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

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

Journal Article · 10 March 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.

View Accepted Manuscript (DOE)

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

Sponsoring Organization:: German Research Foundation (DFG); USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

Grant/Contract Number:: AC04-94AL85000; NA0003525

Other Award/Contract Number:: KU 1530/15-2

OSTI ID:: 1618098

Alternate ID(s):: OSTI ID: 1605331

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

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

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

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On differentiable local bounds preserving stabilization for Euler equations Badia, Santiago; Bonilla, Jesús; Mabuza, Sibusiso arXiv https://doi.org/10.48550/arxiv.1911.06792	text	January 2019

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

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