A new approach to enforcing discrete maximum principles in continuous Galerkin methods for convection-dominated transport equations

This paper presents a set of design principles and new algorithmic tools for enforcing maximum principles and/or positivity preservation in continuous finite element approximations to convection-dominated transport problems. Using a linear first-order advection equation as a model problem, we address the design of first-order artificial diffusion operators and their higherorder counterparts at the element matrix level. The proposed methodology leads to a nonlinear high-resolution scheme capable of resolving moving fronts and internal/boundary layers as sharp localized nonoscillatory features. The amount of numerical dissipation depends on the difference between the solution value at a given node and a local maximum or minimum. The shockcapturing numerical diffusion coefficient is designed to vanish as the nodal values approach a mass-weighted or linearity-preserving average. The universal applicability and simplicity of the element-based limiting procedure makes it an attractive alternative to edge-based algebraic flux correction. Email addresses: [email protected] (Dmitri Kuzmin), [email protected] (John N. Shadid) Preprint submitted to Journal of Computational Physics October 30, 2015 Additionally a Lipschitz continuous version of the limiter is presented that guarantees unique solvability of the nonlinear system associated with the steady state limit of the time-dependent problem. A grid convergence study is performed for two-dimensional test problems.