An analysis of time discretization in the finite element solution of hyperbolic problems
Journal of Computational Physics
Journal of Computational Physics - Keith V. Roberts Memorial Issue
Computational algorithms for aerodynamic analysis and design
Applied Numerical Mathematics
Combined finite element-finite volume solution of compressible flow
Modelling 94 Proceedings of the 1994 international symposium on Mathematical modelling and computational methods
High-resolution conservative algorithms for advection in incompressible flow
SIAM Journal on Numerical Analysis
A well-behaved TVD limiter for high-resolution calculations of unsteady flow
Journal of Computational Physics
The Runge-Kutta discontinuous Galerkin method for conservation laws V multidimensional systems
Journal of Computational Physics
Flux correction tools for finite elements
Journal of Computational Physics
Variable-order finite elements and positivity preservation for hyperbolic PDEs
Applied Numerical Mathematics - Special issue: Workshop on innovative time integrators for PDEs
High-resolution FEM-TVD schemes based on a fully multidimensional flux limiter
Journal of Computational Physics
International Journal of Computing Science and Mathematics
On the design of algebraic flux correction schemes for quadratic finite elements
Journal of Computational and Applied Mathematics
Explicit and implicit FEM-FCT algorithms with flux linearization
Journal of Computational Physics
Journal of Computational Physics
Goal-oriented mesh adaptation for flux-limited approximations to steady hyperbolic problems
Journal of Computational and Applied Mathematics
A monotone finite volume method for advection-diffusion equations on unstructured polygonal meshes
Journal of Computational Physics
Mixed element FEM level set method for numerical simulation of immiscible fluids
Journal of Computational Physics
Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes
Journal of Computational and Applied Mathematics
Journal of Computational Physics
A positivity-preserving finite element method for chemotaxis problems in 3D
Journal of Computational and Applied Mathematics
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The algebraic flux correction (AFC) paradigm is extended to finite element discretizations with a consistent mass matrix. It is shown how to render an implicit Galerkin scheme positivity-preserving and remove excessive artificial diffusion in regions where the solution is sufficiently smooth. To this end, the original discrete operators are modified in a mass-conserving fashion so as to enforce the algebraic constraints to be satisfied by the numerical solution. A node-oriented limiting strategy is employed to control the raw antidiffusive fluxes which consist of a convective part and a contribution of the consistent mass matrix. The former offsets the artificial diffusion due to 'upwinding' of the spatial differential operator and lends itself to an upwind-biased flux limiting. The latter eliminates the error induced by mass lumping and calls for the use of a symmetric flux limiter. The concept of a target flux and a new definition of upper/lower bounds make it possible to combine the advantages of algebraic FCT and TVD schemes introduced previously by the author and his coworkers. Unlike other high-resolution schemes for unstructured meshes, the new algorithm reduces to a consistent (high-order) Galerkin scheme in smooth regions and is designed to provide an optimal treatment of both stationary and time-dependent problems. Its performance is illustrated by application to the linear advection equation for a number of 1D and 2D configurations.