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Investigation of Shock-Capturing with Bound-Preserving Limiters for the Nonlinearly Stable Flux Reconstruction Method

T0 review · 2 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that adding solution-node checks to the positivity-preserving limiter lets the nonlinearly stable flux reconstruction method handle shock-dominated compressible flows without TVD limiting, while keeping high-order…

desk verdict Useful, honestly presented numerical study of NSFR shock-capturing, but the modified limiter's guarantees are asserted rather than proved and the GL flux-node case is not covered by the checked node sets. read the letter →

arxiv 2507.09131 v1 pith:7ZQTGK6D submitted 2025-07-12 math.NA cs.NAphysics.flu-dyn

Sai Shruthi Srinivasan , Siva Nadarajah This is my paper · ORCID
classification math.NAcs.NAphysics.flu-dyn MSC 65M6065M7076N15
keywords nonlinearlystablefluxreconstructionpositivity-preservinglimiterbound-preservingcompressibleEulerequationsshockcapturingparametertwo-pointentropystability
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 argues that a small change to a standard positivity-preserving limiter makes the nonlinearly stable flux reconstruction (NSFR) method reliable for shock-dominated compressible flows. The change is to evaluate minimum density and pressure not only on the two tensor-product quadrature node sets used by the original limiter, but also on the solution nodes themselves. With that addition, the paper claims, the limiter still preserves high-order accuracy, conservation, and positivity, while preventing the negative density or pressure that previously appeared at solution nodes. If true, NSFR can run demanding tests — shock diffraction, double Mach reflection, and a Mach 2000 astrophysical jet — without any TVD-type limiter, at larger time steps than standard DG methods. The paper supports the claim with a suite of one- and two-dimensional Euler tests rather than a proof.

What carries the argument

The load-bearing object is the modified positivity-preserving limiter, given as Algorithm 1. It computes cell-averaged density and pressure from two tensor-product quadrature rules, then forms two scaling factors: $\theta_1$ scales the density toward the cell average so that the minimum density over all three node sets is at least $\varepsilon = 10^{-13}$, and $\theta_2$ scales the whole state vector toward the cell average so that the minimum pressure over the same three sets is positive. The cell average is untouched by both scalings, which is what conserves mass and maintains accuracy; the novel step is including the solution-node set $\xi^{r,3} = \xi^{\alpha} \otimes \eta^{\alpha}$ in the min computations.

What would settle it

Construct or find a cell state where density and pressure are positive on both tensored quadrature node sets and on the solution-node set at the start of a stage, but where after one SSPRK3 stage with the CHRA flux some solution node has negative density or pressure; such a case would refute the claimed guarantee. A direct way to look is to run the shock-diffraction or Mach 2000 jet setup with the modified limiter and monitor the solution-node minima at every Runge-Kutta stage for any negative value.

Watch

Extended reading notes

Core claim

The central discovery is that robustness of the positivity-preserving limiter is governed by where the minima are sampled. The paper modifies the two-stage scaling procedure so that the minimum density $\rho_{\min}$ and minimum pressure $P_{\min}$ are computed over three node sets: the two tensored quadrature sets used to compute cell averages, plus the Gauss-Lobatto solution node set. Because the scaling factors $\theta_1$ and $\theta_2$ are applied directly at the solution nodes, checking those nodes ensures that the limited polynomial is admissible exactly where the scheme evaluates it. The paper demonstrates, case by case, that this modification lets the NSFR scheme complete shock-dominated simulations that fail with the original limiter, at grid resolutions and CFL numbers where the unmodified approach produces nonphysical values.

Load-bearing premise

The central premise is that checking the minimum density and pressure on the solution nodes in addition to the quadrature nodes preserves the limiter's guarantees of positivity, accuracy, and conservation; the paper asserts this and verifies it numerically, but does not prove it.

Editorial extensions

If this is right

  • NSFR with the modified limiter can run the Sod shock tube, Shu-Osher, and strong vortex-shock wave interaction at CFL numbers where standard DG fails, and in some cases without the limiter at all.
  • The limiter preserves the expected order of accuracy: the 2D low-density convergence test reaches the designed rates at $p=2$ and $p=3$ as the grid is refined.
  • Increasing the flux-reconstruction parameter from $c_{DG}$ toward $c_{+}$ damps oscillations and overshoots and raises the maximum stable CFL, at the cost of extra dissipation in smooth regions.
  • The Chandrashekar-Ranocha two-point flux numerically satisfies the CFL condition $\frac{\Delta t}{\Delta x}\max(|u|+c) \le 1$ for positivity, independent of polynomial degree, in both one and two dimensions.
  • With a sufficiently strong FR parameter, the Mach 80 and Mach 2000 astrophysical jet cases run to completion with positivity preserved and no TVD limiting.

Reading between the lines

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

  • The same node-set enlargement could be applied to maximum-principle limiters for scalar conservation laws, where checking solution nodes instead of only quadrature nodes may similarly prevent bound violations at the interpolation points.
  • Because the modification is validated numerically rather than proved, a natural next step is a proof that positivity at the three node sets implies positivity of the scaled polynomial at every solution node under the given CFL condition; that would turn the robustness claim into a theorem.
  • The paper's evidence that larger FR parameters suppress oscillations suggests an adaptive strategy: pick $c$ locally from a shock sensor, using larger values near discontinuities and $c_{DG}$ in smooth flow, which the authors note but do not implement.
  • The numerically established CFL bound for the CHRA flux could be tested for the other two-point fluxes considered in the paper to see whether the same $\Delta t/\Delta x$ condition is flux-independent.
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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 extends the Zhang-Shu/Wang-Shu positivity-preserving limiter to the nonlinearly stable flux reconstruction (NSFR) scheme for the compressible Euler equations. The main modification is to include the solution-node set (xi_{r,3}, the GLL tensor product) in the computation of the minimum density and pressure that enter the two scaling parameters theta_1 and theta_2, in addition to the two mixed quadrature sets xi_{r,1} and xi_{r,2}. The authors assert that this preserves the limiter's properties of positivity, high-order accuracy, and conservativity, and they verify the method on a wide range of one- and two-dimensional problems: a low-density accuracy test, Sod and Shu-Osher shock tubes, the Leblanc shock tube, strong vortex-shock interaction, shock diffraction, double Mach reflection, and Mach 80 and Mach 2000 astrophysical jets. The paper also studies the influence of the two-point flux, the choice of GLL versus GL flux quadrature nodes, the flux reconstruction correction parameter, and numerically investigates a CFL condition for positivity of the CHRA two-point flux.

Significance. If the central claim holds, the paper makes a useful practical contribution: it shows that NSFR, equipped with a modified bound-preserving limiter, can run strongly shocked compressible flows without TVD or subcell limiting, while maintaining high-order accuracy in smooth regions. The numerical evidence is substantial and well organized: convergence tables (Tables 2 and 3) demonstrate the expected order on fine grids, the CFL study (Tables 4, 6, and 7) provides useful quantitative guidance for the CHRA flux, and the parameter study of the FR correction parameter gives a clear picture of the trade-off between oscillation control and accuracy. The comparisons against standard DG are also informative, particularly the entropy behavior in the strong vortex-shock interaction case. The main weakness is that the key property-preservation claim for the modified limiter is asserted rather than proved, and the statement that positivity is 'guaranteed' is stronger than what is actually established for uncollocated GL flux-node runs.

major comments (2)
  1. [§3.1, Algorithm 1] The sentence 'this modification preserves the properties of the limiter' is an assertion, not a proof. The three properties claimed for the enlarged node set {xi_{r,1}, xi_{r,2}, xi_{r,3}}—conservativity, positivity at the checked nodes, and high-order accuracy—are not derived. Conservativity follows immediately from the affine scaling about the cell average, and positivity at the checked nodes is plausible from convexity of the admissible set, but neither is shown. Accuracy requires an argument that the minimum over the enlarged set is not far enough from the cell average in smooth regions to degrade the convergence order. Because the abstract and conclusion state that the modified limiter 'guarantees positivity' and preserves the properties of the limiter, the manuscript should either provide this proof or explicitly downgrade the claim to a numerically verified modification.
  2. [§4.3.3, §4.5.4] The positivity guarantee is only asserted for the checked node sets. When GL flux quadrature is used, the pure GL x GL nodes are not included in any of the sets xi_{r,1}, xi_{r,2}, or xi_{r,3}; these are precisely the volume flux nodes where the solution is evaluated for the two-point flux. The GL runs in Sections 4.3.3 and 4.5.4 therefore rely on the reduced CFL values (0.05 and 0.07, respectively) rather than on a limiter guarantee. The conclusion's statement that the modified limiter 'guarantees positivity for the entire duration of the test' is too strong for these configurations. Please state the guarantee precisely—positivity at the solution nodes and the mixed quadrature sets—and note that the GL flux-node runs are supported by the reduced CFL, not by the limiter alone.
minor comments (6)
  1. [§4.1, Tables 2 and 3] Table 2 uses initial data '1 + 0.999sin(x+y)' in its caption while the text of Section 4.1 specifies '1 + 0.995sin(x+y)' for both the initial condition and exact solution; please reconcile the two values.
  2. [Algorithm 1] Line 3 of Algorithm 1 reads 'rho(xi_{r,})' where it should read 'rho(xi_{r,1})', and line 6 uses 'theta' where the text uses 'theta_1'; please correct these typos so the algorithm can be followed unambiguously.
  3. [Figure 11 caption] The caption for Figure 11 states 'at t=0.2s', but the Leblanc shock tube is run to a final time t=1e-4 in Section 4.4; this is either a typo or an unexplained inconsistency.
  4. [Figure 17 caption] The caption for Figure 17 states 'at t=1.8s', but the strong vortex-shock interaction case is run to t=0.7s in Section 4.5; please correct the time label.
  5. [References] References [1] and [36] appear to be the same paper ('On maximum-principle-satisfying high order schemes for scalar conservation laws'); citing the same work twice with different numbers should be fixed.
  6. [Tables 4, 6, and 7] The CFL verification tables list time steps and computed CFL values but do not indicate which rows failed; please mark the 'last successful' and 'first failing' rows explicitly so the reader can verify the claimed CFL bounds of 1 (Eq. 29) and 0.5 (Eq. 30).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the limiter modification is an algorithmic extension numerically verified against external benchmarks; self-citations to NSFR stability are independent prior published results.

full rationale

The paper's central contribution is a modified positivity-preserving limiter that enlarges the set of nodes used to compute minimum density and pressure from the two mixed quadrature sets (ξr,1, ξr,2) to also include the solution-node set (ξr,3). This is presented as an algorithmic modification in Algorithm 1 and Section 3.1, and it is verified through convergence studies (Section 4.1) and a suite of shock-dominated benchmark problems (Sod, Shu–Osher, Leblanc, SVSW, shock diffraction, double Mach reflection, astrophysical jets) that have exact or independently established reference solutions. No load-bearing step reduces to the paper's own inputs: the limiter's scaling parameters θ1 and θ2 are computed from cell averages and element-local minima, and the success criteria (positivity, order of accuracy, reproduction of known solution features) are measured against external references. The phrase 'this modification preserves the properties of the limiter' (Section 3.1) is an assertion without a proof, and the GL-flux runs in Sections 4.3.3 and 4.5.4 rely on lowered CFL values rather than a proven guarantee at pure GL×GL nodes; these are correctness or support gaps, not circular reasoning. Self-citations to Cicchino et al. [22,23,24] and Cicchino [42] concern the NSFR discretization and its stability proofs, which are prior published results (Journal of Computational Physics) that predate and are independent of the present limiter modification; they are not invoked to derive the limiter claim, and the current tests would fail if those prior results were unsound. The numerical CFL studies (e.g., Tables 4, 6, 7) verify known sufficient conditions for two-point fluxes rather than fitting parameters to force a predicted outcome. The paper is therefore self-contained against external benchmarks and exhibits no circular derivation chain.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central contribution (the modified limiter) rests on an unproved preservation property and an empirically established CFL condition. The NSFR scheme and CHRA flux are taken from prior published work. No new physical entities are introduced.

free parameters (2)
  • epsilon (epsilon) threshold = 1e-13
    Lower bound for the cell-average density and pressure in the limiter (Section 3.1.1). Chosen by hand as a small safety margin; it is not fitted to match data and is a standard tolerance.
  • Flux reconstruction parameter c = 0 (c_DG), 7.44e-6 (c_SD), 1.32e-5 (c_HU), 2.87e-5 (c+), 2.87e-4 (c+ x10) for p=3 (Table 1).
    Tuning parameter of the ESFR/NSFR scheme that controls filtering of the highest mode. The paper does not fit it to data but explores its effect; final recommendation is larger c for shocks, at the cost of smooth-region accuracy.
assumptions (4)
  • domain assumption The NSFR scheme (Cicchino et al. [22,23,24]) is nonlinearly stable for the Euler equations on the discretizations used here.
    The paper takes the NSFR stability proof from prior work and does not re-prove it for the limited scheme. The limiter is applied at each RK stage, so the entropy stability of the overall algorithm is not established.
  • ad hoc to paper Including the solution nodes (xi_r,3) in the computation of rho_min and P_min preserves the positivity-preserving, accuracy, and conservativity properties of the Wang-Shu limiter.
    No proof is given; Section 3.1 states 'this modification preserves the properties of the limiter' and relies on numerical verification. This is the paper's central unproved step.
  • ad hoc to paper The CFL condition (Eq. 29) is sufficient for positivity of density and pressure for the CHRA two-point flux with Roe dissipation.
    Section 3.2 notes that determining a CFL condition for logarithmic-mean two-point fluxes is challenging and the paper 'aims to numerically establish' it. Verified only on Sod, Shu-Osher, and SVSW cases.
  • domain assumption The CHRA two-point flux is entropy conserving, kinetic energy preserving, and pressure-equilibrium-preserving as stated in [45].
    The paper relies on the published properties of the flux; these properties are used to argue entropy stability of the spatial discretization.

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Cite this review

Pith. "Pith review of Investigation of Shock-Capturing with Bound-Preserving Limiters for the Nonlinearly Stable Flux Reconstruction Method." pith.science (2026). https://pith.science/paper/7ZQTGK6D

@misc{pith2026250709131,
  author       = {Pith},
  title        = {Pith review of: Investigation of Shock-Capturing with Bound-Preserving Limiters for the Nonlinearly Stable Flux Reconstruction Method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7ZQTGK6D}},
  note         = {Machine review of arXiv:2507.09131}
}
read the original abstract

Nonlinearly stable flux reconstruction (NSFR) combines the key properties of provable nonlinear stability with the increased time step from energy-stable flux reconstruction. The NSFR scheme has been successfully applied to unsteady compressible flows. Through the use of a bound-preserving limiter, positivity of thermodynamic quantities is preserved, and this enables the extension of NSFR to hyperbolic conservation laws. We extend the limiter of Zhang and Shu [1] to ensure robustness for the proposed scheme. The limiter is modified to consider the minimum density and pressure at the solution nodes when determining the value to scale the solution. The modifications are thoroughly tested with a suite of test cases. In addition to these modifications, this paper conducts a thorough investigation into the shock-capturing capabilities of the NSFR scheme and the advantages it presents over standard discontinuous Galerkin (DG) methods, where, on select variants of the flux reconstruction (FR) scheme, essentially oscillation-free solutions are demonstrated. Various parameters of the scheme are extensively tested and analyzed through several 1D and 2D compressible Euler tests that verify the high-order accuracy, entropy stability, time step advantage and shock-capturing capabilities of the NSFR scheme. These parameters include the two-point flux, quadrature nodes and the strength of the FR parameter. In addition to investigating the impact of the various two-point fluxes, this paper also presents numerical studies to determine the CFL condition required to maintain positivity for the two-point flux of choice. The investigation yields insightful results for all parameters, with the results pertaining to the type of FR scheme being of special interest. The tests showcase increased robustness, time step advantages and oscillation/overshoot mitigation when employing a stronger FR parameter.

Figures

Figures reproduced from arXiv: 2507.09131 by the authors.

Figure 1
Figure 1. [Sod Shock Tube] Entropy over time for initial three test configurations (top) and density and pressure with SSPRK3 Strong DG with TVD limiter (bottom left) and NSFR (bottom right), using p = 3, CF L 0.5 and the PPL at t = 0.2s of the tests is N = 240, 192, 160 respectively to ensure the tests have an equivalent number of degrees of freedom. The three solutions for the different polynomial degrees are compared in [… view at source ↗
Figure 2
Figure 2. [Sod Shock Tube] Density with NSFR-CHRA, Roe dissipation, cDG, p = 3, 4, 5, CF L 0.5, 0.4, 0.3 respectively with PPL at t = 0.2s. Plots of full domain (left) and 0 ≤ x ≤ 3.6 4.2.3. CFL Condition for Two-Point Flux This test is used to numerically verify that the two-point flux used for the tests in this investigation abides by a CF L condition. The number of cells for this test is N = 500. The test is run for polyno… view at source ↗
Figure 3
Figure 3. [Sod Shock Tube] Density and pressure with NSFR, Roe dissipation, cDG, p = 3, CF L 0.5 with PPL at t = 0.2s for CHRA, CH, IR and KG 4.2.5. Investigation of Flux Reconstruction Schemes Here, we investigate variants of the flux reconstruction scheme. The Sod Shock Tube case is run with the cSD, cHU , and c+ schemes with p = 3 and a CF L of 0.5. Using the solution obtained with the cDG scheme in [PITH_FULL_IMAGE:figur… view at source ↗
Figures from the paper (24 more)
Figure 4
Figure 4. Figure 4: [Sod Shock Tube] Density with NSFR-CHRA, Roe dissipation, p = 3, CF L 0.5 respectively with PPL at t = 0.2s for different flux reconstruction parameters. Plots of full domain (left) and 0 ≤ x ≤ 3.6 TVB/TVD limiting. The test also sufficiently verifies the CFL condition…
Figure 5
Figure 5. Figure 5: [Shu Osher Problem] Density solution with NSFR-CHRA, Roe dissipation, cDG, no PPL at t = 1.8s. Plots of full domain (left) and −2.6 ≤ x ≤ 2.6 (right). 4.3.1. CFL Condition for Two-Point Flux This test is also used to numerically verify that the two-point flux used thro…
Figure 6
Figure 6. Figure 6: [Shu Osher Problem] Density solution with NSFR-CHRA, Roe dissipation, cDG, PPL at t = 1.8s. Plots of full domain (left) and −2.6 ≤ x ≤ 2.6 (right). quadratures such as GL, the solution can spike when evaluating the entropy-projected conservative variables at surface po…
Figure 7
Figure 7. Figure 7: [Shu Osher Problem] Density solution with NSFR-CHRA, Roe dissipation, c+, PPL at t = 1.8s. Plots of full domain (left) and −2.6 ≤ x ≤ 2.6 (right). 4.4. Leblanc Shock Tube The modified version of the Leblanc Shock Tube case seen in Zhang and Shu [43] is used to verify t…
Figure 8
Figure 8. Figure 8: [Shu Osher Problem] Density solution with NSFR-CHRA, Roe dissipation, cDG, PPL at t = 1.8s. Plots of GLL nodes (top) and GL nodes (bottom). Plots of full domain (left) and −2.6 ≤ x ≤ 2.6 (right). 27 [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]
Figure 9
Figure 9. Figure 9: [Shu Osher Problem] Density solution with NSFR-CHRA, Roe dissipation, cDG, PPL at t = 1.8s for p = 3, 4. Plots of full domain (left) and −2.6 ≤ x ≤ 2.6 (right). [53]. This test case shows that the PPL can be used in conjunction with NSFR schemes to ensure that the posi…
Figure 10
Figure 10. Figure 10: [Leblanc Shock Tube] Density (left) and pressure (right) with NSFR-CHRA, Roe dissipation, cDG, p = 3, CF L 0.06 and PPL at t = 0.0001s 4 5 6 7 8 9 10 x 0.00 0.02 0.04 0.06 0.08 0.10 Density cDG, CFL 0.06 cSD, CFL 0.28 cHU , CFL 0.29 c+, CFL 0.3 4 5 6 7 8 9 10 x 0.0 0.…
Figure 11
Figure 11. Figure 11: [Leblanc Shock Tube] Density (left) and pressure (right) with NSFR-CHRA, Roe dissipation, p = 3, using the PPL, comparing cDG, cSD, cHU , c+ for 4 ≤ x ≤ 10 at t = 0.2s 29 [PITH_FULL_IMAGE:figures/full_fig_p029_11.png]
Figure 12
Figure 12. Figure 12: [Leblanc Shock Tube] Density (left) and pressure (right) with NSFR-CHRA, Roe dissipation, p = 3, using the PPL, comparing cDG, cSD, cHU , c+ for 4 ≤ x ≤ 10 with CF L = 0.06 at t = 0.0001s ent flux reconstruction parameters are run again with a CF L of 0.06 for all val…
Figure 13
Figure 13. Figure 13: As expected, the strong vortex is split into two separate vortices due to the compression effects [PITH_FULL_IMAGE:figures/full_fig_p031_13.png]
Figure 14
Figure 14. Figure 14: [SVSW] Density solution with NSFR-CHRA, Roe dissipation, cDG, with and without PPL at t = 0.7s. Plot along y = 0.4 (left) and x = 1.05 (right). 4.5.2. Comparison with Strong DG To highlight the difference between the SSPRK3 Strong DG method and the NSFR method, the St…
Figure 15
Figure 15. Figure 15: [SVSW] Top: Density solution with the SSPRK3 Strong DG method at t = 0.7s. Bottom: Plot of entropy over time for the NSFR-CHRA, Roe dissipation, cDG method with and without the PPL and SSPRK3 Strong DG method with the PPL 32 [PITH_FULL_IMAGE:figures/full_fig_p032_15.png]
Figure 16
Figure 16. Figure 16: [SVSW] Density solution with NSFR-CHRA, Roe dissipation, cDG, PPL using GL flux nodes at t = 0.7s The density plots along the lines y = 0.4 and x = 1.05 are also provided for the GL results. To best compare the collocated and uncollocated schemes, the GLL test is reru…
Figure 17
Figure 17. Figure 17: [SVSW] Density solution with NSFR-CHRA, Roe dissipation, cDG, with GLL and GL flux nodes at t = 1.8s. Plot along y = 0.4 (left) and x = 1.05 (right). the KG flux to the three aforementioned fluxes, it is obvious that the results obtained with the KG flux are more osci…
Figure 18
Figure 18. Figure 18: [Extended SVSW] Plot of density along y = 0.425 with NSFR, Roe dissipation, cDG using different entropy conserving fluxes. Top row: t = 0.00s (Left), t = 0.50s (Middle) and t = 1.00s (Right). Bottom row: t = 1.25s (Left), t = 1.50s (Middle) and t = 1.75s (Right) 4.6. …
Figure 19
Figure 19. Figure 19: [Extended SVSW] Plot of entropy over time with NSFR, Roe dissipation, cDG, PPL for the different entropy conserving fluxes. is outflow and the top boundary, y = 11, 0 ≤ x ≤ 13, is a wall boundary. All the schemes are run with a CF L of 0.45. The test fails due to nonp…
Figure 20
Figure 20. Figure 20: [Shock Diffraction] Density solution with NSFR-CHRA, cDG, Roe dissipation, p = 3, PPL at t = 2.3s schemes is difficult due to the presence of the KHI. In the plot focusing on the wake of the shock, we can see that there is a significant amount of oscillations in the c…
Figure 21
Figure 21. Figure 21: [Shock Diffraction] Density solution with NSFR-CHRA, Roe dissipation, p = 3, PPL at t = 2.3s, using different correction parameters - cDG (top left), cSD (top right), cHU (bottom left), c+ (bottom right). 39 [PITH_FULL_IMAGE:figures/full_fig_p039_21.png]
Figure 22
Figure 22. Figure 22: [Shock Diffraction] Plot of density across (1, 0) → (13, 11) with NSFR, Roe dissipation, PPL at t = 2.3s using different flux reconstruction correction parameters (left) with a closer look at the wake of the shock (right) 0 2 4 6 8 10 12 14 16 x 0 1 2 3 4 5 6 7 8 Dens…
Figure 23
Figure 23. Figure 23: [Shock Diffraction] Plot of density across (1, 0) → (13, 11) with NSFR, Roe dissipation, PPL at t = 2.3s (left) with a closer look at the wake of the shock (right) 40 [PITH_FULL_IMAGE:figures/full_fig_p040_23.png]
Figure 24
Figure 24. Figure 24: [DMR] Density solution with NSFR, Roe dissipation, cDG, PPL at t = 0.2s, using different polynomial degrees - 3 (top), 4 (middle), 5 (bottom) 42 [PITH_FULL_IMAGE:figures/full_fig_p042_24.png]
Figure 25
Figure 25. Figure 25: [DMR] Density solution with NSFR, Roe dissipation, PPL at t = 0.2s, using different flux reconstruction parameters - cDG (top), c+ (bottom) In both sets of results, there are two triple points and two slip lines present. The instabilities along the 43 [PITH_FULL_IMAG…
Figure 26
Figure 26. Figure 26: [Astrophysical Jet Mach 80] Density solution with NSFR-CHRA, Roe dissipation, c+, PPL at t = 0.07s. 4.8.2. Mach 2000 Astrophysical Jet The computational domain for the Mach 2000 astrophysical jet is [−0.5, 0.5]2 with the initial condition (ρ, u, v, p) = (0.5, 0, 0, 0.…
Figure 27
Figure 27. Figure 27: [Astrophysical Jet Mach 2000] Density solution with NSFR-CHRA, Roe dissipation, c+ × 10, PPL at t = 0.001s. 5. Conclusion In this paper, the NSFR schemes [22, 23], which have been successfully applied to unsteady compressible flows in arbitrary curvilinear coordinates…

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