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High-Resolution Weighted Essentially Non-Oscillatory Compact Least-Squares Schemes with Implicit Time Integration for Compressible Navier-Stokes Equations on Curvilinear Grids

T0 review · 2 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A weighted compact least-squares reconstruction can preserve designed high-order accuracy in smooth regions and capture shocks and contacts sharply on curvilinear grids.

desk verdict Solid incremental extension of the WCLS framework with broad numerical validation, but the formal order guarantee at smooth extrema rests on an unproved scaling assumption. read the letter →

arxiv 2608.08117 v1 pith:UIZ3BAXA submitted 2026-08-08 math.NA cs.NAphysics.comp-ph

Yongzhi Luo , Huiheng Fan , Wei-Gang Zeng , Yu-Xin Ren , Jianhua Pan This is my paper · ORCID
classification math.NAcs.NAphysics.comp-ph MSC 65M0865M1276N15
keywords weightedcompactleast-squaresfinitevolumemethodcompressibleNavier-Stokesequationsshockcapturinghigh-orderaccuracycurvilineargridsimplicittimeintegrationboundaryvaluedifference
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 develops a family of third- to fifth-order finite-volume schemes for the compressible Navier-Stokes equations and claims that they resolve a long-standing trade-off: high-order accuracy in smooth regions no longer has to be sacrificed for non-oscillatory shock capture. The key move is to enforce the compact least-squares accuracy conditions only along reconstruction lines that are locally smooth, using face weights $\beta_n$ that fade to zero across discontinuities, and to activate a boundary-value-difference penalty $\theta_n$ only at the ends of those smooth lines. In smooth regions $\theta_n$ is $O(\Delta x^k)$, so the designed order is retained; near discontinuities it becomes $O(1)$ and damps Gibbs-type oscillations. Numerical experiments on one-, two-, and three-dimensional inviscid and viscous flows show sharper contacts, lower dissipation, and better small-scale structure preservation than the WENO3-JS, WENO5-Z, and WBAP-CLS reference schemes. If the claims hold, the method offers a practical route to high-resolution compressible turbulence simulations with strong shocks on curvilinear grids.

What carries the argument

The machinery is the weighted compact least-squares reconstruction: a per-control-volume polynomial whose coefficients are determined by minimizing a loss that couples derivatives across neighboring faces. Three ingredients carry the argument: the face weight $\beta_n$, formed from a modified smoothness indicator that uses only von Neumann neighbors, which separates reconstruction lines into smooth segments; the penalty coefficient $\theta_n$, whose boundary-value-difference scaling makes the penalty $O(\Delta x^k)$ in smooth regions and $O(1)$ at discontinuities; and the coupling of the resulting nonlinear reconstruction with implicit time integration through GMRES, so that $\beta$ and $\theta$ are frozen during each Krylov iteration and updated between iterations. The linear weights $W_j$ are tuned by a genetic algorithm to maximize the resolved-wavenumber region, and the penalty parameters are calibrated separately for the third-order and the higher-order versions.

What would settle it

Run the WCLS5 scheme on smooth advection with $\sin^3(\pi x)$ initial data on a sequence of refined grids, record the penalty coefficient in cells adjacent to the extrema, and check both whether it decays like $\Delta x^5$ and whether the $L^\infty$ error slope reaches 5; either failure would contradict the order-preservation claim.

Watch

Extended reading notes

Core claim

The central claim is that the Gibbs phenomenon observed in compact least-squares schemes is not an unavoidable property of compact reconstruction but a consequence of enforcing accuracy constraints across discontinuities. The paper replaces the single global reconstruction line with smooth segments: each interface contribution to the loss is multiplied by $\beta_n \in [0,1]$, computed from a compact smoothness indicator that uses only von Neumann neighbors plus an inexpensive linear reconstruction. When $\beta_n$ drops, the segment endpoint receives a penalty term whose coefficient $\theta_n$ is proportional to the boundary-value difference $|p_i(x_\alpha)-p_n(x_\alpha)|$ divided by the jump in cell averages. Because this quantity is $O(\Delta x^{k+1})/O(\Delta x)=O(\Delta x^k)$ in smooth regions and $O(1)$ near discontinuities, the penalty vanishes asymptotically where the solution is resolved, preserving formal order, and turns on where the solution is under-resolved, suppressing oscillations. The reconstruction system, now nonlinear through $\beta_n$ and $\theta_n$, is solved inside each implicit time step by one GMRES iteration, avoiding the block-tridiagonal solves and the separate limited-polynomial set of earlier compact least-squares schemes.

Load-bearing premise

The accuracy guarantee rests on the assumption that the stabilizing penalty stays at most of order grid-spacing-to-the-k near smooth extrema, where the cell-averaged denominator used to define it can shrink to nearly zero; the paper provides numerical evidence but not a proof for that case.

Editorial extensions

If this is right

  • If the central claim is right, the third-order WCLS scheme resolves contacts and small-scale vortices at a resolution comparable to the fifth-order WENO-Z scheme, so lower-order reconstructions become usable where fifth-order WENO was previously needed.
  • Formal third-, fourth-, and fifth-order convergence is maintained even when the weighting is active everywhere, including flows with first- and second-order smooth extrema, where WENO3-JS and WENO5-Z degrade.
  • The single-set-of-polynomials formulation removes the need for a separate limiter pass, and the GMRES coupling removes block-tridiagonal solves, so the per-iteration cost stays comparable to WENO inside an implicit solver.
  • The method retains low numerical dissipation in long-time compressible vortex simulations, preserving kinetic energy and dissipation peaks better than the WENO and WBAP-CLS references.
  • The penalty formulation remains stable in extreme regimes, including stationary shocks up to Mach 1000 and high-Mach jet flows, where the WBAP-CLS5 reference fails.

Reading between the lines

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

  • Beyond the paper, the compact smoothness indicator built only from von Neumann neighbors suggests that the same weighting could transfer to unstructured grids, but the paper does not demonstrate that extension.
  • Beyond the paper, the spectral-optimization procedure suggests that the free linear weights could be tuned for a given grid or flow to shift the resolved-wavenumber range, a use the paper does not explore.
  • Beyond the paper, the one-GMRES-step-per-iteration coupling suggests a natural synchronization strategy for parallel compact schemes: exchange reconstructed derivatives at halo cells during the Krylov iterations instead of solving a global block-tridiagonal system; the paper states this benefit but does not benchmark it in parallel.
  • Beyond the paper, the BVD-based penalty mechanism could be combined with a positivity-preserving or realizability fix to extend the method to multiphase or very extreme compressible flows.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper introduces a family of third- to fifth-order weighted compact least-squares (WCLS) finite-volume schemes for compressible Navier-Stokes equations on curvilinear grids. The reconstruction weights the accuracy terms so that the compact reconstruction is performed only along smooth lines, adds a boundary-value-difference (BVD) penalty that is active near discontinuities, and embeds the nonlinear reconstruction into an implicit time-marching framework using GMRES. The accuracy theory is stated in Theorem 1, which requires the penalty coefficient theta_n to be O(Delta x^k) in smooth regions. The paper reports designed-order accuracy on smooth problems and improved resolution and lower dissipation than WENO3-JS, WENO5-Z, and WBAP-CLS schemes on a wide range of inviscid and viscous benchmark problems in one, two, and three dimensions.

Significance. If the accuracy claim holds, the WCLS framework offers a practical compromise between compact high-order resolution and robust shock capturing, extending the earlier third-order WCLS work to higher orders with an efficient implicit implementation. The method is novel in its combination of accuracy-preserving weighting, BVD-based penalties, and coupling of reconstruction with implicit iterations. The numerical validation is unusually broad, covering non-uniform curvilinear grids, viscous flows, and 3D turbulent benchmarks, and the reported efficiency comparisons are useful. The principal limitation is that the formal proof of the order-preservation property rests on a scaling assumption that is not rigorously established.

major comments (2)
  1. [Section 2, Eq. (21) and Theorem 1] The theorem's hypothesis that theta_n = O(Delta x^k) is not established at smooth extrema. The argument following Eq. (21) balances the boundary-value difference |p_i(x_alpha)-p_n(x_alpha)| = O(Delta x^{k+1}) against the cell-average difference |u_i-u_m| = O(Delta x), but at a smooth extremum the cell-average difference can be O(Delta x^2) or smaller, and the smoothness indicators of Eq. (12) in the extremal cell and its neighbor can differ by O(1) amounts, making |1/2 - beta_n| = O(1). Under those conditions Eq. (21) gives theta_n = O(Delta x^{k-1}), which is outside the theorem's hypothesis. The accuracy tests in Sec. 4.1 bypass the shock detector and therefore do not close the gap; the central claim that the BVD penalty 'goes to zero ... even near smooth extrema' needs either a rigorous proof or a modification of Eq. (21) that guarantees the required scaling in the active-weighting regime.
  2. [Section 3, Algorithm I and Eq. (11)] The reconstruction system (11) is nonlinear because beta and theta in Eqs. (14) and (21) depend on the unknown reconstruction coefficients. The manuscript freezes these quantities at the beginning of each GMRES iteration (Algorithm I, lines 6-18) and updates them in subsequent inner iterations, but it provides no convergence analysis for this fixed-point iteration: no contractivity estimate, no proof that the inner iteration reaches a solution of the nonlinear system, and no examination of the dependence of the final solution on the iteration path. Since the numerical results in Section 4 rely on the convergence of this coupled process, the paper should provide either a convergence proof or a systematic residual-convergence study to rule out iteration-path-dependent accuracy.
minor comments (6)
  1. [Figure 5 caption] The subfigure labels in the Figure 5 caption are inconsistent: the caption lists (a), (b), (c) and then (c), (d), (f) for the second row; please renumber the panels.
  2. [Section 4.13] The sentence 'The reference result is taken from Li [43]' cites reference [43], which is Zhang and Shu; the citation appears to be misattributed.
  3. [Section 2, Eq. (21)] The piecewise display of Eq. (21) is formatted as a three-branch case but contains only the formula for theta_n and the two sensor definitions S_{i-1} and S_{i+1}; the typesetting should be clarified so that the reader can identify which branches correspond to which variables.
  4. [Section 4.1] The accuracy results are presented as error plots in Fig. 5; reporting the observed convergence rates in a table would make the claimed third-, fourth-, and fifth-order rates easier to verify.
  5. [Section 3] The genetic algorithm used for the spectral optimization is not referenced; please cite the implementation or a standard reference.
  6. [Section 4.7] The grid disturbance type 'ALT-RAND' from Ref. [35] is not defined; a one-sentence definition would help readability.

Circularity Check

1 steps flagged · score 4.0 of 10

Formal-order proof is self-referential: Eq. (21) verifies Theorem 1's hypothesis by assuming Theorem 1's conclusion; numerical benchmarks remain independent.

  1. self definitional [Section 2, Eq. (21) and Theorem 1]
    "Theorem 1. Without considering the boundary conditions, if theta_n = O(Delta x^k), the polynomial p_i(x) reconstructed by the WCLS schemes achieves a prior k-th order of accuracy. ... In smooth regions, the flow is well-resolved and p_i(x_alpha) - p_n(x_alpha) = O(Delta x^{k+1}). ... Therefore, theta_n = O(Delta x^k) considering u_i - u_m = O(Delta x). According to Theorem 1, the prior k-th order of accuracy can be achieved."

    Theorem 1 is a conditional statement: kth-order accuracy follows from theta_n = O(h^k). The paper verifies that hypothesis by asserting p_i(x_alpha) - p_n(x_alpha) = O(h^{k+1}), but that BVD estimate is exactly the kth-order accuracy of the reconstructed polynomials that Theorem 1 is supposed to establish. Since theta_n is defined in Eq. (21) through this same BVD term, the claimed scaling theta_n = O(h^k) in smooth regions is not an independent estimate; it restates the target accuracy as its own hypothesis. At smooth extrema the argument further breaks down because the denominator sum |u_i - u_m| can be O(h^2) rather than O(h) and beta_n need not approach 1/2, so the paper's 'even near smooth extrema' claim is asserted rather than derived.

full rationale

The numerical method is not a fit disguised as a prediction: the spectral weights are optimized for linear advection and then applied to nonlinear flows, and the shock-capturing and resolution claims are tested on external benchmarks (Lax, Shu-Osher, double Mach reflection, TGV, etc.) against WENO and WBAP-CLS reference schemes. The only genuinely circular step is the formal-order argument: Eq. (21) defines theta_n via the boundary-value difference p_i - p_n, and the proof that this difference is O(h^{k+1}) assumes the very kth-order accuracy Theorem 1 is meant to deliver. The independent numerical accuracy tests in Sec. 4.1 (sin, sin^2, sin^3 with m = 1, 2, 3) do show designed order even in the presence of extrema, so the circularity is a proof-level gap rather than a manufactured numerical result. Hence the moderate score.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim rests on several unproven or prior-established components: the k-exact CLS accuracy property, the heuristic smooth-region scaling of the BVD penalty, the reliability of the modified smoothness indicator, and the validity of freezing weights within an iteration. No new physical entities are introduced.

free parameters (6)
  • Penalty coefficients c_h and c_s = c_h=0.1 (WCLS3), 0.05 (WCLS4/5); c_s=0.8
    Eq. (21); hand-set constants controlling baseline and adaptive penalty strength; no derivation; performance depends on them.
  • Shock detector threshold sigma_0 = 0.6
    Section 4; threshold deciding whether flow is treated as continuous and weights are equalized; hand-chosen.
  • Spectral optimization targets kappa_c,Im = 1.0, 1.2, 1.2 for 3rd/4th/5th order
    Section 2 and Table 1; chosen dissipation constraints for the genetic algorithm; not derived from first principles.
  • Spectral weights W1 to W4 = Table 1 values, e.g., 8.238e-02 and 1.072e-02 for 3rd order
    Optimized via genetic algorithm to maximize resolved wavenumber under dissipation constraints; free parameters of the scheme.
  • Boundary reconstruction weights W_bc = W_bc,0=100, W_bc,j=0 for viscous walls; W_bc,0=0, W_bc,j=0.5 for inviscid walls
    Section 4; hand-set to enforce wall boundary conditions; affect solution near boundaries.
  • Smoothness indicator epsilon = max(1e-10*(|u_i|+|u_{i-1}|+|u_{i+1}|)^2/9, 1e-50)
    Eq. (13); small regularization constant; not physically derived.
assumptions (5)
  • domain assumption The k-exact property of the CLS accuracy term I_acc (from prior work by Wang, Ren, Pan, and Li) ensures that M_acc^{-1} b_acc reconstructs polynomials with k-th order accuracy.
    Invoked in the proof of Theorem 1; relies on prior publications [11-13] from the same group; not re-proven here.
  • ad hoc to paper The BVD-based construction makes theta_n=O(Delta x^k) in all smooth regions.
    Eq. (21) and surrounding text; heuristic scaling argument; no proof near smooth extrema; load-bearing for Theorem 1.
  • ad hoc to paper The smoothness indicator of Eq. (12), with the added linear term q_i and the 1/(j!)^2 factors, reliably identifies discontinuities and permits iteration convergence.
    Section 2, Eq. (12); motivated by numerical observations in Fig. 1; no convergence analysis.
  • ad hoc to paper Freezing beta and theta during each GMRES iteration yields a valid linearization of the nonlinear reconstruction system.
    Section 2 and Algorithm 1; no error or convergence analysis provided.
  • domain assumption Perfect gas assumption with gamma=1.4 and Prandtl number Pr=0.72.
    Standard thermodynamics for air; used in the Navier-Stokes discretization of Section 3.

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Cite this review

Pith. "Pith review of High-Resolution Weighted Essentially Non-Oscillatory Compact Least-Squares Schemes with Implicit Time Integration for Compressible Navier-Stokes Equations on Curvilinear Grids." pith.science (2026). https://pith.science/paper/UIZ3BAXA

@misc{pith2026260808117,
  author       = {Pith},
  title        = {Pith review of: High-Resolution Weighted Essentially Non-Oscillatory Compact Least-Squares Schemes with Implicit Time Integration for Compressible Navier-Stokes Equations on Curvilinear Grids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UIZ3BAXA}},
  note         = {Machine review of arXiv:2608.08117}
}
read the original abstract

This paper presents a family of high-resolution weighted essentially non-oscillatory compact least-squares schemes with implicit time integration for the compressible Navier-Stokes equations on curvilinear grids. Compared with the original compact least-squares schemes, the proposed method introduces two main improvements. First, instead of enforcing the accuracy constraints over the entire computational domain, including discontinuous regions, it constructs the reconstruction matrix only along smooth reconstruction lines using an accuracy-preserving weighting strategy. This treatment effectively suppresses the persistent high-frequency oscillations observed in the original compact least-squares schemes and yields sharper profiles near discontinuities. Second, the method simplifies the shock-capturing procedure and improves efficiency by using a single set of polynomials, whereas the original compact least-squares schemes require both unlimited and limited polynomials. Combined with spectral optimization, the proposed method exhibits more favorable spectral properties than conventional weighted essentially non-oscillatory schemes. The smoothness indicators and penalty matrices are constructed through an efficient iterative procedure with modest additional cost. Numerical results for inviscid and viscous one-, two-, and three-dimensional flows demonstrate that the proposed method provides robust shock-capturing capability while maintaining high resolution in smooth regions and across contact discontinuities.

Figures

Figures reproduced from arXiv: 2608.08117 by the authors.

Figure 1
Figure 1. Reconstruction based on the proposed SI of Eq. (12) and SI by Jiang and Shu [20,27]. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Performance of the adopted penalty coefficients for stationary shock problems at different Mach numbers. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Spectral properties for the optimized WCLS schemes. UW3: the third- [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (43 more)
Figure 4
Figure 4. Figure 4: Reconstructed polynomials near smooth high-gradient regions and discontinuities. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Accuracy test for the WCLS schemes. In [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Computational efficiency for the WCLS schemes. [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Results of the WCLS schemes for Gaussian-square-triangle-ellipse waves. [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Close views of the Gaussian-square-triangle-ellipse waves. [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: Acoustic waves for different 𝑘଴ . 4.4 Lax Problem In this section, a shock tube problem, denoted as Lax problem [37], is utilized to further validate the robustness of the proposed WCLS schemes near shocks. The simulation domain is [0,1] discretized by 200 uniform cont…
Figure 10
Figure 10. Figure 10: Lax shock tube problem. N = 200, t = 0.1 and CFL = 1.0. 4.5 Shu-Osher Problem [38] The Shu-Osher problem is a standard benchmark for assessing both the resolution and shock-capturing capability of reconstruction methods for compressible flows. In this problem, a right…
Figure 11
Figure 11. Figure 11: Shu-Osher problem. N = 200, t = 1.8 and CFL = 1.0. 4.6 Isentropic Vortex Problem on Non-uniform Grids A smooth isentropic vortex is first used to assess the performance of the WCLS schemes on two-dimensional non￾uniform curvilinear structured grids. An isentropic dist…
Figure 12
Figure 12. Figure 12: Distorted-rand grids for isentropic vortex problem. [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: Density contours on distorted-rand grids at t = 10. 30 equally spaced contour lines from 0.5085 to 0.9860. [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]
Figure 14
Figure 14. Figure 14: Non-uniform grids for shock-vortex interaction problem. In [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]
Figure 15
Figure 15. Figure 15: Density contours for the shock vortex interaction with 30 lines ranging from 1.01 to 1.22. [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]
Figure 16
Figure 16. Figure 16: Density contours for the 2D Riemann problem with 32 lines ranging from 0.2 to 1.7. [PITH_FULL_IMAGE:figures/full_fig_p020_16.png]
Figure 17
Figure 17. Figure 17: Density contours for the double Mach reflection with 32 lines ranging from 2.2 to 22. [PITH_FULL_IMAGE:figures/full_fig_p020_17.png]
Figure 18
Figure 18. Figure 18: Close view of density contours for the double Mach reflection. [PITH_FULL_IMAGE:figures/full_fig_p021_18.png]
Figure 19
Figure 19. Figure 19: Density contours for the double Mach reflection with 32 lines ranging from 2.2 to 22. [PITH_FULL_IMAGE:figures/full_fig_p022_19.png]
Figure 20
Figure 20. Figure 20: Close view of density contours for the double Mach reflection. [PITH_FULL_IMAGE:figures/full_fig_p022_20.png]
Figure 21
Figure 21. Figure 21: Density contours of the 2D high-Mach-number astrophysical jet computed by different schemes for Mach 80. [PITH_FULL_IMAGE:figures/full_fig_p023_21.png]
Figure 22
Figure 22. Figure 22: Density contours of the 2D high-Mach-number astrophysical jet computed by different schemes for Mach 2000. [PITH_FULL_IMAGE:figures/full_fig_p024_22.png]
Figure 23
Figure 23. Figure 23: Density contours for the viscous shock tube problem. 30 equally spaced contour lines from 22.26 to 121.34. [PITH_FULL_IMAGE:figures/full_fig_p024_23.png]
Figure 24
Figure 24. Figure 24: Distribution of the density along the bottom wall using different schemes. [PITH_FULL_IMAGE:figures/full_fig_p025_24.png]
Figure 25
Figure 25. Figure 25: Density contours by the WENO and WCLS schemes for shock–mixing-layer interaction. 20 equally spaced 0.4 to 2.8. [PITH_FULL_IMAGE:figures/full_fig_p026_25.png]
Figure 26
Figure 26. Figure 26: Pressure contours by the WENO and WCLS schemes for shock–mixing-layer interaction. 20 equally spaced 0.23 to 0.73. [PITH_FULL_IMAGE:figures/full_fig_p026_26.png]
Figure 27
Figure 27. Figure 27: Temperature contours by the WENO and WCLS schemes for shock–mixing-layer interaction. 20 equally spaced 0.2 to 1.1. [PITH_FULL_IMAGE:figures/full_fig_p026_27.png]
Figure 28
Figure 28. Figure 28: Pressure distributions along the line from point (90, 0) to point (200, −6). [PITH_FULL_IMAGE:figures/full_fig_p027_28.png]
Figure 29
Figure 29. Figure 29: The CFL number is 1000 and simulation is continued until converged. [PITH_FULL_IMAGE:figures/full_fig_p027_29.png]
Figure 29
Figure 29. Figure 29: Structured non-uniform grids [PITH_FULL_IMAGE:figures/full_fig_p028_29.png]
Figure 30
Figure 30. Figure 30: Skin friction coefficient (left) and pressure coefficients (right) on the airfoil obtained by different schemes. [PITH_FULL_IMAGE:figures/full_fig_p028_30.png]
Figure 31
Figure 31. Figure 31: Density contours using different schemes. 30 equally spaced lines from 0.88 to 1.13. [PITH_FULL_IMAGE:figures/full_fig_p029_31.png]
Figure 32
Figure 32. Figure 32: Structured non-uniform mesh. (a) WENO3-JS [PITH_FULL_IMAGE:figures/full_fig_p029_32.png]
Figure 33
Figure 33. Figure 33: Mach contours for the flow with Re 10,000 around the SD7003 airfoil using different schemes. 32 equally spaced 0.008 to 0.26. [PITH_FULL_IMAGE:figures/full_fig_p030_33.png]
Figure 34
Figure 34. Figure 34: Vorticity contours for the flow with Re 10,000 around the SD7003 airfoil with different schemes. [PITH_FULL_IMAGE:figures/full_fig_p031_34.png]
Figure 35
Figure 35. Figure 35: Density contours of the 2D Kelvin-Helmholtz instability computed by different schemes [PITH_FULL_IMAGE:figures/full_fig_p032_35.png]
Figure 36
Figure 36. Figure 36: Density contours of the 2D Kelvin-Helmholtz instability computed by different schemes [PITH_FULL_IMAGE:figures/full_fig_p033_36.png]
Figure 37
Figure 37. Figure 37: Density contours of the 2D Kelvin-Helmholtz instability computed by different schemes [PITH_FULL_IMAGE:figures/full_fig_p034_37.png]
Figure 38
Figure 38. Figure 38: Iso-contours (Q = 1.5) obtained for the 3D inviscid TGV problem using different schemes at t = 10. [PITH_FULL_IMAGE:figures/full_fig_p036_38.png]
Figure 39
Figure 39. Figure 39: The evolution of kinetic energy versus dimensionless time obtained by different schemes for 3D inviscid TGV flows. [PITH_FULL_IMAGE:figures/full_fig_p036_39.png]
Figure 40
Figure 40. Figure 40: Iso-surfaces of Q = 2.5, colored by the u-velocity, obtained by different schemes for the viscous TGV problem. [PITH_FULL_IMAGE:figures/full_fig_p038_40.png]
Figure 41
Figure 41. Figure 41: Comparison of different schemes for the viscous TGV problem. (a) Temporal evolution of turbulent kinetic energy; (b) temporal evolution of the turbulent dissipation rate; (c) energy spectrum at t = 20 s. (a) WENO3-JS (b) WENO5-Z [PITH_FULL_IMAGE:figures/full_fig_p038…
Figure 42
Figure 42. Figure 42: Cross-sectional contours of the density-gradient norm at [PITH_FULL_IMAGE:figures/full_fig_p039_42.png]
Figure 43
Figure 43. Figure 43: Iso-surfaces of Q = 0.75, colored by the x-velocity, obtained by different schemes for the viscous TGV problem of Case II. [PITH_FULL_IMAGE:figures/full_fig_p040_43.png]
Figure 44
Figure 44. Figure 44: Comparison of different schemes for the viscous TGV problem of Case II at [PITH_FULL_IMAGE:figures/full_fig_p040_44.png]
Figure 45
Figure 45. Figure 45: Comparison of normalized computational cost for different unsteady test cases. DM-240 and DM-480 denote the double Mach reflection case with [PITH_FULL_IMAGE:figures/full_fig_p041_45.png]

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