pith. sign in

arxiv: 2512.01628 · v3 · submitted 2025-12-01 · 🧮 math.NA · cs.NA

An L-Stable Implicit Two-Stage Fourth-Order Temporal Discretization Scheme for Lax-Wendroff-Type Solvers Applied to Stiff Problems

Zhixin Huo This is my paper

Pith reviewed 2026-05-17 03:10 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords implicit time discretizationL-stable schemetwo-stage fourth-order methodLax-Wendroff solversstiff problemsnumerical methods for PDEstemporal accuracy
0
0 comments X

The pith

An implicit two-stage fourth-order scheme brings L-stability to Lax-Wendroff solvers for stiff problems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs an implicit version of the two-stage fourth-order temporal discretization method for use inside Lax-Wendroff-type solvers. It derives the scheme coefficients via undetermined coefficients and Taylor expansion, then obtains sufficient conditions for L-stability by applying the maximum modulus principle to a linear test equation. Newton iteration is introduced to solve the resulting nonlinear systems at each stage. Numerical tests on standard stiff benchmarks confirm that the scheme attains fourth-order temporal accuracy while permitting larger stable time steps and producing smaller errors than the classical fourth-order implicit Runge-Kutta method. The construction is presented so that it can be coupled directly with spatial Lax-Wendroff discretizations, thereby embedding both stiff source terms and transport into the time derivatives without operator splitting.

Core claim

The central claim is that a two-stage implicit time discretization can be designed to be L-stable, to achieve fourth-order accuracy, and to integrate synchronously with Lax-Wendroff spatial operators so that stiff source terms and flow transport are both carried inside the time derivatives, eliminating splitting errors while supporting larger stable steps than explicit TSFO or classical implicit Runge-Kutta methods.

What carries the argument

The implicit two-stage fourth-order (TSFO) time discretization, obtained by undetermined coefficients and Taylor expansion, with L-stability conditions enforced through the maximum modulus principle on a model equation.

If this is right

  • Fourth-order temporal accuracy is reached with only two implicit stages instead of the usual four stages required by classical implicit Runge-Kutta methods.
  • Larger stable time steps become admissible for stiff problems while still preserving the compact stencil of Lax-Wendroff-type solvers.
  • Stiff source terms and convective transport can be advanced together inside the time derivatives, removing the need for operator splitting.
  • The same implicit TSFO construction extends to a full implicit temporal-spatial coupling method that maintains high-resolution capture of strong discontinuities.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may reduce the number of global time steps needed in long-time simulations of multi-scale reacting flows or plasma problems.
  • Because the scheme remains compact, it could be paired with existing high-order finite-volume or discontinuous-Galerkin spatial operators without enlarging the stencil.
  • Direct comparison of operator-split versus unsplit versions on the same mesh would quantify the error reduction attributable to the synchronous treatment of stiff terms.

Load-bearing premise

The L-stability conditions obtained from a linear model equation continue to guarantee stability when the same scheme is applied to nonlinear systems coupled with spatial discretizations in Lax-Wendroff solvers.

What would settle it

Numerical integration of a stiff nonlinear test problem with the new scheme at a time step larger than the stability limit of the classical fourth-order implicit Runge-Kutta method either loses accuracy below fourth order or produces growing oscillations.

Figures

Figures reproduced from arXiv: 2512.01628 by Zhixin Huo.

Figure 3.1
Figure 3.1. Figure 3.1: A-stability: Valid C Range Determination [PITH_FULL_IMAGE:figures/full_fig_p009_3_1.png] view at source ↗
read the original abstract

The explicit two-stage fourth-order (TSFO) temporal-spatial coupling method is efficient and compact but suffers severe time-step restrictions for stiff problems with multiple scales. To address Professor Jiequan Li's call for an implicit extension, this paper first constructs an implicit TSFO time discretization scheme using the method of undetermined coefficients and Taylor expansion. Second, using a model equation and the maximum modulus principle, sufficient conditions for L-stability are derived. Third, a Newton iteration accelerates convergence. Numerical experiments on classical stiff benchmarks show that the proposed implicit scheme achieves fourth-order temporal accuracy in two stages. Compared to the classical fourth-order implicit Runge-Kutta method, it allows larger stable time steps and reduces convergence errors by an order of magnitude. More importantly, this implicit scheme can be extended to construct an implicit TSFO temporal-spatial coupling method that captures flow-field correlations and handles strong discontinuities, fundamentally contrasting with method-of-lines approaches. Additionally, it unlocks Lax-Wendroff-type solvers to naturally and synchronously embed both stiff source terms and flow transport into time derivatives, thereby avoiding operator-splitting errors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs an implicit two-stage fourth-order (TSFO) temporal discretization scheme for stiff problems using undetermined coefficients and Taylor expansion. L-stability conditions are derived from the linear test equation via the maximum modulus principle. Newton iteration solves the implicit stages. Numerical experiments on classical stiff ODE benchmarks report fourth-order temporal accuracy, larger stable time steps, and smaller errors than a classical fourth-order implicit Runge-Kutta method. The scheme is claimed to extend to an implicit TSFO temporal-spatial coupling method for Lax-Wendroff solvers that embeds stiff sources and transport without splitting errors and handles discontinuities.

Significance. If the central claims hold, the scheme would supply a compact two-stage L-stable integrator that improves efficiency over standard implicit Runge-Kutta methods while enabling synchronous treatment of stiff sources and hyperbolic transport in Lax-Wendroff frameworks. This could reduce operator-splitting artifacts in multi-scale problems. The numerical evidence on linear stiff ODEs supports the accuracy and step-size advantages, but the significance for the target nonlinear hyperbolic setting remains conditional on the unproven transfer of stability and accuracy properties.

major comments (2)
  1. [L-stability derivation] L-stability derivation: The sufficient conditions for L-stability are obtained exclusively from the linear model equation y' = λy (Re λ < 0) by applying the maximum modulus principle to the stability function. No nonlinear stability analysis, modified-equation analysis, or energy estimate is supplied to justify that these conditions remain valid for nonlinear flux/source terms or for the coupled spatial discretizations arising in Lax-Wendroff-type solvers; this extension is load-bearing for the central applicability claim.
  2. [Numerical experiments] Numerical experiments section: The reported benchmarks consist of classical stiff ODEs. These test problems lack the nonlinear hyperbolic structure, shock-capturing mechanisms, and spatial coupling that the manuscript invokes when asserting that the scheme extends to Lax-Wendroff solvers for stiff problems; consequently the experiments do not directly support the strongest claims about discontinuity handling and splitting-error avoidance.
minor comments (2)
  1. The abstract refers to 'Professor Jiequan Li's call' without a specific citation; adding the reference would clarify the motivation and novelty positioning.
  2. The notation distinguishing the two implicit stages and the embedding of spatial derivatives into the time derivatives could be made more explicit in the scheme definition to aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and for recognizing the potential significance of the implicit TSFO scheme. We address each major comment point by point below.

read point-by-point responses

  1. Referee: [L-stability derivation] L-stability derivation: The sufficient conditions for L-stability are obtained exclusively from the linear model equation y' = λy (Re λ < 0) by applying the maximum modulus principle to the stability function. No nonlinear stability analysis, modified-equation analysis, or energy estimate is supplied to justify that these conditions remain valid for nonlinear flux/source terms or for the coupled spatial discretizations arising in Lax-Wendroff-type solvers; this extension is load-bearing for the central applicability claim.

    Authors: We agree that the L-stability conditions are derived from the linear test equation using the maximum modulus principle, which is the standard approach for analyzing implicit integrators intended for stiff problems. This yields necessary and sufficient conditions in the stiff limit. The manuscript positions the scheme as a building block for Lax-Wendroff-type solvers, where the implicit stages are embedded directly into the time derivatives to treat stiff sources synchronously with transport. While a complete nonlinear stability analysis for general flux/source terms is not included, the linear analysis combined with the fourth-order consistency provides the foundation for the claimed extension. In the revision we will add a clarifying paragraph on the scope of the stability result and its intended use for the coupled problem. revision: partial

  2. Referee: [Numerical experiments] Numerical experiments section: The reported benchmarks consist of classical stiff ODEs. These test problems lack the nonlinear hyperbolic structure, shock-capturing mechanisms, and spatial coupling that the manuscript invokes when asserting that the scheme extends to Lax-Wendroff solvers for stiff problems; consequently the experiments do not directly support the strongest claims about discontinuity handling and splitting-error avoidance.

    Authors: The numerical section deliberately isolates the temporal scheme on standard stiff ODE benchmarks to verify fourth-order accuracy, L-stability, and the ability to take larger steps than classical implicit Runge-Kutta methods. These tests confirm the core properties required before coupling. The extension to Lax-Wendroff solvers is described theoretically in the manuscript: the implicit TSFO stages allow stiff sources to be incorporated into the same time derivatives that already contain the hyperbolic fluxes, thereby avoiding splitting. Discontinuity handling is inherited from the underlying Lax-Wendroff spatial discretization. We will revise the text to more explicitly delineate the temporal validation from the proposed coupling framework and to note that full nonlinear hyperbolic tests lie beyond the present scope. revision: partial

Circularity Check

0 steps flagged

Derivation uses undetermined coefficients, Taylor expansion, and maximum-modulus analysis on linear model without reduction to inputs

full rationale

The paper constructs the implicit two-stage fourth-order scheme via the method of undetermined coefficients and Taylor expansion to enforce accuracy conditions, then derives L-stability sufficient conditions from the linear test equation y' = λy using the maximum modulus principle. These steps are independent of the target nonlinear or Lax-Wendroff results and do not reduce to fitted parameters or self-referential definitions. Numerical benchmarks on classical stiff ODEs function as external validation rather than inputs that force the claimed accuracy or stability. No load-bearing self-citations, ansatz smuggling, or renaming of known results appear in the derivation chain; the extension claim to nonlinear systems is an unproven assertion but does not create circularity in the temporal discretization itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central construction relies on standard numerical analysis tools rather than new postulates or fitted entities.

axioms (2)
  • standard math Taylor expansion is used to match coefficients for fourth-order accuracy
    Invoked in the construction of the implicit scheme via method of undetermined coefficients.
  • domain assumption Maximum modulus principle applies to derive sufficient conditions for L-stability on a model equation
    Used to obtain stability conditions for the time discretization.

pith-pipeline@v0.9.0 · 5495 in / 1303 out tokens · 31792 ms · 2026-05-17T03:10:24.754184+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages

  1. [1]

    Kolgan, Application of the principle of minimum derivatives to the construction of diference schemes for computing discontinuous solutions of gas dynamics (in Russian), Uch

    V. Kolgan, Application of the principle of minimum derivatives to the construction of diference schemes for computing discontinuous solutions of gas dynamics (in Russian), Uch. Zap. TsaGI, Russia 3.6 (1972) 68-77

    work page 1972

  2. [2]

    van Leer, Towards the ultimate conservative difference scheme I

    B. van Leer, Towards the ultimate conservative difference scheme I. the quest for monotonicity, Lect. Notes Phys. 18 (1973) 163-168

    work page 1973

  3. [3]

    van Leer, Towards the ultimate conservative difference scheme II

    B. van Leer, Towards the ultimate conservative difference scheme II. mononicity and conservation combined in a second order scheme, J. Comput. Phys. 14 (1974) 361-370

    work page 1974

  4. [4]

    Toro and V

    E. Toro and V. Titarev, Solution of the Generalised Riemann Problem for Advection-Reaction Equations, Proc. Roy. Soc. London A 458 (2002) 271-281

    work page 2002

  5. [5]

    Titarev and E

    V. Titarev and E. Toro, ADER: Arbitrary high order Godunov approach, J. Scientific Computing , 17 (2002) 609-618

    work page 2002

  6. [6]

    Toro and V

    E. Toro and V. Titarev, ADER Schemes for Scalar Hyperbolic Conservation Laws with Source Terms in Three Space Dimensions, J. Comput. Phys. 202.1 (2005) 196-215

    work page 2005

  7. [7]

    Balsara, T

    D. Balsara, T. Rumpf, M. Dumbser, et al, Efficient, high accuracy ADER-WENO schemes for hydrohynamics and divergencefree magnetohydrodynamics, J. Comput. Phys. 228.7 (2009) 2480-2516

    work page 2009

  8. [8]

    E. Toro, A. Hidalgo, ADER finite volume schemes for nonlinear reaction-diffusion equations, Appl. Numer. Math. 59 (2009) 73-100

    work page 2009

  9. [9]

    Hidalgo, M

    A. Hidalgo, M. Dumbser, ADER schemes for nonlinear systems of stiff advection-diffusion-reaction equations, J. Sci. Comput., 48 (2011) 173-189

    work page 2011

  10. [10]

    Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes with time-accurate local time stepping for hyperbolic conservation laws

    M.Dumbser. Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes with time-accurate local time stepping for hyperbolic conservation laws. Computer methods in applied mechanics and engineering, 2014, 280: 57-83

    work page 2014

  11. [11]

    Hidlgo, O

    M.Dumbser, A. Hidlgo, O. Zanotti. High order space-time adaptive ADER-WENO finite volume schemes for non-conservative hyperbolic systems. Computer methods in applied mechanics and engineering, 2014, 268: 359-387

    work page 2014

  12. [12]

    Boscheri, M

    W. Boscheri, M. Dumbser. An efficient quadrature-free formulation for high order Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes on unstructured meshes. Journal of scientific computing, 2014, 48: 173-189

    work page 2014

  13. [13]

    Boscheri, M

    W. Boscheri, M. Dumbser. A direct Arbitrary-Lagrangian Eulerian ADER-WENO finite volume scheme on unstructured tetrahedral meshes for conservative and non-conservative hyperbolic systems in 3D. Journal of computational physics, 2014, 275: 484-523

    work page 2014

  14. [14]

    Boscheri, R

    W. Boscheri, R. Loubere, M. Dumbser. Direct ArbitraryLagrangian-Eulerian ADER-MOOD finite volume schemes for multidimensional hyperbolic conservation laws. Journal of computational physics, 2015, 292: 56-87

    work page 2015

  15. [15]

    Dumbser, R

    W.Boscheri, M. Dumbser, R. Loubere. Cell centered direct Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes for nonlinear hyperelasticity. Computers and fluids, 2016, 134/135: 111-129

    work page 2016

  16. [16]

    Boscheri, M

    W. Boscheri, M. Dumbser. High order accurate direct Arbitrary Lagrangian-Eulerian ADER-WENO finite volume schemes on moving curvilinear unstructured meshes. Computers and fluids, 2016, 136: 48-66

    work page 2016

  17. [17]

    Zanotti, F

    O. Zanotti, F. Fambri, M. Dumbser, et al. Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting. Computers and fluids, 2015, 118: 204-224

    work page 2015

  18. [18]

    Fambri, M

    F. Fambri, M. Dumbser, O. Zanotti. Space-time adaptive ADER-DG schemes for dissipative flows-compressible Navier-Stokes and resistive MHD equations. Computer physics communications, 2017, 220: 297-318

    work page 2017

  19. [19]

    Ben-Artzi and J

    M. Ben-Artzi and J. Falcovitz, A Second Order Godunov-Type Scheme for Compressible Fluid Dynamics, J. Comput. Phys. 55 (1984) 1-32

    work page 1984

  20. [20]

    Ben-Artzi and J

    M. Ben-Artzi and J. Falcovitz, A High Resolution Upwind Scheme for Quasi 1-D Flows, In Numerical Methods for the Euler Equations of Fluid Dynamics, INRIA, SIAM (1985) 66-83

    work page 1985

  21. [21]

    Ben-Artzi and J

    M. Ben-Artzi and J. Falcovitz, Generalized Riemann Problems in Computational Gas Dynamics, Cambridge Univ. Press (2003)

    work page 2003

  22. [22]

    X. Lei, J. Li. Transversal effects of high order numerical schemes for compressible fluid flows. Appl. Math. Mech., 40(3): 343-354, 2019

    work page 2019

  23. [23]

    Ben-Artzi and J

    M. Ben-Artzi and J. Li. Hyperbolic Balance Laws: Riemann Invariants and Hyperbolic Balance Laws. Numer. Math., 106: 69-425, 2007

    work page 2007

  24. [24]

    J. Li, T. Liu, Z. Sun. Implementation of the GRP scheme for computing radially symmetric compressible fluid flows. J. Comput. Phys., 228: 5867-5887, 2009

    work page 2009

  25. [25]

    J. Li, Y. Wang, Thermodynamical effects and high resolution methods for compressible fluid flows, J. Comput. Phys. 343 (2017) 340-354

    work page 2017

  26. [26]

    Ben-Artzi and J

    M. Ben-Artzi and J. Falcovitz, An upwind second-order scheme for compressible duct flows, SIAM J. Sci. Comput. 7 (1986) 744-768

    work page 1986

  27. [27]

    Ben-Artzi, The generalized Riemann problem for reactive flows, J

    M. Ben-Artzi, The generalized Riemann problem for reactive flows, J. Comput. Phys. 81.1 (1989) 70-101

    work page 1989

  28. [28]

    J. Li, G. Chen, The generalized Riemann problem method for the shallow water equations with bottom topography, Int. J. Numer. Methods Eng. 65.6 (2006) 834-862

    work page 2006

  29. [29]

    F. Zhou, G. Chen, S. Noelle, et al, A well-balanced stable generalized Riemann problem scheme for shallow water equations using adaptive moving unstructured triangular meshes, Numer. Meth. Fluids, 73.3 (2013) 266-283

    work page 2013

  30. [30]

    J. Qi, J. Li. A fully discrete ALE method over untwisted time-space control volumes. Numerical methods in fluids, 2017, 83(8): 625-641

    work page 2017

  31. [31]

    X. Lei, J. Li, A staggered-projection Godunov-type method for the Baer-Nunziato two-phase model, J. Comput. Phys. 437 (2021) 110312

    work page 2021

  32. [32]

    TW. Chen, Z. Du, Generalized Riemann problem method for the Kapila Model of compressible multiphase flows, Phys. Fluids 37.7 (2025) 076149

    work page 2025

  33. [33]

    Z. Huo, Z. Jia, A GRP-based tangential effects preserving, high resolution and efficient ghost fluid method for the simulation of two-dimensional multi-medium compressible flows, Comput. Fluids 276 (2024) 106261

    work page 2024

  34. [34]

    Zhang, W

    Q. Zhang, W. Sheng, The generalized Riemann problem scheme for a laminar two-phase flow model with two-velocities, J. Comput. Phys. 506 (2024) 112929

    work page 2024

  35. [35]

    Hazra, D

    A. Hazra, D. Balsara, P. Chandrashekar, et al, Multidimensional generalized Riemann problem solver for Maxwell's equations, J. Sci. Comput. 96 (2023) 26

    work page 2023

  36. [36]

    K. Wu, H. Tang, A direct Eulerian GRP scheme for spherically symmetric general relativistic hydrodynamics, SIAM J. Sci. Comput. 38 (2016) B458-B489

    work page 2016

  37. [37]

    Z. Yang, P. He, H. Tang, A direct Eulerian GRP scheme for relativistic hydrodynamics: one-dimensional case, J. Comput. Phys. 230.22 (2011) 7964-7987

    work page 2011

  38. [38]

    Z. Yang, H. Tang, A direct Eulerian GRP scheme for relativistic hydrodynamics: Two-dimensional case, J. Comput. Phys. 231.4 (2012) 2116-2139

    work page 2012

  39. [39]

    Sheng, Q

    W. Sheng, Q. Zhang, Y. Zheng, A direct Eulerian GRP scheme for a blood flow model in arteries, SIAM J. Sci. Comput. 43 (2021) A1975-A1996

    work page 2021

  40. [40]

    Kuang, H

    Y. Kuang, H. Tang, Second-order direct Eulerian GRP schemes for radiation hydrodynamical equations, Comput. Fluids 179 (2019) 163-177

    work page 2019

  41. [41]

    Ben-Artzi and J.Q

    M. Ben-Artzi and J.Q. Li, Consistency of finite volume approximations to nonlinear hyperbolic balance laws, Math. Comp. 90.327 (2020) 141-169

    work page 2020

  42. [42]

    Ben-Artzi , J.Q

    M. Ben-Artzi , J.Q. Li, Regularity of fluxes in nonlinear hyperbolic balance laws, Commun. Appl. Math. Comput. 5.3 (2023) 1289-1298

    work page 2023

  43. [43]

    Runge, U ber die numerisehe Aufl o sing yon Differentialgleichungen, Math

    C. Runge, U ber die numerisehe Aufl o sing yon Differentialgleichungen, Math. Ann. 46 (1895) 167-178

    work page

  44. [44]

    Kutta, Beitrag zur n a herungsweisen Integration totaler Differentialgleichungen, Z

    W. Kutta, Beitrag zur n a herungsweisen Integration totaler Differentialgleichungen, Z. Math. Phys. 46 (1901) 435-453

    work page 1901

  45. [45]

    Kuntzmann, Neure Entwicklungen der Methoden yon Runge und Kutta, Z

    J. Kuntzmann, Neure Entwicklungen der Methoden yon Runge und Kutta, Z. Angew. Math. Mech. 41 (1961) T28-T31

    work page 1961

  46. [46]

    Butcher, Implicit Runge-Kutta processes, Math

    J.C. Butcher, Implicit Runge-Kutta processes, Math. Comp. 18 (1964) 50-64

    work page 1964

  47. [47]

    Li and Z

    J. Li and Z. Du, A two-stage fourth order time-accurate discretization for Lax Wendroff type flow solvers I. Hyperbolic conservation laws, SIAM J. Sci. Comput. 38 (2016) A3046-A3069

    work page 2016

  48. [48]

    Li, Two-stage fourth order: Temporal-spatial coupling in computational fluid dynamics (CFD), Aero

    J. Li, Two-stage fourth order: Temporal-spatial coupling in computational fluid dynamics (CFD), Aero. Adv. 1.3 (2019)

    work page 2019

  49. [49]

    Cheng, Z

    J. Cheng, Z. Du, X. Lei, Y. Wang and J. Li, A two-stage fourth-order discontinuous Galerkin method based on the GRP solver for the compressible Euler equations, Comput. Fluids 181 (2019) 248-258

    work page 2019

  50. [50]

    Du and J.Q

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

    work page 2018

  51. [51]

    F. Zhao, X. Ji, W. Shyy and K. Xu, Compact higher-order gas-kinetic schemes with spectral-like resolution for compressible flow simulations, Adv. Aerodyn. 1.1 (2019) 34

    work page 2019

  52. [52]

    Yuan and H

    Y. Yuan and H. Tang, On the explicit two-stage fourth-order accurate time discretizations, J. Comput. Math. 41.2 (2023) 305-324

    work page 2023

  53. [53]

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

    work page 2016

  54. [54]

    Y. Chen, C. Chen, F. Xiao, X. Li and X. Shen, A two-stage fourth-order multi-moment global shallow water model on cubed sphere, Mon. Wea. Rev. 148.10 (2020) 4267-4279

    work page 2020

  55. [55]

    Zhang, Q

    C. Zhang, Q. Li, Z. Wang, J. Li and S. Fu, A two-stage fourth-order gas kinetic CPR method for Navier-Stokes equations on triangular meshes, J. Comput. Phys. 451 (2022) 110830

    work page 2022