Dortmund Univ. (Germany)

A stabilized continuous Galerkin (CG) method for magnetohydrodynamics (MHD) is presented herein. Ideal, compressible inviscid MHD equations are discretized in space on unstructured meshes using piecewise linear or bilinear finite element bases to get a semi-discrete scheme. Stabilization is then introduced to the semi-discrete method in a strategy that follows the algebraic flux correction paradigm. This involves adding some artificial diffusion to the high order, semi-discrete method and mass lumping in the time derivative term. The result is a low order method that provides local extremum diminishing properties for hyperbolic systems. The difference between the low order method and the high order method is scaled element-wise using a limiter and added to the low order scheme. The limiter is solution dependent and computed via an iterative linearity preserving nodal variation limiting strategy. The stabilization also involves an optional consistent background high order dissipation that reduces phase errors. The resulting stabilized scheme is a semi-discrete method that can be applied to inviscid shock MHD problems and may be even extended to resistive and viscous MHD problems. To satisfy the divergence free constraint of the MHD equations, we add parabolic divergence cleaning to the system. Various time integration methods can be used to discretize the scheme in time. We demonstrate the robustness of the scheme by solving several shock MHD problems.

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Research Organization:: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); German Research Foundation (DFG)

Grant/Contract Number:: AC04-94AL85000; NA0003525; KU 1530/15-2

OSTI ID:: 1618098

Alternate ID(s):: OSTI ID: 1605331

Report Number(s):: SAND-2019-3661J; 674276; TRN: US2106794

Journal Information:: Journal of Computational Physics, Vol. 410, Issue C; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 10 works

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References (34)

Divergence-Free Adaptive Mesh Refinement for Magnetohydrodynamics Balsara, Dinshaw S. Journal of Computational Physics, Vol. 174, Issue 2 https://doi.org/10.1006/jcph.2001.6917	journal	December 2001
High-resolution FEM–FCT schemes for multidimensional conservation laws Kuzmin, D.; Möller, M.; Turek, S. Computer Methods in Applied Mechanics and Engineering, Vol. 193, Issue 45-47 https://doi.org/10.1016/j.cma.2004.05.009	journal	November 2004
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
Local bounds preserving stabilization for continuous Galerkin discretization of hyperbolic systems Mabuza, Sibusiso; Shadid, John N.; Kuzmin, Dmitri Journal of Computational Physics, Vol. 361 https://doi.org/10.1016/j.jcp.2018.01.048	journal	May 2018
Flux Correction Tools for Finite Elements Kuzmin, D.; Turek, S. Journal of Computational Physics, Vol. 175, Issue 2 https://doi.org/10.1006/jcph.2001.6955	journal	January 2002
Notes on the Eigensystem of Magnetohydrodynamics Roe, P. L.; Balsara, D. S. SIAM Journal on Applied Mathematics, Vol. 56, Issue 1 https://doi.org/10.1137/S003613999427084X	journal	February 1996
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
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
Positivity-Preserving Analysis of Numerical Schemes for Ideal Magnetohydrodynamics Wu, Kailiang SIAM Journal on Numerical Analysis, Vol. 56, Issue 4 https://doi.org/10.1137/18M1168017	journal	January 2018
Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws Kuzmin, Dmitri Computer Methods in Applied Mechanics and Engineering, Vol. 361 https://doi.org/10.1016/j.cma.2019.112804	journal	April 2020
Gradient-based nodal limiters for artificial diffusion operators in finite element schemes for transport equations: Gradient-based nodal limiters for artificial diffusion operators in finite element schemes for transport equations Kuzmin, Dmitri; Shadid, John N. International Journal for Numerical Methods in Fluids, Vol. 84, Issue 11 https://doi.org/10.1002/fld.4365	journal	February 2017
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
The group finite element formulation Fletcher, C. A. J. Computer Methods in Applied Mechanics and Engineering, Vol. 37, Issue 2 https://doi.org/10.1016/0045-7825(83)90122-6	journal	April 1983
Limiting and divergence cleaning for continuous finite element discretizations of the MHD equations Kuzmin, Dmitri; Klyushnev, Nikita Journal of Computational Physics, Vol. 407 https://doi.org/10.1016/j.jcp.2020.109230	journal	April 2020
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
Sequential limiting in continuous and discontinuous Galerkin methods for the Euler equations Dobrev, V.; Kolev, Tz.; Kuzmin, D. Journal of Computational Physics, Vol. 356 https://doi.org/10.1016/j.jcp.2017.12.012	journal	March 2018
Convergence of locally divergence-free discontinuous-Galerkin methods for the induction equations of the 2D-MHD system Besse, Nicolas; Kröner, Dietmar ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 39, Issue 6 https://doi.org/10.1051/m2an:2005051	journal	November 2005
Fully multidimensional flux-corrected transport algorithms for fluids Zalesak, Steven T. Journal of Computational Physics, Vol. 31, Issue 3 https://doi.org/10.1016/0021-9991(79)90051-2	journal	June 1979
A Provably Positive Discontinuous Galerkin Method for Multidimensional Ideal Magnetohydrodynamics Wu, Kailiang; Shu, Chi-Wang SIAM Journal on Scientific Computing, Vol. 40, Issue 5 https://doi.org/10.1137/18M1168042	journal	January 2018
Second-Order Invariant Domain Preserving Approximation of the Euler Equations Using Convex Limiting Guermond, Jean-Luc; Nazarov, Murtazo; Popov, Bojan SIAM Journal on Scientific Computing, Vol. 40, Issue 5 https://doi.org/10.1137/17M1149961	journal	January 2018
A novel high-order, entropy stable, 3D AMR MHD solver with guaranteed positive pressure Derigs, Dominik; Winters, Andrew R.; Gassner, Gregor J. Journal of Computational Physics, Vol. 317 https://doi.org/10.1016/j.jcp.2016.04.048	journal	July 2016
Efficient MHD Riemann solvers for simulations on unstructured triangular grids Wesenberg, M. Journal of Numerical Mathematics, Vol. 10, Issue 1 https://doi.org/10.1515/JNMA.2002.37	journal	January 2002
A second-order unsplit Godunov scheme for cell-centered MHD: The CTU-GLM scheme Mignone, Andrea; Tzeferacos, Petros Journal of Computational Physics, Vol. 229, Issue 6 https://doi.org/10.1016/j.jcp.2009.11.026	journal	March 2010
Differentiable monotonicity-preserving schemes for discontinuous Galerkin methods on arbitrary meshes Badia, Santiago; Bonilla, Jesús; Hierro, Alba Computer Methods in Applied Mechanics and Engineering, Vol. 320 https://doi.org/10.1016/j.cma.2017.03.032	journal	June 2017
High order ADER schemes for a unified first order hyperbolic formulation of Newtonian continuum mechanics coupled with electro-dynamics Dumbser, Michael; Peshkov, Ilya; Romenski, Evgeniy Journal of Computational Physics, Vol. 348 https://doi.org/10.1016/j.jcp.2017.07.020	journal	November 2017
Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization Badia, Santiago; Bonilla, Jesús Computer Methods in Applied Mechanics and Engineering, Vol. 313 https://doi.org/10.1016/j.cma.2016.09.035	journal	January 2017
A Higher-Order Godunov Method for Multidimensional Ideal Magnetohydrodynamics Zachary, Andrew L.; Malagoli, Andrea; Colella, Phillip SIAM Journal on Scientific Computing, Vol. 15, Issue 2 https://doi.org/10.1137/0915019	journal	March 1994
Comparison of Some Flux Corrected Transport and Total Variation Diminishing Numerical Schemes for Hydrodynamic and Magnetohydrodynamic Problems Tóth, Gábor; Odstrčil, Dušan Journal of Computational Physics, Vol. 128, Issue 1 https://doi.org/10.1006/jcph.1996.0197	journal	October 1996
Provably positive high-order schemes for ideal magnetohydrodynamics: analysis on general meshes Wu, Kailiang; Shu, Chi-Wang Numerische Mathematik, Vol. 142, Issue 4 https://doi.org/10.1007/s00211-019-01042-w	journal	May 2019
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
An FCT finite element scheme for ideal MHD equations in 1D and 2D Basting, Melanie; Kuzmin, Dmitri Journal of Computational Physics, Vol. 338 https://doi.org/10.1016/j.jcp.2017.02.051	journal	June 2017
Linearity-preserving monotone local projection stabilization schemes for continuous finite elements Kuzmin, Dmitri; Basting, Steffen; Shadid, John N. Computer Methods in Applied Mechanics and Engineering, Vol. 322 https://doi.org/10.1016/j.cma.2017.04.030	journal	August 2017
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
Edge-based nonlinear diffusion for finite element approximations of convection–diffusion equations and its relation to algebraic flux-correction schemes Barrenechea, Gabriel R.; Burman, Erik; Karakatsani, Fotini Numerische Mathematik, Vol. 135, Issue 2 https://doi.org/10.1007/s00211-016-0808-z	journal	May 2016

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