A hyperbolic relaxation system of the incompressible Navier-Stokes equations with artificial compressibility

Christian Rohde; Qian Huang; Ruixi Zhang; Wen-an Yong

A hyperbolic relaxation system of the incompressible Navier-Stokes equations with artificial compressibility

T0 review · 0 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read A new two-parameter hyperbolic approximation is proved to converge to the incompressible Navier-Stokes equations, with explicit pressure error bounds when the relaxation parameter is much smaller than the square root of the…

desk verdict Solid, new two-parameter convergence result with the first pressure error estimate; the central proof closes under the stated well-prepared assumptions, with only minor typos to fix. read the letter →

arxiv 2411.15575 v1 pith:PCL5PENJ submitted 2024-11-23 math.AP

Qian Huang , Christian Rohde , Wen-An Yong , Ruixi Zhang This is my paper · ORCID

classification math.AP MSC 35Q3076D05

keywords incompressibleNavier-Stokesequationshyperbolicrelaxationapproximationartificialcompressibilitytwo-parametersingularlimitpressureconvergenceenergyestimatesinitial-layercorrection

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 introduces a two-parameter hyperbolic approximation to the two-dimensional incompressible Navier-Stokes equations, obtained by combining the artificial compressibility method (parameter $\varepsilon$) with a first-order relaxation of the velocity gradient (parameter $\delta$). The central claim is that, as $\varepsilon$ and $\delta$ both go to zero, the smooth solutions of this new system converge to the Navier-Stokes solution, with explicit error bounds for both velocity and pressure, provided $\delta = O(\sqrt{\varepsilon})$ and the initial data are well prepared. Velocity convergence is obtained by direct energy estimates for a symmetrized residual system; pressure convergence, which the one-parameter methods in the literature did not deliver, is obtained by inserting a linear auxiliary system and estimating the two error legs separately. A sympathetic reader should care because the approximating system is genuinely hyperbolic, hence has finite signal speeds and access to hyperbolic numerical methods, while still provably reproducing incompressible flow.

What carries the argument

The load-bearing object is the linear auxiliary system (4.2): it is obtained from the two-parameter model by replacing the quadratic term $u\otimes u$ with the known $u^{NS}\otimes u^{NS}$, keeping the same initial data. The argument first bounds the difference between the auxiliary system and the incompressible Navier-Stokes solution, then bounds the difference between the original approximation and the auxiliary system, and combines the two. The auxiliary pressure is further controlled through the derived scalar PDE $\varepsilon\delta\,\partial_t^3 p' - (\varepsilon+\delta)\partial_t\Delta p' + \varepsilon\,\partial_t^2 p' - \Delta(p'-p^{NS})=0$, reformulated as a first-order system in $f=\delta\partial_t p'+p'-p^{NS}$ and $g=\sqrt{\varepsilon\delta}\,\partial_t\nabla p'$ with an explicit energy; the velocity difference is split into divergence and curl parts, with the vorticity leg requiring an initial-layer correction for the auxiliary vorticity variables.

What would settle it

Solve the approximate system numerically on the two-dimensional periodic square for a smooth test flow with $\delta=\sqrt{\varepsilon}$ (so $\delta$ is not $o(\sqrt{\varepsilon})$) and well-prepared initial data, and measure $\sup_{0<t<T}\|p^{\varepsilon,\delta}-p^{NS}\|_{H^1}$ as $\varepsilon\to0$. The theorem only guarantees $\sqrt{\varepsilon}\|p^{\varepsilon,\delta}-p^{NS}\|_{H^1}=O(\varepsilon+\delta)$, so the unweighted error could diverge like $1/\sqrt{\varepsilon}$; observing it instead vanish would show the $\delta=o(\sqrt{\varepsilon})$ threshold is not necessary. Conversely, keeping $\delta=o(\sqrt{\varepsilon})$ but taking data with $\|\nabla\,\mathrm{div}\,u_0^{\varepsilon,\delta}\|_{L^2}$ of order one rather than $O(\varepsilon+\delta)$ would test whether the well-prepared condition is needed for pressure convergence.

Watch

Extended reading notes

Core claim

On the unit periodic square, for smooth Navier-Stokes solutions and smooth solutions of the new system, the paper proves (Theorem 2.4) that if $\delta \leq C\sqrt{\varepsilon}$ and the initial data satisfy the stated $H^1$ and divergence-preparation conditions, then for any fixed time $T$ before the approximation's lifespan, $\sup_{0<t<T}\big(\|u^{\varepsilon,\delta}(t)-u^{NS}(t)\|_{H^1}+\sqrt{\varepsilon}\,\|p^{\varepsilon,\delta}(t)-p^{NS}(t)\|_{H^1}\big) \leq C_T(\varepsilon+\delta)$. Corollary 2.5 then yields unweighted pressure convergence in $H^1$ with rate $O(\sqrt{\varepsilon}+\delta/\sqrt{\varepsilon})$ whenever $\delta=o(\sqrt{\varepsilon})$. Earlier theorems (2.1 and 2.2) establish $L^2$ velocity convergence at rate $O(\varepsilon+\delta)$ and show the lifespan tends to infinity as $\varepsilon+\delta\to0$. The proof of pressure convergence relies on a linear intermediate system (4.2) and energy estimates for its differences to both the original approximation and Navier-Stokes; a key vorticity estimate uses an initial-layer correction for the auxiliary vorticity variables.

Load-bearing premise

The weakest link is the combination of the parameter relation $\delta \leq C\sqrt{\varepsilon}$ with the well-prepared initial data conditions (in particular $\|\nabla\,\mathrm{div}\,u_0^{\varepsilon,\delta}\|_{L^2} \leq C(\varepsilon+\delta)$ and $\delta(\|\Delta p_0^{\varepsilon,\delta}\|_{L^2}+\|\nabla\,\mathrm{div}\,U_0^{\varepsilon,\delta}\|_{L^2}) \leq C(\varepsilon+\delta)$); if either fails, the proof gives no convergence of the pressure.

Editorial extensions

If this is right

When $\delta=o(\sqrt{\varepsilon})$, the approximate pressure $p^{\varepsilon,\delta}$ converges to $p^{NS}$ in $H^1$ at rate $O(\sqrt{\varepsilon}+\delta/\sqrt{\varepsilon})$, so the model can be used as a provably consistent pressure proxy, not just a velocity proxy.
For any fixed finite time horizon, smooth solutions of the approximate system exist for all sufficiently small parameters, so the asymptotic statement is uniform in time up to $T$.
Since the approximation is hyperbolic with finite propagation speed, it can be discretized with established hyperbolic balance-law schemes, and the proven rates give a concrete parameter guideline: pick $\delta$ much smaller than $\sqrt{\varepsilon}$ to keep pressure errors small.
Adding a linear friction term to the Navier-Stokes target does not destroy any of the convergence results (Proposition 2.6).
The proof gives a template for two-parameter singular limits in hyperbolic relaxation systems: use an intermediate linear system to transfer convergence to a variable, here pressure, that direct energy estimates cannot control.

Reading between the lines

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

The restriction $\delta=O(\sqrt{\varepsilon})$ is likely essential to the pressure argument, because the $\sqrt{\varepsilon}$ in front of the pressure error forces the auxiliary system to track $p^{NS}$ at a faster rate; testing $\delta\approx\sqrt{\varepsilon}$ would reveal whether the unweighted pressure error genuinely fails to vanish in that regime, or whether another intermediate system could
The techniques rely on the 2D periodic torus through Ladyzhenskaya-type interpolation, Poincar\'e inequalities, Helmholtz decomposition, and constant mean-zero properties; extending the result to 3D or to bounded domains with physical boundary conditions will require new boundary-layer and interpolation arguments.
The explicit rates suggest a testable numerical prediction: for well-prepared initial data and $\delta\ll\sqrt{\varepsilon}$, the $H^1$ pressure error over a fixed time interval should scale like $\sqrt{\varepsilon}$; a modest numerical experiment on a periodic shear flow could check whether the predicted rate appears.
For non-well-prepared initial data the paper proves nothing about pressure; a natural extension would be to introduce initial-layer corrections in the original system itself, not only in the auxiliary vorticity equations, to remove the preparation conditions.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

This is a strong analytical paper. It introduces the two-parameter relaxation–artificial compressibility system (1.3), which is new, and proves the first joint convergence result for both parameters, including the first pressure error estimate. The main theorems (2.4 and Corollary 2.5) give H1 convergence of velocity and pressure to the incompressible Navier–Stokes solution under δ = O(√ε) and well-prepared initial data. The proof strategy is sensible: energy estimates for the velocity, then an auxiliary linear system to control the pressure. I went through the key steps, particularly Section 4, and the argument closes. The assumptions are stated explicitly, and the paper does not overclaim.

The genuinely new content is the pressure estimate. Prior work on the Brenier–Natalini–Puel relaxation did not characterize the pressure, and the classical artificial compressibility estimates for pressure did not have rates. The auxiliary system (4.2) and the decomposition into divergence and curl in Section 4.2 are the right tools, and the proof is careful.

Soft spots are real but not damning. First, the pressure result only works for δ = O(√ε), and the convergent rate for pressure is √ε after fixing δ = o(√ε). Other parameter regimes are not analyzed—the paper says so. Second, the well-prepared initial data conditions are fairly strong, e.g. ||∇div u0|| ≤ C(ε+δ) and δ(||Δp0||+||∇div U0||) ≤ C(ε+δ). They are stated clearly, but they limit the practical applicability. Third, the lower bound on the existence time is logarithmic in the parameters, which means for a fixed time horizon the parameters have to be exponentially small; that is a practical caveat for any numerical use. Fourth, there is no numerical verification, and the proofs are not machine-checked; however, the estimates are detailed enough that I found no hidden assumption.

There are two typos that should be fixed: in (4.11a) the sign in front of ε∂t²p^NS appears to be wrong, and in Corollary 2.5 the parenthetical should read δ/√ε → 0, not δ√ε → 0. Neither affects the results.

Who is this for? Researchers working on hyperbolic relaxation approximations to incompressible flows, singular limits, and numerical methods based on such systems. It deserves a serious referee; I expect it to be accepted after the minor typos are corrected. I would engage with it.

Referee Report

0 major / 4 minor

Summary. The manuscript proposes a two-parameter hyperbolic relaxation approximation (1.3) of the two-dimensional incompressible Navier-Stokes equations, combining a first-order relaxation term with the artificial compressibility method. The main results are: (i) Theorem 2.1 and Theorem 2.2, which prove that smooth solutions exist on a time interval that grows to infinity as (ε, δ) → 0 and that the velocity component converges to the Navier-Stokes velocity in L2, with rates O(ε + δ), under suitable initial-data assumptions; and (ii) Theorem 2.4 and Corollary 2.5, which, under the scaling δ = O(√ε) and well-prepared initial data, prove convergence of the velocity in H1 and of the pressure in H1 with rate √ε + δ/√ε. The key technical device is an auxiliary linear system (4.2). The paper estimates the difference between the original system and the auxiliary system, and separately the difference between the auxiliary system and Navier-Stokes, using energy estimates, an auxiliary pressure equation (4.11), and a vorticity initial-layer correction.

Significance. If the main theorem is correct, this is a valuable contribution: it provides a rigorous two-parameter singular limit for a hyperbolic approximation of the incompressible Navier-Stokes equations, and it gives pressure convergence, which the earlier one-parameter relaxation results did not provide. The proof is self-contained and follows standard energy methods; all assumptions are stated explicitly, and no parameter is fitted to make the convergence work. The auxiliary- system argument is coherent and the term-by-term estimates are plausible. The main limitation is the regime δ = O(√ε) and the well-prepared initial-data hypotheses; the paper explicitly acknowledges that more general parameter relations are left to future work. The manuscript does not provide machine-checked proofs, but the estimates are detailed enough for a human referee to trace. I judge the central claim sound.

minor comments (4)

[§2, Corollary 2.5] The displayed convergence rate and the parenthetical equivalence are incorrect as printed: δ = o(√ε) means δ/√ε → 0, not δ√ε → 0, and the proof of Theorem 2.4 gives ||p^{ε,δ} − p^{NS}||_{H1} ≤ C_T(√ε + δ/√ε), not C_T(√ε + δ√ε). The final conclusion that the pressure converges is still true, but the displayed rate should be corrected.
[§4, Eq. (4.11a)] There is a sign error in (4.11a): independently re-deriving it from Proposition 4.2 and the definitions (4.10) gives (ε∂_t^2 − Δ)f − √(ε/δ) ∇·g = −ε∂_t^2 p^{NS}, not +ε∂_t^2 p^{NS}. The sign is irrelevant for the subsequent absolute-value and energy estimates, but the displayed equation should be corrected.
[§4.2, text after Eq. (4.22)] In the energy estimate for the vorticity system, the expression 'δtΩ^{NS}' should read '∂_tΩ^{NS}'.
[§4, Remark 4.4] Remark 4.4 contains a LaTeX artifact: 'δ /greaterorsimilar√ε' should read 'δ \gtrsim √ε' or 'with δ not smaller than √ε'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence proof targets an external benchmark (the incompressible Navier-Stokes equations) and no fitted parameter or self-citation chain is load-bearing.

full rationale

The derivation is self-contained. The target system (1.1) is an external benchmark; the approximating system (1.3) is not constructed from the target by fitting parameters. The convergence is proven by direct energy estimates on residuals (Sections 3 and 4), with the auxiliary linear system (4.2) as an analytical device whose difference from both (1.1) and (1.3) is estimated independently. The hypotheses of Theorem 2.4 are stated well-prepared initial-data and scaling assumptions (e.g., δ≤C√ε) that enter the estimates explicitly (e.g., (4.16)-(4.17) and the vorticity argument of Section 4.2); they are not disguised outputs. Citations to previous work by co-author Yong ([33], [5], [31]) are contextual descriptions of a general framework, but the paper does not invoke any of those results as a black box for the main theorem; the energy estimates are carried out here. No uniqueness theorem is imported to force the choice of the auxiliary system. I find no step where Eq. X is Eq. Y by construction or where a fitted input is renamed as a prediction. The only irregularities are minor typos—a sign in (4.11a) that is irrelevant to the absolute-value estimates, and the parenthetical 'δ√ε→0' in Corollary 2.5, which should read δ/√ε→0—neither of which affects the validity or circularity status of the proof.

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

The model contains no fitted parameters or new physical entities. The only 'invention' is the auxiliary linear system (4.2), which is a mathematical device used to bridge the error estimates; it is not an independent physical postulate. The central claim rests on standard PDE theory plus the explicitly stated well-prepared initial data and the scaling delta=O(sqrt(epsilon)).

assumptions (6)

standard math Kato's local existence and uniqueness theory for quasilinear symmetric hyperbolic systems applies to (1.3) for fixed (epsilon, delta).
Invoked in Section 2 to guarantee existence/uniqueness of smooth solutions of (1.3) and in Lemma 2.8 via Theorem 1 of [20].
standard math The Gagliardo-Nirenberg and Ladyzhenskaya interpolation inequalities (2.1a)-(2.1c) hold on T^2.
Used throughout Sections 3 and 4 to bound nonlinear and higher-order terms; (2.1a) is cited to [22], (2.1b)-(2.1c) are proved in Appendix A.
standard math The incompressible Navier-Stokes equations (1.1) on T^2 have a smooth solution for smooth initial data, with the pressure of zero spatial mean.
The target solution is used to define U^{NS} and to derive (4.9) and (4.11a); smoothness is cited to [22,19].
standard math The Helmholtz decomposition identity ||nabla u||^2 = ||nabla dot u||^2 + ||nabla cross u||^2 holds for u in H^1(T^2).
Lemma 2.9, proved directly; used in Section 4.2 to split u' - u^{NS} into a divergence part and a curl part.
domain assumption The initial data for (1.3) satisfy the well-prepared conditions in Theorem 2.4, including ||nabla div u_0^{epsilon,delta}|| <= C(epsilon+delta) and delta(||Delta p_0|| + ||nabla div U_0||) <= C(epsilon+delta).
These hypotheses are used in the energy estimates of Section 4, e.g., in (4.16) and (4.17), and are essential for the stated pressure convergence result.
domain assumption For the pressure convergence, the relaxation parameter delta must be asymptotically smaller than sqrt(epsilon) (Theorem 2.4 assumes delta <= C sqrt(epsilon); Corollary 2.5 takes delta = o(sqrt(epsilon))).
This scaling is stated in Theorem 2.4 and Remark 4.4; it is used to convert the auxiliary-system energy estimate into the pressure error bound, e.g., the term Ct*delta in (4.14).

how reviews work

0 comments

Cite this review

Pith. "Pith review of A hyperbolic relaxation system of the incompressible Navier-Stokes equations with artificial compressibility." pith.science (2026). https://pith.science/paper/PCL5PENJ

@misc{pith2026241115575,
  author       = {Pith},
  title        = {Pith review of: A hyperbolic relaxation system of the incompressible Navier-Stokes equations with artificial compressibility},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PCL5PENJ}},
  note         = {Machine review of arXiv:2411.15575}
}

read the original abstract

We introduce a new hyperbolic approximation to the incompressible Navier-Stokes equations by incorporating a first-order relaxation and using the artificial compressibility method. With two relaxation parameters in the model, we rigorously prove the asymptotic limit of the system towards the incompressible Navier-Stokes equations as both parameters tend to zero. Notably, the convergence of the approximate pressure variable is achieved by the help of a linear `auxiliary' system and energy-type error estimates of its differences with the two-parameter model and the Navier-Stokes equations.

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

A Note on Hyperbolic Relaxation of the Navier-Stokes-Cahn-Hilliard system for incompressible two-phase flow
math.AP 2024-12 conditional novelty 6.0 of 10

The relaxed friction-type approximation of the Navier-Stokes-Cahn-Hilliard system has a hyperbolic first-order subsystem in 1D, proved via a convex entropy-entropy flux pair.

Reference graph

Works this paper leans on

32 extracted references · 31 canonical work pages · cited by 1 Pith paper

[1]

Denise Aregba-Driollet and Roberto Natalini, Discrete kinetic schemes for multidimensional systems of c onservation laws, SIAM J. Numer. Anal. 37 (2000), no. 6, 1973–2004

work page 2000
[2]

Hajer Bahouri, Jean-Yves Chemin, and Rapha¨ el Danchin, Fourier analysis and nonlinear partial diﬀerential equa- tions, Springer, 2011

work page 2011
[3]

Roberta Bianchini, Strong convergence of a vector-BGK model to the incompressi ble navier-stokes equations via the relative entropy method , J. Math. Pure Appl. 132 (2019), 280–307

work page 2019
[4]

Yann Brenier, Roberto Natalini, and Marjolaine Puel, On a relaxation approximation of the incompressible Navier - Stokes equations , Proc. Am. Math. Soc. 132 (2004), 1021–1028

work page 2004
[5]

Yann Brenier and W en-An Yong, Derivation of particle, string, and membrane motions from t he Born–Infeld electromagnetism, J. Math. Phys. 46 (2005), no. 6, 062305

work page 2005
[6]

Alexandre Joel Chorin, A numerical method for solving incompressible viscous ﬂow p roblems, J. Comput. Phys. 2 (1967), no. 1, 12–26

work page 1967
[7]

, The numerical solution of the Navier-Stokes equations for a n incompressible ﬂuid , B. Am. Math. Soc. 73 (1967), no. 6, 928–931

work page 1967
[8]

, Numerical solution of the Navier-Stokes equations , Math. Comput. 22 (1968), no. 104, 745–762

work page 1968

Show all 32 references

[9]

Pure Appl

Anna De Masi, Raﬀaele Esposito, and Joel-Louis Lebowitz, Incompressible Navier-Stokes and Euler limits of the Boltzmann equation , Commun. Pure Appl. Math. 42 (1989), no. 8, 1189–1214

1989
[10]

Donatella Donatelli and Pierangelo Marcati, Leray weak solutions of the incompressible Navier Stokes sy stem on exterior domains via the artiﬁcial compressibility method , Indiana Univ. Math. J. 59 (2010), 1831–1852

2010
[11]

Rudolf Friedrich, Anton Daitche, Oliver Kamps, Johanne s L¨ ulﬀ, Michel Voßkuhle, and Michael Wilczek, The Lundgren–Monin–Novikov hierarchy: kinetic equations for turbulence, C. R. Phys. 13 (2012), no. 9-10, 929–953

2012
[12]

Fran¸ cois Golse and Laure Saint-Raymond, The Navier–Stokes limit of the Boltzmann equation for bound ed collision kernels, Invent. Math. 155 (2004), 81–161

2004
[13]

Meth- ods Appl

Philip M Gresho, Some current CFD issues relevant to the incompressible Navi er-Stokes equations, Comput. Meth- ods Appl. Mech. Eng. 87 (1991), no. 2-3, 201–252

1991
[14]

Theory Methods Appl

Im` ene Hachicha, Global existence for a damped wave equation and convergence towards a solution of the Navier– Stokes problem , Nonlinear Anal. Theory Methods Appl. 96 (2014), 68–86

2014
[15]

Alexei Ilyin, Yuri Rykov, and Sergey Zelik, Hyperbolic relaxation of the 2D Navier–Stokes equations in a bounded domain, Physica D 376 (2018), 171–179

2018
[16]

Pure Appl

Shi Jin and Zhouping Xin, The relaxation schemes for systems of conservation laws in a rbitrary space dimensions , Commun. Pure Appl. Math. 48 (1995), no. 3, 235–276

1995
[17]

Michael Junk and W en-An Yong, Rigorous Navier–Stokes limit of the lattice Boltzmann equa tion, Asymptot. Anal. 35 (2003), no. 2, 165–185

2003
[18]

Yoshiyuki Kagei, On the mathematical analysis of the artiﬁcial compressibil ity method , Partial Diﬀer. Equ. Appl. 2 (2021), 1–19

2021
[19]

Tosio Kato, Nonstationary ﬂows of viscous and ideal ﬂuids in R3, J. Func. Anal. 9 (1972), no. 3, 296–305

1972
[20]

, The Cauchy problem for quasi-linear symmetric hyperbolic s ystems, Arch. Ration. Mech. Anal. 58 (1975), no. 3, 181–205

1975
[21]

Pure Appl

Sergiu Klainerman and Andrew Majda, Singular limits of quasilinear hyperbolic systems with lar ge parameters and the incompressible limit of compressible ﬂuids , Commun. Pure Appl. Math. 34 (1981), no. 4, 481–524. 18 QIAN HUANG, CHRISTIAN ROHDE, WEN-AN YONG, AND RUIXI ZHANG

1981
[22]

Olga Aleksandrovna Ladyzhenskaya, The mathematical theory of viscous incompressible ﬂow , Gordon & Breach, New York, 1969

1969
[23]

T. S. Lundgren, Distribution functions in the statistical theory of turbul ence, Phys. Fluids. 10 (1967), no. 5, 969–975

1967
[24]

Massa, L

F. Massa, L. Ostrowski, F. Bassi, and C. Rohde, An artiﬁcial equation of state based Riemann solver for a dis con- tinuous Galerkin discretization of the incompressible Nav ier-Stokes equations , J. Comput. Phys. 448 (2022), Paper No. 110705, 25. MR 4319340

2022
[25]

P Nithiarasu, An eﬃcient artiﬁcial compressibility (AC) scheme based on t he characteristic based split (CBS) method for incompressible ﬂows , Int. J. Numer. Methods Eng. 56 (2003), no. 13, 1815–1845

2003
[26]

21, EDP Sciences, 2007, pp

Marius Paicu and Genevi` eve Raugel, Une perturbation hyperbolique des ´ equations de Navier-Stokes, ESAIM, Proc., vol. 21, EDP Sciences, 2007, pp. 65–87

2007
[27]

Reinhard Racke and J¨ urgen Saal, Hyperbolic Navier-Stokes equations II: Global existence o f small solutions , Evol. Equ. Control Theory 1 (2012), no. 1, 217–234

2012
[28]

Roger Temam, Sur l’approximation de la solution des ´ equations de Navier -Stokes par la m´ ethode des pas fraction- naires (I) , Arch. Ration. Mech. Anal. 32 (1969), 135–153. 29. , Sur l’approximation de la solution des ´ equations de Navier -Stokes par la m´ ethode des pas f...

1969
[30]

343, American Mathematical Soc., 2001

, Navier-Stokes equations: Theory and numerical analysis , vol. 343, American Mathematical Soc., 2001

2001
[31]

Jiang Xu and W en-An Yong, Zero-relaxation limit of non-isentropic hydrodynamic mod els for semiconductors , Discrete Contin. Dyn. Syst. Ser. A 25 (2009), no. 4, 1319

2009
[32]

W en-An Yong, Singular perturbations of ﬁrst-order hyperbolic systems w ith stiﬀ source terms , J. Diﬀer. Equations 155 (1999), no. 1, 89–132

1999
[33]

259–305, Birkh¨ auser Boston, Boston, MA, 2001

W en-An Yong, Basic aspects of hyperbolic relaxation systems , pp. 259–305, Birkh¨ auser Boston, Boston, MA, 2001. Institute of Applied Analysis and Numerical Simulation, Univer sity of Stuttgart, 70569 Stuttgart, Germany Email address : qian.huang@mathematik.uni-stuttgart.de;...

2001

Pith tools

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

[1] [1]

Denise Aregba-Driollet and Roberto Natalini, Discrete kinetic schemes for multidimensional systems of c onservation laws, SIAM J. Numer. Anal. 37 (2000), no. 6, 1973–2004

work page 2000

[2] [2]

Hajer Bahouri, Jean-Yves Chemin, and Rapha¨ el Danchin, Fourier analysis and nonlinear partial diﬀerential equa- tions, Springer, 2011

work page 2011

[3] [3]

Roberta Bianchini, Strong convergence of a vector-BGK model to the incompressi ble navier-stokes equations via the relative entropy method , J. Math. Pure Appl. 132 (2019), 280–307

work page 2019

[4] [4]

Yann Brenier, Roberto Natalini, and Marjolaine Puel, On a relaxation approximation of the incompressible Navier - Stokes equations , Proc. Am. Math. Soc. 132 (2004), 1021–1028

work page 2004

[5] [5]

Yann Brenier and W en-An Yong, Derivation of particle, string, and membrane motions from t he Born–Infeld electromagnetism, J. Math. Phys. 46 (2005), no. 6, 062305

work page 2005

[6] [6]

Alexandre Joel Chorin, A numerical method for solving incompressible viscous ﬂow p roblems, J. Comput. Phys. 2 (1967), no. 1, 12–26

work page 1967

[7] [7]

, The numerical solution of the Navier-Stokes equations for a n incompressible ﬂuid , B. Am. Math. Soc. 73 (1967), no. 6, 928–931

work page 1967

[8] [8]

, Numerical solution of the Navier-Stokes equations , Math. Comput. 22 (1968), no. 104, 745–762

work page 1968