Matrix-free subcell residual distribution for Bernstein finite element discretizations of linear advection equations
Abstract
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.
- Authors:
-
[1];
- 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
- Publication Date:
- Research Org.:
- Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA); German Research Foundation (DFG); SNF
- OSTI Identifier:
- 1836210
- Alternate Identifier(s):
- OSTI ID: 1580794
- Report Number(s):
- LLNL-JRNL-768125
Journal ID: ISSN 0045-7825; 958809
- Grant/Contract Number:
- AC52-07NA27344; KU 1530/23-1; 200020 175784
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Computer Methods in Applied Mechanics and Engineering
- Additional Journal Information:
- Journal Volume: 359; Journal ID: ISSN 0045-7825
- Publisher:
- Elsevier
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; advection equation; discrete maximum principles; Bernstein finite elements; matrix-free methods; residual distribution; flux-corrected transport
Citation Formats
Hajduk, Hennes, Kuzmin, Dmitri, Kolev, Tzanio, and Abgrall, Remi. Matrix-free subcell residual distribution for Bernstein finite element discretizations of linear advection equations. United States: N. p., 2019.
Web. doi:10.1016/j.cma.2019.112658.
Hajduk, Hennes, Kuzmin, Dmitri, Kolev, Tzanio, & Abgrall, Remi. Matrix-free subcell residual distribution for Bernstein finite element discretizations of linear advection equations. United States. https://doi.org/10.1016/j.cma.2019.112658
Hajduk, Hennes, Kuzmin, Dmitri, Kolev, Tzanio, and Abgrall, Remi. Wed .
"Matrix-free subcell residual distribution for Bernstein finite element discretizations of linear advection equations". United States. https://doi.org/10.1016/j.cma.2019.112658. https://www.osti.gov/servlets/purl/1836210.
@article{osti_1836210,
title = {Matrix-free subcell residual distribution for Bernstein finite element discretizations of linear advection equations},
author = {Hajduk, Hennes and Kuzmin, Dmitri and Kolev, Tzanio and Abgrall, Remi},
abstractNote = {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.},
doi = {10.1016/j.cma.2019.112658},
journal = {Computer Methods in Applied Mechanics and Engineering},
number = ,
volume = 359,
place = {United States},
year = {Wed Oct 09 00:00:00 EDT 2019},
month = {Wed Oct 09 00:00:00 EDT 2019}
}
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Works referencing / citing this record:
A subcell-enriched Galerkin method for advection problems
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- Computers & Mathematics with Applications, Vol. 93
Entropy stabilization and property-preserving limiters for discontinuous Galerkin discretizations of nonlinear hyperbolic equations
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