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

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.

View Accepted Manuscript (DOE)

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

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

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (28)

Multidimensional upwinding: Its relation to finite elements Carette, J. -C.; Deconinck, H.; Paillere, H. International Journal for Numerical Methods in Fluids, Vol. 20, Issue 8-9 https://doi.org/10.1002/fld.1650200815	journal	April 1995
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
Multidimensional upwind schemes based on fluctuation-splitting for systems of conservation laws Deconinck, H.; Paill�re, H.; Struijs, R. Computational Mechanics, Vol. 11, Issue 5-6 https://doi.org/10.1007/BF00350091	journal	January 1993
Further variation diminishing properties of Bernstein polynomials on triangles Goodman, T. N. T. Constructive Approximation, Vol. 3, Issue 1 https://doi.org/10.1007/BF01890572	journal	December 1987
Fast inversion of the simplicial Bernstein mass matrix Kirby, Robert C. Numerische Mathematik, Vol. 135, Issue 1 https://doi.org/10.1007/s00211-016-0795-0	journal	February 2016
An Example of High Order Residual Distribution Scheme Using non-Lagrange Elements Abgrall, R.; Trefilík, J. Journal of Scientific Computing, Vol. 45, Issue 1-3 https://doi.org/10.1007/s10915-010-9405-y	journal	July 2010
A Posteriori Analysis of a Positive Streamwise Invariant Discretization of a Convection-Diffusion Equation Bernardi, Christine; Chacón Rebollo, Tomás; Restelli, Marco Journal of Scientific Computing, Vol. 51, Issue 2 https://doi.org/10.1007/s10915-011-9514-2	journal	August 2011
Construction of a p-Adaptive Continuous Residual Distribution Scheme Abgrall, R.; Viville, Q.; Beaugendre, H. Journal of Scientific Computing, Vol. 72, Issue 3 https://doi.org/10.1007/s10915-017-0399-6	journal	March 2017
A vertex-based hierarchical slope limiter for p-adaptive discontinuous Galerkin methods Kuzmin, Dmitri Journal of Computational and Applied Mathematics, Vol. 233, Issue 12 https://doi.org/10.1016/j.cam.2009.05.028	journal	April 2010
A partition of unity approach to adaptivity and limiting in continuous finite element methods Kuzmin, Dmitri; Quezada de Luna, Manuel; Kees, Christopher E. Computers & Mathematics with Applications, Vol. 78, Issue 3 https://doi.org/10.1016/j.camwa.2019.03.021	journal	August 2019
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
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
Improved detection criteria for the Multi-dimensional Optimal Order Detection (MOOD) on unstructured meshes with very high-order polynomials Diot, S.; Clain, S.; Loubère, R. Computers & Fluids, Vol. 64 https://doi.org/10.1016/j.compfluid.2012.05.004	journal	July 2012
Essentially non-oscillatory Residual Distribution schemes for hyperbolic problems Abgrall, R. Journal of Computational Physics, Vol. 214, Issue 2 https://doi.org/10.1016/j.jcp.2005.10.034	journal	May 2006
Limiters for high-order discontinuous Galerkin methods Krivodonova, Lilia Journal of Computational Physics, Vol. 226, Issue 1 https://doi.org/10.1016/j.jcp.2007.05.011	journal	September 2007
On maximum-principle-satisfying high order schemes for scalar conservation laws Zhang, Xiangxiong; Shu, Chi-Wang Journal of Computational Physics, Vol. 229, Issue 9 https://doi.org/10.1016/j.jcp.2009.12.030	journal	May 2010
A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws Dumbser, Michael; Zanotti, Olindo; Loubère, Raphaël Journal of Computational Physics, Vol. 278 https://doi.org/10.1016/j.jcp.2014.08.009	journal	December 2014
Embedded discontinuous Galerkin transport schemes with localised limiters Cotter, C. J.; Kuzmin, D. Journal of Computational Physics, Vol. 311 https://doi.org/10.1016/j.jcp.2016.02.021	journal	April 2016
High-order local maximum principle preserving (MPP) discontinuous Galerkin finite element method for the transport equation Anderson, R.; Dobrev, V.; Kolev, Tz. Journal of Computational Physics, Vol. 334 https://doi.org/10.1016/j.jcp.2016.12.031	journal	April 2017
Flux-corrected transport algorithms for continuous Galerkin methods based on high order Bernstein finite elements Lohmann, Christoph; Kuzmin, Dmitri; Shadid, John N. Journal of Computational Physics, Vol. 344 https://doi.org/10.1016/j.jcp.2017.04.059	journal	September 2017
A posteriori correction of high-order discontinuous Galerkin scheme through subcell finite volume formulation and flux reconstruction Vilar, François Journal of Computational Physics, Vol. 387 https://doi.org/10.1016/j.jcp.2018.10.050	journal	June 2019
High-Resolution Conservative Algorithms for Advection in Incompressible Flow LeVeque, Randall J. SIAM Journal on Numerical Analysis, Vol. 33, Issue 2 https://doi.org/10.1137/0733033	journal	April 1996
A Second-Order Maximum Principle Preserving Lagrange Finite Element Technique for Nonlinear Scalar Conservation Equations Guermond, Jean-Luc; Nazarov, Murtazo; Popov, Bojan SIAM Journal on Numerical Analysis, Vol. 52, Issue 4 https://doi.org/10.1137/130950240	journal	January 2014
Analysis of Algebraic Flux Correction Schemes Barrenechea, Gabriel R.; John, Volker; Knobloch, Petr SIAM Journal on Numerical Analysis, Vol. 54, Issue 4 https://doi.org/10.1137/15M1018216	journal	January 2016
Staggered Grid Residual Distribution Scheme for Lagrangian Hydrodynamics Abgrall, Rémi; Tokareva, Svetlana SIAM Journal on Scientific Computing, Vol. 39, Issue 5 https://doi.org/10.1137/16M1078781	journal	January 2017
High-Order Multi-Material ALE Hydrodynamics Anderson, Robert W.; Dobrev, Veselin A.; Kolev, Tzanio V. SIAM Journal on Scientific Computing, Vol. 40, Issue 1 https://doi.org/10.1137/17M1116453	journal	January 2018
Strong Stability-Preserving High-Order Time Discretization Methods Gottlieb, Sigal; Shu, Chi-Wang; Tadmor, Eitan SIAM Review, Vol. 43, Issue 1 https://doi.org/10.1137/S003614450036757X	journal	January 2001
A Review of Residual Distribution Schemes for Hyperbolic and Parabolic Problems: The July 2010 State of the Art Abgrall, Remi Communications in Computational Physics, Vol. 11, Issue 4 https://doi.org/10.4208/cicp.270710.130711s	journal	April 2012

Cited By (2)

A subcell-enriched Galerkin method for advection problems Rupp, Andreas; Hauck, Moritz; Aizinger, Vadym Computers & Mathematics with Applications, Vol. 93 https://doi.org/10.1016/j.camwa.2021.04.010	journal	July 2021
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

Similar Records

Matrix-free subcell residual distribution for Bernstein finite elements: Monolithic limiting

Journal Article · Tue Jan 21 23:00:00 EST 2020 · Computers and Fluids · OSTI ID:1769913

Sparse invariant domain preserving discontinuous Galerkin methods with subcell convex limiting

Journal Article · Thu Apr 29 00:00:00 EDT 2021 · Computer Methods in Applied Mechanics and Engineering · OSTI ID:1806422

Related Subjects

97 MATHEMATICS AND COMPUTING
Bernstein finite elements
advection equation
discrete maximum principles
flux-corrected transport
matrix-free methods
residual distribution

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

Citation Formats

References (28)

Cited By (2)

Similar Records

Related Subjects