Pith. sign in

REVIEW 3 major objections 5 minor 43 references

High-order Discontinuous Galerkin solver based on Jacobi polynomial expansion for compressible flows on unstructured meshes

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper constructs an arbitrary-high-order discontinuous Galerkin solver for the compressible Euler equations from Jacobi-polynomial modal bases on mixed unstructured meshes, and verifies design-order convergence and shock-resolving…

desk verdict A solid, clearly written modal DG engineering paper that delivers expected convergence on straight-sided meshes but overclaims 'arbitrary order' and leaves row-sum mass lumping on non-affine quads unvalidated. read the letter →

arxiv 2411.15699 v1 pith:7MFPJOJW submitted 2024-11-24 physics.comp-ph physics.flu-dyn

Yu-Xiang Peng , Biao Wang , Peng-Nan Sun , A-Man Zhang This is my paper · ORCID
classification physics.comp-phphysics.flu-dyn MSC 65M6065M7076M1076N15 PACS 47.11.Fg47.40.-x
keywords DiscontinuousGalerkinmethodJacobipolynomialsmodalexpansioncompressibleEulerequationsunstructuredmeshHLLCRiemannsolvershockcapturinghigh-orderconvergence
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes a Discontinuous Galerkin solver for the compressible Euler equations whose spatial basis of arbitrary polynomial order is built from one-dimensional Jacobi polynomials, extended by tensor products to quadrilaterals and hexahedra and by collapsed-coordinate variants to triangles, tetrahedra, and wedges. The discretization is derived by requiring the residual of the Euler equations to be orthogonal to these basis functions, with HLLC numerical fluxes at element boundaries, Gauss-Jacobi quadrature for integration, and a shock-detection plus slope-limiting procedure for discontinuities. The authors report design convergence rates of roughly 2, 3, and 4 for polynomial orders 1, 2, and 3 on both triangle and quadrilateral meshes, and show that the high-order versions reach a given error level in less CPU time than lower-order versions on finer grids. They then demonstrate the solver on the forward step, double Mach reflection, transonic NACA0012 flow, Schardin's problem, and a Mach-4 blunt-cylinder flow, comparing favorably with reference computations and experiments. The point of the paper is that a single Jacobi-based modal framework can support arbitrary order and mixed element types without rebuilding the basis.

What carries the argument

The central object is the set of normalized Jacobi polynomials $P^{(0,0)}_n = \sqrt{(2n+1)/2}\,J^{(0,0)}_n$, with $\alpha=\beta=0$, used as an orthonormal modal basis on the reference element. Tensor products $\phi_{ijk}=P_iP_jP_k$ give the square and cube bases, while a collapsed-coordinate Jacobi construction for triangles and tetrahedra uses the factors $(1-\eta)^i$ and $(1-\zeta)^{i+j}$ to maintain orthogonality on simplices. This basis carries the argument because orthogonality makes the mass matrix diagonal on simplices, reduces the spatial discretization to the matrix-vector form of Eq. (30), and fixes the Gauss-Jacobi quadrature that evaluates the volume and face integrals. The HLLC flux supplies the inter-element coupling, and the trouble-cell detector plus slope limiter supplies the shock capturing.

What would settle it

Run the Section 4 smooth-flow convergence test on strongly stretched or curved quadrilateral and hexahedral meshes, comparing the row-sum lumped mass matrix with the consistent one. If the lumped solver's $L^1$ error converges at a lower order than the consistent solver's, or below $N+1$ for polynomial order $N$, the central accuracy claim is falsified.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that an arbitrary-high-order modal DG method for compressible flows can be constructed uniformly from normalized 1D Jacobi polynomials, with the Legendre case $\alpha=\beta=0$, and that this construction works across triangles, quadrilaterals, tetrahedra, hexahedra, and wedges, giving discrete Euler equations with a diagonal mass matrix on simplices and a row-sum lumped mass matrix on quadrilaterals and hexahedra. The numerical flux is the HLLC Riemann solver applied after rotating the velocity into the boundary-normal local frame, volume and face integrals use Gauss-Jacobi quadrature with simplex rules for triangles and tetrahedra, and time stepping uses strong-stability-preserving schemes with a CFL limit of $C \le 0.4/(2N+1)$. Discontinuous cells are flagged by a troubled-cell indicator and smoothed by a vertex-based slope limiter. With this machinery, the paper reports convergence orders close to $N+1$ for $N=1,2,3$ and benchmark results for shocks that match reference solutions and experiments.

Load-bearing premise

For quadrilateral and hexahedral cells, the solver replaces the mass matrix by its row-sum lumped version and assumes this preserves the claimed order of accuracy, even though no error analysis, no comparison with the consistent mass matrix, and no study of curved or stretched cells is given.

Editorial extensions

If this is right

  • The same Jacobi-based code can switch spatial order by changing one integer $N$; no new basis functions need to be derived.
  • On smooth flows, orders $N=2$ and $N=3$ reach a given error level on coarser meshes and in less CPU time than $N=1$ on fine meshes, so high order is not inherently more expensive.
  • Mixed-element meshes, including hexahedra combined with tetrahedra and wedges, can be used in one solver, allowing geometry-flexible meshing without sacrificing high order.
  • The shock-detection plus local limiting confines limiting to genuinely discontinuous cells, so smooth regions keep the high-order accuracy.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension the paper does not explore is choosing nonzero Jacobi parameters $\alpha,\beta$ to tune resolution toward boundaries or interfaces.
  • The untested interaction of row-sum mass lumping with curved boundary elements is the most direct place to probe whether design accuracy persists at very high order.
  • The same basis and quadrature machinery should carry over to the Navier-Stokes equations, since only the viscous fluxes would need adding; the paper itself treats only the inviscid Euler system.
  • The troubled-cell threshold $C_k = 0.015 \times 2^{k-1}$ is used across all examples; whether it needs retuning for higher Mach numbers or higher orders is an open question beyond the paper's tests.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript presents a discontinuous Galerkin solver for the compressible Euler equations using orthonormal Jacobi-polynomial bases (modal expansion) on mixed unstructured meshes. The basis is constructed via tensor products for quadrilaterals/hexahedra and via Proriol-Koornwinder-Dubiner polynomials for triangles/tetrahedra/wedges. The discretization follows the standard DG weak form with HLLC fluxes, SSP-RK time stepping, and a shock-capturing scheme combining a troubled-cell indicator and a vertex-based slope limiter. Convergence rates for polynomial orders N=1,2,3 are verified on Cartesian quadrilateral and triangular meshes, and several 2D/3D benchmark flows are compared with literature and experimental data.

Significance. If the method performs as claimed, the paper provides a reusable modal DG framework that can in principle be extended to arbitrary order on mixed elements. The strengths include a clean construction of orthonormal bases, convergence tables that match expected orders N+1 on affine meshes, CPU-cost comparisons showing high-order efficiency, and validation against experimental shock patterns. The main limitation is that the design-order claim is not demonstrated on the non-affine quadrilateral/hexahedral elements for which the paper introduces row-sum mass lumping; the benchmark cases on those element types are only qualitative. Thus the central claim is plausible but not yet fully supported.

major comments (3)
  1. [2.3, Eq. (29)] Row-sum mass lumping for quadrilaterals and hexahedra is introduced without justification. For non-affine elements the Jacobian determinant J(ξ) in Eq. (27) is not constant, so the exact mass matrix is not diagonal and the lumped diagonal matrix changes the L2 projection. Table 1 only uses Cartesian quads and straight-sided triangles, where J is constant and lumping is exact, so the verified convergence rates do not test Eq. (29). The NACA0012 (Sec. 5.3) and Schardin (Sec. 5.4) simulations use non-affine quads, but their accuracy is only evaluated qualitatively. The authors should either (i) prove or cite that row-sum lumping preserves the DG order for tensor-product Legendre bases on bilinear/trilinear elements, (ii) add a convergence study on a smooth problem with non-affine quadrilaterals/hexahedra, or (iii) use the consistent mass matrix. This is load-bearing for the central arbitrary-order claim.
  2. [2.2, Eq. (15)] The derivative formula is missing the normalization factor. Since φ_n = P_n^{(0,0)} = J_n^{(0,0)}/√γ_n with γ_n = 2/(2n+1), the correct identity is dφ_n/dξ = √(n(n+1)) P_{n-1}^{(1,1)} = √(n(n+1)) J_{n-1}^{(1,1)}/√γ_{n-1}^{(1,1)}. As written, the RHS uses the unnormalized J_{n-1}^{(1,1)} and is off by a factor. The convergence results indicate the implementation uses the correct formula, but the published equation must be corrected.
  3. [3.4] The shock-capturing scheme, described in the abstract as 'improved', is specified only by citing Fu & Shu and Kuzmin, with no equations or algorithmic details for the troubled-cell indicator, the gradient reconstruction from adjacent elements, or the limiting procedure. Because this is advertised as a contribution, the section should be expanded to make the scheme reproducible.
minor comments (5)
  1. [2.2, after Eqs. (18)-(20)] The sentence 'the isoparametric quadrilateral element is a square' in the paragraph after Eq. (18) should refer to the triangle; the text also appears to say 'cube' when discussing the tetrahedron. Please correct the element-type names.
  2. [5.3] The problem statement is inconsistent: it says 'supersonic flow' and M∞ = 0.8, but later states 'inflow Mach number of 0.85'; the computational domain is given as '[−4,−4] × [6,4]'. Please correct these values and clarify the free-stream condition.
  3. [4, Table 1] The notation p for polynomial order in Table 1 is not clearly tied to N used elsewhere; the text refers to 'p1' and N=1 interchangeably. Define the relationship explicitly.
  4. [3.3] The statement that the Runge-Kutta order should be higher than the polynomial order is imprecise; for an N-th order spatial discretization the time integrator should be at least order N+1, and the three-stage third-order scheme in Eq. (44) is not sufficient for N=3. Please clarify which RK scheme is used for each N.
  5. [Throughout] There are numerous typos (e.g., 'Langende', 'increaselingly', 'simulaition', 'quadridual', 'prssure', 'arifoil', 'discription') and inconsistent symbols (e.g., 'ϕ' vs 'φ', 'P' vs 'J'). A thorough proofreading is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: basis construction, DG discretization, and benchmark validation are self-contained; the only self-citation is a non-load-bearing application reference.

full rationale

The derivation chain runs from standard Jacobi/PKD orthonormal bases (Eqs. 9–20) through the Galerkin residual projection (Eqs. 22–26) to the semi-discrete equation (Eq. 30); no step defines a quantity in terms of the target result. The convergence claim is tested against an exact manufactured solution (Eq. 46 and Table 1), and the flow benchmarks are compared with independent literature and experiment, so the reported orders are outputs, not inputs. The row-sum mass lumping in Eq. (29) and the derivative-normalization typo in Eq. (15) are correctness concerns, not circularity: they are neither fitted parameters renamed as predictions nor inputs that force the validation. The only author self-citation, reference [13], is a routine application reference in the introduction and carries none of the derivation or validation load.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The derivation is not circular: the convergence test is against an exact solution and flow cases against literature or experiment. The solver depends on known polynomial bases, the HLLC flux, and external limiter and indicator methods. The main unexamined ingredient is mass lumping on bilinear elements, and the shock indicator and CFL constants are free parameters without sensitivity analysis. No invented physical entities are introduced.

free parameters (2)
  • Troubled-cell indicator threshold C_k = 0.015 × 2^k − 1
    Used in the shock detector (Fig. 6 caption); no sensitivity study is given, so the result depends on this hand-chosen constant.
  • CFL coefficient C = 0.4/(2N+1)
    Global time-step coefficient adopted from literature for linear stability; standard but hand-chosen, and central to the time-increment formula Eq. (45).
assumptions (5)
  • domain assumption Euler equations and stiffened equation of state p=(gamma−1)rho*e + gamma*B close the system.
    Section 2.1, Eqs. (1)-(2); the solver is only intended for inviscid compressible flows with this EOS.
  • standard math Jacobi and Proriol-Koornwinder-Dubiner polynomials form orthogonal bases on reference elements.
    Section 2.2, Eqs. (9)-(20); standard results from refs [27-30] and [22].
  • ad hoc to paper Row-sum lumped mass matrix for quadrilaterals and hexahedra preserves design accuracy.
    Eq. (29); no error analysis or consistent-mass comparison is provided, yet the solver uses this matrix inversion.
  • domain assumption HLLC flux and the Fu-Shu indicator plus Kuzmin limiter give stable, accurate shock capturing.
    Sections 2.4 and 3.4; relies on external methods [33], [23], and [39] and is only checked qualitatively.
  • standard math Gauss-Jacobi and Grundmann-Moller quadratures integrate the weak form sufficiently exactly.
    Section 3.1; standard numerical integration, exactness claimed for polynomials of degree 2n-1.

how reviews work

0 comments
Cite this review

Pith. "Pith review of High-order Discontinuous Galerkin solver based on Jacobi polynomial expansion for compressible flows on unstructured meshes." pith.science (2026). https://pith.science/paper/7MFPJOJW

@misc{pith2026241115699,
  author       = {Pith},
  title        = {Pith review of: High-order Discontinuous Galerkin solver based on Jacobi polynomial expansion for compressible flows on unstructured meshes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7MFPJOJW}},
  note         = {Machine review of arXiv:2411.15699}
}
read the original abstract

Based on the Jacobi polynomial expansion, an arbitrary high-order Discontinuous Galerkin solver for compressible flows on unstructured meshes is proposed in the present work. First, we construct orthogonal polynomials for 2D and 3D isoparametric elements using the 1D Jacobi polynomials. We perform modal expansions of the state variables using the orthogonal polynomials, enabling arbitrary high-order spatial discretization of these variables. Subsequently, the discrete governing equations are derived by considering the orthogonality of the Euler equations' residuals and the test functions. On this basis, we develop a high-order Discontinuous Galerkin solver that supports various element types, including triangles, quadrilaterals, tetrahedra, hexahedra, etc. An improved shock-capturing scheme has been adopted to capture shock discontinuities within the flow field. The variable's gradients at the discontinuous elements are reconstructed by its adjacent elements, and the slope limiter is applied to modify the state variables, smoothing the state variables and enhancing the robustness of the solver. The convergence rates of solvers of different orders have been verified by a benchmark case, and the CPU costs are given to prove that high-precision algorithms have higher computational efficiency under the same error level. Finally, several two- and three-dimensional compressible fluid dynamics problems are studied, compared with literature and experimental results, the effectiveness and accuracy of the solver were verified.

Figures

Figures reproduced from arXiv: 2411.15699 by the authors.

Figure 1
Figure 1. The mapping for triangle element. xi denotes the coordinates of the element vertexs, Ni(ξ, η) is the FEM shape function for triangular element. For the integration on the element boundary in n dimension, the coordinates and weights of the boundary integration points are obtained by the n − 1 dimensional Gauss quadrature. For instance, the boundary integration of triangle elements can be obtained by adopting the one￾… view at source ↗
Figure 2
Figure 2. Meshes for the simulation of 2D Euler equations, [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Forward step problem. Left: uniform distributed quadrilateral mesh with cell length [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: The fluid variables are reset on the elements, which are marked red. Left: the uniform quadrilateral mesh [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: Density contours of the fluid domain. The fluid density changes from 0 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Forward step problem. Troubled cells are marked black, the highest polynomial order (a) [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: Density contours of the fluid domain that discretized by the unstructured triangle element. The fluid [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Double Mach reflection problem. The fluid density changes from 0.562 to 20.9 on 30 contours. Top to [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Fluid pressure obtained by different mesh size (top: ∆ [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: NACA0012 arifoil Schematic Diagram. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: Velocity contour history calculated using the present DG solver with [PITH_FULL_IMAGE:figures/full_fig_p021_11.png]
Figure 13
Figure 13. Figure 13: Pressure contours at the end of the simulation, [PITH_FULL_IMAGE:figures/full_fig_p022_13.png]
Figure 14
Figure 14. Figure 14: Schardin’s problem Schematic Diagram. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]
Figure 15
Figure 15. Figure 15: The pressure contour history obtained by the present DGM solver, upper: [PITH_FULL_IMAGE:figures/full_fig_p023_15.png]
Figure 16
Figure 16. Figure 16: Schardin’s Problem: Comparison of Experimental and Numerical Results [PITH_FULL_IMAGE:figures/full_fig_p024_16.png]
Figure 17
Figure 17. Figure 17: The triple point trajectory of the shock [PITH_FULL_IMAGE:figures/full_fig_p025_17.png]
Figure 19
Figure 19. Figure 19: Problem description and mesh discretization, (a) dimension of the computational domain; (b) hexahedron [PITH_FULL_IMAGE:figures/full_fig_p025_19.png]
Figure 20
Figure 20. Figure 20: The experimental and numerical results are presented as follows: (a)-(c) show the Schlieren images [PITH_FULL_IMAGE:figures/full_fig_p026_20.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 38 canonical work pages

  1. [1]

    J. Tu, G. H. Yeoh, C. Liu, Y. Tao, Computational fluid dynamics: a practical approach, Elsevier, 2023

    2023

  2. [2]

    Z. J. Wang, K. Fidkowski, R. Abgrall, F. Bassi, D. Caraeni, A. Cary, H. Deconinck, R. Hart- mann, K. Hillewaert, H. T. Huynh, et al., High-order cfd methods: current status and per- spective, International Journal for Numerical Methods in Fluids 72 (8) (2013) 811–845

    work page 2013

  3. [3]

    Z. Wang, J. Zhu, Y. Yang, N. Zhao, A new fifth-order alternative finite difference multi- resolution weno scheme for solving compressible flow, Computer Methods in Applied Mechanics and Engineering 382 (2021) 113853

    work page 2021

  4. [4]

    Y. Chen, K. Wu, A physical-constraint-preserving finite volume weno method for special rela- tivistic hydrodynamics on unstructured meshes, Journal of Computational Physics 466 (2022) 111398

    work page 2022

  5. [5]

    Zhang, J

    Y. Zhang, J. Zhu, Finite difference and finite volume ghost multi-resolution weno schemes with increasingly higher order of accuracy, Journal of Computational Physics 504 (2024) 112890

    work page 2024

  6. [6]

    C. D. Cantwell, D. Moxey, A. Comerford, A. Bolis, G. Rocco, G. Mengaldo, D. De Grazia, S. Yakovlev, J.-E. Lombard, D. Ekelschot, et al., Nektar++: An open-source spectral/hp element framework, Computer physics communications 192 (2015) 205–219. 27

    work page 2015

  7. [7]

    Cockburn, G

    B. Cockburn, G. E. Karniadakis, C.-W. Shu, Discontinuous Galerkin methods: theory, com- putation and applications, Vol. 11, Springer Science & Business Media, 2012

    work page 2012

  8. [8]

    Y. Liu, J. Lu, C.-W. Shu, An essentially oscillation-free discontinuous galerkin method for hyperbolic systems, SIAM Journal on Scientific Computing 44 (1) (2022) A230–A259

    work page 2022

Show all 43 references
  1. [9]

    D. A. Di Pietro, A. Ern, Mathematical aspects of discontinuous Galerkin methods, Vol. 69, Springer Science & Business Media, 2011

    2011

  2. [10]

    Silveira, R

    A. Silveira, R. Moura, A. Silva, M. Ortega, Higher-order surface treatment for discontinu- ous galerkin methods with applications to aerodynamics, International Journal for Numerical Methods in Fluids 79 (7) (2015) 323–342

    2015

  3. [11]

    Fernandez, N

    P. Fernandez, N. C. Nguyen, J. Peraire, The hybridized discontinuous galerkin method for im- plicit large-eddy simulation of transitional turbulent flows, Journal of Computational Physics 336 (2017) 308–329

    2017

  4. [12]

    Owkes, O

    M. Owkes, O. Desjardins, A discontinuous galerkin conservative level set scheme for interface capturing in multiphase flows, Journal of Computational Physics 249 (2013) 275–302

    2013

  5. [13]

    Ge, A.-M

    L. Ge, A.-M. Zhang, S.-P. Wang, Investigation of underwater explosion near composite struc- tures using a combined rkdg-fem approach, Journal of Computational Physics 404 (2020) 109113

    2020

  6. [14]

    Zhang, J

    F. Zhang, J. Cheng, Analysis on physical-constraint-preserving high-order discontinuous galerkin method for solving kapila’s five-equation model, Journal of Computational Physics 492 (2023) 112417

    2023

  7. [15]

    W. H. Reed, T. Hill, Triangular mesh methods for the neutron transport equation, Tech. rep., Los Alamos Scientific Lab., N. Mex.(USA) (1973)

    1973

  8. [16]

    Cockburn, C.-W

    B. Cockburn, C.-W. Shu, Tvb runge-kutta local projection discontinuous galerkin finite el- ement method for conservation laws. ii. general framework, Mathematics of computation 52 (186) (1989) 411–435

    1989

  9. [17]

    Cockburn, C.-W

    B. Cockburn, C.-W. Shu, The runge–kutta discontinuous galerkin method for conservation laws v: multidimensional systems, Journal of computational physics 141 (2) (1998) 199–224

    1998

  10. [18]

    Yan, C.-W

    J. Yan, C.-W. Shu, A local discontinuous galerkin method for kdv type equations, SIAM Journal on Numerical Analysis 40 (2) (2002) 769–791. 28

    2002

  11. [19]

    H. Luo, L. Luo, R. Nourgaliev, V. A. Mousseau, N. Dinh, A reconstructed discontinuous galerkin method for the compressible navier–stokes equations on arbitrary grids, Journal of Computational Physics 229 (19) (2010) 6961–6978

    2010

  12. [20]

    M. E. Danis, J. Yan, A new direct discontinuous galerkin method with interface correction for two-dimensional compressible navier-stokes equations, Journal of Computational Physics 452 (2022) 110904

    2022

  13. [21]

    C.-W. Shu, Discontinuous galerkin method for time-dependent problems: survey and recent developments, Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations: 2012 John H Barrett Memorial Lectures (2014) 25–62

    2014

  14. [22]

    Karniadakis, S

    G. Karniadakis, S. J. Sherwin, Spectral/hp element methods for computational fluid dynamics, Oxford University Press, USA, 2005

    2005

  15. [23]

    Fu, C.-W

    G. Fu, C.-W. Shu, A new troubled-cell indicator for discontinuous galerkin methods for hy- perbolic conservation laws, Journal of Computational Physics 347 (2017) 305–327

    2017

  16. [24]

    J. D. Anderson, J. Wendt, Computational fluid dynamics, Vol. 206, Springer, 1995

    1995

  17. [25]

    O. C. Zienkiewicz, R. L. Taylor, J. Z. Zhu, The finite element method: its basis and funda- mentals, Elsevier, 2005

    2005

  18. [26]

    R. D. Cook, et al., Concepts and applications of finite element analysis, John wiley & sons, 2007

    2007

  19. [27]

    Szeg¨ o, Orthogonal polynomials, American Mathematical Society, Providence, RI, 1939

    G. Szeg¨ o, Orthogonal polynomials, American Mathematical Society, Providence, RI, 1939

    1939

  20. [28]

    Proriol, Sur une famille de polynomes ´ a deux variables orthogonaux dans un triangle, Comptes Rendus Hebdomadaires Des Seances De L Academie Des Sciences 245 (26) (1957) 2459–2461

    J. Proriol, Sur une famille de polynomes ´ a deux variables orthogonaux dans un triangle, Comptes Rendus Hebdomadaires Des Seances De L Academie Des Sciences 245 (26) (1957) 2459–2461

    1957

  21. [29]

    Koornwinder, Two-variable analogues of the classical orthogonal polynomials, in: Theory and application of special functions, Elsevier, 1975, pp

    T. Koornwinder, Two-variable analogues of the classical orthogonal polynomials, in: Theory and application of special functions, Elsevier, 1975, pp. 435–495

    1975

  22. [30]

    Dubiner, Spectral methods on triangles and other domains, Journal of Scientific Computing 6 (4) (1991) 345–390

    M. Dubiner, Spectral methods on triangles and other domains, Journal of Scientific Computing 6 (4) (1991) 345–390

    1991

  23. [31]

    Bangerth, R

    W. Bangerth, R. Rannacher, Adaptive finite element methods for differential equations, Birkh¨ auser, 2013. 29

    2013

  24. [32]

    Li, An hllc riemann solver for magneto-hydrodynamics, Journal of computational physics 203 (1) (2005) 344–357

    S. Li, An hllc riemann solver for magneto-hydrodynamics, Journal of computational physics 203 (1) (2005) 344–357

    2005

  25. [33]

    E. F. Toro, M. Spruce, W. Speares, Restoration of the contact surface in the hll-riemann solver, Shock waves 4 (1) (1994) 25–34

    1994

  26. [34]

    G. H. Golub, J. H. Welsch, Calculation of gauss quadrature rules, Mathematics of computation 23 (106) (1969) 221–230

    1969

  27. [35]

    N. Hale, A. Townsend, Fast and accurate computation of gauss–legendre and gauss–jacobi quadrature nodes and weights, SIAM Journal on Scientific Computing 35 (2) (2013) A652– A674

    2013

  28. [36]

    Grundmann, H.-M

    A. Grundmann, H.-M. M¨ oller, Invariant integration formulas for the n-simplex by combinato- rial methods, SIAM Journal on Numerical Analysis 15 (2) (1978) 282–290

    1978

  29. [37]

    J. S. Hesthaven, T. Warburton, Nodal discontinuous Galerkin methods: algorithms, analysis, and applications, Springer Science & Business Media, 2007

    2007

  30. [38]

    H. Luo, J. D. Baum, R. L¨ ohner, A hermite weno-based limiter for discontinuous galerkin method on unstructured grids, Journal of Computational Physics 225 (1) (2007) 686–713

    2007

  31. [39]

    Kuzmin, A vertex-based hierarchical slope limiter for p-adaptive discontinuous galerkin methods, Journal of computational and applied mathematics 233 (12) (2010) 3077–3085

    D. Kuzmin, A vertex-based hierarchical slope limiter for p-adaptive discontinuous galerkin methods, Journal of computational and applied mathematics 233 (12) (2010) 3077–3085

    2010

  32. [40]

    Woodward, P

    P. Woodward, P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks, Journal of computational physics 54 (1) (1984) 115–173

    1984

  33. [41]

    Schardin, High frequency cinematography in the shock tube, The Journal of Photographic Science 5 (2) (1957) 17–19

    H. Schardin, High frequency cinematography in the shock tube, The Journal of Photographic Science 5 (2) (1957) 17–19

    1957

  34. [42]

    Chang, K.-S

    S.-M. Chang, K.-S. Chang, On the shock–vortex interaction in schardin’s problem, Shock Waves 10 (5) (2000) 333–343

    2000

  35. [43]

    Houtman, W

    E. Houtman, W. Bannink, B. Timmerman, Experimental and numerical investigation of the high-supersonic flow around an axi-symmetric blunt-cylinder-flare model, in: Aerothermody- namics for space vehicles, Proceedings of the 2nd European Symposium held in ESTEC, No- ordwijk, The...

    1994

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.