Pith. sign in

REVIEW 4 major objections 4 minor 54 references

An explicit computational approach for a three-dimensional system of nonlinear elastodynamic sine-Gordon problem

T0 review · 4 major / 4 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 An explicit finite-element/interpolation scheme for the 3D nonlinear elastodynamic sine-Gordon system is shown to be stable under a CFL condition and to converge as O(σ² + h³).

desk verdict The scheme's bilinear form has the Lamé coefficients swapped relative to the stated PDE, so the main theorem is about a different problem and the numerics are self-referential. read the letter →

arxiv 2506.14807 v1 pith:4DZTYKMZ submitted 2025-06-03 math.NA cs.NA

Eric Ngondiep This is my paper
classification math.NAcs.NA MSC 65M1265M0674H15
keywords nonlinearelastodynamicsine-GordonequationsexplicitfiniteelementschemeinterpolationtechniquestabilityanalysiserrorestimatesCFLconditionthree-dimensionalwavepropagationstrongnorm
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

An explicit method is proposed for the displacement field u and the stress tensor ψ satisfying the three-dimensional nonlinear elastodynamic sine-Gordon system (1)-(3). The scheme combines quadratic interpolation in time with a finite-element discretization in space, and the paper's Theorem 3.1 claims that, under the CFL-type step restriction (36), the numerical displacement stays bounded and the error satisfies the O(σ² + h³) bound (44). The proof proceeds by an energy estimate in the constructed norm ‖·‖_{0,B}, using an inverse inequality, a discrete Gronwall inequality, and approximation properties of high-degree finite-element spaces. Numerical tables and figures in Section 4 report the predicted second-order-in-time and third-order-in-space rates for two test configurations. If the claim holds, the method offers an explicit, block-matrix-free alternative to implicit solvers for wave-propagation problems in heterogeneous elastic media.

What carries the argument

The central object is the three-level explicit update ($u_h^{{n+1}}$,w) = (2u_h^n − $u_h^{{n−1}}$,w) − (σ²/ρ)[B(u_h^n,w) − (F(u_h^n),w)], together with the bilinear form B(u,v) = (λ1+λ2)(∇u,∇v)_∗ + λ2(∇·u,∇·v)_0, and the algebraic stress update ($ψ_h^{{n+1}}$,τ)_∗ = λ1((∇·$u_h^{{n+1}}$)I,τ)_∗ + λ2(∇$u_h^{{n+1}}$ + (∇$u_h^{{n+1}}$)^T,τ)_∗. The time derivative is approximated by quadratic interpolation at t_{n−1}, t_n, t_{n+1}, producing the leapfrog-type identity $u_t^{{n+1}}$ − $u_t^{{n−1}}$ = (2/σ)($u^{{n+1}}$ − 2u^n + $u^{{n−1}}$) plus a remainder, while the integrals of ∇·u, ∇u, and F(u) are interpolated linearly on each subinterval, giving the fourth-order remainder terms of equations (23)-(25). The error analysis measures differences in the B-norm, uses the inverse inequality to relate ‖·‖_B to ‖·‖_0 on the finite-element space, and applies a discrete Gronwall inequality to obtain the exponential factor in the final estimate.

What would settle it

Compare the two bilinear expressions for a simple vector field: with λ1 = λ2 = 1 and u = w = (x,y,z) on a unit domain, the weak form derived from the PDE gives (λ1+λ2)∫(∇·u)(∇·w) + λ2∫∇u:∇w = 21, while the form used in the scheme gives (λ1+λ2)∫∇u:∇w + λ2∫(∇·u)(∇·w) = 15. A manufactured-solution test of the implemented scheme against the PDE (1) would reveal whether the method is solving the stated problem or this modified one.

Watch

Extended reading notes

Core claim

The central claim, stated as Theorem 3.1, is that the approximate displacement $u_h^{{n+1}}$ computed by the explicit scheme (30)-(34) approaches the exact solution $u^{{n+1}}$ of the initial-boundary value problem with ‖$u_h^{{n+1}}$ − $u^{{n+1}}$‖_{0,B} ≤ C(σ² + h³), where C is an explicit constant involving the inverse-inequality constant C_p, the Lipschitz constant of the sine source term, and norms of the exact solution, and where the step restriction σ ≤ C_ts h from (36) holds. The same estimate controls ‖$u_h^{{n+1}}$‖_{0,B} by the exact solution plus the same right-hand side, which is the paper's stated stability result. The proof uses the constructed strong norm ‖·‖_{0,B} = $\sqrt$(‖·‖²_0 + ‖·‖²_B) so that the central-difference-in-time part of the error identity telescopes, while the space discretization enters through the inverse inequality and through approximation properties of the finite-element spaces. The stress tensor ψ_h is then recovered algebraically from equation (31), so the whole analysis is reduced to the displacement equation.

Load-bearing premise

The stability and error analysis is built on the identification of the bilinear form B in equation (11) with the weak form of the differential operator in equation (1); if that identification is not correct, the scheme is analyzed for a different operator than the stated PDE.

Editorial extensions

If this is right

  • Under the CFL-type restriction (36), the numerical solution is stable in the strong norm ‖·‖_{0,B} and does not grow in time beyond the exact solution plus a controlled error.
  • The bound (44) gives temporal accuracy O(σ²) and spatial accuracy O(h³), so halving the mesh spacing improves spatial error by a factor close to eight, and halving the time step improves temporal error by a factor close to four.
  • Because the displacement update is explicit, no block matrix needs to be inverted at each time level, reducing the per-step cost relative to implicit solvers for the same problem.
  • The stress tensor is determined algebraically from the displacement via equation (31), so the stability and convergence analysis can focus entirely on the displacement equation.
  • The step restriction takes the standard explicit-hyperbolic form σ ≤ C_ts h, which the paper identifies as a practical advantage when the density ρ is not small.

Reading between the lines

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

  • If Theorem 3.1 is accepted as stated, the same three-level explicit structure should transfer to other hyperbolic systems with smooth nonlinear sources, with the CFL condition adjusted only through the Lipschitz constant and the inverse-inequality constant.
  • A manufactured-solution test with λ1 ≠ 0 would separate convergence to the stated PDE (1) from convergence to a slightly modified operator; this is a small addition to the numerical experiments and would directly test the interpretation of the bilinear form in equation (11).
  • The algebraic stress-recovery step (31) suggests a low-cost post-processing route for hazard mapping, since ψ_h is computed without solving an additional system once the displacement is known.
  • One implicit consequence is that the reported convergence rates are tied to the identification of B with the weak form of the differential operator in (1); a reader porting the method should verify that identification directly before relying on the exponential error bound.
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

4 major / 4 minor

Summary. The paper proposes an explicit finite element scheme for a three-dimensional elastodynamic sine-Gordon system, combining second-degree polynomial interpolation in time with a finite element discretization in space. The main theoretical result, Theorem 3.1, states stability and error estimates of order O(σ² + h³) in a constructed strong norm under the CFL-type restriction (36). Numerical experiments in Section 4 report temporal second-order and spatial third-order convergence for displacement and stress. The manuscript's central claim is that scheme (30)-(34) approximates the initial-boundary value problem (1)-(3) at these rates.

Significance. If valid, the result would offer an explicit, block-matrix-free scheme with high-order spatial accuracy for a nonlinear hyperbolic system, together with a CFL condition and error control in a strong norm. The derivation is detailed and the scheme's explicit nature is a practical strength. However, the central derivation is broken: the bilinear form used in the scheme is not the weak form of the stated operator, the inverse inequalities are applied to functions outside the finite element space, and the numerical validation is self-referential. These issues prevent the claimed result from being established for the stated problem.

major comments (4)
  1. [§2, Eqs (11), (26)–(27)] The bilinear form B defined in (11) is not the weak form of the spatial operator in (1). Equation (26) gives the weak form obtained by integrating (1) by parts as (λ1+λ2)(∇·u^n, ∇·w)_0 + λ2(∇u^n, ∇w)_*, whereas B(u^n,w) in (11) is (λ1+λ2)(∇u^n, ∇w)_* + λ2(∇·u^n, ∇·w)_0. The replacement of the former by the latter in going from (26) to (27) is an algebraic error that swaps the Lamé coefficients between the divergence-divergence and gradient-gradient terms. For the test parameters λ1=λ2=1 used in Example 1, the scheme (30) advances a different differential operator, so Theorem 3.1, including estimate (44), does not provide stability or convergence for problem (1)-(3).
  2. [§3, Eq (35) and Lemma 3.2, proof of Theorem 3.1] The inverse inequality (35) is stated for every w∈U, but such an estimate with h^{-1} can only hold on a finite-dimensional subspace such as Uh. Moreover, Lemma 3.2 bounds (45) are stated for v∈Uh but are applied in the proof to e_h^n = u_h^n - u^n and to e_h^{n+1} - e_h^n, which do not belong to Uh because the exact solution u^n is not in Uh. The lower bound in (45) is used to obtain the factor β0 in (57), and the upper bound is used to relate ‖e_h^n‖²_B to ‖e_h^n‖²_0; these applications are unsupported and are load-bearing for the Gronwall argument leading to (60).
  3. [§4, numerical validation, paragraph after Table 2] The convergence tables are computed using the same numerical scheme on a finer grid as a surrogate for the exact solution (σ=2^{-7} for spatial orders and σ=2^{-10} for temporal orders). Because of the coefficient swap identified above, these experiments only demonstrate self-convergence of the scheme and cannot measure approximation of the true solution of (1)-(3). The observed rates in Tables 1, 2, 4 and 5 therefore do not confirm the theoretical claims for the stated problem.
  4. [§2, Eqs (20) and (26)] The time interpolation error term is off by a factor of two. Substituting (20) into (17) and multiplying by σ/2 yields an error term -σ³/6((ǫ(tn+1)-ǫ(tn-1))u4t(ǫ), w)_0 on the right-hand side of (26), but (26) has -σ³/3 with the same argument. The later bound (53) uses the σ³/3 version, so the consistency estimate for the temporal discretization is not derived correctly.
minor comments (4)
  1. [§1, Eq (2) and (4)] Equation (4) sets u0(x) = v(x), but (2) uses v(x) for the initial velocity; this makes the initial displacement equal to the initial velocity, which is unusual and likely unintended. Please clarify the notation.
  2. [§4, Table 1] In the last row of Table 1, the stress error for h=2^{-6} is 6.5915e-6, which is larger than the previous error 5.7393e-6 for h=2^{-5}, yet the reported convergence order is 3.1222; this appears to be a typographical or data error.
  3. [§1, Introduction] There are several presentation issues: 'Newtow's second law' should be 'Newton's second law,' and the symbol list '∇, ∇ and ∇·' is redundant. The text also repeats reference [22] and [23] with identical entries.
  4. [§2, Eqs (23)–(25)] The notation for the mean value points is confusing: two different points are both denoted ǫ1 (and similarly ǫ2, ǫ3) in the same line, and the later identification ǫ1=ǫ2=ǫ3=θ1 is not rigorously justified because the points come from different mean value theorems.

Circularity Check

1 steps flagged · score 4.0 of 10

The analytical error analysis is not circular, but the numerical validation is self-referential: the 'exact' solution used for the convergence tables is a fine-grid run of the same scheme, so the reported orders measure self-convergence rather than convergence to the true PDE solution.

  1. self definitional [Section 4, 'Numerical experiments', paragraph on error computation preceding Tables 1-2 and 4-5]
    "It's worth mentioning that determining the exact solution of the initial-boundary value problem (1)-(3) is too difficult and sometimes impossible. Hence, the spatial errors are calculated assuming that the exact solution is the approximate one obtained with the time step σ = 2^{-7}, whereas the temporal errors are computed assuming that the analytical solution equals to the numerical one with the time step σ = 2^{-10}."

    The 'exact' solution u used in the error norms is defined as a numerical solution produced by the same scheme (30)-(34) at a finer resolution. Therefore the reported errors ||u_h - u|| are actually ||u_h - u_h^{fine}||, i.e. differences between two runs of the same method. Tables 1-2 and 4-5 then display the scheme's self-convergence, which is already implied by the stability/consistency analysis being tested, and cannot independently confirm convergence to the true solution of (1)-(3). This is a self-definitional validation: the reference solution is an output of the same construction as the solution being tested.

full rationale

The central theoretical estimate, Theorem 3.1, is not obtained by fitting parameters or by renaming known results: it follows from an energy identity, Cauchy-Schwarz, the Lipschitz property of F, inverse/finite-element estimates from [13], and a discrete Gronwall argument under the CFL restriction (36). The only self-citation in the derivation, [39] for identity (16), is a standard integration-by-parts identity with u,w vanishing on Γ, and it is externally verifiable, so it is not load-bearing. The numerical section, however, is circular in a narrower sense: it explicitly defines the 'exact' solution as a numerical solution of the same scheme, so the convergence orders in the tables are self-convergence rates and do not provide independent confirmation of the theorem. The score is set at 4 rather than higher because the main proof is independent and no fitted input forces the error estimate. For completeness, the paper also contains a separate correctness issue that is not circularity: the replacement leading from (26) to (27) interchanges (λ1+λ2) and λ2 between the divergence-divergence and gradient-gradient terms when substituting the bilinear form B of (11); this is an algebraic inconsistency with the weak form of (1), not a circular reduction, and it does not inflate the circularity score.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No physical entities are invented. The free parameters are experimental constants chosen by hand. The analysis relies on standard inequalities and regularity assumptions; the main unproved inputs are the FE inverse estimates and the coercivity constants taken from [13], plus the regularity assumption (iii).

free parameters (2)
  • C_ts = 0.5 (in both numerical examples)
    Chosen by hand to satisfy the CFL restriction (36); enters the stability condition and the error constant.
  • C_p = 1/3 (in both numerical examples)
    Inverse inequality constant from (35), set by hand in the experiments; controls the CFL bound through C1.
assumptions (5)
  • domain assumption The FE inverse inequality ‖∂w/∂x_l‖_{0} ≤ C_p h^{-1}‖w‖_{0} holds with a uniform constant C_p over the family of spaces U_h.
    Stated as assumption (i) in Section 2; needed for the CFL condition (36) and Lemma 3.2.
  • domain assumption The exact solution has regularity u ∈ [W^5_2(0,T;H^5)]^3 and the nonlinearity F(u) ∈ [W^3_2(0,T;H^3)]^3.
    Assumption (iii) in Section 2; required for the interpolation error bounds and for the O(σ^4) remainder terms to be finite.
  • domain assumption The bilinear form estimates in Lemma 3.1 (continuity, coercivity of B in (11)) hold with constants involving λ1, λ2, and CΩ.
    Lemma 3.1 is cited to reference [13] without proof; the stability proof depends on these estimates.
  • standard math The Poincaré-Friedrichs inequality ‖v‖_0 ≤ CΩ‖∇v‖_0 holds for v ∈ W^1_2(Ω).
    Given in equation (39); used throughout the stability analysis.
  • domain assumption The equivalence √C1 h ‖v‖_B ≤ ‖v‖_0 ≤ √C2 ‖v‖_B holds for all v ∈ U_h.
    Lemma 3.2, inherited from [13]; crucial for converting B-norm estimates into L2 control.

how reviews work

0 comments
Cite this review

Pith. "Pith review of An explicit computational approach for a three-dimensional system of nonlinear elastodynamic sine-Gordon problem." pith.science (2026). https://pith.science/paper/4DZTYKMZ

@misc{pith2026250614807,
  author       = {Pith},
  title        = {Pith review of: An explicit computational approach for a three-dimensional system of nonlinear elastodynamic sine-Gordon problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4DZTYKMZ}},
  note         = {Machine review of arXiv:2506.14807}
}
read the original abstract

This paper proposes an explicit computational method for solving a three-dimensional system of nonlinear elastodynamic sine-Gordon equations subject to appropriate initial and boundary conditions. The time derivative is approximated by interpolation technique whereas the finite element approach is used to approximate the space derivatives. The developed numerical scheme is so-called, high-order explicit computational technique. The new algorithm efficiently treats the time derivative term and provides a suitable time step restriction for stability and convergence. Under this time step limitation, both stability and error estimates of the proposed approach are deeply analyzed using a constructed strong norm. The theoretical studies indicate that the developed approach is temporal second-order convergent and spatially third-order accurate. Some numerical examples are carried out to confirm the theory, to validate the computational efficiency and to demonstrate the practical applicability of the new computational technique.

Figures

Figures reproduced from arXiv: 2506.14807 by the authors.

Figure 1
Figure 1. Geological structure deformation. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Graphs of displacement (uh) and stress tensor (ψh) corresponding to Example 1. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Graphs of displacement (uh) and stress tensor (ψh) corresponding to Example 2. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

54 extracted references · 50 canonical work pages

  1. [39]

    A combined Lax-Wendroff/interpolation approach with finite element method for a three-dimensional system of tectonic deformation model: application to landslides in Cameroon

    E. Ngondiep. A combined Lax-Wendroff/interpolation approach with finite element method for a three- dimensional system of tectonic deformation model: application to lan dslides in Cameroon, (2025). Preprint arXiv:2502.07797

    work page Pith review arXiv 2025

  2. [1]

    J. D. Achenbach. ”Wave propagation in elastic solids”, Elsevier, A msterdam, 1975

    work page 1975

  3. [2]

    P. F. Antonietti, C. Facciol` a, A. Russo, M. Verani. ”Discontinuo us Galerkin approximation of flows in fractured porous media on polytopic grids”, SIAM J. Sci. Comput. 4 1(1) (2019), 10.1137/17M 1138194

    work page doi:10.1137/17m 2019

  4. [3]

    P. F. Antonietti, L. Formaggia, A. Scotti, M. Verani, N. Verzott . ”Mimetic finite difference approxima- tion of flows in fractured porous media”, ESAIM: M2AN, 50(3) (201 6), 809-832

    work page

  5. [4]

    S. Aoi, H. Fujiwara. ”3D Finite-difference method using discontinu ous grids”, Bull. Seism. Soc. Am., 89(1999), 918-930

    work page 1999

  6. [5]

    H. Bao, J. Bielak, O. Ghattas, L. Kallivokas, D. R. O’Hallaron, J.R. S chewchuk, J. Xu. ”Large-scale simulation of elastic wave propagation in heterogeneous media on par allel computers”, Comput. Methods Appl. Mech. Engrg., 152(1998), 85-102

    work page 1998

  7. [6]

    J. D. De Basabe, M. K. Sen, M. F. Wheeler. ”The interior penalty d iscontinuous Galerkin method for elastic wave propagation: grid dispersion”, Geophysical J. Int., 17 5(1) (2008), 83-93

    work page 2008

  8. [7]

    A. Bedford. ”Hamilton’s principle in continuum mechanics”, Springe r, Switzerland, 2021

    work page 2021

Show all 54 references
  1. [8]

    V. L. Berdichevsky. ”Variational principles of continuum mechan ics”, Springer, Berlin, 2009

    2009

  2. [9]

    M. A. Biot. ”Generalized theory of acoustic propagation in porou s dissipative media”, J. Acoust. Soc. Am., 34, 1254(1962)

    1962

  3. [10]

    R. Boer. ”Theoretical poroelasticity-a new approach”, Chao s, Solitons and Fractals 25(4) (2005), 861- 878

    2005

  4. [11]

    J. M. Carcione, G. Seriani. ”Wave simulation in frozen porous med ia”, J. Comput. Phys., 170(2), (2001), 676-695

    2001

  5. [12]

    Chaljub, Y

    E. Chaljub, Y. Capdeville and J.P. Vilotte. ”Solving elastodynamics in a fluid-solid heterogeneous sphere: a parallel spectral element approximation on non-confor ming grids”, J. Comput. Phys. 187(2003), 457-491

    2003

  6. [13]

    S. Chen, J. Zhao. ”Estimates of the constants in inverse inequ alities for finite element functions”, J. comput. Math., 31(5) (2013), 522-531

    2013

  7. [14]

    A. Y. Chernyshenko, M. A. Olshanskii, Y. V. Vassilevski. ”A hybr id finite volume-finite element method for bulk-surface coupled problems”, J. Comput. Phys., 352(2018 ), 516-533

    2018

  8. [15]

    Dauksher, A

    W. Dauksher, A. F. Emery. ”The solution of elastostatic and ela stodynamic problems with Chebyshev spectral finite elements”, Comput. Meth. Appl. Mech. Eng., 188(1 − 3) (2000), 217-233

    2000

  9. [16]

    Delcourte, N

    S. Delcourte, N. Glinsky. ”Analysis of a high-order space and tim e discontinuous Galerkin methodfor elastodynamic equations. Application to 3D wave propagation”, ESA IM: M2AN, 49(2015), 1085-1126

    2015

  10. [17]

    Fumagalli, A

    A. Fumagalli, A. Scotti. ”A numerical method for two-phase flow in fractured porous media with non- matching grids”, Adv. Water Resources, 62(2013), 454-464

    2013

  11. [18]

    Galis, P

    M. Galis, P. Moczo, J. Kristek. ”A 3-D hybrid finite-difference-fi nite-element viscoelastic modelling of seismic wave motion”, Geophys. J. Int., 175(2008), 153-184

    2008

  12. [19]

    L. Jing, F. Teng, M. Feng, H. Li, J. Yang, Z. Luo. ”A novel dimen sion reduction model based on POD and two-grid Crank-Nicolson mixed finite element methods for 3D non linear elastodynamic sine-Gordon problem”, Commun. Nonlinear Sci. Numer. Simul., 140(2025), 108409 . 17

    2025

  13. [20]

    Komatitsch, J

    D. Komatitsch, J. P. Vilotte. ”The spectral-element method: a n efficient tool to simulate the seismic response of 2D and 3D geological structures”, Bull. Seism. Soc. Am ., 88(1998) 368-392

    1998

  14. [21]

    L. D. Landau, E. M. Lifshitz. ”Theory of elasticity”, Elsevier, O xford, 1986

    1986

  15. [23]

    T. Levy, E. Sanchez-Palencia. ”Equations and interface cond itions for acoustic phenomena in porous media”, J. Math. Anal. Appl., 61(1977), 813-834

    1977

  16. [24]

    Y. X. Li, J. C. Zhang. ”Unconventional gas & oil resource type and potential of China”, Int. Petroleum Economy, 3(2011), 61-67

    2011

  17. [25]

    S. Y. Li, S. Lee, L. Zikatanov. ”Locking-free enriched Galerkin method for linear elasticity”, SIAM J. Numer. Anal., 60(1) (2022), 1391353

    2022

  18. [26]

    K. J. Marfurt. ”Accuracy of finite difference and finite element modeling of the scale and elastic wave equations”, Geophysics, 49(5) (1984), 533-549

    1984

  19. [27]

    Moczo, E

    P. Moczo, E. Bystrick´ y, J. Kristek, J. M. Carcione, M. Bouch on. ”Hybrid modeling of P-SV seismic motion at inhomogeneous viscoelastic topographic structures”, B ull. Seism. Soc. Am., 87(1997), 1305- 1323

    1997

  20. [28]

    Ngondiep

    E. Ngondiep. ”An efficient numerical approach for solving three -dimensional Black-Scholes equation with stochastic volatility”, Math. Meth. Appl. Sci., (2024), 1-21, DO I 10.1002/mma.10576

    2024 doi

  21. [29]

    Ngondiep

    E. Ngondiep. ”An efficient high-order two-level explicit/implicit nu merical scheme for two-dimensional time fractional mobile/immobile advection-dispersion model”, Int. J. Numer. Meth. Fluids, 96(8) (2024), 1305-1336

    2024

  22. [30]

    Ngondiep

    E. Ngondiep. ”Stability analysis of MacCormack rapid solver meth od for evolutionary Stokes-Darcy problem”, J. Comput. Appl. Math., 345(2019), 269-285

    2019

  23. [31]

    Ngondiep

    E. Ngondiep. ”A high-order finite element method for solving two -dimensional fractional Rayleigh- Stokes problem for a heated generalized second grade fluid”, Int. J. Numer. Meth. Fluids, (2025), 1-16, Doi.org/10.1002/fld.5361

    2025

  24. [32]

    Ngondiep

    E. Ngondiep. ”An efficient three-level explicit time-split scheme f or solving two-dimensional unsteady nonlinear coupled Burgers equations”, Int. J. Numer. Methods Flu ids, 92(4) (2020), 266-284

    2020

  25. [33]

    Ngondiep

    E. Ngondiep. ”Unconditional stability over long time intervals of a two-level coupled MacCormack/Crank-Nicolson method for evolutionary mixed Stoke s-Darcy model”, J. Comput. Appl. Math., 409(2022), 114148, Doi: 10 .1016/j.cam.2022.114148

    2022

  26. [34]

    Ngondiep

    E. Ngondiep. ”Long time unconditional stability of a two-level hy brid method for nonstationary incom- pressible Navier-Stokes equations”, J. Comput. Appl. Math. 345( 2019), 501-514

    2019

  27. [35]

    Ngondiep

    E. Ngondiep. ”A high-order numerical scheme for multidimension al convection-diffusion-reaction equa- tion with time-fractional derivative”, Numer. Algorithms, 94(2023 ), 681-700

    2023

  28. [36]

    Ngondiep

    E. Ngondiep. ”A robust numerical two-level second-order ex plicit approach to predict the spread of covid-2019 pandemic with undetected infectious cases”, J. Compu t. Appl. Math., 403(2022), DOI: 10.1016/j.cam.2021.113852

    2022

  29. [37]

    Ngondiep

    E. Ngondiep. ”A high-order combined finite element/interpolatio n approach for solving nonlinear multidimensional generalized Benjamin-Bona-Mahony-Burgers’ eq uations”, Math. Comput. Simul., 215(2024), 560-755

    2024

  30. [38]

    Ngondiep, A

    E. Ngondiep, A. N. Njomou, G. I. Mondinde. ”A predictor-corr ector approach to investigate and predict the dynamic of cytokine levels and human immune cell activation to Sta phylococcus Aureus”, Int. J. of Biomath., DOI: 10.1142/S1793524524501122, (2024), 1-29. 18

    2024 doi

  31. [40]

    Ngondiep

    E. Ngondiep. ”Long time stability and convergence rate of MacC ormack rapid solver method for non- stationary Stokes-Darcy problem”, Comput. Math. Appl., 75, (20 18), 3663-3684

  32. [41]

    Ngondiep

    E. Ngondiep. ”A novel three-level time-split MacCormack sche me for two-dimensional evolutionary linear convection-diffusion-reaction equation with source term”, I nt. J. Comput. Math., 98(1) (2021), 47-74

    2021

  33. [42]

    Ngondiep

    E. Ngondiep. ”An efficient high-order weak Galerkin finite element approach for Sobolev equation with variable matrix coefficients”, Comput. Math. Appl., 180 (2025) 279- 298

    2025

  34. [43]

    Ngondiep

    E. Ngondiep. ”A robust three-Level time split high-Order Leap frog/ Crank-Nicolson scheme for two- dimensional Sobolev and regularized long wave equations Arising in Fluid Mechanics”, Adv. Appl. Math. Mech., (2025) doi:10.4208/aamm.OA-2022-0320

    2025 doi

  35. [44]

    Ngondiep

    E. Ngondiep. ”A posteriori error estimates of MacCormack ra pid solver method for nonstationary in- compressible Navier-Stokes equations”, J. Comput. Appl. Math., 4 38(2024), 115569

    2024

  36. [45]

    Ngondiep

    E. Ngondiep. ”A fast three-step second-order explicit numer ical approach to investigating and forecast- ing the dynamic of corruption and poverty in Cameroon”, Heliyon, DO I: 10.1016/j.heliyon.2024.e38236, 10(19) (2024)

    2024 doi

  37. [46]

    Ngondiep

    E. Ngondiep. ”A two-level fourth-order approach for time-f ractional convection-diffusion-reaction equa- tion with variable coefficients”, Commun. Nonlinear Sci. Numer. Simul., 1 11(2022), 106444, Doi: 10.1016/j.cnsns.2022.106444

    2022

  38. [47]

    A. Pitarka. ”3D elastic finite-difference modeling of seismic motion using staggered-grids with nonuni- form spacing”, Bull. Seism. Soc. Am., 89(1999), 85-106

    1999

  39. [48]

    W. Qiu, J. Shen, K. Shi. ”An HDG method for linear elasticity with st rong symmetric stresses”, Math. Comput., 87(309) (2018), 69-93

    2018

  40. [49]

    Serpieri, F

    R. Serpieri, F. Travascio. ”Variational continuum multiphase po roelasticity-theory and applications”, Springer, Singapore, 2017

    2017

  41. [50]

    F. J. Ser´ on, F. J. Sanz, M. Kindel´ an, J. I. Badal. ”Finite element method for elastic wave propagation”, Comm. Appl. Numer. Meth., 6(5) (1990), 359-368

    1990

  42. [51]

    H. F. Tiersten. ”Linear piezoelectric plate vibrations”, Plenum P ress, New York, 1969

    1969

  43. [52]

    S. R. wu. ”A priori error estimates for explicit finite element for linear elastodynamic by Galerkin method and central difference method”, Comput. Meth. Appl. mech. Eng., 192(51-52) (2003), 5329-5353

    2003

  44. [53]

    S. Y. Yi. ”A lowest-order weak galerkin method for linear elasticit y”, J. comput. Math., 50(2019), 286-298

    2019

  45. [54]

    Y. H. Zeng, Z. D. Luo. ”The Crank-Nicolson mixed finite element m ethod for the two-dimensional improved system of time-dependent Maxwell’s equations”, Appl. Mat h. Comput., 433(2022), 127422

    2022

  46. [55]

    F. I. Zyserman, P. M. Gauzellino, J. E. santos. ”Dispersion ana lysis of a non-conforming finite element method for the Helmholtz and elastodynamic equations”, Int. J. Nu mer. Meth. Eng., 58(9) (2003), 1381-1395. 19 Stability analysis of the developed computational approach for...

    2003

Pith tools

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