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    Subcell flux limiting for high-order Bernstein finite element discretizations of scalar hyperbolic conservation laws

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    DK_MQL_subcell_flux_limiting_for_Bernstein_FEM.pdf
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    Type
    Article
    Authors
    Kuzmin, Dmitri
    Quezada de Luna, Manuel
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Date
    2020-03-21
    Online Publication Date
    2020-03-21
    Print Publication Date
    2020-06
    Submitted Date
    2019-09-16
    Permanent link to this record
    http://hdl.handle.net/10754/662387
    
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    Abstract
    This work extends the concepts of algebraic flux correction and convex limiting to continuous high-order Bernstein finite element discretizations of scalar hyperbolic problems. Using an array of adjustable diffusive fluxes, the standard Galerkin approximation is transformed into a nonlinear high-resolution scheme which has the compact sparsity pattern of the piecewise-linear or multilinear subcell discretization. The representation of this scheme in terms of invariant domain preserving states makes it possible to prove the validity of local discrete maximum principles under CFL-like conditions. In contrast to predictor-corrector approaches based on the flux-corrected transport methodology, the proposed flux limiting strategy is monolithic, i.e., limited antidiffusive terms are incorporated into the well-defined residual of a nonlinear (semi-)discrete problem. A stabilized high-order Galerkin discretization is recovered if no limiting is performed. In the limited version, the compact stencil property prevents direct mass exchange between nodes that are not nearest neighbors. A formal proof of sparsity is provided for simplicial and box elements. The involved element contributions can be calculated efficiently making use of matrix-free algorithms and precomputed element matrices of the reference element. Numerical studies for Q2 discretizations of linear and nonlinear two-dimensional test problems illustrate the virtues of monolithic convex limiting based on subcell flux decompositions.
    Citation
    Kuzmin, D., & Quezada de Luna, M. (2020). Subcell flux limiting for high-order Bernstein finite element discretizations of scalar hyperbolic conservation laws. Journal of Computational Physics, 411, 109411. doi:10.1016/j.jcp.2020.109411
    Sponsors
    The work of Dmitri Kuzmin was supported by the German Research Association (DFG) under grant KU 1530/23-1. The work of Manuel Quezada de Luna was supported by King Abdullah University of Science and Technology (KAUST) in Thuwal, Saudi Arabia. The authors would like to thank Prof. David I. Ketcheson (KAUST) and Christoph Lohmann (TU Dortmund University) for helpful discussions.
    Publisher
    Elsevier BV
    Journal
    Journal of Computational Physics
    DOI
    10.1016/j.jcp.2020.109411
    arXiv
    1909.03328
    Additional Links
    https://linkinghub.elsevier.com/retrieve/pii/S0021999120301856
    ae974a485f413a2113503eed53cd6c53
    10.1016/j.jcp.2020.109411
    Scopus Count
    Collections
    Articles; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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