Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

An $rp$-adaptive method for accurate resolution of shock-dominated viscous flow based on implicit shock tracking

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read Viscous shock-dominated flows can be resolved without nonlinear stabilization by letting the mesh nodes move as part of the discrete solve, compressing elements into shocks and boundary layers.

desk verdict A solid, honest extension of HOIST to viscous flows; the cylinder heat-flux claim is credible but lacks an independent reference. read the letter →

arxiv 2504.15177 v1 pith:QVF4LDRS submitted 2025-04-21 math.NA cs.NAmath.OCphysics.flu-dyn

Huijing Dong , Masayuki Yano , Tianci Huang , Matthew J. Zahr This is my paper · ORCID
classification math.NAcs.NAmath.OCphysics.flu-dyn MSC 65N3065N5076M1076K05
keywords implicitshocktrackingrp-adaptivitydiscontinuousGalerkinhypersonicflowviscousboundarylayerheatfluxpredictionmeshoptimization
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 claims that viscous, shock-dominated flows can be resolved accurately without nonlinear stabilization if the computational mesh is allowed to move while the flow is being solved. The method treats the mesh nodal coordinates and the discrete flow variables as unknowns of a constrained optimization: the objective is an enriched discontinuous Galerkin residual, and the constraint is the IPDG equations, so minimizing drives elements into the viscous shocks and boundary layers. A p-adaptivity loop then raises the polynomial degree where the enriched residual is largest, reducing how much mesh compression is required. On hypersonic flow over a cylinder, the method predicts pointwise heat flux profiles, and for the integrated heat flux it reaches sub-1% error with fewer degrees of freedom than high-order h-adaptation.

What carries the argument

The load-bearing object is a PDE-constrained optimization problem whose objective is the two-norm of the enriched IPDG residual plus a small mesh-distortion penalty, with the IPDG equations imposed as constraints. The enriched residual uses a test space two degrees higher than the trial space (enrichment degree $\Delta=2$), making the objective sensitive to under-resolved viscous transitions. A constant amplification weight $\rho=\lambda$ on wall-adjacent elements forces boundary layers to compete with stronger shocks in the objective; viscosity and Reynolds-number continuation supplies robust initializations that prevent carbuncles; and a sequential quadratic programming solver with step-length limits, pseudo-transient step modifications, and enriched-residual-based Hessian regularization drives elements to extreme compression. The p-adaptive loop uses the enriched residual as an error indicator to raise polynomial degree locally in under-resolved regions.

What would settle it

Apply the method with the paper's fixed parameters ($\lambda=10$, $\Delta=2$, same continuation schedule) to hypersonic flow over a different blunt geometry or Mach number: if the final pseudo-transient solve cannot drive the DG residual to about $10^{-9}$ without stabilization, or the heat flux profile departs by more than 1% from a high-fidelity reference, the central claim fails outside the demonstrated case.

Watch

Extended reading notes

Core claim

The central claim is that aggressive r-adaptation via implicit shock tracking can resolve the internal structure of viscous shocks and boundary layers on coarse meshes, making nonlinear stabilization unnecessary. The discrete solution and nodal coordinates are found together as the minimizer of the norm of an enriched interior-penalty DG residual (a test space two polynomial degrees above the trial space), subject to the DG equations; the minimizer compresses elements into the steep transitions. With viscosity continuation to avoid carbuncles, boundary-layer residual weighting to balance weak and strong features, and p-refinement driven by the same enriched residual, the method reproduces the Blasius boundary-layer profile and skin friction on a flat plate, and yields accurate heat flux profiles for $M_\infty=5$ flow over a cylinder. In the cylinder test, the rp-adaptive method achieves sub-1% integrated heat-flux error with fewer degrees of freedom than h-adaptation at constant polynomial degrees 1, 2, and 3.

Load-bearing premise

The stabilization-free resolution rests on the hand-tuned choices of the boundary-layer amplification factor $\lambda$, enrichment degree $\Delta=2$, the viscosity continuation schedule, and the enriched-residual Hessian regularization producing sufficient mesh compression on any new viscous, shock-dominated problem; the paper demonstrates failures when these are changed, including a carbuncle without continuation and poor boundary-layer resolution with $\Delta=1$ or $\lambda=1$.

Editorial extensions

If this is right

  • The hypersonic cylinder example indicates that heat flux, a notoriously sensitive output, can be predicted on coarse grids without artificial viscosity or limiting after several p-refinement rounds.
  • Because p-adaptivity shares the resolution burden, the required mesh compression is less extreme than in pure r-adaptation, so element stretching and polynomial degree can be balanced to save degrees of freedom.
  • Reynolds-number continuation is a necessary ingredient of the stabilization-free claim: without it, carbuncles form and the moving mesh tracks them.
  • Using the enriched residual as the p-adaptation indicator also keeps the shock-aligned mesh that allows deep pseudo-transient convergence without stabilization, something the dual-weighted-residual and feature-based indicators do not achieve.
  • On the flat-plate drag comparison, the rp-adaptive method is not more efficient than h-adaptation; its efficiency advantage appears in the cylinder heat-flux problem.

Reading between the lines

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

  • If the hand-set parameters ($\lambda=10$ for the cylinder, $\lambda=100$ for the flat plate, $\Delta=2$, and the continuation schedule) could be chosen adaptively, the method would transfer more readily to new geometries; the paper's sensitivity experiments show these choices are load-bearing.
  • The reference-domain visualization, where compressed features are expanded, suggests the same machinery could be coupled with output-based error estimation to target specific quantities such as heating rates rather than the residual norm.
  • The approach may extend to three-dimensional and turbulent hypersonic flows, provided the optimizer can compress elements in three dimensions and the continuation path remains carbuncle-free; these extensions are not demonstrated here.
  • A stricter cost comparison that counts the expense of the continuation and optimization iterations, rather than only degrees of freedom, would place the rp-adaptive method and h-adaptation on more equal footing.
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 proposes an rp-adaptive extension of the high-order implicit shock tracking (HOIST) method for viscous, shock-dominated flows. The method minimizes an enriched IPDG residual simultaneously over flow variables and mesh nodal coordinates, thereby compressing elements into viscous shocks and boundary layers; robustness is achieved through residual weighting, step constraints and modifications, viscosity continuation, and Hessian regularization. A p-adaptivity loop locally increases the polynomial degree using residual-based, dual-weighted, or feature-based indicators. The method is tested on viscous Burgers problems, a laminar flat-plate boundary layer, and hypersonic flow over a cylinder, and is compared with anisotropic h-adaptation in terms of accuracy per degree of freedom.

Significance. If the central claims hold, this is a substantial contribution: it extends implicit shock tracking from inviscid discontinuities to viscous transitions, removes the need for nonlinear stabilization in the final DG solve, and couples aggressive r-adaptivity with p-adaptivity in an optimization-based framework. The paper contains several genuinely independent checks: the Burgers solutions are compared with reference finite volume solutions, the flat-plate boundary-layer profile and skin friction are compared with the Blasius solution, and the stability/accuracy benefit of each proposed component is demonstrated through ablation studies. These elements give credibility to the algorithmic framework. However, the headline quantitative claim — accurate heat flux prediction for hypersonic cylinder flow — is supported mainly by self-convergence of the heat flux profile and by a 'sub-1% error' assertion whose reference solution is not disclosed. That gap, together with the apparent sensitivity of the method to hand-set parameters, prevents the paper from being accepted in its present form.

major comments (3)
  1. [§4.2.2, Figures 25 and 27] The headline claim of accurate heat flux prediction is not backed by an independent converged reference solution. Figure 25 shows that successive p-adaptation iterates of the pointwise heat flux become smoother and stabilize, which demonstrates self-consistency but not accuracy relative to the true heat flux. Figure 27 claims that both HOIST and the h-adaptive comparator reach 'sub-1% error' in integrated heat flux, but the reference value defining that 1% is not identified anywhere in the text. If the reference is generated by the same IPDG/DWR-h-adaptive method family, agreement could reflect shared discretization bias rather than independent accuracy. I request that the authors overlay a grid-converged external reference (e.g., an independently computed fine-mesh solution) in Figure 25 and report the integrated heat flux error relative to that reference in Figure 27.
  2. [§3.5.1, §3.5.3, §4.2.4, §4.2.5] The method's central behavior depends on user-set parameters — enrichment degree Δ=2 and boundary-layer residual scaling λ=10 for the cylinder and λ=100 for the flat plate — and the ablation studies in §4.2.4 and §4.2.5 show that Δ=1 or λ=1 substantially degrades boundary-layer resolution and heat-flux prediction. No criterion or automatic procedure is provided for selecting these values for a new geometry or flow regime, and the values were evidently tuned on the cylinder problem used for the headline heat flux claim. This makes the 'stabilization-free' resolution claim contingent on parameters that may not transfer. I ask for either a priori selection rules or a sensitivity study over Δ and λ on an independent configuration.
  3. [§4.2.6] The p-adaptivity indicator comparison shows that a dual-weighted-residual indicator tailored to the integrated heat flux produces a trial space for which the PTC solver cannot converge without stabilization, while the feature-based indicator yields inaccurate integrated heat flux; the enriched-residual indicator is then declared to combine the best aspects of both. This is a plausible conclusion, but the supporting evidence is mostly qualitative degree distributions. A quantitative comparison of the three indicators in terms of heat-flux profile error against the same external reference requested above would strengthen the claim that the enriched-residual indicator is the right choice for the proposed rp-adaptive framework.
minor comments (5)
  1. [§1] There is a typo on page 2: 'whiche amounts to a savings factor' should be 'which amounts to a savings factor.'
  2. [References] References [53] and [54] are duplicates of the same paper (Zahr, Shi, and Persson, J. Comput. Phys. 410:109385, 2020); one should be removed and the citations renumbered. Also, reference [9] lists 'AIAA Paper 2025-153' despite the 2009 conference date; the paper number appears incorrect.
  3. [§4.2.1, Figure 19] In the degree-of-freedom comparison, flow and mesh degrees of freedom are counted for HOIST while only flow degrees of freedom are counted for the h-adaptive method. This is conservative for the HOIST efficiency claim, but the asymmetry should be stated explicitly in the caption or text so readers do not misread the comparison as apples-to-apples.
  4. [§4.2.2] The initialization for the cylinder problem uses a shock-capturing DG solution with PDE-based artificial viscosity, yet the abstract claims the method operates 'without nonlinear stabilization.' This is not inconsistent — the final solution phase is stabilization-free — but the sentence should be qualified to clarify that an artificial-viscosity solution is used only to initialize the optimization.
  5. [§3.5.6] The notation in (48)–(50) overloads C: it denotes both the scalar function of the residual logarithm and the range parameter in the pair (Cl, Cu). Using a different symbol for the log-residual field would avoid ambiguity.

Circularity Check

0 steps flagged · score 2.0 of 10

No construction-level circularity: the rp-adaptive solution is produced by solving the IPDG equations on an optimized mesh and is independently anchored by Burgers and Blasius benchmarks; the main weakness is an unspecified reference for the cylinder heat-flux accuracy claim.

full rationale

No step in the paper reduces to its own inputs by the quoted-equation standard. The central outputs are obtained by minimizing the enriched DG residual subject to the IPDG constraint (Eqs. 29-33), and the final cylinder solution is subsequently driven to a deeply converged DG residual of about 1e-9 on a fixed mesh without stabilization, so the heat flux is a post-processing of a discrete solution rather than an injected fit. The enriched residual is used both as the shock-tracking objective (Eq. 31) and as one p-adaptivity indicator (Eq. 24); this is a legitimate residual-based error indicator, not a logical identity between the objective and the claimed physical accuracy, especially because the paper provides external benchmarks for viscous Burgers problems and the Blasius flat-plate boundary layer. The boundary-layer amplification lambda and enrichment degree Delta are hand-set and their impact is demonstrated on the same cylinder case (Sections 4.2.4-4.2.5); this is a parameter-selection and validation limitation rather than a circular derivation, since the heat flux does not enter the objective or the parameter fitting. Self-citations to prior HOIST work supply the SQP framework and optimization formulation, but the viscous extension, continuation, residual weighting, and numerical studies are new and testable. The most serious evidence gap is that the 'sub-1% error' claim for the cylinder integrated heat flux (Section 4.2.2, Figure 27) does not identify the reference defining that error; if that reference is from the same IPDG/DWR h-adaptive family, the comparison could share discretization bias. That is missing independent validation, not circularity.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a large set of hand-set algorithmic parameters (lambda, Delta, continuation schedules, Hessian regularization), which are tuned on the very problems used for validation, plus standard numerical analysis results for IPDG and DWR. No new physical entities are introduced. The tuned parameters weaken the generality claim but do not, on their own, invalidate the demonstrated behavior.

free parameters (7)
  • lambda (boundary layer residual scaling) = 10 (cylinder), 100 (flat plate)
    Section 3.5.3; chosen by hand to amplify wall-adjacent element residuals, and shown in Section 4.2.5 to be critical for heat flux accuracy. The value changes between problems.
  • Delta (test space enrichment degree) = 2
    Section 3.5.1 and Section 4.2.4; Delta=1 gives insufficient boundary layer compression, so Delta=2 is selected based on the cylinder experiments.
  • kappa (mesh quality penalty) = 1e-6
    Section 3.5.2; fixed small value so the tracking term dominates while inverted meshes are still penalized.
  • Viscosity/Reynolds continuation schedule = varying stages per problem (11, 20, 15, 10)
    Sections 4.1.1, 4.1.3, 4.2.1, 4.2.2; the number of continuation stages and SQP iterations per stage are set per problem and shown essential for avoiding carbuncles in Section 4.2.3.
  • Hessian regularization constants (gamma_hat, eta1, eta2, Cl, Cu) = gamma_hat=100 or 1e4; eta1, eta2 in {0.8, 1}; Cl=1, Cu=10
    Section 3.5.6; the Young's modulus range and decay rates are chosen by hand to stabilize the optimization and reported per experiment.
  • Step length and modification parameters (theta_l, theta_u, epsilon, nptc, omega_mod) = theta_l=0.1, theta_u=10, epsilon=0.9, nptc=5, omega_mod=5 or 10
    Sections 3.5.4 and 3.5.5; fixed values chosen for robustness; omega_mod varies between problems.
  • p-adaptivity threshold tau and termination parameters (Jmax, eps_hat) = tau not numerically specified; Jmax varies (3, 4, 5)
    Section 2.4 and Algorithm 3; refinement threshold and stopping criteria are user-defined and set per experiment.
assumptions (6)
  • standard math IPDG discretization is consistent, stable, and high-order accurate for elliptic and compressible Navier-Stokes problems.
    Invoked in Sections 2.3 and 3.1, relying on Arnold et al. [2] and Hartmann-Houston [23]; the paper inherits these properties for the constrained problem (29).
  • domain assumption The viscous flux is linear in the state gradient (Fvisc(W,Q) = D(W):Q).
    Equation (4), used for the transformed viscous tensor and the IPDG formulation; valid for the constant-viscosity ideal gas model used in Section 4.2.
  • domain assumption The domain mapping G is a diffeomorphism with a boundary parametrization phi that keeps nodes on original boundaries for any y.
    Section 2.2 and Section 3.1, following [54, 26]; the existence of such phi and positivity of Jacobians is assumed throughout.
  • ad hoc to paper Minimizing the enriched residual (with Delta=2 and boundary-layer weighting) aligns and compresses the mesh into viscous transitions better than Delta=1 or the unweighted residual.
    Core methodological premise of Sections 3.5.1 and 3.5.3; supported only by the numerical experiments in Sections 4.2.4 and 4.2.5, not by an a priori analysis.
  • standard math The dual-weighted residual estimator and enriched residual estimator provide reliable local error indicators for p-adaptivity.
    Section 2.4 cites the DWR theory of Becker-Rannacher [5] and Fidkowski-Darmofal [18]; the enriched residual indicator is used heuristically and empirically compared in Section 4.2.6.
  • domain assumption Ideal gas, constant viscosity, Fourier heat conduction, and Prandtl number Pr=0.72 model the hypersonic cylinder flow.
    Section 4.2, equations (57)-(61); the accuracy of the physical model is standard for laminar perfect-gas CFD but limits generality to real-gas effects.

how reviews work

0 comments
Cite this review

Pith. "Pith review of An $rp$-adaptive method for accurate resolution of shock-dominated viscous flow based on implicit shock tracking." pith.science (2026). https://pith.science/paper/QVF4LDRS

@misc{pith2026250415177,
  author       = {Pith},
  title        = {Pith review of: An $rp$-adaptive method for accurate resolution of shock-dominated viscous flow based on implicit shock tracking},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QVF4LDRS}},
  note         = {Machine review of arXiv:2504.15177}
}
abstract

This work introduces an optimization-based $rp$-adaptive numerical method to approximate solutions of viscous, shock-dominated flows using implicit shock tracking and a high-order discontinuous Galerkin discretization on traditionally coarse grids without nonlinear stabilization (e.g., artificial viscosity or limiting). The proposed method adapts implicit shock tracking methods, originally developed to align mesh faces with solution discontinuities, to compress elements into viscous shocks and boundary layers, functioning as a novel approach to aggressive $r$-adaptation. This form of $r$-adaptation is achieved naturally as the minimizer of the enriched residual with respect to the discrete flow variables and coordinates of the nodes of the grid. Several innovations to the shock tracking optimization solver are proposed to ensure sufficient mesh compression at viscous features to render stabilization unnecessary, including residual weighting, step constraints and modifications, and viscosity-based continuation. Finally, $p$-adaptivity is used to locally increase the polynomial degree with three clear benefits: (1) lessens the mesh compression requirements near shock waves and boundary layers, (2) reduces the error in regions where $r$-adaptivity is not sufficient with the given grid topology, and (3) reduces computational cost by performing a majority of the $r$-adaptivity iterations on the coarsest discretization. A series of numerical experiments show the proposed method effectively resolves viscous, shock-dominated flows, including accurate prediction of heat flux profiles produced by hypersonic flow over a cylinder, and compares favorably in terms of accuracy per degree of freedom to $h$-adaptation with a high-order discretization.

Figures

Figures reproduced from arXiv: 2504.15177 by the authors.

Figure 1
Figure 1. Elementwise magnitude of the enriched DG residual (∆ [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Distribution of the Young’s modulus field [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Mesh and solution (first-order finite volume with [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (32 more)
Figure 4
Figure 4. Figure 4: HOIST solution (pure r-adaptation) in the physical domain GpΩ0q to the viscous Burgers’ problem with straight shock at ν “ 10´3 , 5 ˆ 10´4 , 10´4 (left-to-right) with and without mesh edges shown. Colorbar in [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: HOIST solution (pure r-adaptation) in the reference domain Ω0 to the viscous Burgers’ problem with straight shock at ν “ 10´3 , 5 ˆ 10´4 , 10´4 (left-to-right) with and without mesh edges shown. Colorbar in [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Temporal slices of the HOIST solution (pure [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: HOIST solution (pure r-adaptation) for the steady Burgers’ problem for four families of polynomial distributions pˆ2, ˆp3, ˆp4, ˆp5 (top-to-bottom) for parameter ˆp “ 1, 2, . . . , 9 and viscosities ν “ 0.1 (left) and ν “ 0.01 (right). The vertical lines delineate the …
Figure 8
Figure 8. Figure 8: HOIST error (pure r-adaptation) for the steady Burgers’ problem for the five families of polynomial distributions pˆ1 ( ), ˆp2 ( ), ˆp3 ( ), ˆp4 ( ), ˆp5 ( ) and viscosities ν “ 0.1 (left) and ν “ 0.01 (right) as a function of the number of solution degrees of freedom.…
Figure 9
Figure 9. Figure 9: Mesh and solution (first-order finite volume with [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: HOIST solution (rp-adaptation) (left), the enriched residual error indicator (middle), and polynomial degree distri￾bution (right) in the physical domain GpΩ0q to the viscous Burgers’ problem with curved shock. These quantities are provided prior to p-adaptation and a…
Figure 11
Figure 11. Figure 11: HOIST solution (rp-adaptation) (left), the enriched residual error indicator (middle), and polynomial degree distri￾bution (right) in the reference domain to the viscous Burgers’ problem with curved shock. These quantities are provided prior to p-adaptation and after …
Figure 12
Figure 12. Figure 12: Temporal slice of the HOIST solution (rp-adaptation) at t “ 0.8 ( ) and a reference solution computed with a highly refined second-order finite volume method ( ) for the viscous Burgers’ problem with curved shock. 4.2. Navier-Stokes equations Next, we consider steady …
Figure 13
Figure 13. Figure 13: Convergence of the DG residual › › ›r pjq pu pjq k , ϕpy pjq k q; Ξiq › › › ( ) across all viscosity continuation stages tΞiu 20 i“1 and p-adaptivity iterations tpj u 3 j“0 . Vertical lines indicate the end of a continuation state ( ) (i.e., transition from Ξi to Ξi`1…
Figure 14
Figure 14. Figure 14: Schematic of flat plate domain and boundary conditions. Boundary conditions: subsonic inlet ( [PITH_FULL_IMAGE:figures/full_fig_p027_14.png]
Figure 15
Figure 15. Figure 15: Mesh and solution (DG solution with Re “ 104 ) used to initialize the rp-adaptive HOIST method for the flat plate problem. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_15.png]
Figure 16
Figure 16. Figure 16: HOIST solution (rp-adaptation) (Mach number) (left), the drag-based dual-weighted residual error estimate (middle), and the polynomial degree distribution (right) in the physical domain GpΩ0q to the flat plate problem. These quantities are provided prior to p-adaptati…
Figure 17
Figure 17. Figure 17: HOIST solution (rp-adaptation) (Mach number) (left), the drag-based dual-weighted residual error estimate (middle), and the polynomial degree distribution (right) in the reference domain to the flat plate problem. These quantities are provided prior to p-adaptation an…
Figure 18
Figure 18. Figure 18: The skin friction coefficient (left) and boundary layer profile (right) computed from the Blasius solution ( ) and the HOIST solution (rp-adaptation) prior to p-adaptation ( ) and after the first ( ), second ( ), third ( ), and fourth ( ) p-adaptation iterations for t…
Figure 19
Figure 19. Figure 19: Convergence of the error in the drag coefficient for the HOIST method ( [PITH_FULL_IMAGE:figures/full_fig_p030_19.png]
Figure 20
Figure 20. Figure 20: The h-adapted mesh (left) and corresponding solution (Mach number) (right) for the flat plate problem using drag-based dual-weighted residual adaptation used for the comparative study with the HOIST method (rp-adaptation). 4.2.2. Hypersonic flow over cylinder: rp- vs.…
Figure 21
Figure 21. Figure 21: Schematic of cylinder domain and boundary conditions. Boundary conditions: supersonic inlet ( [PITH_FULL_IMAGE:figures/full_fig_p032_21.png]
Figure 22
Figure 22. Figure 22: Mesh and solution (shock capturing for Re [PITH_FULL_IMAGE:figures/full_fig_p032_22.png]
Figure 23
Figure 23. Figure 23: HOIST solution (rp-adaptation) (Mach number) (left), the enriched residual error estimate (middle), and the polynomial degree distribution (right) in the physical domain GpΩ0q to the cylinder problem. These quantities are provided prior to p-adaptation and after each …
Figure 24
Figure 24. Figure 24: HOIST solution (rp-adaptation) (Mach number) (left), the enriched residual error estimate (middle), and the polynomial degree distribution (right) in the reference domain to the cylinder problem. These quantities are provided prior to p-adaptation and after each p-ada…
Figure 25
Figure 25. Figure 25: Pointwise heat flux over the cylinder surface computed from the HOIST solution ( [PITH_FULL_IMAGE:figures/full_fig_p034_25.png]
Figure 26
Figure 26. Figure 26: Pointwise heat flux profile over the cylinder surface computed from the HOIST solution ( [PITH_FULL_IMAGE:figures/full_fig_p035_26.png]
Figure 27
Figure 27. Figure 27: Convergence of the integrated heat flux for the HOIST method ( [PITH_FULL_IMAGE:figures/full_fig_p035_27.png]
Figure 28
Figure 28. Figure 28: The h-adapted mesh (left) and corresponding solution (Mach number) (right) for the cylinder problem using integrated heat flux-based dual-weighted residual adaptation used for the comparative study with the HOIST method (rp￾adaptation). 0 50 100 150 200 250 300 350 40…
Figure 29
Figure 29. Figure 29: Convergence of the DG residual › › ›r pjq pu pjq k , ϕpy pjq k q; Ξiq › › › ( ) and enriched DG residual › › › R pjq ρ pu pjq k , ϕpy pjq k q; Ξiq › › › ( ) across all Reynolds number continuation stage tΞiu 10 i“1 and p-adaptivity iterations tpj u 5 j“0 . Vertical li…
Figure 30
Figure 30. Figure 30: HOIST solution (Mach number) (r-adaptation) with the Reynolds number continuation strategy of Section 3.3 (left) and without continuation (Ξ1 “ Re “ 1000) (right) for the cylinder problem. Colorbar in [PITH_FULL_IMAGE:figures/full_fig_p036_30.png]
Figure 31
Figure 31. Figure 31: HOIST solution (temperature) (rp-adaptation) based on test space enrichment ∆ “ 2 (left) and ∆ “ 1 (right) for the cylinder problem. 4.2.5. Hypersonic flow over cylinder: impact of boundary residual weighting Next, we consider the hypersonic cylinder problem in Sectio…
Figure 32
Figure 32. Figure 32: Pointwise heat flux profile over the cylinder surface computed from the HOIST solution ( [PITH_FULL_IMAGE:figures/full_fig_p038_32.png]
Figure 33
Figure 33. Figure 33: HOIST solution (temperature) (rp-adaptation) based on boundary residual scaling λ “ 10 (left) and λ “ 1 (unweighted) (right). Colorbar in [PITH_FULL_IMAGE:figures/full_fig_p038_33.png]
Figure 34
Figure 34. Figure 34: Pointwise heat flux profile over the cylinder surface computed from the HOIST solution ( [PITH_FULL_IMAGE:figures/full_fig_p039_34.png]
Figure 35
Figure 35. Figure 35: Distribution of the polynomial degree for the HOIST method ( [PITH_FULL_IMAGE:figures/full_fig_p040_35.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Shape optimisation for adaptive $r$-refinement: the one-dimensional case with residual based error estimators

    math.NA 2026-07 conditional novelty 6.0 of 10

    For 1D Poisson, an Armijo-type mesh-moving algorithm driven by the residual estimator converges, and numerical tests show optimized meshes outperform uniform meshes.

Reference graph

Works this paper leans on

54 extracted references · 53 canonical work pages · cited by 1 Pith paper

  1. [1]

    Douglas N. Arnold. An interior penalty finite element method with discontinuous elements. SIAM Journal on Numerical Analysis , 19(4):742–760, 1982

    work page 1982

  2. [2]

    Arnold, Franco Brezzi, Bernardo Cockburn, and L

    Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems.SIAM Journal on Numerical Analysis, 39(5):1749– 1779, 2002

    work page 2002

  3. [3]

    Numerical simulation of shock/boundary-layer interaction using an unstructured shock-fitting technique

    Alessia Assonitis, Renato Paciorri, and Aldo Bonfiglioli. Numerical simulation of shock/boundary-layer interaction using an unstructured shock-fitting technique. Computers & Fluids , 228:105058, 2021

    work page 2021

  4. [4]

    Barter and David L

    Garrett E. Barter and David L. Darmofal. Shock capturing with PDE-based artificial viscosity for DGFEM: Part I. Formulation. Journal of Computational Physics , 229(5):1810–1827, 2010

    work page 2010

  5. [5]

    Becker and R

    R. Becker and R. Rannacher. An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica, 10:1–102, may 2001

    work page 2001

  6. [6]

    A feed-back approach to error control in finite element methods: basic analysis and examples

    Roland Becker and Rolf Rannacher. A feed-back approach to error control in finite element methods: basic analysis and examples. East-West J. Numer. Math , 4:237–264, 1996

    work page 1996

  7. [7]

    Paciorri, and L

    Aldo Bonfiglioli, R. Paciorri, and L. Campoli. Unsteady shock-fitting for unstructured grids. Interna- tional Journal for Numerical Methods in Fluids , 81(4):245–261, 2016

    work page 2016

  8. [8]

    Graham V. Candler. Next-generation CFD for hypersonic and aerothermal flows. In 22nd AIAA Computational Fluid Dynamics Conference , Dallas, Texas, 6/22/2015 – 6/26/2015. American Institute of Aeronautics and Astronautics, AIAA Paper 2015-3048

    work page 2015

Show all 54 references
  1. [9]

    Candler, Dimitri J

    Graham V. Candler, Dimitri J. Mavriplis, and Loretta Trevino. Current status and future prospects for the numerical simulation of hypersonic flows. In 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition, Orlando, Florida, 1/5/2009 – 1...

    2009

  2. [10]

    Candler, Pramod K

    Graham V. Candler, Pramod K. Subbareddy, and Joseph M. Brock. Advances in computational fluid dynamics methods for hypersonic flows. Journal of Spacecraft and Rockets, 52(1):17–28, 2015

    2015

  3. [11]

    Ching, Andrew D

    Eric J. Ching, Andrew D. Kercher, and Andrew Corrigan. The moving discontinuous Galerkin method with interface condition enforcement for the simulation of hypersonic, viscous flows. Computer Methods in Applied Mechanics and Engineering , 427:117045, 2024

    2024

  4. [12]

    Ching, Yu Lv, Peter Gnoffo, Michael Barnhardt, and Matthias Ihme

    Eric J. Ching, Yu Lv, Peter Gnoffo, Michael Barnhardt, and Matthias Ihme. Shock capturing for discontinuous Galerkin methods with application to predicting heat transfer in hypersonic flows.Journal of Computational Physics , 376:54–75, 2019

    2019

  5. [13]

    Extrapolated shock tracking: bridging shock-fitting and embedded boundary methods

    Mirco Ciallella, Mario Ricchiuto, Renato Paciorri, and Aldo Bonfiglioli. Extrapolated shock tracking: bridging shock-fitting and embedded boundary methods. Journal of Computational Physics, 412:109440, 2020

    2020

  6. [14]

    Kercher, and David A

    Andrew Corrigan, Andrew D. Kercher, and David A. Kessler. A moving discontinuous Galerkin finite element method for flows with interfaces. International Journal for Numerical Methods in Fluids , 89(9):362–406, 2019

    2019

  7. [15]

    Huijing Dong, Alexander Perez-Reyes, and Matthew J. Zahr. A p-adaptive high-order implicit shock tracking method for compressible, inviscid flows. In AIAA Science and Technology Forum and Expo- sition (SciTech2025), Orlando, Florida, 1/6/2025 – 1/10/2025. American Institute of...

    2025

  8. [16]

    Economon, Francisco Palacios, Sean R

    Thomas D. Economon, Francisco Palacios, Sean R. Copeland, Trent W. Lukaczyk, and Juan J. Alonso. SU2: An open-source suite for multiphysics simulation and design. AIAA Journal, 54(3):828–846, 2016. 41

    2016

  9. [17]

    A physics-based shock capturing method for unsteady laminar and turbulent flows

    Pablo Fernandez, Cuong Nguyen, and Jaime Peraire. A physics-based shock capturing method for unsteady laminar and turbulent flows. In 2018 AIAA Aerospace Sciences Meeting, Kissimmee, Florida, 1/8/2018 – 1/12/2018. American Institute of Aeronautics and Astronautics, AIAA Paper ...

    2018

  10. [18]

    Fidkowski and David L

    Krzysztof J. Fidkowski and David L. Darmofal. Review of output-based error estimation and mesh adaptation in computational fluid dynamics. AIAA Journal, 49(4):673–694, 2011

    2011

  11. [19]

    An extended discontinuous Galerkin method for high-order shock-fitting

    Markus Geisenhofer, Florian Kummer, and Martin Oberlack. An extended discontinuous Galerkin method for high-order shock-fitting. arXiv preprint, 2020

    2020

  12. [20]

    Multi-dimensional inviscid flux reconstruction for simulation of hypersonic heating on tetrahedral grids

    Peter Gnoffo. Multi-dimensional inviscid flux reconstruction for simulation of hypersonic heating on tetrahedral grids. In 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition, Orlando, Florida, 1/5/2009 – 1/8/2009. American Institute ...

    2009

  13. [21]

    Chakravarthy

    Ami Harten, Bjorn Engquist, Stanley Osher, and Sukumar R. Chakravarthy. Uniformly high order accurate essentially non-oscillatory schemes, III. Journal of Computational Physics , 131(1):3–47, 1997

    1997

  14. [22]

    Adjoint consistency analysis of discontinuous Galerkin discretizations

    Ralf Hartmann. Adjoint consistency analysis of discontinuous Galerkin discretizations. SIAM Journal on Numerical Analysis, 45(6):2671–2696, 2007

    2007

  15. [23]

    An optimal order interior penalty discontinuous Galerkin discretiza- tion of the compressible Navier–Stokes equations.Journal of Computational Physics, 227(22):9670–9685, 2008

    Ralf Hartmann and Paul Houston. An optimal order interior penalty discontinuous Galerkin discretiza- tion of the compressible Navier–Stokes equations.Journal of Computational Physics, 227(22):9670–9685, 2008

    2008

  16. [24]

    Higher order and adaptive DG methods for compressible flows

    Ralf Hartmann and Tobias Leicht. Higher order and adaptive DG methods for compressible flows. 37th Advanced CFD Lecture Series: Recent developments in higher order methods and industrial application in aeronautics, 2014(3):1–156, 2014

    2014

  17. [25]

    Naudet, and Matthew J

    Tianci Huang, Charles J. Naudet, and Matthew J. Zahr. High-order implicit shock tracking boundary conditions for flows with parametrized shocks. Journal of Computational Physics , 495:112517, 2023

    2023

  18. [26]

    Tianci Huang and Matthew J. Zahr. A robust, high-order implicit shock tracking method for simulation of complex, high-speed flows. Journal of Computational Physics , 454:110981, 2022

    2022

  19. [27]

    Efficient implementation of weighted ENO schemes

    Guang-Shan Jiang and Chi-Wang Shu. Efficient implementation of weighted ENO schemes. Journal of Computational Physics, 126(1):202–228, 1996

    1996

  20. [28]

    Bhagatwala, William H Cabot, Parviz Moin, Britton J

    Eric Johnsen, Johan Larsson, Ankit V. Bhagatwala, William H Cabot, Parviz Moin, Britton J. Olson, Pradeep S. Rawat, Santhosh K. Shankar, Bj¨ orn Sj¨ ogreen, Helen C. Yee, et al. Assessment of high- resolution methods for numerical simulations of compressible turbulence with sh...

    2010

  21. [29]

    Carl Timothy Kelley and David E. Keyes. Convergence analysis of pseudo-transient continuation. SIAM Journal on Numerical Analysis , 35(2):508–523, 1998

    1998

  22. [30]

    Kercher and Andrew Corrigan

    Andrew D. Kercher and Andrew Corrigan. A least-squares formulation of the moving discontinuous Galerkin finite element method with interface condition enforcement. Computers & Mathematics with Applications, 95:143–171, 2021

    2021

  23. [31]

    Kercher, Andrew Corrigan, and David A

    Andrew D. Kercher, Andrew Corrigan, and David A. Kessler. The moving discontinuous Galerkin finite element method with interface condition enforcement for compressible viscous flows. International Journal for Numerical Methods in Fluids , 93(5):1490–1519, 2021

    2021

  24. [32]

    Spurious numerical oscillations in simulation of supersonic flows using shock-capturing schemes

    Theodore Kai Lee and Xiaolin Zhong. Spurious numerical oscillations in simulation of supersonic flows using shock-capturing schemes. AIAA journal, 37(3):313–319, 1999

    1999

  25. [33]

    Weighted essentially non-oscillatory schemes

    Xu-Dong Liu, Stanley Osher, and Tony Chan. Weighted essentially non-oscillatory schemes. Journal of Computational Physics, 115(1):200–212, 1994. 42

    1994

  26. [34]

    An a posteriori error control framework for adaptive precision optimization using discontin- uous Galerkin finite element method

    James Lu. An a posteriori error control framework for adaptive precision optimization using discontin- uous Galerkin finite element method . PhD thesis, Massachusetts Institute of Technology, 2005

    2005

  27. [35]

    I Do Like CFD , volume 1

    Katate Masatsuka. I Do Like CFD , volume 1. Lulu.com, 2009

    2009

  28. [36]

    Modisette

    James M. Modisette. An automated reliable method for two-dimensional Reynolds-averaged Navier- Stokes simulations. PhD thesis, Massachusetts Institute of Technology, 2011

    2011

  29. [37]

    Thirty-six years of shock fitting

    Gino Moretti. Thirty-six years of shock fitting. Computers & Fluids , 31(4-7):719–723, 2002

    2002

  30. [38]

    Gino Moretti and Manuel D. Salas. Numerical analysis of viscous one-dimensional flows. Journal of Computational Physics, 5(3):487–506, 1970

    1970

  31. [39]

    Naudet and Matthew J

    Charles J. Naudet and Matthew J. Zahr. A space-time high-order implicit shock tracking method for shock-dominated unsteady flows. Journal of Computational Physics , 501:112792, 2024

    2024

  32. [40]

    Numerical Optimization

    Jorge Nocedal and Stephen Wright. Numerical Optimization. Springer Science & Business Media, 2006

    2006

  33. [41]

    Alexander Perez-Reyes and Matthew J. Zahr. An implicit shock tracking method for simulation of shock- dominated flow over complex domains using mesh-based parametrizations. Journal of Computational Physics, in review 2025

    2025

  34. [42]

    Perry and S

    K. Perry and S. Imlay. Blunt-body flow simulations. In 24th Joint Propulsion Conference , Boston, Massachusetts, 7/11/1988 – 7/13/1988. American Institute of Aeronautics and Astronautics, AIAA Paper 1988-2904

    1988

  35. [43]

    Sub-cell shock capturing for discontinuous Galerkin methods

    Per-Olof Persson and Jaime Peraire. Sub-cell shock capturing for discontinuous Galerkin methods. In 44th AIAA Aerospace Sciences Meeting and Exhibit , Reno, Nevada, 1/9/2006 – 1/12/2006. American Institute of Aeronautics and Astronautics, AIAA Paper 2006-112

    2006

  36. [44]

    Robinet, J´ ermie Gressier, Gr´ egoire Casalis, and J.-M

    J.-Ch. Robinet, J´ ermie Gressier, Gr´ egoire Casalis, and J.-M. Moschetta. Shock wave instability and the carbuncle phenomenon: same intrinsic origin? Journal of Fluid Mechanics , 417:237–263, 2000

    2000

  37. [45]

    Manuel D. Salas. A Shock-Fitting Primer . CRC Press, 2009

    2009

  38. [46]

    Andrew Shi, P-O Persson, and Matthew J. Zahr. Implicit shock tracking for unsteady flows by the method of lines. Journal of Computational Physics , 454:110906, 2022

    2022

  39. [47]

    Towards the ultimate conservative difference scheme

    Bram Van Leer. Towards the ultimate conservative difference scheme. V. a second-order sequel to Godunov’s method. Journal of Computational Physics , 32(1):101–136, 1979

    1979

  40. [48]

    Venditti and David L

    David A. Venditti and David L. Darmofal. Grid adaptation for functional outputs: application to two-dimensional inviscid flows. Journal of Computational Physics , 176(1):40–69, 2002

    2002

  41. [49]

    Venditti and David L

    David A. Venditti and David L. Darmofal. Anisotropic grid adaptation for functional outputs: appli- cation to two-dimensional viscous flows. Journal of Computational Physics , 187(1):22–46, 2003

    2003

  42. [50]

    The importance of mesh adaptation for higher- order discretizations of aerodynamic flows

    Masayuki Yano, James Modisette, and David Darmofal. The importance of mesh adaptation for higher- order discretizations of aerodynamic flows. In 20th AIAA Computational Fluid Dynamics Conference , pages 2011–3852, 2011

    2011

  43. [51]

    Zahr and Per-Olof Persson

    Matthew J. Zahr and Per-Olof Persson. An optimization-based approach for high-order accurate dis- cretization of conservation laws with discontinuous solutions.Journal of Computational Physics, 365:105 – 134, 2018

    2018

  44. [52]

    Zahr and Joseph M

    Matthew J. Zahr and Joseph M. Powers. High-order resolution of multidimensional compressible reactive flow using implicit shock tracking. AIAA Journal, 59(1):150–164, December 2020

    2020

  45. [53]

    Zahr, Andrew Shi, and P-O Persson

    Matthew J. Zahr, Andrew Shi, and P-O Persson. Implicit shock tracking using an optimization-based high-order discontinuous Galerkin method. Journal of Computational Physics , 410:109385, 2020

    2020

  46. [54]

    Zahr, Andrew Shi, and Per-Olof Persson

    Matthew J. Zahr, Andrew Shi, and Per-Olof Persson. Implicit shock tracking using an optimization- based high-order discontinuous Galerkin method. Journal of Computational Physics , 410:109385, June 2020. 43

    2020

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.