Subcell flux limiting for high-order Bernstein finite element discretizations of scalar hyperbolic conservation laws
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2020-03-21Online Publication Date
2020-03-21Print Publication Date
2020-06Submitted Date
2019-09-16Permanent link to this record
http://hdl.handle.net/10754/662387
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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.2020.109411Sponsors
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 BVJournal
Journal of Computational PhysicsarXiv
1909.03328Additional Links
https://linkinghub.elsevier.com/retrieve/pii/S0021999120301856ae974a485f413a2113503eed53cd6c53
10.1016/j.jcp.2020.109411