Monolithic convex limiting in discontinuous Galerkin discretizations of hyperbolic conservation laws

Hajduk, Hennes

doi:10.1016/j.camwa.2021.02.012

Cited by 24 publications

(13 citation statements)

References 44 publications

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“…We consider Ω = (0, 1) with periodic boundaries. The analytical solution at any time t ∈ N coincides with the initial condition [8] (10) exp( 1 0.5−x ) exp( 1 x−0.9 ) if 0.5 < x < 0.9, 0 otherwise, (…”

Section: Numerical Examplesmentioning

confidence: 83%

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Analysis of algebraic flux correction for semi-discrete advection problems

Hajduk,

Rupp,

Kuzmin

2021

Preprint

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We present stability and error analysis for algebraic flux correction schemes based on monolithic convex limiting. For a continuous finite element discretization of the time-dependent advection equation, we prove global-in-time existence and the worstcase convergence rate of 1 2 w. r. t. the L 2 error of the spatial semidiscretization. Moreover, we address the important issue of stabilization for raw antidiffusive fluxes. Our a priori error analysis reveals that their limited counterparts should satisfy a generalized coercivity condition. We introduce a limiter for enforcing this condition in the process of flux correction. To verify the results of our theoretical studies, we perform numerical experiments for simple one-dimensional test problems. The methods under investigation exhibit the expected behavior in all numerical examples. In particular, the use of stabilized fluxes improves the accuracy of numerical solutions and coercivity enforcement often becomes redundant.

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Section: Numerical Examplesmentioning

confidence: 83%

“…Remark 3.4. Antidiffusive fluxes of AFC schemes based on discontinuous Galerkin (DG) methods (as proposed, e.g., in [1,8,14,29]) do not require stabilization of uA i via uD i because DG discretizations of the linear advection equation are inherently stable.…”

Section: Algebraic Flux Correction Toolsmentioning

confidence: 99%

Analysis of algebraic flux correction for semi-discrete advection problems

Hajduk,

Rupp,

Kuzmin

2021

Preprint

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“…In the next step, we would like to extend our investigation to entropy conservative/stable methods. Here, various approaches exist as described inter alia in [2,4,15,23,24,51], but from our perspective the convex limiting strategies seems the most promising one [30,37].…”

Section: Discussionmentioning

confidence: 99%

An Arbitrary High Order and Positivity Preserving Method for the Shallow Water Equations

Ciallella¹,

Lorenzo²,

Öffner³

et al. 2021

Preprint

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In this paper, we develop and present an arbitrary high order well-balanced finite volume WENO method combined with the modified Patankar Deferred Correction (mPDeC) time integration method for the shallow water equations. Due to the positivity-preserving property of mPDeC, the resulting scheme is unconditionally positivity preserving for the water height. To apply the mPDeC approach, we have to interpret the spatial semi-discretization in terms of production-destruction systems. Only small modifications inside the classical WENO implementation are necessary and we explain how it can be done. In numerical simulations, focusing on a fifth order method, we demonstrate the good performance of the new method and verify the theoretical properties.

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“…One particular application of bounds-constrained approximation comes in the numerical solution of partial differential equations, especially hyperbolic equations [30]. Some approaches to limiting in discontinuous Galerkin (DG) methods for hyperbolic PDE explicitly utilize the geometric properties of Bernstein polynomials to enforce maximum principles and other invariant properties [18,23]. These methods are monolithic, with a built-in limiting process.…”

Section: Introductionmentioning

confidence: 99%

Bounds-constrained polynomial approximation using the Bernstein basis

Allen¹,

Kirby²

2021

Preprint

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A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In this paper, we consider the problem of approximating functions by polynomials whose Bernstein coefficients with respect to a given degree satisfy such bounds, which implies such bounds on the approximant. We frame the problem as an inequality-constrained optimization problem and give an algorithm for finding the Bernstein coefficients of the exact solution. Additionally, our method can be modified slightly to include equality constraints such as mass preservation. It also extends naturally to multivariate polynomials over a simplex.

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Monolithic convex limiting in discontinuous Galerkin discretizations of hyperbolic conservation laws

Cited by 24 publications

References 44 publications

Analysis of algebraic flux correction for semi-discrete advection problems

Analysis of algebraic flux correction for semi-discrete advection problems

An Arbitrary High Order and Positivity Preserving Method for the Shallow Water Equations

Bounds-constrained polynomial approximation using the Bernstein basis

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