Gradient‐based nodal limiters for artificial diffusion operators in finite element schemes for transport equations

• Published in: International journal for numerical methods in fluids Vol. 84; no. 11; pp. 675 - 695
• Main Authors: Kuzmin, Dmitri, Shadid, John N.
• Format: Journal Article
• Language: English
• Published: Bognor Regis Wiley Subscription Services, Inc 20-08-2017
• Subjects: Accuracy; Algebra; algebraic flux correction; Anisotropy; Continuity (mathematics); Convection; convective transport; Diffusion; Dimensions; discrete maximum principles; Discretization; Distortion; Dye dispersion; Finite element method; finite element schemes; Fluctuations; Fluxes; Fronts; Gradients; limiters; Linearity; Long-term depression; Mathematical analysis; Mathematical models; Maximum principle; Operators (mathematics); Preservation; special t4ht@.anisotropic diffusion; Transport; Transport equations
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.4365

Summary
This paper presents new linearity‐preserving nodal limiters for enforcing discrete maximum principles in continuous (linear or bilinear) finite element approximations to transport problems with steep fronts. In the process of algebraic flux correction, the oscillatory antidiffusive part of a high‐order base discretization is decomposed into a set of internodal fluxes and constrained to be local extremum dim inishing. The proposed nodal limiter functions are designed to be continuous and satisfy the principle of linearity preservation that implies the preservation of second‐order accuracy in smooth regions. The use of limited nodal gradients makes it possible to circumvent angle conditions and guarantee that the discrete maximum principle holds on arbitrary meshes. A numerical study is performed for linear convection and anisotropic diffusion problems on uniform and distorted meshes in two space dimensions. Copyright © 2017 John Wiley & Sons, Ltd.
Edge‐based limiting techniques are proposed for enforcing local maximum principles in continuous finite element schemes for transport equations. Different generalizations of a one‐dimensional jump and average limiter are considered and improved step by step. The use of limited nodal gradients makes it possible to circumvent angle conditions that apply to other local extremum diminishing and linearity‐preserving limiters.