A unified analysis of algebraic flux correction schemes for convection–diffusion equations

Barrenechea, Gabriel R.; John, Volker; Knobloch, Petr; Rankin, Richard

doi:10.1007/s40324-018-0160-6

Cited by 42 publications

(56 citation statements)

References 42 publications

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“…Since the matricesM and M L are diagonal, the lagged treatment ofġ H is particularly advantageous in the context of explicit schemes for numerical solution of (7). Invoking (4) and (5), we find that right-hand side vectors…”

Section: Enriched Galerkin Methodsmentioning

confidence: 99%

“…The P 0 enrichment of a CG-P 1 or CG-Q 1 discretizations has a positive impact on the rates of convergence to smooth solutions but may be insufficient to prevent spurious undershoots and overshoots in the neighborhood of steep gradients. This unsatisfactory behavior of EG solutions can be cured using the algebraic flux correction (AFC) methodology [3,5,25,32], a general framework for constraining a high-order discretization to be LED. As a first step toward that end, we need to write our EG scheme in the equivalent form…”

Section: Algebraic Splittingmentioning

confidence: 99%

“…Preservation of local bounds for implicit time discretizations of semi-discrete LED schemes and the validity of generalized discrete maximum principles at steady state can be shown using more sophisticated proof techniques. For details, we refer the interested reader to [5,32].…”

Section: Remarkmentioning

confidence: 99%

“…To enforce the validity of discrete maximum principles, numerical methods for conservation laws are commonly equipped with flux or slope limiters. Recent years have witnessed significant advances in the analysis and design of algebraic flux correction (AFC) tools for nonlinear high-resolution finite element schemes [3,5,12,25,32]. However, no bound-preserving limiters are currently available for locally conservative enriched Galerkin approximations, which restricts their practical applicability to problems with smooth solutions.…”

Section: Introductionmentioning

confidence: 99%

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Locally bound-preserving enriched Galerkin methods for the linear advection equation

2020

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In this work, we introduce algebraic flux correction schemes for enriched (P 1 ⊕ P 0 and Q 1 ⊕ P 0 ) Galerkin discretizations of the linear advection equation. The piecewise-constant component stabilizes the continuous Galerkin approximation without introducing free parameters. However, violations of discrete maximum principles are possible in the neighborhood of discontinuities and steep fronts. To keep the cell averages and the degrees of freedom of the continuous P 1 /Q 1 component in the admissible range, we limit the fluxes and element contributions, the complete removal of which would correspond to first-order upwinding. The first limiting procedure that we consider in this paper is based on the flux-corrected transport (FCT) paradigm. It belongs to the family of predictor-corrector algorithms and requires the use of small time steps. The second limiting strategy is monolithic and produces nonlinear problems with well-defined residuals. This kind of limiting is well suited for stationary and time-dependent problems alike. The need for inverting consistent mass matrices in explicit strong stability preserving Runge-Kutta time integrators is avoided by reconstructing nodal time derivatives from cell averages. Numerical studies are performed for standard 2D test problems.

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Section: Enriched Galerkin Methodsmentioning

confidence: 99%

Section: Algebraic Splittingmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Locally bound-preserving enriched Galerkin methods for the linear advection equation

2020

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“…If the set on the right-hand side is empty, there exists no meaningful maximum principle. However, corresponding estimates regarding strongly enforced (Dirichlet) boundary conditions as considered in the scalar case in [8] remain valid. The corresponding proof is very similar to the one presented below, which is inspired by [8,15].…”

Section: Example Of a Tensorial Eigenvalue Range Limitermentioning

confidence: 99%

Algebraic flux correction schemes preserving the eigenvalue range of symmetric tensor fields

Lohmann

2019

ESAIM: M2AN

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This work extends the algebraic flux correction (AFC) paradigm to finite element discretizations of conservation laws for symmetric tensor fields. The proposed algorithms are designed to enforce discrete maximum principles and preserve the eigenvalue range of evolving tensors. To that end, a continuous Galerkin approximation is modified by adding a linear artificial diffusion operator and a nonlinear antidiffusive correction. The latter is decomposed into edge-based fluxes and constrained to prevent violations of local bounds for the minimal and maximal eigenvalues. In contrast to the flux-corrected transport (FCT) algorithm developed previously by the author and existing slope limiting techniques for stress tensors , the admissible eigenvalue range is defined implicitly and the limited antidiffusive terms are incorporated into the residual of the nonlinear system. In addition to scalar limiters that use a common correction factor for all components of a tensor-valued antidiffusive flux, tensor limiters are designed using spectral decompositions. The new limiter functions are analyzed using tensorial extensions of the existing AFC theory for scalar convection-diffusion equations. The proposed methodology is backed by rigorous proofs of eigenvalue range preservation and Lipschitz continuity. Convergence of pseudo time-stepping methods to stationary solutions is demonstrated in numerical studies. 1991 Mathematics Subject Classification. 65N12, 65N30.The dates will be set by the publisher.

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Discrete strong extremum principles for finite element solutions of advection‐diffusion problems with nonlinear corrections

Wang,

Yuan

2024

Numerical Methods in Fluids

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A nonlinear correction technique for finite element methods of advection‐diffusion problems on general triangular meshes is introduced. The classic linear finite element method is modified, and the resulting scheme satisfies discrete strong extremum principle unconditionally, which means that it is unnecessary to impose the well‐known restrictions on diffusion coefficients and geometry of mesh‐cell (e.g., “acute angle” condition), and we need not to perform upwind treatment on the advection term separately. Moreover, numerical example shows that when a discrete scheme does not satisfy the strong extremum principle, even if it maintains the global physical bound, non‐physical numerical oscillations may still occur within local regions where no numerical result is beyond the physical bound. Thus, it is worth to point out that our new nonlinear finite element scheme can avoid non‐physical oscillations around sharp layers in advection‐dominate regions, due to maintaining discrete strong extremum principle. Convergence rates are verified by numerical tests for both diffusion‐dominate and advection‐dominate problems.

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A unified analysis of algebraic flux correction schemes for convection–diffusion equations

Cited by 42 publications

References 42 publications

Locally bound-preserving enriched Galerkin methods for the linear advection equation

Locally bound-preserving enriched Galerkin methods for the linear advection equation

Algebraic flux correction schemes preserving the eigenvalue range of symmetric tensor fields

Discrete strong extremum principles for finite element solutions of advection‐diffusion problems with nonlinear corrections

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