Analysis of algebraic flux correction for semi-discrete advection problems

Hajduk, Hennes; Rupp, Andreas; Kuzmin, Dmitri

doi:10.48550/arxiv.2104.05639

Cited by 1 publication

(8 citation statements)

References 31 publications

(62 reference statements)

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“…The contents of this paper are to a large degree based on [14,Ch. 5], which improves upon the analysis presented in our preprint [15]. In particular, we improved both the theoretical parts and the numerical examples by properly addressing the treatment of boundary conditions and presenting numerical results not just for simple 1D problems but also in the 2D case.…”

Section: Introductionmentioning

confidence: 82%

“…5.4, the standard MCL scheme using low order time derivatives (2.14) for stabilization purposes is also prone to producing compatible pairs. In [15,Sec. 3.3] we present a modified MCL procedure with which (3.1) can be guaranteed.…”

Section: Energy Estimatementioning

confidence: 99%

“…Note that as a consequence of Lemma 2 and of the nonnegativity of basis functions, all terms appearing on the left hand side of inequality (3.2) are nonnegative. To guarantee that the assumptions of Proposition 1 are satisfied in practice, one can use the scheme proposed in [15,Sec. 3.3], which enforces (3.1) for user defined values of γ .…”

Section: Lemma 3 Any Functionmentioning

confidence: 99%

“…For analytical purposes, we make an assumption regarding compatibility of the semi-discrete approximations and corresponding time derivatives. As shown in [15,Sec. 3.3], it is possible to enforce this condition by adapting the limiting procedure of the standard MCL approach.…”

Section: Introductionmentioning

confidence: 98%

“…5.4. Therefore, we do not discuss enforcement of the compatibility condition in this work and instead refer the interested reader to [15,Sec. 3.3].…”

Section: Introductionmentioning

confidence: 99%

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Analysis of algebraic flux correction schemes for semi-discrete advection problems

Hajduk

Rupp

2023

Bit Numer Math

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Based on recent developments regarding the analysis of algebraic flux correction schemes, we consider a locally bound-preserving discretization of the time-dependent advection equation. Specifically, we analyze a monolithic convex limiting scheme based on piecewise (multi-)linear continuous finite elements in the semi-discrete formulation. To stabilize the discretization, we use low order time derivatives in the definition of raw antidiffusive fluxes. Our analytical investigation reveals that their limited counterparts should satisfy a certain compatibility condition. The conducted numerical experiments suggest that this prerequisite is satisfied unless the size of mesh elements is vastly different. We prove global-in-time existence of semi-discrete approximations and derive an a priori error estimate for finite time intervals with a worst-case convergence rate of $$\frac{1}{2}$$ 1 2 w. r. t. the $${\textrm{L}}^2$$ L 2 error. This rate is optimal in the setting under consideration because we allow all correction factors of the flux-corrected scheme to become zero. In this case, the algorithm reduces to the bound-preserving discrete upwinding method but the limited counterpart of this scheme converges much faster, in practice. Additional numerical experiments are performed to verify the provable convergence rate for a few variants of the scheme.

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Section: Introductionmentioning

confidence: 82%

Section: Energy Estimatementioning

confidence: 99%

Section: Lemma 3 Any Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

“…5.4. Therefore, we do not discuss enforcement of the compatibility condition in this work and instead refer the interested reader to [15,Sec. 3.3].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations