Matrix-free subcell residual distribution for Bernstein finite element discretizations of linear advection equations

Hajduk, Hennes; Kuzmin, Dmitri; Kolev, Tzanio; Abgrall, Remi

doi:10.1016/j.cma.2019.112658

Title: Matrix-free subcell residual distribution for Bernstein finite element discretizations of linear advection equations

Journal Article · Wed Oct 09 00:00:00 EDT 2019 · Computer Methods in Applied Mechanics and Engineering

DOI:https://doi.org/10.1016/j.cma.2019.112658· OSTI ID:1836210

^[1]; Kuzmin, Dmitri ^[1]; Kolev, Tzanio ^[2]; Abgrall, Remi ^[3]

Technische Univ. of Dortmund (Germany). Inst. of Applied Mathematics
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing
Univ. Zurich (Switzerland). Inst. of Computational Sciences

In this work, we introduce a new residual distribution (RD) framework for the design of bound-preserving high-resolution finite element schemes. The continuous and discontinuous Galerkin discretizations of the linear advection equation are modified to construct local extremum diminishing (LED) approximations. To that end, we perform mass lumping and redistribute the element residuals in a manner which guarantees the LED property. The hierarchical correction procedure for high-order Bernstein finite element discretizations involves localization to subcells and definition of bound-preserving weights for subcell contributions. Using strong stability preserving (SSP) Runge–Kutta methods for time integration, we prove the validity of discrete maximum principles under CFL-like time step restrictions. The low-order version of our method has roughly the same accuracy as the one derived from a piecewise (multi)-linear approximation on a submesh with the same nodal points. In high-order extensions, we use an element-based flux-corrected transport (FCT) algorithm which can be interpreted as a nonlinear RD scheme. The proposed LED corrections are tailor-made for matrix-free implementations which avoid the rapidly growing cost of matrix assembly for high-order Bernstein elements. The results for 1D, 2D, and 3D test problems compare favorably to those obtained with the best matrix-based approaches.

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Research Organization:: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA); German Research Foundation (DFG); SNF

Grant/Contract Number:: AC52-07NA27344; KU 1530/23-1; 200020 175784

OSTI ID:: 1836210

Alternate ID(s):: OSTI ID: 1580794

Report Number(s):: LLNL-JRNL-768125; 958809

Journal Information:: Computer Methods in Applied Mechanics and Engineering, Vol. 359; ISSN 0045-7825

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 10 works

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Entropy stabilization and property-preserving limiters for discontinuous Galerkin discretizations of nonlinear hyperbolic equations Kuzmin, Dmitri arXiv https://doi.org/10.48550/arxiv.2004.03521	preprint	January 2020

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