Flux-corrected transport algorithms for continuous Galerkin methods based on high order Bernstein finite elements

JOURNAL OF COMPUTATIONAL PHYSICS

卷:344

Article

Lohmann, Christoph¹  Kuzmin, Dmitri¹  Shadid, John N.^2,3  Mabuza, Sibusiso² 

[1] TU Dortmund Univ, Inst Appl Math LS 3, Vogelpothsweg 87, D-44227 Dortmund, Germany

[2] Sandia Natl Labs, Computat Math Dept, POB 5800 MS 1321, Albuquerque, NM 87185 USA

[3] Univ New Mexico, Dept Math & Stat, MSC01 1115, Albuquerque, NM 87131 USA

关键词: Bernstein-Bezier finite elements;    Continuous Galerkin method;    Flux-corrected transport;    Artificial diffusion;    Local discrete maximum principles;    Total variation diminishing property;   

DOI  :  10.1016/j.jcp.2017.04.059

来源: Elsevier

【 摘 要 】

This work extends the flux-corrected transport (FCT) methodology to arbitrary order continuous finite element discretizations of scalar conservation laws on simplex meshes. Using Bernstein polynomials as local basis functions, we constrain the total variation of the numerical solution by imposing local discrete maximum principles on the Bezier net. The design of accuracy-preserving FCT schemes for high order Bernstein-Bezier finite elements requires the development of new algorithms and/or generalization of limiting techniques tailored for linear and multilinear Lagrange elements. In this paper, we propose (i) a new discrete upwinding strategy leading to local extremum bounded low order approximations with compact stencils, (ii) high order variational stabilization based on the difference between two gradient approximations, and (iii) new localized limiting techniques for antidiffusive element contributions. The optional use of a smoothness indicator, based on a second derivative test, makes it possible to potentially avoid unnecessary limiting at smooth extrema and achieve optimal convergence rates for problems with smooth solutions. The accuracy of the proposed schemes is assessed in numerical studies for the linear transport equation in 1D and 2D. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free