Adaptive hierarchic transformations for dynamically p-enriched slope-limiting over discontinuous Galerkin systems of generalized equations

“…Since conditions (2)- (3) are violated at the local maxima and minima of , all cells containing these extrema are marked as "troubled" [11]. To distinguish between smooth peaks and spurious undershoots/overshoots, the regularity estimator must be applied to each component of the gradient ∇ = ( ) T .…”

Section: Letmentioning

confidence: 99%

“…To prevent nonphysical oscillations in these regions, the numerical scheme must contain a mechanism for adaptively reducing the order of approximation and/or generating shock-capturing artificial diffusion. In discontinuous Galerkin (DG) methods, this task is commonly accomplished by means of slope limiters that switch to a piecewise-constant approximation in "troubled" cells [1,2,4,6,11,16]. Obviously, this approach is not an option in the context of continuous finite element schemes.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, it is generally safe to use the standard Galerkin method and higher-order basis functions in elements where the numerical solution exhibits sufficient regularity. For this reason, many strategies for identifying "smooth" and "troubled" cells have been proposed in the literature [2,4,[11][12][13]16]. A typical troubled cell marker for DG methods measures the interelement jumps or highest-order components of the shape function.…”

Section: Introductionmentioning

confidence: 99%

“…However, there is no simple way to find the threshold for a troubled cell marker. The use of fixed tolerances for the absolute or relative values of smoothness sensors may result in a poor estimate of local regularity [11]. The failure to detect a trouble cell may cause a violation of the maximum principle and give rise to numerical instabilities.…”

Section: Introductionmentioning

confidence: 99%

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A parameter-free smoothness indicator for high-resolution finite element schemes

Kuzmin

Schieweck

2013

Open Mathematics

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This paper presents a postprocessing technique for estimating the local regularity of numerical solutions in highresolution finite element schemes. A derivative of degree ≥ 0 is considered to be smooth if a discontinuous linear reconstruction does not create new maxima or minima. The intended use of this criterion is the identification of smooth cells in the context of -adaptation or selective flux limiting. As a model problem, we consider a 2D convection equation discretized with bilinear finite elements. The discrete maximum principle is enforced using a linearized flux-corrected transport algorithm. The deactivation of the flux limiter in regions of high regularity makes it possible to avoid the peak clipping effect at smooth extrema without generating spurious undershoots or overshoots elsewhere. MSC:65N30

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Slope limiting for discontinuous Galerkin approximations with a possibly non‐orthogonal Taylor basis

Kuzmin

2012

Numerical Methods in Fluids

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SUMMARYThe use of high‐order polynomials in discontinuous Galerkin (DG) approximations to convection‐dominated transport problems tends to cause a violation of the maximum principle in regions where the derivatives of the solution are large. In this paper, we express the DG solution in terms of Taylor basis functions associated with the cell average and derivatives at the center of the cell. To control the (derivatives of the) discontinuous solution, the values at the vertices of each element are required to be bounded by the means. This constraint is enforced using a hierarchical vertex‐based slope limiter to constrain the coefficients of the Taylor polynomial in a conservative manner starting with the highest‐order terms. The loss of accuracy at smooth extrema is avoided by taking the maximum of the correction factors for derivatives of order p and higher. No free parameters, oscillation detectors, or troubled cell markers are involved. In the case of a non‐orthogonal Taylor basis, the same limiter is applied to the vector of discretized time derivatives before the multiplication by the off‐diagonal part of the consistent mass matrix. This strategy leads to a remarkable gain of accuracy, especially in the case of simplex meshes. A numerical study is performed for a 2D convection equation discretized with linear and quadratic finite elements. Copyright © 2012 John Wiley & Sons, Ltd.

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Adaptive hierarchic transformations for dynamically p-enriched slope-limiting over discontinuous Galerkin systems of generalized equations

Cited by 28 publications

References 37 publications

A discontinuous Galerkin method for modeling flow in networks of channels

A discontinuous Galerkin method for modeling flow in networks of channels

A parameter-free smoothness indicator for high-resolution finite element schemes

Slope limiting for discontinuous Galerkin approximations with a possibly non‐orthogonal Taylor basis

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