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arxiv: 2605.00619 · v1 · submitted 2026-05-01 · 🧮 math.NA · cs.NA

Continuous 3D Finite Element Subgrid Basis Functions for Discontinuous Galerkin Methods on Polyhedral Meshes

Sixtine Michel , Lorenzo Diazzi , Walter Boscheri This is my paper

Pith reviewed 2026-05-09 18:36 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords discontinuous Galerkin methodspolyhedral meshesagglomerated finite elementsquadrature-freeADER predictorartificial viscosityhyperbolic PDEsNavier-Stokes equations
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The pith

Agglomerated basis functions on tetrahedral subgrids inside polyhedra enable quadrature-free high-order DG methods on arbitrary unstructured 3D meshes.

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

The paper develops a high-order nodal discontinuous Galerkin method for nonlinear hyperbolic PDEs on fully unstructured three-dimensional polyhedral meshes. It generates an admissible tetrahedral subgrid within each polyhedral element and constructs the discrete solution using piecewise continuous polynomials of degree N defined on that subgrid. By agglomerating standard finite element basis functions from the sub-tetrahedra, the method allows precomputation of local mass and stiffness matrices on the reference unit tetrahedron. This results in a quadrature-free scheme that remains efficient on highly irregular polyhedra, combined with an ADER spacetime predictor for time accuracy and an artificial viscosity limiter for robustness at discontinuities.

Core claim

The discrete solution within each polyhedral element is represented using piecewise continuous polynomials of degree N on an internal tetrahedral subgrid, with AFE basis functions obtained by agglomerating standard finite element basis functions on each sub-tetrahedron, enabling precomputation of universal local matrices on the unit tetrahedron for a quadrature-free implementation.

What carries the argument

The agglomerated finite element (AFE) basis functions, built by combining standard finite element basis functions across the sub-tetrahedra of the internal grid within each polyhedral cell.

If this is right

  • The scheme achieves high-order accuracy in space and time independently within each polyhedral element using the ADER approach.
  • Local matrices can be precomputed on the reference tetrahedron, making the method efficient regardless of mesh irregularity.
  • An artificial viscosity limiter provides stabilization at shocks while preserving accuracy in smooth regions.
  • The approach is validated on three-dimensional benchmarks for the compressible Euler and Navier-Stokes equations.

Where Pith is reading between the lines

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

  • This subgrid construction could allow DG methods to handle complex geometries more flexibly without needing specialized polyhedral quadrature rules.
  • The quadrature-free property might extend to other types of mesh elements or higher dimensions if similar subgrids can be defined.
  • Potential for integration with adaptive mesh refinement strategies on polyhedral grids to focus resolution where needed.

Load-bearing premise

That an admissible tetrahedral subgrid can always be generated inside every polyhedral element in such a way that the resulting agglomerated basis functions maintain stability and the required approximation properties for high-order convergence.

What would settle it

A computation on a polyhedral mesh containing highly irregular elements where the numerical solution fails to converge at the expected high-order rate or exhibits instability without the limiter.

Figures

Figures reproduced from arXiv: 2605.00619 by Lorenzo Diazzi, Sixtine Michel, Walter Boscheri.

Figure 1
Figure 1. Figure 1: Two examples of LPR. Left: two-dimensional case showing how the local polyhedral view at source ↗
Figure 2
Figure 2. Figure 2: This example shows a concave central polyhedron. On the left only the front polyhedral view at source ↗
Figure 3
Figure 3. Figure 3: Tetrahedrization of a truncated tetrahedron view at source ↗
Figure 4
Figure 4. Figure 4: This two-dimensional example depicts a comprehensive picture of how grid vertices (blue) view at source ↗
Figure 5
Figure 5. Figure 5: The picture shows the two domains used to collect the data reported in Table 1. The view at source ↗
Figure 6
Figure 6. Figure 6: Representation of a two-dimensional AFE-DG discretization, with second (left) and third view at source ↗
Figure 7
Figure 7. Figure 7: Example of a polyhedral element, divided into a set of tetrahedral elements with second view at source ↗
Figure 8
Figure 8. Figure 8: Left: Density and velocity field of the stationary vortex at view at source ↗
Figure 9
Figure 9. Figure 9: Left: Density and velocity field of the travelling vortex at view at source ↗
Figure 10
Figure 10. Figure 10: Comparison between exact and numerical solution for the first problem of Stokes at view at source ↗
Figure 11
Figure 11. Figure 11: Comparison between exact and numerical solution for the first problem of Stokes at view at source ↗
Figure 12
Figure 12. Figure 12: Left: zoom on the AFE mesh, composed of tetrahedral elements, resulting from the view at source ↗
Figure 13
Figure 13. Figure 13: Two-dimensional contour plots of density, velocity magnitude, and pressure in the plane view at source ↗
Figure 14
Figure 14. Figure 14: One-dimensional cuts along the radial direction ( view at source ↗
Figure 15
Figure 15. Figure 15: Two-dimensional Taylor-Green vortex problem at view at source ↗
Figure 16
Figure 16. Figure 16: Elementary tetrahedral element: with a second (top left), third (top right), fourth view at source ↗
read the original abstract

We present a novel high-order accurate nodal discontinuous Galerkin (DG) method for solving nonlinear hyperbolic systems of partial differential equations (PDEs) on fully unstructured three-dimensional polyhedral meshes. A mesh generator is firstly discussed in detail, which ensures the generation of admissible control volumes. For the first time, we then extend the concept of agglomerated finite element (AFE) basis functions to polyhedral grids. In this context, the discrete solution is represented within each polyhedral element using piecewise continuous polynomials of degree N, defined on an internal tetrahedral subgrid. The AFE basis functions are therefore constructed by agglomerating standard finite element basis functions on each sub-tetrahedron of the computational cell. This allows for the precomputation of universal local matrices (mass and stiffness) on the reference element given by the unit tetrahedron, enabling a quadrature-free implementation that remains efficient even on highly irregular polyhedral meshes. High-order of accuracy in time is achieved using a local spacetime Galerkin predictor as part of the ADER approach, applied independently within each polyhedral element. To ensure robustness in the presence of discontinuities such as shocks, an artificial viscosity limiter is embedded into the numerical scheme, allowing for controlled dissipation and stabilization without compromising the overall accuracy in smooth regions. To demonstrate the robustness and accuracy of the method, we validate it through different three-dimensional benchmark problems for the compressible Euler and Navier-Stokes equations.

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 / 1 minor

Summary. The paper presents a novel high-order nodal discontinuous Galerkin (DG) method for nonlinear hyperbolic systems on fully unstructured 3D polyhedral meshes. It first details a mesh generator producing admissible control volumes, then extends agglomerated finite element (AFE) basis functions to polyhedra by representing the solution as piecewise continuous degree-N polynomials on an internal tetrahedral subgrid within each polyhedron. Standard FE bases on the sub-tetrahedra are agglomerated to enable precomputed universal mass and stiffness matrices on the reference unit tetrahedron, supporting a quadrature-free implementation. High-order time accuracy uses an ADER local spacetime Galerkin predictor per element, with an artificial viscosity limiter for shocks. Validation is performed on 3D compressible Euler and Navier-Stokes benchmarks.

Significance. If the subgrid construction reliably preserves polynomial reproduction, stability, and conditioning, the method offers a practical route to efficient high-order DG on complex polyhedral meshes without quadrature costs, which would be valuable for unstructured-grid CFD. The quadrature-free property via reference-tetrahedron precomputation and the ADER predictor are clear efficiency strengths when the underlying assumptions hold.

major comments (2)
  1. [Mesh generator and AFE basis construction] The description of the mesh generator and AFE basis construction (abstract and associated sections): the central claim that an admissible tetrahedral subgrid can always be generated such that the agglomerated space contains the full P_N polynomials, the summed matrices remain well-conditioned, and the reference mapping preserves formal order is load-bearing for both the high-order accuracy and quadrature-free efficiency assertions, yet no explicit algorithm, admissibility criteria, or proof is supplied that these properties hold uniformly for non-convex, high-face-count, or highly distorted polyhedra.
  2. [Numerical results / validation] Validation section: the abstract states that the method is validated on 3D Euler and Navier-Stokes benchmarks, but the absence of quantitative error tables, observed convergence rates versus polynomial degree N, or comparisons against standard DG on tetrahedral meshes leaves the actual accuracy and efficiency claims unverified, particularly on irregular polyhedra.
minor comments (1)
  1. [Abstract] The abstract could usefully indicate the range of polynomial degrees N employed in the benchmark computations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's thorough review and valuable feedback on our manuscript. We address the major comments point by point below, providing clarifications and indicating revisions where the manuscript will be updated to strengthen the presentation.

read point-by-point responses

  1. Referee: [Mesh generator and AFE basis construction] The description of the mesh generator and AFE basis construction (abstract and associated sections): the central claim that an admissible tetrahedral subgrid can always be generated such that the agglomerated space contains the full P_N polynomials, the summed matrices remain well-conditioned, and the reference mapping preserves formal order is load-bearing for both the high-order accuracy and quadrature-free efficiency assertions, yet no explicit algorithm, admissibility criteria, or proof is supplied that these properties hold uniformly for non-convex, high-face-count, or highly distorted polyhedra.

    Authors: We thank the referee for highlighting this important aspect. While the manuscript discusses the mesh generator in detail, we agree that an explicit algorithm, admissibility criteria, and a brief justification for the polynomial reproduction and conditioning are not fully detailed. In the revised version, we will include a pseudocode description of the subgrid generation procedure, specify the admissibility criteria (e.g., requirements on sub-tetrahedron quality and polyhedron convexity handling), and add a remark explaining that the full P_N space is reproduced by construction as the agglomerated functions include all nodal basis functions from the sub-tetrahedra. We will also discuss the conditioning of the summed matrices and the preservation of formal order under the reference mapping. These revisions will make the claims more transparent. revision: yes

  2. Referee: [Numerical results / validation] Validation section: the abstract states that the method is validated on 3D Euler and Navier-Stokes benchmarks, but the absence of quantitative error tables, observed convergence rates versus polynomial degree N, or comparisons against standard DG on tetrahedral meshes leaves the actual accuracy and efficiency claims unverified, particularly on irregular polyhedra.

    Authors: We agree with the referee that quantitative error tables, convergence rates, and comparisons would provide stronger verification of the accuracy and efficiency claims. Although the validation section presents results on 3D Euler and Navier-Stokes benchmarks to demonstrate robustness and accuracy, we will enhance it in the revised manuscript by adding tables of L2 errors and observed convergence orders for different polynomial degrees N on polyhedral meshes. We will also include a comparison with standard DG on tetrahedral meshes for at least one test case to quantify the efficiency gains from the quadrature-free approach. These additions will be incorporated into the numerical results section. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation builds on explicit mesh generator and standard FE agglomeration without reducing claims to inputs by construction.

full rationale

The paper first describes a mesh generator that produces admissible tetrahedral subgrids inside each polyhedron, then defines the discrete solution as piecewise continuous degree-N polynomials on that subgrid and agglomerates standard FE basis functions from the sub-tetrahedra. Universal mass and stiffness matrices are precomputed once on the reference unit tetrahedron, and the ADER predictor plus artificial viscosity limiter are applied element-wise. None of these steps equates a derived quantity (e.g., the agglomerated basis or the quadrature-free property) to its own input by definition, nor does the text invoke a self-citation chain or uniqueness theorem whose validity depends on the present work. The construction therefore remains self-contained against external benchmarks of DG and FE theory.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The method rests on standard finite-element approximation theory and the existence of admissible tetrahedral subgrids inside arbitrary polyhedra; no new physical entities are introduced.

free parameters (1)
  • Polynomial degree N
    User-chosen integer that sets the formal order of accuracy; not fitted to data.
axioms (2)
  • domain assumption Piecewise continuous polynomials of degree N on an internal tetrahedral subgrid can be agglomerated to form a valid basis for the polyhedral element that preserves DG stability and accuracy.
    Invoked when the AFE basis functions are defined and when quadrature-free matrices are asserted to remain accurate.
  • domain assumption The mesh generator produces admissible control volumes for which the subgrid construction is always possible.
    Stated at the beginning of the abstract as a prerequisite.

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Reference graph

Works this paper leans on

61 extracted references · 61 canonical work pages

  1. [1]

    Antonietti, M

    P.F. Antonietti, M. Caldana, I. Mazzieri, and A. Fraschini. Magnet: an open-source library for mesh agglomeration by graph neural networks.Engineering with Computers, 41:4825–4850, 2025

    work page 2025

  2. [2]

    Indirect predicates for geometric constructions.Computer-Aided Design, 126:102856, 2020

    Marco Attene. Indirect predicates for geometric constructions.Computer-Aided Design, 126:102856, 2020

    work page 2020

  3. [3]

    Dinshaw S. Balsara. Self-adjusting, positivity preserving high order schemes for hydro- dynamics and magnetohydrodynamics.Journal of Computational Physics, 231(22):7504– 7517, 2012

    work page 2012

  4. [4]

    Bassi, F

    F. Bassi, F. Massa, L. Botti, and A. Colombo. Artificial compressibility godunov fluxes for variable density incompressible flows.Computers & Fluids, 169:186–200, 2018. Recent progress in nonlinear numerical methods for time-dependent flow & transport problems. 29

    work page 2018

  5. [5]

    Beir˜ ao da Veiga, F

    L. Beir˜ ao da Veiga, F. Brezzi, A. Cangiani, L.D. Marini, G. Manzini, and A. Russo. The basic principles of virtual elements methods.Math. Models Methods Appl. Sci., 23(1):199– 214, 2013

    work page 2013

  6. [6]

    Beir˜ ao da Veiga, F

    L. Beir˜ ao da Veiga, F. Brezzi, and L.D. Marini. Virtual elements for linear elasticity problems.SIAM Journal on Numerical Analysis, 51(2):794–812, 2013

    work page 2013

  7. [7]

    Beir˜ ao da Veiga, F

    L. Beir˜ ao da Veiga, F. Dassi, and A. Russo. High-order virtual element method on poly- hedral meshes.Computer Methods in Applied Mechanics and Engineering, 74:1110–1122., 2017

    work page 2017

  8. [8]

    Beir˜ ao da Veiga and G

    L. Beir˜ ao da Veiga and G. Manzini. A virtual element method with arbitrary regularity. IMA Journal of Numerical Analysis, 34(2):759–781, 2013

    work page 2013

  9. [9]

    Walter Boscheri and Dinshaw S. Balsara. High order direct arbitrary-lagrangian-eulerian (ale) pnpm schemes with weno adaptive-order reconstruction on unstructured meshes. Journal of Computational Physics, 398:108899, 2019

    work page 2019

  10. [10]

    Walter Boscheri and Michael Dumbser. Arbitrary-lagrangian–eulerian discontinuous galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes.Journal of Computational Physics, 346, 12 2016

    work page 2016

  11. [11]

    Walter Boscheri, Michael Dumbser, and Elena Gaburro. Continuous finite element sub- grid basis functions for discontinuous galerkin schemes on unstructured polygonal voronoi meshes.Communications in Computational Physics, 32:259–298, 06 2022

    work page 2022

  12. [12]

    Central weno subcell finite volume limiters for ader discontinuous galerkin schemes on fixed and moving unstructured meshes.Communications in Computational Physics, 25, 01 2019

    Walter Boscheri, Matteo Semplice, and Michael Dumbser. Central weno subcell finite volume limiters for ader discontinuous galerkin schemes on fixed and moving unstructured meshes.Communications in Computational Physics, 25, 01 2019

    work page 2019

  13. [13]

    High order semi-implicit schemes for viscous com- pressible flows in 3d.Applied Mathematics and Computation, 434:127457, 2022

    Walter Boscheri and Maurizio Tavelli. High order semi-implicit schemes for viscous com- pressible flows in 3d.Applied Mathematics and Computation, 434:127457, 2022

    work page 2022

  14. [14]

    Busto, E.F

    S. Busto, E.F. Toro, and M.E. V´ azquez-Cend´ on. Design and analysis of ader-type schemes for model advection–diffusion–reaction equations.Journal of Computational Physics, 327:553–575, 2016

    work page 2016

  15. [15]

    High order ader schemes for continuum mechanics.Frontiers in Physics, 8, 12 2019

    Saray Busto Ulloa, Simone Chiocchetti, Michael Dumbser, Elena Gaburro, and Ilya Peshkov. High order ader schemes for continuum mechanics.Frontiers in Physics, 8, 12 2019

    work page 2019

  16. [16]

    Karniadakis, and Chi-Wang Shu.Discontinuous Galerkin Methods: Theory, Computation and Applications, volume 11 ofLecture Notes in Compu- tational Science and Engineering

    Bernardo Cockburn, George E. Karniadakis, and Chi-Wang Shu.Discontinuous Galerkin Methods: Theory, Computation and Applications, volume 11 ofLecture Notes in Compu- tational Science and Engineering. Springer, Berlin, Heidelberg, 2000

    work page 2000

  17. [17]

    Runge–kutta discontinuous galerkin methods for convection-dominated problems.Journal of Scientific Computing, 16(3):173–261, 2001

    Bernardo Cockburn and Chi-Wang Shu. Runge–kutta discontinuous galerkin methods for convection-dominated problems.Journal of Scientific Computing, 16(3):173–261, 2001

    work page 2001

  18. [18]

    Generation of polyhedral delaunay meshes

    David Contreras and Nancy Hitschfeld-Kahler. Generation of polyhedral delaunay meshes. Procedia Engineering, 82:291–300, 2014. 23rd International Meshing Roundtable (IMR23)

    work page 2014

  19. [19]

    Convex polyhedral meshing for robust solid modeling

    Lorenzo Diazzi and Marco Attene. Convex polyhedral meshing for robust solid modeling. ACM Trans. Graph., 40(6), December 2021

    work page 2021

  20. [20]

    Constrained delaunay tetrahedrization: A robust and practical approach.ACM Trans

    Lorenzo Diazzi, Daniele Panozzo, Amir Vaxman, and Marco Attene. Constrained delaunay tetrahedrization: A robust and practical approach.ACM Trans. Graph., 42(6), December 2023

    work page 2023

  21. [21]

    Arbitrary high order pnpm schemes on unstructured meshes for the compressible navier–stokes equations.Computers & Fluids, 39(1):60–76, 2010

    Michael Dumbser. Arbitrary high order pnpm schemes on unstructured meshes for the compressible navier–stokes equations.Computers & Fluids, 39(1):60–76, 2010. 30

    work page 2010

  22. [22]

    Balsara, Eleuterio F

    Michael Dumbser, Dinshaw S. Balsara, Eleuterio F. Toro, and Claus-Dieter Munz. A unified framework for the construction of one-step finite volume and discontinuous galerkin schemes on unstructured meshes.Journal of Computational Physics, 227(18):8209–8253, 2008

    work page 2008

  23. [23]

    A posteriori subcell limiting of the discontinuous galerkin finite element method for hyperbolic conservation laws.Journal of Computational Physics, 278:47–75, 2014

    Michael Dumbser, Olindo Zanotti, Rapha¨ el Loub` ere, and Steven Diot. A posteriori subcell limiting of the discontinuous galerkin finite element method for hyperbolic conservation laws.Journal of Computational Physics, 278:47–75, 2014

    work page 2014

  24. [24]

    Florian Frank, Andreas Rupp, and Dmitri Kuzmin. Bound-preserving flux limiting schemes for dg discretizations of conservation laws with applications to the cahn–hilliard equation.Computer Methods in Applied Mechanics and Engineering, 359:112665, 2020

    work page 2020

  25. [25]

    Elena Gaburro, Walter Boscheri, Simone Chiocchetti, and Mario Ricchiuto. Discontinuous galerkin schemes for hyperbolic systems in non-conservative variables: Quasi-conservative formulation with subcell finite volume corrections.Computer Methods in Applied Mechan- ics and Engineering, 431:117311, 2024

    work page 2024

  26. [26]

    Garimella, Jibum Kim, and Markus Berndt

    Rao V. Garimella, Jibum Kim, and Markus Berndt. Polyhedral mesh generation and optimization for non-manifold domains. In Josep Sarrate and Matthew Staten, editors, Proceedings of the 22nd International Meshing Roundtable, pages 313–330, Cham, 2014. Springer International Publishing

    work page 2014

  27. [27]

    A contribution to the construc- tion of diffusion fluxes for finite volume and discontinuous galerkin schemes.Journal of Computational Physics, 224:1049–1063, 06 2007

    Gregor Gassner, Frieder L¨ orcher, and Claus-Dieter Munz. A contribution to the construc- tion of diffusion fluxes for finite volume and discontinuous galerkin schemes.Journal of Computational Physics, 224:1049–1063, 06 2007

    work page 2007

  28. [28]

    A space–time adaptive discontinuous galerkin scheme.Computers & Fluids, 117:247–261, 2015

    Gregor Gassner, Marc Staudenmaier, Florian Hindenlang, Muhammed Atak, and Claus- Dieter Munz. A space–time adaptive discontinuous galerkin scheme.Computers & Fluids, 117:247–261, 2015

    work page 2015

  29. [29]

    Sigal Gottlieb and Steven J. Ruuth. Optimal strong-stability-preserving time-stepping schemes with fast downwind spatial discretizations.Journal of Scientific Computing, 27:289–303, 2006

    work page 2006

  30. [30]

    Hu and C.W

    C. Hu and C.W. Shu. Weighted essentially non-oscillatory schemes on triangular meshes. Journal of Computational Physics, 150:97–127, 1999

    work page 1999

  31. [31]

    On the eigenvalues of the ader-weno galerkin predictor.Journal of Com- putational Physics, 333:409–413, 2017

    Haran Jackson. On the eigenvalues of the ader-weno galerkin predictor.Journal of Com- putational Physics, 333:409–413, 2017

    work page 2017

  32. [32]

    Hesthaven

    Andreas Kl¨ ockner, Timothy Warburton, and Jan S. Hesthaven. Viscous shock captur- ing in a time-explicit discontinuous galerkin method.Mathematical Modelling of Natural Phenomena, 6(3):57–83, 2011

    work page 2011

  33. [33]

    D. Kuzmin. Entropy conservation property and stabilization procedures for high-order continuous galerkin methods. 2020

    work page 2020

  34. [34]

    A vertex-based hierarchical slope limiter for p-adaptive discontinuous galerkin methods.Journal of Computational and Applied Mathematics, 233(12):3077– 3085, 2010

    Dmitri Kuzmin. A vertex-based hierarchical slope limiter for p-adaptive discontinuous galerkin methods.Journal of Computational and Applied Mathematics, 233(12):3077– 3085, 2010. Finite Element Methods in Engineering and Science (FEMTEC 2009)

    work page 2010

  35. [35]

    Polyhedral mesh generation and a treatise on concave geometrical edges

    Sang Yong Lee. Polyhedral mesh generation and a treatise on concave geometrical edges. Procedia Engineering, 124:174–186, 2015. 24th International Meshing Roundtable

    work page 2015

  36. [36]

    A sub-element adaptive shock capturing approach for discontinuous galerkin methods.Communications on Applied Mathematics and Computation, 2021

    Johannes Markert, Gregor Gassner, and Stefanie Walch. A sub-element adaptive shock capturing approach for discontinuous galerkin methods.Communications on Applied Mathematics and Computation, 2021. 31

    work page 2021

  37. [37]

    Springer, Cham, 03 2020

    Francesco Massa, Francesco Bassi, Lorenzo Botti, and Alessandro Colombo.An Implicit High-Order Discontinuous Galerkin Approach for Variable Density Incompressible Flows, pages 191–202. Springer, Cham, 03 2020

    work page 2020

  38. [38]

    Springer, New York, 01 2009

    Nikos Mastorakis and John Sakellaris.Advances in Numerical Methods, volume 11. Springer, New York, 01 2009

    work page 2009

  39. [39]

    Sixtine Michel, Davide Torlo, Mario Ricchiuto, and R´ emi Abgrall. Spectral analysis of high order continuous fem for hyperbolic pdes on triangular meshes: Influence of approximation, stabilization, and time-stepping.Journal of Scientific Computing, 94, 01 2023

    work page 2023

  40. [40]

    Hybrid dg/fv schemes for magne- tohydrodynamics and relativistic hydrodynamics.Computer Physics Communications, 222:113–135, 10 2017

    Jonatan Nunez de la Rosa and Claus-Dieter Munz. Hybrid dg/fv schemes for magne- tohydrodynamics and relativistic hydrodynamics.Computer Physics Communications, 222:113–135, 10 2017

    work page 2017

  41. [41]

    Per-Olof Persson and J. Peraire. Sub-cell shock capturing for discontinuous galerkin meth- ods.AIAA paper, 2, 01 2006

    work page 2006

  42. [42]

    W. H. Reed and T. R. Hill. Triangular mesh methods for the neutron transport equation. Los Alamos Scientific Laboratory Report LA-UR-73-479, 1973

    work page 1973

  43. [43]

    Global optimization of explicit strong-stability-preserving runge–kutta methods.Math

    Steven Ruuth. Global optimization of explicit strong-stability-preserving runge–kutta methods.Math. Comput., 75:183–207, 01 2006

    work page 2006

  44. [44]

    Grenzschicht-theorie springer-verlag

    H Schlichting, K Gersten, E Krause, and H Oertel. Grenzschicht-theorie springer-verlag. Auflage, Berlin, Heidelberg, 1997

    work page 1997

  45. [45]

    Efficient implementation of essentially non-oscillatory shock-capturing schemes.Journal of Computational Physics, 77(2):439–471, 1988

    Chi-Wang Shu and Stanley Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes.Journal of Computational Physics, 77(2):439–471, 1988

    work page 1988

  46. [46]

    Shock capturing for discontinuous galerkin methods using finite volume subcells.Finite Volumes for Complex Applications VII, 78:945–953, 01 2014

    Matthias Sonntag and Claus-Dieter Munz. Shock capturing for discontinuous galerkin methods using finite volume subcells.Finite Volumes for Complex Applications VII, 78:945–953, 01 2014

    work page 2014

  47. [47]

    A new class of optimal high-order strong-stability- preserving time discretization methods.SIAM Journal on Numerical Analysis, 40:469–491, 07 2006

    Raymond Spiteri and Steven Ruuth. A new class of optimal high-order strong-stability- preserving time discretization methods.SIAM Journal on Numerical Analysis, 40:469–491, 07 2006

    work page 2006

  48. [48]

    George Gabriel Stokes. On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids.Transactions of the Cambridge Philosophical Society, 8:287–319, 1845

    work page

  49. [49]

    Mechanism of the production of small eddies from large ones.Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, 158:499–521, 1937

    Geoffrey Ingram Sir Taylor and Albert Edward Green. Mechanism of the production of small eddies from large ones.Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, 158:499–521, 1937

    work page 1937

  50. [50]

    Titarev and E.F

    V.A. Titarev and E.F. Toro. Ader schemes for three-dimensional non-linear hyperbolic systems.Journal of Computational Physics, 204(2):715–736, 2005

    work page 2005

  51. [51]

    Ader: Arbitrary high order godunov approach.J

    Vladimir Titarev and Eleuterio Toro. Ader: Arbitrary high order godunov approach.J. Sci. Comput., 17:609–618, 12 2002

    work page 2002

  52. [52]

    Toro and V.A

    E.F. Toro and V.A. Titarev. Derivative riemann solvers for systems of conservation laws and ader methods.Journal of Computational Physics, 212(1):150–165, 2006

    work page 2006

  53. [53]

    Ader schemes for scalar non-linear hyperbolic con- servation laws with source terms in three-space dimensions.Journal of Computational Physics, 202:196–215, 01 2005

    Eleuterio Toro and Vladimir Titarev. Ader schemes for scalar non-linear hyperbolic con- servation laws with source terms in three-space dimensions.Journal of Computational Physics, 202:196–215, 01 2005

    work page 2005

  54. [54]

    The ader approach for approximating hyperbolic equations to very high accuracy

    Eleuterio Toro, Vladimir Titarev, Michael Dumbser, Armin Iske, Claus Goetz, Crist´ obal Castro, Gino Montecinos, and Riccardo Dematt` e. The ader approach for approximating hyperbolic equations to very high accuracy. InHyperbolic Problems: Theory, Numerics, Applications, pages 83–105. Springer, Cham, 05 2024. 32

    work page 2024

  55. [55]

    Toro.Riemann Solvers and Numerical Methods for Fluid Dynamics: A Prac- tical Introduction

    Eleuterio F. Toro.Riemann Solvers and Numerical Methods for Fluid Dynamics: A Prac- tical Introduction. Springer, Berlin, Heidelberg, 3rd edition, 2009

    work page 2009

  56. [56]

    Local subcell monolithic dg/fv convex property preserving scheme on un- structured grids and entropy consideration.Journal of Computational Physics, 521:113535, 2025

    Fran¸ cois Vilar. Local subcell monolithic dg/fv convex property preserving scheme on un- structured grids and entropy consideration.Journal of Computational Physics, 521:113535, 2025

    work page 2025

  57. [57]

    Efficient computation of clipped voronoi diagram for mesh generation.Computer-Aided Design, 45(4):843–852,

    Dong-Ming Yan, Wenping Wang, Bruno L´ evy, and Yang Liu. Efficient computation of clipped voronoi diagram for mesh generation.Computer-Aided Design, 45(4):843–852,

    work page

  58. [58]

    Geometric Modeling and Processing 2010

    work page 2010

  59. [59]

    Space–time adaptive ader discontinuous galerkin finite element schemes with a posteriori sub-cell finite volume limiting.Computers & Fluids, 118:204–224, 2015

    Olindo Zanotti, Francesco Fambri, Michael Dumbser, and Arturo Hidalgo. Space–time adaptive ader discontinuous galerkin finite element schemes with a posteriori sub-cell finite volume limiting.Computers & Fluids, 118:204–224, 2015

    work page 2015

  60. [60]

    Zelkowitz, editor.Advances in Computers, volume 75

    Marvin V. Zelkowitz, editor.Advances in Computers, volume 75. Elsevier, Amsterdam, 2009

    work page 2009

  61. [61]

    High-order runge-kutta discontinuous galerkin methods with a new type of multi-resolution weno limiters.Journal of Computational Physics, 404:109105, 2020

    Jun Zhu, Jianxian Qiu, and Chi-Wang Shu. High-order runge-kutta discontinuous galerkin methods with a new type of multi-resolution weno limiters.Journal of Computational Physics, 404:109105, 2020. 33

    work page 2020