Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

Multicontinuum Modeling of Time-Fractional Diffusion-Wave Equation in Heterogeneous Media

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read A new coarse-grid model approximates time-fractional diffusion-wave solutions in high-contrast media with relative L2 errors below about five percent in tested cases.

desk verdict A useful extension of multicontinuum homogenization to fractional diffusion-wave equations, with solid numerics, but the key truncation step in the derivation is not justified when the RVE is the coarse block. read the letter →

arxiv 2502.09428 v1 pith:VBFBTYDI submitted 2025-02-13 math.NA cs.NA

Huiran Bai , Dmitry Ammosov , Yin Yang , Wei Xie , Mohammed Al Kobaisi This is my paper
classification math.NAcs.NA MSC 35B2735R1165M60
keywords multicontinuumhomogenizationtime-fractionaldiffusion-waveequationheterogeneousmediahigh-contrastcoefficientCaputofractionalderivativemultiscalemodelreductioncoarse-gridsimulation
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 tries to show that a time-fractional diffusion-wave equation with a strongly heterogeneous diffusion coefficient can be replaced, on a coarse grid, by a small set of macroscopic multicontinuum equations without losing the fine-scale behavior that matters. Direct simulation of such problems is expensive because the coefficient varies by orders of magnitude on scales much finer than the computational grid. The authors derive the coarse model by solving constrained cell problems in small sampling regions, then substituting a truncated expansion of the fine solution into the variational form and using smoothness of the macroscopic variables to obtain effective coefficients. In two-dimensional tests with crossed and layered high-contrast inclusions, the coarse model reproduces the per-region averaged fine solution with relative $L^2$ errors mostly below five percent.

What carries the argument

The load-bearing mechanism is the truncated expansion $u\approx \phi_iU_i + \phi_i^m\nabla_m U_i$ in each representative volume element, together with the constrained cell problems (22)--(23) that determine $\phi_i$ and $\phi_i^m$. Those cell problems are posed in oversampled regions and impose constraints on the averages and first moments of each continuum, which suppresses boundary effects and yields the scale estimates $\|\phi_i\|=O(1)$, $\|\nabla\phi_i\|=O(1/\epsilon)$, $\|\phi_i^m\|=O(\epsilon)$, and $\|\nabla\phi_i^m\|=O(1)$. These estimates justify keeping only the leading term in the fractional-derivative part of the variational form and define the scaled effective coefficients $cC_{ji}$, $dB^{mn}_{ji}$, $dB^m_{ji}$, and $dB_{ji}$; integration by parts then removes the first-order terms and produces the macroscopic model (32). The Caputo time derivative is discretized with the fully discrete scheme (4), which provides temporal accuracy of order $3-\alpha$.

What would settle it

Take a high-contrast coefficient whose finest heterogeneity scale equals the coarse-grid size $H$, so no interior sample region can represent a whole coarse block, solve the fine-scale equation (1), and compare with the multicontinuum model; if the relative $L^2$ errors remain below five percent, the scale-separation assumption is not necessary, whereas the derivation predicts the errors should grow.

Watch

Extended reading notes

Core claim

The central claim is that the multicontinuum homogenization construction carries over to the time-fractional diffusion-wave equation (1). In each representative volume element the fine solution is expanded as $u\approx \phi_iU_i + \phi_i^m\nabla_m U_i$, where $U_i$ is the macroscopic average in continuum $i$ and the auxiliary functions are defined by the constrained cell problems (22)--(23) in oversampled regions. Substituting this expansion into the weak formulation and dropping terms whose auxiliary functions are of order $\epsilon$ relative to the leading term gives the strong-form model $$cC_{ji}\frac{\$partial^{{\alpha}}$U_i}{\partial $t^{{\alpha}}$} - \nabla_n($dB^{{mn}}$_{ji}\nabla_m U_i) + \$epsilon^{{-2}}$dB_{ji}U_i = f_j,$$ with effective coefficients computed from cell-problem integrals; for mixed fractional orders the same construction gives $$dC_{jip}\frac{\$partial^{{\alpha_p}}$U_i}{\partial $t^{{\alpha_p}}$} - \nabla_n($dB^{{mn}}$_{ji}\nabla_m U_i) + \$epsilon^{{-2}}$dB_{ji}U_i = f_j,$$ so that each continuum equation contains a linear combination of all time-fractional derivatives. The reported numerical experiments support the claim: relative $L^2$ errors of the averaged multicontinuum solution versus the fine-scale reference stay below about five percent in all tested configurations and typically fall below one percent on the finer coarse grid.

Load-bearing premise

The derivation assumes a clean separation of scales: the averaged state in a small sample region (the RVE) represents the whole coarse block, and the macroscopic variables are smooth enough that they and their gradients can be pulled out of the RVE integrals; if the solution varies on the coarse scale, the effective model has no guaranteed error bound.

Editorial extensions

If this is right

  • Coarse-grid simulations with model (32) reproduce per-continuum averaged fine-scale solutions for fractional orders $1<\alpha<2$ in high-contrast two-dimensional media.
  • The mixed-derivative model (39) makes it possible to upscale problems in which different regions follow different memory laws, because each continuum equation contains a linear combination of all fractional time derivatives.
  • The $\epsilon^{-2}$ reaction term in the model is dominant unless diffusion is large, so the high-contrast structure enters through computed effective coefficients rather than through fine-grid degrees of freedom.
  • Refining the coarse mesh from $H=1/20$ to $H=1/40$ consistently reduces the relative $L^2$ errors in all reported experiments.

Reading between the lines

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

  • Beyond the tested two-dimensional, two-continuum settings, the derivation itself does not restrict the number of dimensions or continua, so a plausible inference is that the construction extends to three dimensions and to three or more continua; the paper does not test this.
  • A natural stress test would apply the method to a coefficient whose heterogeneity period is comparable to the coarse-grid size, so that no interior sample region can represent the coarse block; the scale-separation assumption then fails and the error should grow.
  • The mixed-derivative variant also suggests a way to upscale coupled subdiffusion/superdiffusion processes in composite materials, where one constituent has memory closer to diffusion and another closer to wave propagation; this connection is not explored in the paper.
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.

Referee Report

3 major / 5 minor

Summary. The paper extends the multicontinuum homogenization method to the time-fractional diffusion-wave equation (1) with a high-contrast heterogeneous coefficient. The authors formulate constrained cell problems in oversampled regions, obtain multicontinuum expansions of the fine-scale solution, and substitute these expansions into the variational formulation to derive a macroscopic multicontinuum model, both for a single fractional order and for mixed fractional derivatives in different continua. Numerical experiments in two-dimensional high-contrast crossed and layered media report relative L2 errors in the range of roughly 0.3% to 5% for the averaged multiscale solutions, with smaller errors on finer coarse grids.

Significance. If the derivation can be made rigorous, the paper offers a useful extension of multicontinuum homogenization to fractional-order diffusion-wave problems with memory, including a genuinely new mixed-derivative formulation in which different continua have different fractional orders. The numerical evidence is extensive and encouraging: the effective coefficients come from cell problems rather than parameter fitting, the fine-grid reference solutions are independent, and the reported errors are small for several fractional orders and two qualitatively different heterogeneity geometries. The main weakness is that the central asymptotic derivation rests on smoothness and smallness assumptions that are not verified in the numerical regime actually tested, so the paper currently overstates the strength of its analytical claim.

major comments (3)
  1. [§3.1, Eqs. (20)-(28) and (31)] The cancellation of the second, third, and fourth terms in (28) is not justified by the stated comparison. The paper drops these terms because φ_j^n is O(ε), but each dropped term also contains a macroscopic gradient ∇_n V_j or ∇_m U_i. For a coarse finite element test function on a mesh of size H, |∇V_j| ~ 1/H, and generically |∇U_i| ~ 1/H as well, so the dropped terms are O(ε/H) relative to the retained term. Since Section 4 explicitly states that each coarse-grid block is taken as the RVE, we have ε = H in all numerical experiments, making the dropped terms O(1) rather than O(ε). Thus the formal derivation of (32) does not establish the model in the parameter regime used for validation; an alternative argument, such as orthogonality or cancellation arising from the constrained cell problems (22)-(23), is needed. The same issue affects the mixed-derivative model (39) via the analogous step in Eq. (37).
  2. [§4, first paragraph] The abstract and Section 3 claim that the multicontinuum model is 'rigorously derived', but the argument is formal: the expansion (20) is truncated after the first-gradient term without quantifying the remainder; macroscopic variables Ui and their gradients are pulled out of RVE integrals using smoothness assumptions; and the O(ε) estimates in (24) concern norms of φ_i and φ_i^m, not products with macroscopic gradients. The statement after (31) that integration by parts makes the sum of the 1/ε terms negligible is also asserted without proof and is not obvious when the effective coefficients are computed independently on each RVE. For the paper to claim rigor, it needs either a convergence or residual estimate controlling the neglected terms, or it should explicitly present the derivation as a formal homogenization procedure followed by empirical validation.
  3. [§3.2, Eq. (37)] The numerical experiments are performed exclusively in the regime ε = H, i.e., the RVE coincides with the coarse block, which is precisely the regime where the smallness argument used in (28) breaks down. The paper should either test a regime with ε ≪ H, such as RVEs that are genuinely smaller than the coarse blocks, or supply a separate analytical explanation for why the model remains accurate when the dropped terms are not small by the stated order estimates. As it stands, the numerical results provide empirical support for the method, but they do not validate the asymptotic derivation as written.
minor comments (5)
  1. [§2.2, Eq. (9)] There is a typo in the sentence following Eq. (6): 'symbolk is the Einstein summation convention' should read 'where the index k follows the Einstein summation convention'.
  2. [§4, Eq. (40)] The notation for representative volumes is inconsistent: the text and Figure 1 use Rω and R+ω interchangeably with R_ω and R^+_ω. Please standardize the notation, especially in the integrals and summations in Eqs. (9) and (25).
  3. [§4, Figures 4-15] The error measure in Eq. (40) should specify how the integrals over K and K ∩ Ω_i are computed numerically, and how the case K ∩ Ω_i = ∅ is handled; currently the formula is ambiguous because the denominator involves an integral over K ∩ Ω_i that may be zero for some coarse blocks.
  4. [§4, Tables 1-6] The figure captions say 'from top to bottom' but do not explicitly label the rows as reference and multiscale solutions; adding row labels directly on the panels would make the comparison clearer.
  5. [§1, Introduction] The error tables report four decimal places, which is unnecessarily precise for relative errors of this magnitude; reporting two or three significant digits would be sufficient and easier to read.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: effective properties are computed from cell problems without fitting, and the benchmark is an independently computed fine-grid reference; only minor non-load-bearing self-citations appear.

full rationale

The claimed derivation is self-contained against the fine-grid benchmark. The multicontinuum expansion (20)-(21) is an explicit ansatz, and the cell problems (22)-(23) are solved for the auxiliary basis functions from the coefficient κ and the continuum geometry alone; they do not involve the fine-scale solution u. The effective properties (30) are integrals of these precomputed basis functions, so the macroscopic models (32) and (39) are not obtained by fitting any parameter to the reference solution. The error metric (40) compares the coarse model solution U_i with a separately computed 400x400 fine-grid reference average, so the reported 0.3-5% relative errors are an independent numerical check. The only self-referential element is the introductory statement that the multicontinuum method 'has already been successfully applied to different problems [27,28,29,30]', where [27]-[29] include overlapping authors; this is not load-bearing for the derivation, which relies on the general theory [24,25]. A separate rigor concern, not a circularity, is the dropping of the second-fourth terms in (28) using O(epsilon) estimates for phi_j^n: in the tested regime the RVE coincides with the coarse block (epsilon = H), and coarse test-function gradients scale as 1/H, so the asymptotic comparison is not obviously valid. This affects the strength of the 'rigorous derivation' claim but does not make the prediction equivalent to its inputs by construction.

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

The model has no fitted free parameters; the effective coefficients are computed from the constrained cell problems. The derivation rests on prior multicontinuum homogenization theory [24,25] and on scale-separation and smoothness assumptions for the macroscopic variables, which are not proved here.

assumptions (5)
  • domain assumption The constrained cell problems (22)-(23) have solutions satisfying the estimates in (24): ||φ_i||=O(1), ||∇φ_i||=O(1/ε), ||φ_i^m||=O(ε), ||∇φ_i^m||=O(1).
    Equation (24) states these estimates and attributes them to reference [24]; the present paper does not prove them.
  • domain assumption Macroscopic variables U_i are smooth and slowly varying over each coarse block, so they can be taken outside RVE integrals and the truncated expansion (21) is accurate.
    Invoked in Section 3 immediately before eq (28) and in the integral-localization step (25).
  • standard math The fully discrete time-fractional scheme (4) from Sun and Wu [11] is stable and convergent with order O(τ^{3-α}).
    Used without proof for temporal discretization; cited from reference [11].
  • domain assumption The sum of the 1/ε terms in (31) is negligible after integration by parts.
    Stated after (31) with a citation to [24] and no calculation shown in this paper.
  • domain assumption The RVE average over Rω represents the whole coarse block in terms of heterogeneities.
    Assumed in the integral-localization step (25) and illustrated in Figure 1.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Multicontinuum Modeling of Time-Fractional Diffusion-Wave Equation in Heterogeneous Media." pith.science (2026). https://pith.science/paper/VBFBTYDI

@misc{pith2026250209428,
  author       = {Pith},
  title        = {Pith review of: Multicontinuum Modeling of Time-Fractional Diffusion-Wave Equation in Heterogeneous Media},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VBFBTYDI}},
  note         = {Machine review of arXiv:2502.09428}
}
read the original abstract

This paper considers a time-fractional diffusion-wave equation with a high-contrast heterogeneous diffusion coefficient. A numerical solution to this problem can present great computational challenges due to its multiscale nature. Therefore, in this paper, we derive a multicontinuum time-fractional diffusion-wave model using the multicontinuum homogenization method. For this purpose, we formulate constraint cell problems considering various homogenized effects. These cell problems are implemented in oversampled regions to avoid boundary effects. By solving the cell problems, we obtain multicontinuum expansions of fine-scale solutions. Then, using these multicontinuum expansions and supposing the smoothness of the macroscopic variables, we rigorously derive the corresponding multicontinuum model. Finally, we present numerical results for two-dimensional model problems with different time-fractional derivatives to verify the accuracy of our proposed approach.

Figures

Figures reproduced from arXiv: 2502.09428 by the authors.

Figure 1
Figure 1. Illustration of computation domain Ω, coarse block [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. The coefficient κ (Ω1: blue regions; Ω2: red regions). Crossed field. The source term f is given as f(x) = e −40((x1−0.5)2+(x2−0.5)2 ) for any x = (x1, x2) ∈ Ω. We set the initial condition u0 = u(x, 0) = 0, and the following Dirichlet boundary condition u(x, t) = 0, x ∈ ∂Ω. (42) 11 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. The fine-grid reference solution with α = 1.5 at t = 0.1, 0.5, 1 for Case 1 in Example 1 (from left to right). Figures 4-5 presents distributions of the average solutions with H = 1/40 in Ω1 and Ω2 subregions, respectively. From top to bottom, we depict the reference and multiscale average solutions. One can see that the solutions are very similar, which indicates that our proposed multicontinuum approach can approx… view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Average solution with α = 1.5 at t = 0.1, 0.5, 1 for Case 1 in Example 1 (from left to right), H = 1/40. First row: reference averaged solution in Ω1. Second row: multiscale solution in Ω1 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Average solution with α = 1.5 at t = 0.1, 0.5, 1 for Case 1 in Example 1 (from left to right), H = 1/40. First row: reference averaged solution in Ω2. Second row: multiscale solution in Ω2 [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: The fine-grid reference solution with α1 = 1.1, α2 = 1.9 at t = 0.1, 0.5, 1 for Case 2 in Example 1 (from left to right) [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Average solution with α1 = 1.1, α2 = 1.9 at t = 0.1, 0.5, 1 for Case 2 in Example 1 (from left to right), H = 1/40. First row: reference averaged solution in Ω1. Second row: multiscale solution in Ω1 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: Average solution with α1 = 1.1, α2 = 1.9 at t = 0.1, 0.5, 1 for Case 2 in Example 1 (from left to right), H = 1/40. First row: reference averaged solution in Ω2. Second row: multiscale solution in Ω2. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: The coefficient κ (Ω1: blue regions; Ω2: red regions). Layered field. Again, we consider both regular and mixed cases of time-fractional derivative orders in the following numerical experiments. 4.2.1 Case 1: Regular time derivatives In this case, we set the time fract…
Figure 10
Figure 10. Figure 10: The fine-grid reference solution with α = 1.2 at t = 0.1, 0.5, 1 for Case 1 in Example 2 (from left to right). Next, let us consider the average solutions of the reference and multiscale solutions. In Figures 11 and 12, we present distributions of the average solution…
Figure 11
Figure 11. Figure 11: Average solution with α = 1.2 at t = 0.1, 0.5, 1 for Case 1 in Example 2 (from left to right), H = 1/40. First row: reference averaged solution in Ω1. Second row: multiscale solution in Ω1. Let us consider the errors of the multiscale solution. In [PITH_FULL_IMAGE:fi…
Figure 12
Figure 12. Figure 12: Average solution with α = 1.2 at t = 0.1, 0.5, 1 for Case 1 in Example 2 (from left to right), H = 1/40. First row: reference averaged solution in Ω2. Second row: multiscale solution U2 in Ω2. 4.2.2 Case 2: Mixed time derivatives Finally, let us consider the case with…
Figure 13
Figure 13. Figure 13: shows distributions of the fine-grid solution at different time steps. One can see the changes in the solution distributions compared to the previous case due to the mixed time-fractional derivatives. It is especially noticeable in the high-conductive channels at the …
Figure 14
Figure 14. Figure 14: Average solution with α1 = 1.1, α2 = 1.9 at t = 0.1, 0.4, 1 for Case 2 in Example 2 (from left to right), H = 1/40. First row: reference averaged solution in Ω1. Second row: multiscale solution in Ω1. Next, let us consider the errors of our multicontinuum approach [P…
Figure 15
Figure 15. Figure 15: Average solution with α1 = 1.1, α2 = 1.9 at t = 0.1, 0.4, 1 for Case 2 in Example 2 (from left to right), H = 1/40. First row: reference averaged solution in Ω2. Second row: multiscale solution in Ω2. geneities that can result in isotropic and anisotropic propagation.…

Discussion (0). Sign in 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. Robust space-time multiscale upscaling via multicontinuum homogenization for evolving perforated media

    math.NA 2025-06 conditional novelty 6.0 of 10

    A space-time multicontinuum homogenization method is introduced for parabolic equations in shrinking perforated domains, validated by three numerical experiments with errors mostly below 10 percent.

Reference graph

Works this paper leans on

50 extracted references · 41 canonical work pages · cited by 1 Pith paper

  1. [1]

    A legendre spectral method on graded meshes for the two-dimensional multi-term time-fractional diffusion equation with non-smooth solutions

    Rumeng Zheng, Fawang Liu, and Xiaoyun Jiang. A legendre spectral method on graded meshes for the two-dimensional multi-term time-fractional diffusion equation with non-smooth solutions. Applied Mathematics Letters , 104:106247, 2020

    work page 2020

  2. [2]

    Some temporal second order difference schemes for fractional wave equations

    Hong Sun, Zhi-Zhong Sun, and Guang-Hua Gao. Some temporal second order difference schemes for fractional wave equations. Numerical Methods for Partial Differential Equations , 32(3):970–1001, 2016

    work page 2016

  3. [3]

    Fractional calculus: some basic problems in continuum and statistical mechanics

    Francesco Mainardi. Fractional calculus: some basic problems in continuum and statistical mechanics. Springer, 1997

    work page 1997

  4. [4]

    The random walk’s guide to anomalous diffusion: a fractional dynamics approach

    Ralf Metzler and Joseph Klafter. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics reports, 339(1):1–77, 2000

    2000

  5. [5]

    Pattern formation in a fractional reaction–diffusion system

    VV Gafiychuk and B Yo Datsko. Pattern formation in a fractional reaction–diffusion system. Physica A: Statistical Mechanics and its Applications , 365(2):300–306, 2006

    work page 2006

  6. [6]

    Applications of fractional calculus in physics

    Rudolf Hilfer. Applications of fractional calculus in physics . World scientific, 2000

    work page 2000

  7. [7]

    Ralf Metzler and Joseph Klafter. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics.Journal of Physics A: Mathematical and General, 37(31):R161, 2004

    work page 2004

  8. [8]

    universal response

    RR Nigmatullin. To the theoretical explanation of the “universal response”. physica status solidi (b), 123(2):739–745, 1984

    work page 1984

Show all 50 references
  1. [9]

    The realization of the generalized transfer equation in a medium with fractal geometry

    RR Nigmatullin. The realization of the generalized transfer equation in a medium with fractal geometry. Physica status solidi (b) , 133(1):425–430, 1986

    1986

  2. [10]

    Fractional diffusive waves

    Francesco Mainardi and Paolo Paradisi. Fractional diffusive waves. Journal of Computational Acous- tics, 9(04):1417–1436, 2001

    2001

  3. [11]

    A fully discrete difference scheme for a diffusion-wave system

    Zhi-zhong Sun and Xiaonan Wu. A fully discrete difference scheme for a diffusion-wave system. Applied Numerical Mathematics , 56(2):193–209, 2006

    2006

  4. [12]

    A compact difference scheme for the fractional diffusion-wave equation

    Rui Du, WR Cao, and ZZ Sun. A compact difference scheme for the fractional diffusion-wave equation. Applied Mathematical Modelling , 34(10):2998–3007, 2010

    2010

  5. [13]

    Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation

    Ya-Nan Zhang, Zhi-zhong Sun, and Xuan Zhao. Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation. SIAM Journal on Numerical Analysis , 50(3):1535–1555, 2012

    2012

  6. [14]

    A fast element-free galerkin method for the fractional diffusion-wave equation

    Xiaolin Li and Shuling Li. A fast element-free galerkin method for the fractional diffusion-wave equation. Applied Mathematics Letters , 122:107529, 2021

    2021

  7. [15]

    A second-order difference scheme for the nonlinear time-fractional diffusion-wave equation with gener- alized memory kernel in the presence of time delay

    Anatoly A Alikhanov, Mohammad Shahbazi Asl, Chengming Huang, and Aslanbek Khibiev. A second-order difference scheme for the nonlinear time-fractional diffusion-wave equation with gener- alized memory kernel in the presence of time delay. Journal of Computational and Applied Ma...

    2024

  8. [16]

    A generalized multiscale finite element method for poroelastic- ity problems i: Linear problems

    Donald L Brown and Maria Vasilyeva. A generalized multiscale finite element method for poroelastic- ity problems i: Linear problems. Journal of Computational and Applied Mathematics , 294:372–388, 2016

    2016

  9. [17]

    A one-equation turbulence transport model for high reynolds number wall-bounded flows

    Barrett Baldwin and Timothy Barth. A one-equation turbulence transport model for high reynolds number wall-bounded flows. In 29th aerospace sciences meeting, page 610, 1991

    1991

  10. [18]

    Analysis of upscaling absolute permeability

    Xiao-Hui Wu, Yalchin Efendiev, and Thomas Y Hou. Analysis of upscaling absolute permeability. Discrete and Continuous Dynamical Systems Series B , 2(2):185–204, 2002

    2002

  11. [19]

    Homogenization and porous media , volume 6

    Ulrich Hornung. Homogenization and porous media , volume 6. Springer Science & Business Media, 2012

    2012

  12. [20]

    On a question about the propagation of heat in heterogeneous media.(russian) izvestiya akad

    LI Rubinˇ steın. On a question about the propagation of heat in heterogeneous media.(russian) izvestiya akad. Nauk SSSR. Ser. Geograf. Geofiz , 12:27–45, 1948

    1948

  13. [21]

    Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata]

    Grigory I Barenblatt, Iu P Zheltov, and IN Kochina. Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata]. Journal of applied mathematics and mechanics , 24(5):1286–1303, 1960

    1960

  14. [22]

    Derivation of the double porosity model of single phase flow via homogenization theory

    Todd Arbogast, Jim Douglas, Jr, and Ulrich Hornung. Derivation of the double porosity model of single phase flow via homogenization theory. SIAM Journal on Mathematical Analysis , 21(4):823– 836, 1990

    1990

  15. [23]

    Multicontinuum wave propagation in a laminated beam with contrasting stiffness and density of layers

    GP Panasenko. Multicontinuum wave propagation in a laminated beam with contrasting stiffness and density of layers. Journal of Mathematical Sciences , 232:503–515, 2018

    2018

  16. [24]

    Multicontinuum homogenization and its relation to nonlocal multicontinuum theories

    Yalchin Efendiev and Wing Tat Leung. Multicontinuum homogenization and its relation to nonlocal multicontinuum theories. Journal of Computational Physics , 474:111761, 2023

    2023

  17. [25]

    Multicontinuum homogenization

    E Chung, Yalchin Efendiev, Juan Galvis, and Wing Tat Leung. Multicontinuum homogenization. general theory and applications. Journal of Computational Physics , 510:112980, 2024

    2024

  18. [26]

    Some convergence analysis for multicontinuum homogenization

    Wing Tat Leung. Some convergence analysis for multicontinuum homogenization. arXiv preprint arXiv:2401.12799, 2024

    2024 arXiv

  19. [27]

    Multicontinuum homogenization in perforated domains

    Wei Xie, Yalchin Efendiev, Yunqing Huang, Wing Tat Leung, and Yin Yang. Multicontinuum homogenization in perforated domains. arXiv preprint arXiv:2404.17471 , 2024

    2024 arXiv

  20. [28]

    Multicontinuum homogenization for coupled flow and transport equations

    Dmitry Ammosov, WT Leung, Buzheng Shan, and Jian Huang. Multicontinuum homogenization for coupled flow and transport equations. arXiv preprint arXiv:2405.14572 , 2024

    2024 arXiv

  21. [29]

    Multicontinuum homog- enization for richards’ equation: The derivation and numerical experiments

    Dmitry Ammosov, Sergei Stepanov, Denis Spiridonov, and Wenyuan Li. Multicontinuum homog- enization for richards’ equation: The derivation and numerical experiments. Russian Journal of Numerical Analysis and Mathematical Modelling , 38(4):207–218, 2023

    2023

  22. [30]

    Multicontinuum splitting scheme for multiscale flow problems

    Yalchin Efendiev, Wing Tat Leung, Buzheng Shan, and Min Wang. Multicontinuum splitting scheme for multiscale flow problems. arXiv preprint arXiv:2410.05253 , 2024

    2024 arXiv

  23. [31]

    Multi-scale finite-volume method for elliptic problems in subsurface flow simulation

    Patrick Jenny, SH Lee, and Hamdi A Tchelepi. Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. Journal of computational physics , 187(1):47–67, 2003

    2003

  24. [32]

    Multiscale finite-volume method for compressible multiphase flow in porous media

    Ivan Lunati and Patrick Jenny. Multiscale finite-volume method for compressible multiphase flow in porous media. Journal of Computational Physics , 216(2):616–636, 2006

    2006

  25. [33]

    Algorithmic monotone multiscale finite volume methods for porous media flow

    Omar Chaabi and Mohammed Al Kobaisi. Algorithmic monotone multiscale finite volume methods for porous media flow. Journal of Computational Physics , 499:112739, 2024. 23

    2024

  26. [34]

    A multiscale finite element method for elliptic problems in composite materials and porous media

    Thomas Y Hou and Xiao-Hui Wu. A multiscale finite element method for elliptic problems in composite materials and porous media. Journal of computational physics , 134(1):169–189, 1997

    1997

  27. [35]

    Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients

    Thomas Hou, Xiao-Hui Wu, and Zhiqiang Cai. Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Mathematics of computation, 68(227):913–943, 1999

    1999

  28. [36]

    Multiscale finite element methods: theory and applications , volume 4

    Yalchin Efendiev and Thomas Y Hou. Multiscale finite element methods: theory and applications , volume 4. Springer Science & Business Media, 2009

    2009

  29. [37]

    Reduced multiscale computation on adapted grid for the convection-diffusion robin problem

    Shan Jiang, Meiling Sun, and Yin Yang. Reduced multiscale computation on adapted grid for the convection-diffusion robin problem. Journal of Applied Analysis and Computation , 7(4):1488–1502, 2017

    2017

  30. [38]

    Generalized multiscale finite element methods (gmsfem)

    Yalchin Efendiev, Juan Galvis, and Thomas Y Hou. Generalized multiscale finite element methods (gmsfem). Journal of computational physics , 251:116–135, 2013

    2013

  31. [39]

    Generalized multiscale finite element methods for wave propagation in heterogeneous media

    Eric T Chung, Yalchin Efendiev, and Wing Tat Leung. Generalized multiscale finite element methods for wave propagation in heterogeneous media. Multiscale Modeling & Simulation , 12(4):1691–1721, 2014

    2014

  32. [40]

    Generalized multiscale finite element method for elasticity equations

    Eric T Chung, Yalchin Efendiev, and Shubin Fu. Generalized multiscale finite element method for elasticity equations. GEM-International Journal on Geomathematics , 5:225–254, 2014

    2014

  33. [41]

    Adaptive multiscale model reduction with generalized multiscale finite element methods

    Eric Chung, Yalchin Efendiev, and Thomas Y Hou. Adaptive multiscale model reduction with generalized multiscale finite element methods. Journal of Computational Physics , 320:69–95, 2016

    2016

  34. [42]

    On time integrators for generalized multiscale finite element methods applied to advection–diffusion in high-contrast multiscale media

    Wei Xie, Juan Galvis, Yin Yang, and Yunqing Huang. On time integrators for generalized multiscale finite element methods applied to advection–diffusion in high-contrast multiscale media. Journal of Computational and Applied Mathematics , 460:116363, 2025

    2025

  35. [43]

    Constraint energy minimizing general- ized multiscale finite element method

    Eric T Chung, Yalchin Efendiev, and Wing Tat Leung. Constraint energy minimizing general- ized multiscale finite element method. Computer Methods in Applied Mechanics and Engineering , 339:298–319, 2018

    2018

  36. [44]

    Convergence of the cem-gmsfem for stokes flows in heterogeneous perforated domains

    Eric Chung, Jiuhua Hu, and Sai-Mang Pun. Convergence of the cem-gmsfem for stokes flows in heterogeneous perforated domains. Journal of Computational and Applied Mathematics , 389:113327, 2021

    2021

  37. [45]

    Cem-gmsfem for poisson equations in hetero- geneous perforated domains

    Wei Xie, Yin Yang, Eric Chung, and Yunqing Huang. Cem-gmsfem for poisson equations in hetero- geneous perforated domains. Multiscale Modeling & Simulation , 22(4):1683–1708, 2024

    2024

  38. [46]

    A computational macroscale model for the time fractional poroelasticity problem in fractured and heterogeneous media

    Aleksei Tyrylgin, Maria Vasilyeva, Anatoly Alikhanov, and Dongwoo Sheen. A computational macroscale model for the time fractional poroelasticity problem in fractured and heterogeneous media. Journal of Computational and Applied Mathematics , 418:114670, 2023

    2023

  39. [47]

    Partially explicit time discretization for nonlinear time fractional diffusion equations.Communications in Nonlinear Science and Numerical Simulation , 113:106440, 2022

    Wenyuan Li, Anatoly Alikhanov, Yalchin Efendiev, and Wing Tat Leung. Partially explicit time discretization for nonlinear time fractional diffusion equations.Communications in Nonlinear Science and Numerical Simulation , 113:106440, 2022

    2022

  40. [48]

    Multiscale model reduction for the time fractional thermoporoelasticity problem in fractured and heterogeneous media

    Anatoly Alikhanov, Huiran Bai, Jian Huang, Aleksei Tyrylgin, and Yin Yang. Multiscale model reduction for the time fractional thermoporoelasticity problem in fractured and heterogeneous media. Journal of Computational and Applied Mathematics , 455:116157, 2025

    2025

  41. [49]

    Automated solution of differential equations by the finite element method: The FEniCS book , volume 84

    Anders Logg, Kent-Andre Mardal, and Garth Wells. Automated solution of differential equations by the finite element method: The FEniCS book , volume 84. Springer Science & Business Media, 2012

    2012

  42. [50]

    Paraview: An end-user tool for large data visualiza- tion

    James Ahrens, Berk Geveci, and Charles Law. Paraview: An end-user tool for large data visualiza- tion. The visualization handbook , 717, 2005. 24

    2005

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.