Pith. sign in

REVIEW 3 major objections 5 minor 42 references

High-order gas-kinetic scheme with three-dimensional WENO reconstruction for the Euler and Navier-Stokes solutions

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read The paper claims that a WENO-AO reconstruction combining one quadratic big-stencil polynomial with 24 linear sub-stencils at fixed positive weights, fed into a two-stage fourth-order gas-kinetic scheme, gives a third-order finite-volume…

desk verdict A genuinely useful 3D WENO-AO/HGKS extension with clean third-order inviscid convergence; the Navier-Stokes order claim needs a viscous convergence test before I'd take it as established. read the letter →

arxiv 1909.01580 v1 pith:DCBGF4M6 submitted 2019-09-04 math.NA cs.NA

Liang Pan , Kun Xu This is my paper · ORCID
classification math.NAcs.NA MSC 65M0876N1576P05
keywords WENO-AOgas-kineticschemethird-orderfinitevolumemethodthree-dimensionalEulerequationsNavier-StokesmovingmeshgeometricconservationlawBGKequation
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

The paper sets out to show that a genuinely three-dimensional third-order finite-volume scheme for compressible flow need not pay the classical WENO price: solving for linear weights at every Gaussian quadrature point and coping with negative or topology-dependent weights. Its proposal is a WENO-AO reconstruction in which one quadratic big-stencil polynomial and twenty-four linear sub-stencil polynomials are combined with fixed positive weights that sum to one, with shock detection supplied by smoothness-indicator-based nonlinear weights. Coupled to the two-stage fourth-order gas-kinetic scheme, the method is claimed to be third-order accurate in space for smooth three-dimensional Euler and Navier-Stokes flows, well behaved across strong discontinuities, and exactly conservative on moving meshes. If true, this removes the main efficiency and robustness obstacles to three-dimensional high-order WENO and simplifies the route to moving-mesh and eventually unstructured computations.

What carries the argument

The load-bearing object is the WENO-AO combination formula P̃0(xG) = ω0($γ0^{{-1}}$P0(xG) − ∑_{m=1}^{24}(γm/γ0)Pm(xG)) + ∑_{m=1}^{24}ωmPm(xG), built from a 19-cell big stencil (the 3×3×3 block minus its eight corner cells) supporting a quadratic polynomial P0 and 24 five-cell sub-stencils supporting linear polynomials Pm. The linear weights γm are fixed positive constants summing to one, and the nonlinear weights ωm = γm(1 + τ/(βm + ε)) with τ = ∑|β0 − βm|/24 convert the combination into a shock-detecting selector. This reconstruction replaces the classical per-quadrature-point linear-weight solves; the gas-kinetic flux then reads reconstructed point values and slopes directly off the same polynomials at each Gaussian quadrature point per face, and the two-stage fourth-order time stepping needs only the flux and its temporal derivative at each cell interface.

What would settle it

Perform a three-dimensional viscous convergence test—for example, the same density-perturbation advection with Re = 100 viscosity or a manufactured Navier-Stokes solution with a smooth source term—and record the observed L1 order on nested meshes. If the order saturates near two rather than three, the third-order Navier-Stokes claim is false. A direct check of the mechanism: measure the error of the reconstructed slopes ∂P̃0/∂x at a Gaussian quadrature point on smooth data; the viscous flux inherits its order from those slopes, which must be second-order accurate for a third-order flux.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a WENO-AO reconstruction with one quadratic big-stencil polynomial and 24 linear sub-stencil polynomials, combined with fixed positive linear weights γ0 = 1 − 24Γ and γm = Γ (Γ = 0.0025) that are independent of local mesh topology, attains third-order point-value accuracy for three-dimensional Euler and Navier-Stokes solutions provided the nonlinear weights satisfy ωm = γm(1 + O(h)). The accuracy mechanism is the separation of smoothness-indicator scales: the quadratic indicator carries an O(|Ω|^{2/3}) error while the linear indicators carry O(|Ω|^{1/3}), so the parameter τ defined by averaging the 24 differences |β0 − βm| yields the required first-order deviation. Coupled with the two-stage fourth-order gas-kinetic scheme, the method shows third-order convergence on uniform, non-coplanar, and moving hexahedral meshes, reproduces Sod, blast-wave, spherical-Sod, and Sedov shock profiles, matches Taylor-Green vortex and isotropic-turbulence reference data up to turbulent Mach number 1.2, and preserves the geometric conservation law to machine precision on moving meshes.

Load-bearing premise

The Navier-Stokes accuracy claim assumes the spatial slopes built from the least-squares linear sub-stencil polynomials are accurate enough for a third-order viscous flux, an assumption the paper does not test since its convergence study covers an inviscid case only.

Editorial extensions

If this is right

  • On uniform, non-coplanar, and moving hexahedral meshes, the smooth advection test converges at approximately third order, with measured L1 orders between 2.85 and 3.00 across the meshes tested.
  • Fixed positive linear weights with γ0 = 1 − 24Γ and γm = Γ avoid the negative and topology-dependent weights of classical 3D WENO, eliminating per-quadrature-point linear-system solves and improving efficiency, especially for moving meshes.
  • The scheme resolves strong shocks—Sod, Woodward-Collella blast wave, spherical Sod, and Sedov—without spurious oscillation, and the Mach 5 and Mach 8 sphere test shows no carbuncle phenomenon.
  • On moving meshes, the uniform-flow test holds L1 errors near 10^{-14}, meaning the geometric conservation law is satisfied to machine precision.
  • The scheme reproduces reference kinetic-energy and dissipation histories for the Taylor-Green vortex at Re = 280 and for compressible isotropic turbulence at turbulent Mach numbers 0.3 to 1.2, including the supersonic regime.

Reading between the lines

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

  • The topology-independent fixed weights suggest the reconstruction transfers to unstructured hexahedral meshes without re-deriving weights; the paper names this as future work, and that transfer is the natural test of the construction's main selling point.
  • Because the accuracy analysis proves third-order point-value reconstruction but the viscous flux depends on reconstructed slopes from first-order-accurate linear sub-stencils, a viscous manufactured-solution test would settle whether the Navier-Stokes claim holds; the paper contains no such test.
  • The parameter τ, chosen as a 24-term average of |β0 − βm|, may behave differently near critical points where the solution gradient vanishes; whether third-order accuracy survives there—the classical WENO critical-point problem—is left untested.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 paper develops a three-dimensional finite-volume WENO-AO reconstruction for high-order gas-kinetic schemes (HGKS). The reconstruction uses one quadratic big-stencil polynomial and 24 linear sub-stencil polynomials with fixed positive linear weights, avoiding topology-dependent and possibly negative linear weights at the quadrature points of a hexahedron. The resulting scheme is coupled with the two-stage fourth-order gas-kinetic solver and is extended to moving meshes through a trilinear parameterization of hexahedra. The authors claim third-order accuracy for both the Euler and Navier-Stokes equations, robustness for strong discontinuities, and exact preservation of the geometric conservation law on moving meshes. Numerical evidence includes inviscid three-dimensional advection convergence tests on uniform, non-coplanar, and moving meshes, geometric conservation-law tests to machine precision, Riemann and Sedov problems, a hypersonic sphere impingement test, and qualitative comparisons for the Taylor-Green vortex and compressible isotropic turbulence.

Significance. The WENO-AO construction is a genuine efficiency improvement: the linear weights are fixed positive numbers independent of the local mesh topology, and the scheme avoids solving weight systems at 24 quadrature points. The inviscid convergence tests in Tables 1-3 are clean and credible, and the moving-mesh geometric conservation-law results in Table 4 are a strong positive feature. However, the claimed third-order accuracy for the Navier-Stokes equations is not established by the evidence presented: the accuracy analysis in Section 2 concerns point values only, and the viscous tests in Sections 4.6 and 4.7 are single-resolution qualitative comparisons. This is a load-bearing gap because the viscous flux in the gas distribution function depends on reconstructed slopes whose accuracy is neither analyzed nor tested.

major comments (3)
  1. [Section 2, Eq. (15)] The accuracy analysis in Section 2 establishes only that the reconstructed point value tilde-P_0(x_G) is third-order under the assumed weight condition (6). The spatial derivatives of tilde-P_0, defined immediately after Eq. (5), are used in the gas distribution function (15) through the coefficients a_l and a_r, and these coefficients enter the viscous flux through the tau = mu/p term. The 24 linear sub-stencil polynomials P_m are one-sided least-squares fits, so their individual gradients are only first-order accurate for general smooth data. No Taylor analysis or numerical experiment shows that the WENO combination raises the slope accuracy to the level needed for a third-order viscous flux. Consequently, the abstract and conclusion's claim of third-order accuracy for Navier-Stokes solutions is not supported. A viscous convergence study on multiple meshes, for example with a manufactured solution, and preferably a short Taylor analysis of the reconstructed slopes, should be added.
  2. [Eq. (6), Section 2] The sufficient condition omega_m = gamma_m(1 + O(h^k)) with k = 1 is stated as an assumption rather than proved. The Taylor expansions of the smoothness indicators beta_m are given only to leading order, and no argument is supplied for points where the leading gradient term of Q vanishes; at such critical points the order k in (6) could in principle degrade. The inviscid advection tests suggest the condition holds for point values, but the same condition is needed for the reconstructed slopes that enter Eq. (15). The authors should either prove (6) under appropriate smoothness assumptions or provide a numerical verification that includes critical points.
  3. [Sections 4.6-4.7] The Taylor-Green vortex and compressible isotropic turbulence tests are single-resolution comparisons with reference data: the TGV computation uses 192^3 cells and the turbulence computation uses 128^3 cells, and neither reports errors or observed order on a sequence of meshes. These tests demonstrate qualitative agreement and robustness, but they cannot verify the order of accuracy of the Navier-Stokes discretization. A grid-refinement study for a viscous flow with a known or manufactured solution is necessary to support the central third-order claim for viscous flows.
minor comments (5)
  1. [Section 3, after Eq. (7)] There are two typos: 'can ba obtained' should read 'can be obtained', and in Section 2 the phrase 'smooth indicator' should read 'smoothness indicator'.
  2. [Section 4.2] In the geometric conservation-law paragraph, 'the above moment of computational mesh' should presumably be 'the above motion of the computational mesh'.
  3. [Eq. (3)] The symbols omega_m and omega_m are used both for the unnormalized and the normalized nonlinear weights, which is confusing; please use a tilde or a superscript for the normalized quantity.
  4. [Figures 7-10] The scheme name is written inconsistently as 'HGKS-3D-WENO' in Figure 7 and as 'HGKS-WENO-3D' in Figures 9 and 10; please make the notation uniform.
  5. [References] References [27] and [30] appear to be the same preprint with different version labels; please unify them and cite the published version if available.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the WENO-AO accuracy argument is a self-contained Taylor-expansion derivation, and the numerical claims are checked against exact and reference solutions.

full rationale

The paper's central claim is that a 3D finite-volume gas-kinetic scheme with WENO-AO reconstruction is third-order accurate for smooth Euler flows and robust for discontinuities. The accuracy argument in Section 2 is not circular: it constructs a big-stencil quadratic polynomial and 24 sub-stencil linear polynomials, defines nonlinear weights from smoothness indicators, and proves via Taylor expansion that the reconstructed point values are third-order provided the weights satisfy omega_m = gamma_m(1+O(h)), which the preceding expansion of beta_m and tau establishes. No fitted parameter is later renamed as a prediction; the only constants, such as Gamma = 0.0025 and CFL = 0.35, are fixed and are not tuned to the convergence tables. The convergence tests in Tables 1-3 compare against the exact advection solution on uniform, non-coplanar, and moving meshes, while the shock and turbulence tests are compared with exact or reference data or with a WENO-Z scheme, so the validation is independent of the derivation. The paper does cite prior GKS and WENO-AO work, including by the same authors, but those citations supply standard evolution-model and reconstruction ingredients and are not used to forbid alternatives; the fourth-order two-stage time discretization is a cited external framework, and the present spatial reconstruction is separately tested. The absence of a viscous convergence study means that the statement of third-order accuracy for Navier-Stokes solutions is not fully demonstrated, but that is an evidentiary gap rather than a circular reduction: the Navier-Stokes flux is not fitted to the DNS reference data. No step in the paper defines its output in terms of its input, fits a parameter and then calls the result a prediction, or imports an unverified uniqueness theorem from prior work. Therefore no circularity is found.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central accuracy claim rests on standard polynomial approximation theory, the BGK gas-kinetic model, and hand-chosen scheme constants (Gamma, epsilon, C, CFL). No new physical entities are introduced. The most fragile axiom is the uniform O(h) closeness of nonlinear weights to linear weights in smooth regions, which is assumed in the third-order proof.

free parameters (3)
  • sub-stencil linear weight Gamma = 0.0025
    Chosen by hand in Section 2; gamma0 = 1 - 24*Gamma = 0.94. Controls how much weight the 24 linear sub-stencils carry relative to the big quadratic. Not fitted to benchmarks, but affects nonlinear weight response and robustness.
  • inviscid collision-time parameters epsilon and C = epsilon=0.01, C=1
    Used in tau = epsilon*dt + C*|p_l-p_r|/(p_l+p_r)*dt in Section 4. Standard artificial dissipation choices; no parameter study is provided.
  • CFL number = 0.35
    Used for all computations in Section 4. Standard stability choice, not analyzed.
assumptions (4)
  • domain assumption The BGK equation with the BGK collision operator is an adequate description of Euler and Navier-Stokes flows in the continuum limit via Chapman-Enskog expansion.
    Invoked in Section 3 to derive the gas distribution function and numerical flux; standard for gas-kinetic schemes.
  • standard math The two-stage fourth-order temporal discretization is fourth-order accurate for hyperbolic conservation laws.
    Used in Section 3; proved in cited works [23,28], not reproved here.
  • standard math Least-squares polynomial reconstruction over the chosen stencils yields pointwise errors O(h^3) for the quadratic and O(h^2) for the linear polynomials for smooth data.
    Basis of the accuracy analysis in Section 2; standard finite-volume approximation theory, assuming sufficient smoothness and stencil geometry.
  • ad hoc to paper The nonlinear weights satisfy omega_m = gamma_m (1 + O(h)) uniformly in smooth regions, including near critical points where gradients vanish.
    Assumed in the third-order proof in Section 2. The paper derives this from Eq. (5) for generic smooth data, but does not discuss critical points where beta_0 and beta_m differences may be smaller, potentially degrading the O(h) closeness.

how reviews work

0 comments
Cite this review

Pith. "Pith review of High-order gas-kinetic scheme with three-dimensional WENO reconstruction for the Euler and Navier-Stokes solutions." pith.science (2026). https://pith.science/paper/DCBGF4M6

@misc{pith2026190901580,
  author       = {Pith},
  title        = {Pith review of: High-order gas-kinetic scheme with three-dimensional WENO reconstruction for the Euler and Navier-Stokes solutions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DCBGF4M6}},
  note         = {Machine review of arXiv:1909.01580}
}
read the original abstract

In this paper, a simple and efficient third-order weighted essentially non-oscillatory (WENO) reconstruction is developed for three-dimensional flows, in which the idea of two-dimensional WENO-AO scheme on unstructured meshes \cite{WENO-ao-3} is adopted. In the classical finite volume type WENO schemes, the linear weights for the candidate stencils are obtained by solving linear systems at Gaussian quadrature points of cell interface. For the three-dimensional scheme, such operations at twenty-four Gaussian quadrature points of a hexahedron would reduce the efficiency greatly, especially for the moving-mesh computation. Another drawback of classical WENO schemes is the appearance of negative weights with irregular local topology, which affect the robustness of spatial reconstruction. In such three-dimensional WENO-AO scheme, a simple strategy of selecting big stencil and sub-stencils for reconstruction is proposed. With the reconstructed quadratic polynomial from big stencil and linear polynomials from sub-stencils, the linear weights are chosen as positive numbers with the requirement that their sum equals one and be independent of local mesh topology. With such WENO reconstruction, a high-order gas-kinetic scheme (HGKS) is developed for both three-dimensional inviscid and viscous flows. Taken the grid velocity into account, the scheme is extended into the moving-mesh computation as well. Numerical results are provided to illustrate the good performance of such new finite volume WENO schemes. In the future, such WENO reconstruction will be extended to the unstructured meshes.

Figures

Figures reproduced from arXiv: 1909.01580 by the authors.

Figure 1
Figure 1. Stencils of cell Ωijk (red cubic) for three-dimensional reconstruction. Twenty-four candidate sub-stencils are selected from the big stencil as well, and one third 4 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Riemann problem: the density, velocity and pressure distr [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Riemann problem: the density and pressure distributions a [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Sedov problem: the three-dimensional density and press [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]
Figure 5
Figure 5. Figure 5: Sedov problem: the density and pressure profiles long [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: Flow impinging on sphere: the mesh and density distribution, [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: Taylor-Green Vortex: the time history of kinetic energy [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]
Figure 8
Figure 8. Figure 8: Taylor-Green Vortex problem: the iso-surfaces of [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]
Figure 9
Figure 9. Figure 9: Compressible isotropic turbulence: time history of [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]
Figure 10
Figure 10. Figure 10: Compressible isotropic turbulence: time history of [PITH_FULL_IMAGE:figures/full_fig_p025_10.png]
Figure 11
Figure 11. Figure 11: Compressible isotropic turbulence: the contours of dilat [PITH_FULL_IMAGE:figures/full_fig_p026_11.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

42 extracted references · 40 canonical work pages

  1. [1]

    Abgrall, On essentially non-oscillatory schemes on unstructur ed meshes: Analysis and implementa- tion, J

    R. Abgrall, On essentially non-oscillatory schemes on unstructur ed meshes: Analysis and implementa- tion, J. Comput. Phys. 114 (1994) 45-58

    work page 1994

  2. [2]

    Balsara, S

    D.S. Balsara, S. Garain, C.W. Shu. An efficient class of WENO scheme s with adaptive order. Journal of Computational Physics, 326 (2016) 780C804

    work page 2016

  3. [3]

    Ben-Artzi, J

    M. Ben-Artzi, J. Li, Hyperbolic conservation laws: Riemann invaria nts and the generalized Riemann problem, Numerische Mathematik. 106 (2007) 369-425

    work page 2007

  4. [4]

    Bhatnagar, E.P

    P.L. Bhatnagar, E.P. Gross, M. Krook, A Model for Collision Proce sses in Gases I: Small Amplitude Processes in Charged and Neutral One-Component Systems, Phy s. Rev. 94 (1954) 511-525

    work page 1954

  5. [5]

    Borges, M

    R. Borges, M. Carmona, B. Costa, W. S. Don, An improved weight ed essentially non-oscillatory scheme for hyperbolic conservation laws, J. Comput. Phys. 227 (2008) 31 91-3211

    work page 2008

  6. [6]

    J. R. Bull, A. Jameson, Simulation of the compressible Taylor-Gree n vortex using high-order flux reconstruction schemes, AIAA 2014-3210

    work page 2014

  7. [7]

    G.Y. Cao, L. Pan, K. Xu, Three dimensional high-order gas-kinet ic scheme for supersonic isotropic turbulence I: criterion for direct numerical simulation, Computers & Fluids 192 (2019) 104273

    work page 2019

  8. [8]

    Chapman, T.G

    S. Chapman, T.G. Cowling, The Mathematical theory of non-unifo rm gases, third edition, Cambridge University Press, (1990)

    work page 1990

Show all 42 references
  1. [9]

    Cockburn, C.W

    B. Cockburn, C.W. Shu, TVB Runge-Kutta local projection disco ntinuous Galerkin finite element method for conservation laws II: general framework, Mathemat ics of Computation, 52 (1989) 411-435

    1989

  2. [10]

    Cockburn, C.W

    B. Cockburn, C.W. Shu, The Runge-Kutta discontinuous Galerk in method for conservation laws V: multidimensional systems, J. Comput. Phys. 141 (1998) 199-224

    1998

  3. [11]

    Dumbser, M

    M. Dumbser, M. Kaser, Arbitrary high order non-oscillatory fin ite volume schemes on unstructured meshes for linear hyperbolic systems, J. Comput. Phys. 221 (2007 ) 693C723

    2007

  4. [12]

    Debonis, Solutions of the Taylor-Green vortex problem using high-resolution explicit finite difference methods, AIAA Paper (2013) 2013-0382

    J. Debonis, Solutions of the Taylor-Green vortex problem using high-resolution explicit finite difference methods, AIAA Paper (2013) 2013-0382

    2013

  5. [13]

    Z.F. Du, J.Q. Li, A Hermite WENO reconstruction for fourth orde r temporal accurate schemes based on the GRP solver for hyperbolic conservation laws, J. Comput. Phy s. 355 (2018) 385-396

    2018

  6. [14]

    Gottlieb, C.W

    S. Gottlieb, C.W. Shu, Total variation diminishing Runge-Tutta sc hemes, Mathematics of computation, 67 (1998) 73-85

    1998

  7. [15]

    Harten, B

    A. Harten, B. Engquist, S. Osher and S. R. Chakravarthy, Un iformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71 (1987) 231-3 03

    1987

  8. [16]

    A. K. Henrick, T. D. Aslam, J. M. Powers, Mapped weighted esse ntially non-oscillatory schemes: achieving optimal order near critical points, J. Comput. Phys. 207 (2005) 542-567

    2005

  9. [17]

    C. Hu, C.W. Shu, Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys. 150 (1999) 97-127

    1999

  10. [18]

    X. Ji, L. Pan, W. Shyy, K. Xu, A compact fourth-order gas-kin etic scheme for the Euler and Navier- Stokes equations, J. Comput. Phys. 372 (2018) 446-472

    2018

  11. [19]

    X. Ji, K. Xu, Performance Enhancement for High-order Gas-k inetic Scheme Based on WENO-adaptive- order Reconstruction, arXiv:1905.08489v1

    1905 arXiv

  12. [20]

    Jiang, C

    G.S. Jiang, C. W. Shu, Efficient implementation of weighted ENO sch emes, J. Comput. Phys. 126 (1996) 202-228

    1996

  13. [21]

    Kamm, F.X

    J.R. Kamm, F.X. Timmes, On efficient generation of numerically robu st Sedov solutions, Technical Report LA-UR-07-2849, Los Alamos National Laboratory, (2007 )

    2007

  14. [22]

    Lele, Compact finite difference schemes with spectral-like re solution, J

    S.K. Lele, Compact finite difference schemes with spectral-like re solution, J. Comput. Phys. 103 (1992) 16-42

    1992

  15. [23]

    J.Q. Li, Z.F. Du, A two-stage fourth order time-accurate discr etization for Lax-Wendroff type flow solvers I. hyperbolic conservation laws, SIAM J. Sci. Computing, 38 (2016) 3046-3069

    2016

  16. [24]

    Q.B. Li, K. Xu, S. Fu, A high-order gas-kinetic Navier-Stokes flo w solver, J. Comput. Phys. 229 (2010) 6715-6731

    2010

  17. [25]

    X.D. Liu, S. Osher, T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys. 115 (1994) 200-212. 28

    1994

  18. [26]

    Liu, H.Z

    N. Liu, H.Z. Tang, A high-order accurate gas-kinetic scheme fo r one- and two-dimensional flow simu- lation, Commun. Comput. Phys. 15 (2014) 911-943

    2014

  19. [28]

    L. Pan, K. Xu, Q.B. Li, J.Q. Li, An efficient and accurate two-stag e fourth-order gas-kinetic scheme for the Navier-Stokes equations, J. Comput. Phys. 326 (2016), 197-221

    2016

  20. [29]

    L. Pan, K. Xu, Two-stage fourth-order gas-kinetic scheme f or three-dimensional Euler and Navier- Stokes solutions, Int. J. Comput. Fluid Dynamics, 32 (2018) 395-4 11

    2018

  21. [30]

    Pan, F.X

    L. Pan, F.X. Zhao, K. Xu, High-order ALE gas-kinetic scheme wit h unstructured WENO reconstruc- tion, arXiv:1905.07837v1

    1905 arXiv

  22. [31]

    Samtaney, D.I

    R. Samtaney, D.I. Pullin, B. Kosovic, Direct numerical simulation o f decaying compressible turbulence and shocklet statistics. Physiscs of Fluids 13 (2001) 1415-1430

    2001

  23. [32]

    J. Shi, C. Hu, C.W. Shu, A technique of treating negative weights in WENO schemes, J. Comput. Phys. 175 (2002) 108-127

    2002

  24. [33]

    C.W. Shu, S. Osher, Efficient implementation of essentially nonosc illatory shock-capturing schemes II, J. Comput. Phys. 83 (1989) 32-78

    1989

  25. [34]

    Thomas, C

    P. Thomas, C. Lombard, Geometric conservation law and its app lication to flow computations on moving grids, AIAA J. 17 (1979) 1030-1037

    1979

  26. [35]

    Toro, Riemann Solvers and Numerical Methods for Fluid Dyna mics, Third Edition, Springer (2009)

    E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dyna mics, Third Edition, Springer (2009)

    2009

  27. [36]

    L. Wang, W. K. Anderson, T. Erwin, S. Kapadia, High-order disc ontinuous Galerkin method for computation of turbulent flows, AIAA Journal 53 (2015) 1157-11 71

    2015

  28. [37]

    Wang, L.P

    J.C. Wang, L.P. Wang, Z.L. Xiao, Y. Shi, S.Y. Chen, A hybrid numeric al simulation of isotropic compressible turbulence, J. Comput. Phys. 229 (2010) 5257-527 9

    2010

  29. [38]

    Xu, Direct modeling for computational fluid dynamics: constr uction and application of unfied gas kinetic schemes, World Scientific (2015)

    K. Xu, Direct modeling for computational fluid dynamics: constr uction and application of unfied gas kinetic schemes, World Scientific (2015)

    2015

  30. [39]

    Xu, A gas-kinetic BGK scheme for the Navier-Stokes equatio ns and its connection with artificial dissipation and Godunov method, J

    K. Xu, A gas-kinetic BGK scheme for the Navier-Stokes equatio ns and its connection with artificial dissipation and Godunov method, J. Comput. Phys. 171 (2001) 289 -335

    2001

  31. [40]

    F.X. Zhao, L. Pan, S.H. Wang, Weighted essentially non-oscillator y scheme on unstructured quadrilat- eral and triangular meshes for hyperbolic conservation laws, J. Co mput. Phys. 374 (2018) 605-624

    2018

  32. [41]

    F.X. Zhao, X. Ji, W. Shyy, K. Xu, Compact higher-order gas-kin etic schemes with spectral-like resolu- tion for compressible flow simulations, Advances in Aerodynamics 1:13 (2019)

    2019

  33. [42]

    Zhu, J.X

    J. Zhu, J.X. Qiu, A new fifth order finite difference weno scheme f or solving hyperbolic conservation laws. J. Comput. Phys. 318 (2016) 110-121

    2016

  34. [43]

    Zhu, J.X

    J. Zhu, J.X. Qiu, New finite volume weighted essentially non-oscillat ory scheme on triangular meshes, SIAM J. Sci. Computing, 40 (2018) 903-928. 29

    2018

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.