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arxiv: 1907.04939 · v1 · pith:AQT665O3new · submitted 2019-07-10 · 🧮 math.NA · cs.NA· physics.comp-ph

Multi-element SIAC filter for shock capturing applied to high-order discontinuous Galerkin spectral element methods

Marvin Bohm , Sven Schermeng , Andrew R. Winters , Gregor J. Gassner , Gustaaf B. Jacobs This is my paper

Pith reviewed 2026-05-24 23:15 UTC · model grok-4.3

classification 🧮 math.NA cs.NAphysics.comp-ph
keywords SIAC filtershock capturingdiscontinuous Galerkin spectral element methodconservation lawsEuler equationsmagnetohydrodynamicshigh-order methods
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The pith

A multi-element SIAC filter applied to DGSEM captures shocks adaptively while preserving high-order accuracy for conservation laws.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a multi-element extension of the SIAC shock capturing filter originally for single-element spectral methods and applies it to the nodal discontinuous Galerkin spectral element method. This extension is designed so the filter can be used adaptively across the domain and across element interfaces without loss of the underlying high-order accuracy. The construction is carried out for general systems of conservation laws. A sympathetic reader would care because high-order schemes for nonlinear conservation laws produce spurious oscillations near discontinuities, and this approach supplies a targeted way to control them.

Core claim

The authors construct a multi-element variant of the SIAC filtering technique for the DGSEM on general systems of conservation laws. The filter is designed so that the numerical scheme remains high-order accurate and shock capturing can be applied adaptively throughout the domain. They demonstrate the approach on the two-dimensional Euler and ideal magnetohydrodynamics equations for standard test problems with a variety of boundary conditions.

What carries the argument

The multi-element SIAC filtering technique, an extension of single-element smoothness-increasing accuracy-conserving Dirac-delta polynomial kernels to multiple elements that preserves those properties when applied adaptively across interfaces.

If this is right

  • The numerical scheme remains high-order accurate with adaptive application of the filter.
  • The shock capturing method applies to general systems of conservation laws.
  • The approach is demonstrated on two-dimensional Euler and ideal MHD equations with various boundary conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The adaptive application could be combined with existing sensors to localize filtering effort.
  • The same multi-element construction might be tested on other high-order discretizations beyond DGSEM.

Load-bearing premise

The multi-element SIAC construction preserves the smoothness-increasing and accuracy-conserving properties of the original single-element SIAC while remaining high-order accurate when applied adaptively across element interfaces in general systems of conservation laws.

What would settle it

A numerical experiment on a smooth solution showing that the formal order of accuracy drops when the multi-element SIAC filter crosses element interfaces would falsify the central claim.

Figures

Figures reproduced from arXiv: 1907.04939 by Andrew R. Winters, Gregor J. Gassner, Gustaaf B. Jacobs, Marvin Bohm, Sven Schermeng.

Figure 1
Figure 1. Figure 1: Visualization of the Dirac-delta kernel for ε = 1. For compact notation we introduce the filter matrix Φ and approximate its values with LGL quadrature by mapping the corresponding integration area [x − ε, x + ε] into the reference ele￾ment E = [−1, 1] (3.6) Φij = xZi+ε xi−ε ψj (τ )δ m,k ε (xi − τ ) dτ = ε Z 1 −1 ψj (εx + xi)δ m,k ε (εx) dx ≈ ε X N∗ ν=0 ωνψj (εxν + xi)δ m,k ε (εxν), where {xν} N∗ ν=0 and {… view at source ↗
Figure 2
Figure 2. Figure 2: Density of the 2D explosion problem at T = 0.25 for N = 7, NQ = 802 , CFL = 0.1 filtered adaptively with (m, k) = (1, 6), Nd = 0.6, σmin = −7 and σmax = −3. For the simulation result illustrated in [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Density of the 2D explosion problem at T = 0.25 for N = 7, NQ = 802 , CFL = 0.1 filtered adaptively with (m, k) = (3, 6), Nd = 2.5, σmin = −8 and σmax = −5. In contrast, [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Density slices of the 2D explosion problem at x = y, T = 0.25 for N = 7, NQ = 802 , CFL = 0.1. The four primitive variable states of the initial conditions are then assigned to these four quad￾rants: (%, v1, v2, p)(x, y, 0) =    (%t`, v1,t`, v2,t`, pt`), if (x, y) ∈ Ωt`, (%b`, v1,b`, v2,b`, pb`), if (x, y) ∈ Ωb`, (%tr, v1,tr, v2,tr, ptr), if (x, y) ∈ Ωtr, (%br, v1,br, v2,br, pbr), if (x, y) ∈ Ωbr.… view at source ↗
Figure 5
Figure 5. Figure 5: Density (left) and pressure (right) of the four state Riemann test configuration 17 at T = 0.3 for N = 7, NQ = 602 , CFL = 0.1 filtered adaptively with (m, k) = (5, 7), Nd = 4.5, σmin = −8 and σmax = −3. Configuration 19: The initial conditions for the primitive variables of configuration 19 are defined as %t` = 2, v1,t` = 0, v2,t` = −0.3, pt` = 1 %b` = 1.0625, v1,b` = 0, v2,b` = 0.2145, pb` = 0.4 %tr = 1,… view at source ↗
Figure 6
Figure 6. Figure 6: DGSEM approximation of density %, left, and pressure p, right, adaptively filtered after every time step at t = 0.3 for (m, k) = (5, 7), Nd = 4.5, σmin = 10−8 and σmax = 10−3 . separates a left and right state defined by the primitive variables, i.e. %L(x, y, 0) = 8, %R(x, y, 0) = 1.4, v1,L(x, y, 0) = 8.25 · π 6 , v1,R(x, y, 0) = 0, v2,L(x, y, 0) = −8.25 · π 6 , v2,R(x, y, 0) = 0, pL(x, y, 0) = 116.5, pR(x… view at source ↗
Figure 7
Figure 7. Figure 7: Density of the double Mach reflection at T = 0.2 for N = 7, CFL = 0.1 on 325 × 100 elements filtered adaptively with (m, k) = (3, 6), Nd = 2.5, σmin = −7 and σmax = −2. 4.2. Ideal MHD tests. Next, we consider the ideal magneto-hydrodynamic (MHD) equations, which can be expressed in a compact form as a system of conservation laws (4.8) Ut + Fx + Gy = 0 with (4.9) U =   % %~v %e B~   , F =  … view at source ↗
Figure 8
Figure 8. Figure 8: Time evolution of the Orszag-Tang vortex density on 40×40 elements with N = 5 and CFL = 0.5 filtered adaptively with m = 3, k = 8, ε = 1.4 and σmin = σmax = −8. In order to assess the performance of the two-dimensional Dirac-delta filter, we compare the simulation results to a reference solution obtained by the publicly available high performance application code FLASH (http://flash.uchicago.edu/site/flash… view at source ↗
Figure 9
Figure 9. Figure 9: Time evolution of the convex parameter λ on 40 × 40 elements with N = 5 and CFL = 0.5 filtered adaptively with m = 3, k = 8, ε = 1.4 and σmin = σmax = −8. T = 0.5, in which we cut through the density distribution at x = y (left) and y = 0.3 (right) to compare the profiles obtained by the multi-element SIAC filter against a reference solution. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 r 0.05 0.10 0.15 0.20 0.25 0.30 … view at source ↗
Figure 10
Figure 10. Figure 10: Orszag-Tang-Vortex density slices at x = y (left) and y = 0.3 (right) for T = 0.5, CFL = 0.5, N = 5 and 40 × 40 elements. We see that the oscillations are smoothed out by the multi-element SIAC filter and the approx￾imation matches the reference solution quite well, but the filtering technique still produces little overshoots at shocks. However, taking into account the coarse resolution of 40 ×40 elements… view at source ↗
Figure 11
Figure 11. Figure 11: Due to the strong circular shocks combined with the high-order DG approximation [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
read the original abstract

We build a multi-element variant of the smoothness increasing accuracy conserving (SIAC) shock capturing technique proposed for single element spectral methods by Wissink et al. (B.W. Wissink, G.B. Jacobs, J.K. Ryan, W.S. Don, and E.T.A. van der Weide. Shock regularization with smoothness-increasing accuracy-conserving Dirac-delta polynomial kernels. Journal of Scientific Computing, 77:579--596, 2018). In particular, the baseline scheme of our method is the nodal discontinuous Galerkin spectral element method (DGSEM) for approximating the solution of systems of conservation laws. It is well known that high-order methods generate spurious oscillations near discontinuities which can develop in the solution for nonlinear problems, even when the initial data is smooth. We propose a novel multi-element SIAC filtering technique applied to the DGSEM as a shock capturing method. We design the SIAC filtering such that the numerical scheme remains high-order accurate and that the shock capturing is applied adaptively throughout the domain. The shock capturing method is derived for general systems of conservation laws. We apply the novel SIAC filter to the two-dimensional Euler and ideal magnetohydrodynamics (MHD) equations to several standard test problems with a variety of boundary conditions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper extends the single-element SIAC filter of Wissink et al. (2018) to a multi-element variant for shock capturing within the nodal DGSEM discretization of systems of conservation laws. The central claim is that the resulting filter can be applied adaptively across element interfaces while preserving high-order accuracy and the smoothness-increasing/accuracy-conserving properties of the original kernel; the method is derived for general systems and demonstrated on standard 2-D Euler and ideal MHD test problems with various boundary conditions.

Significance. If the multi-element construction is shown to satisfy the requisite moment conditions at interfaces without order reduction for nonlinear systems, the technique would supply a practical, adaptive shock-capturing tool compatible with existing high-order DGSEM codes. The explicit treatment of general conservation laws and the MHD examples constitute concrete strengths.

major comments (2)
  1. [§3] §3 (multi-element kernel construction): the manuscript must explicitly verify that the moment conditions of the SIAC kernel remain satisfied when the filter support straddles DGSEM element interfaces. Because the underlying polynomial basis and quadrature are element-local, the extension from the single-element case is not automatic; without this verification the high-order claim for general nonlinear systems rests on an unproven step.
  2. [§4] §4 (numerical results, Euler/MHD tests): the reported convergence studies and adaptive-filter runs do not include a direct check that the observed order matches the design order in smooth sub-regions once the multi-element filter has been activated near discontinuities. Such a check is load-bearing for the claim that accuracy is conserved under adaptive application.
minor comments (2)
  1. Notation for the multi-element kernel support and the characteristic decomposition should be introduced with a single consistent symbol set rather than re-defined in each subsection.
  2. Figure captions for the MHD examples should state the polynomial degree, filter width, and sensor threshold used in each run.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. We address each major comment below and will revise the paper to strengthen the presentation of the multi-element SIAC construction and its accuracy properties.

read point-by-point responses

  1. Referee: [§3] §3 (multi-element kernel construction): the manuscript must explicitly verify that the moment conditions of the SIAC kernel remain satisfied when the filter support straddles DGSEM element interfaces. Because the underlying polynomial basis and quadrature are element-local, the extension from the single-element case is not automatic; without this verification the high-order claim for general nonlinear systems rests on an unproven step.

    Authors: We agree that an explicit verification of the moment conditions for kernels whose support crosses element interfaces is required to rigorously justify the high-order claim. While §3 constructs the multi-element kernel by extending the single-element SIAC approach to satisfy the requisite moments by design (matching the polynomial reproduction properties across the support), the manuscript does not include a dedicated calculation or table confirming the conditions hold when the support straddles DGSEM interfaces for general nonlinear systems. In the revised version we will add a short subsection (or appendix) that explicitly verifies the moment conditions for the straddling case, including the relevant quadrature and basis considerations. revision: yes

  2. Referee: [§4] §4 (numerical results, Euler/MHD tests): the reported convergence studies and adaptive-filter runs do not include a direct check that the observed order matches the design order in smooth sub-regions once the multi-element filter has been activated near discontinuities. Such a check is load-bearing for the claim that accuracy is conserved under adaptive application.

    Authors: We acknowledge that isolating the convergence rate in smooth sub-regions after the adaptive filter has been activated near discontinuities would provide stronger evidence for accuracy conservation. The current numerical section demonstrates overall performance on the Euler and MHD tests and shows that the filter does not degrade global accuracy, but does not report a dedicated local-order study in smooth zones post-activation. In the revision we will add such a check (e.g., L2-error tables restricted to smooth subdomains) for at least one test case to confirm that the design order is retained away from discontinuities. revision: yes

Circularity Check

0 steps flagged

No circularity: extension constructed independently of inputs

full rationale

The paper explicitly frames the multi-element SIAC as a designed extension of the single-element kernel from the 2018 citation, with the high-order accuracy and adaptive application stated as properties achieved by the new construction for DGSEM on general conservation laws. No equation or claim reduces a 'prediction' to a fitted parameter or prior result by definition; the abstract and described method present the preservation of smoothness-increasing/accuracy-conserving properties as a design goal rather than an automatic consequence of self-referential inputs. The overlapping author on the cited baseline does not make the load-bearing step circular because the present work supplies the multi-element adaptation and tests rather than invoking a uniqueness theorem or ansatz from self-work. This is the common case of an honest non-finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are identifiable. The approach presumably inherits standard DGSEM assumptions (e.g., polynomial basis, quadrature rules) and SIAC kernel properties from the cited prior work.

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Reference graph

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