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Thermodynamical extension of a symplectic numerical scheme with half space and time shift demonstrated on rheological waves in solids

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Pith's one-line read This paper introduces a staggered half-space, half-time finite-difference scheme for the Poynting–Thomson–Zener rheological solid and argues that at $\alpha=1/2$, $\hat C=1$ it is second-order accurate, stable, and free of dissipative and…

desk verdict A solid staggered finite-difference scheme for PTZ waves with rigorous periodic stability analysis, but the optimal C-hat=1 claim rests on an unproven boundary-condition assumption. read the letter →

arxiv 1908.07975 v4 pith:4SM3PCRI submitted 2019-08-21 physics.class-ph

Tamás Fülöp , Róbert Kovács , Mátyás Szücs , Mohammad Fawaier This is my paper · ORCID
classification physics.class-ph
keywords Poynting–Thomson–ZenermodelstaggeredgridsymplecticEulerrheologicalwavesplane-wavestabilityanalysisnumericaldissipationdispersionfinitedifferencescheme
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 establish that a finite-difference scheme built on staggering each field by half a space step and half a time step relative to the fields it couples to can solve the Poynting–Thomson–Zener (PTZ) rheological model both accurately and fast. Its central claim is that this placement, chosen from the spacetime roles of the quantities, turns the computationally identical symplectic Euler method into a second-order scheme and, at the interpolation parameter $\alpha=1/2$ with the Courant condition based on the fastest wave speed $\hat C\le 1$, removes both dissipative and dispersive numerical error. The evidence is a plane-wave stability analysis, truncation-error calculations, and numerical experiments in which the scheme reproduces a stress-pulse shape with a few dozen spatial cells while a commercial finite-element package shows artificial damping, oscillations, or instability and needs run times 100 to 10,000 times longer. If the claim holds, the scheme provides a thermodynamics-motivated discretization recipe for continuum systems and shows that a widely used commercial finite-element package can fail qualitatively on the simplest elastic limit.

What carries the argument

The load-bearing mechanism is the staggered spacetime grid: velocities live at half-integer positions in both space and time relative to stress, strain shares stress's nodes, and temperature is half-shifted in time afterwards, so every discrete derivative is evaluated at the midpoint of the quantity it couples to. This arrangement makes the Hooke-case scheme computationally identical to symplectic Euler, the standard first-order geometric integrator, while raising its accuracy to second order by reflection symmetry, and it turns the PTZ constitutive equation into the explicit weighted update (33) with parameter $\alpha$. Stability is controlled by the eigenvalues of the $3\times 3$ iteration (transfer) matrix, which the paper tests through two classical polynomial root-location criteria; those criteria isolate the parameter-free thermodynamic condition $\hat\tau>\tau$ and the fastest-wave Courant condition, and at $\alpha=1/2$, $\hat C=1$ the eigenvalues collapse to $1$ and $e^{\pm ik\Delta x}$, eliminating numerical dissipation and dispersion up to $O(\Delta t/\tau)$.

What would settle it

For the actual finite sample (stress pulse at one end, free end at the other), assemble the full iteration matrix and compute its eigenvalues at $\alpha=1/2$, $\hat C=1$; any eigenvalue with modulus above 1, or a numerical experiment showing unbounded growth of total energy over many bounces, would falsify the claimed boundary-value stability.

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Extended reading notes

Core claim

The central discovery is that the PTZ wave system admits a staggered finite-difference realization in which stress, strain, and velocity sit at mutually half-shifted positions dictated by the equations containing them: velocity is half-shifted in both space and time from stress, strain sits with stress, and temperature is then half-shifted in time. The rheological equation, which couples each quantity to its own time derivative, is discretized by an $\alpha$-weighted average that is explicit and second-order accurate at $\alpha=1/2$. A plane-wave stability analysis of the resulting iteration matrix yields three conditions: the thermodynamic requirement $\hat\tau>\tau$, a relation between $\alpha$ and $\Delta t$, and the Courant-type bound $\hat C<1$ (extendable to $\hat C\le 1$ with boundary conditions), where $\hat C=\hat c\,\Delta t/\Delta x$ uses the fast wave speed $\hat c=\sqrt{\hat E/(\tau\varrho)}$. At the special point $\alpha=1/2$, $\hat C=1$, the three eigenvalues that multiply each Fourier mode per time step are exactly $1$ and $e^{\pm ik\Delta x}$, so all wavelengths travel at the same discrete speed and the dissipative and dispersive errors are of order $O(\Delta t/\tau)$. The same scheme applied to the elastic limit conserves total energy over many bounces and produces clean wave pulses where the commercial finite-element software COMSOL, with several tuned time-stepping methods, gives damped, oscillatory, or unstable results and takes 100 to 10,000 times longer.

Load-bearing premise

The load-bearing premise is that stability proved for waves on an infinitely long periodic medium also holds for the finite sample with a stress pulse at one end and a free end, a boundary-value extension the paper treats as a rule of thumb rather than a proof.

Editorial extensions

If this is right

  • Setting $\alpha=1/2$ and $\hat C\le 1$ is enough for stability of the PTZ scheme, and with $\hat C=1$ the discrete dispersion branches are linear, so the scheme has no numerical dissipation or dispersion up to $O(\Delta t/\tau)$.
  • In the Hooke limit the scheme reduces to symplectic Euler, is stable for $C\le 1$, conserves elastic plus kinetic energy over long times, and needs $C=1$ to avoid both dissipative and dispersive artifacts.
  • For the Kelvin–Voigt limit ($\tau=0$) the stability conditions become $\alpha<1/2$ (relaxed to $\alpha\le 1/2$ with boundary conditions) together with a mixed parabolic-hyperbolic Courant bound combining $\Delta t^2$ and $\hat\tau\Delta t$ terms.
  • The stability analysis is not purely numerical: it reproduces the thermodynamic stability condition $\hat\tau>\tau$ as a scheme-independent requirement, so numerical stability criteria can teach something about the underlying continuum model.
  • Practically, the $\alpha=1/2$ scheme gives a reliable PTZ stress-signal shape with as few as 25 to 50 spatial cells, whereas $\alpha=0$ needs more than 1000 cells for comparable quality.

Reading between the lines

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

  • A natural extension, listed by the authors as future work but not demonstrated, is that the same half-shift recipe transfers to other members of the Kluitenberg–Verhás family and to non-Fourier heat conduction, with parabolic limits likely requiring mixed Courant conditions analogous to the Kelvin–Voigt case.
  • Because the elastic limit coincides with symplectic Euler, a plausible conjecture the paper does not prove is that the full PTZ scheme inherits a discrete variational or symplectic structure, which would explain the observed total-energy conservation.
  • A direct testable extension would be to compare the scheme's temperature histories with an analytic PTZ solution in the force-equilibrial limit; the paper leaves analytic comparison as future work and does not yet provide a convergence study of the thermal field.
  • The COMSOL comparison suggests a broader benchmarking lesson: commercial finite-element packages may be unreliable for viscoelastic wave propagation unless the time stepper and tolerances are carefully chosen, so benchmark suites should include dissipative wave problems rather than only static or quasi-static cases.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper presents a staggered finite difference scheme for the one-dimensional Poynting–Thomson–Zener (PTZ) rheological model, building on the symplectic Euler method and extending it to a dissipative continuum system. The authors derive the scheme from a spacetime-staggered arrangement of stress, strain, and velocity, analyze its von Neumann stability for both the Hooke and PTZ cases, and study dissipative and dispersive errors analytically and numerically. They then compare the scheme with COMSOL finite element simulations on a Hookean wave-propagation problem, reporting large run-time and accuracy advantages. The central claims are that the scheme is second-order accurate for α=1/2, that stability is governed by the Courant condition based on the fastest wave speed, and that with α=1/2 and the Courant number at the boundary of the stability region the scheme has linear dispersion branches and suppressed dissipative error.

Significance. If the claims hold, the scheme is a useful, simple, and fast alternative to standard finite element tools for linear rheological wave problems, with the attractive feature that its design follows from the spacetime structure of the governing equations rather than from fitted parameters. The paper contributes a clean, self-contained derivation: no constants are fitted, the Hooke-case stability analysis is rigorous and explicit, and the PTZ stability conditions are derived in closed form via both Jury and Routh–Hurwitz criteria. The explicit demonstration that the thermodynamic condition τ̂>τ emerges from the numerical stability analysis is a nice conceptual point. The COMSOL comparison, though limited to a linear elastic setup, is concrete and reproducible in its setup.

major comments (2)
  1. [Section 4.2.2, Eq. (66)–(67)] The transition from the strict stability condition Ĉ<1 derived in Eq. (60)/(66) to the claimed Ĉ≤1 in Eq. (67) is not proven. The paper explicitly labels this as a rule-of-thumb extension: the von Neumann analysis treats plane waves on an infinite periodic domain, while the actual simulations in Section 5.2 use a finite sample with a stress pulse at one end and a free boundary at the other. The exceptional mode with S=1 (kΔx=π) is precisely where the Jury inequality (60) fails, and the paper does not show that this mode is inadmissible for the pulse/free-boundary problem. Since the headline numerical results are run at Ĉ=1, the advertised stability of the scheme for the reported boundary-value problem rests on an unproven assumption. Please either prove that the exceptional mode is excluded by the boundary conditions, perform a boundary-mode analysis, or revise the stability claim to Ĉ<1 and rerun the key simulations accordingly.
  2. [Section 3, Eq. (32)–(33)] The claim that α=1/2 renders the PTZ update (33) second-order accurate is asserted without proof. The sentence "Second order accuracy of (33) for α=1/2 is then straightforward to verify" is not sufficient, especially because the update combines a finite difference ratio with an interpolation in σ and ε. The paper's stated advantage over the first-order symplectic Euler method depends on this accuracy claim, so a local truncation error derivation for the rheological update should be included or explicitly referenced. Without it, the accuracy comparison in Section 5.2 and the error discussion in Section 6.2 lack a rigorous basis.
minor comments (4)
  1. [Section 4.1, after Eq. (46)] There is a typo in the sentence "this affects only one mode, S=1, k=π/k"; this should read "kΔx=π".
  2. [Section 6.1, Figure 8] The caption of Figure 8 does not indicate whether the upper and lower rows correspond to C=1 and C=1/2, respectively, as stated in the text; the caption should be self-contained.
  3. [Section 7] The comparison with COMSOL would be more convincing if the exact COMSOL settings (mesh element order, solver tolerances, time-stepping parameters) were listed in a table, since the runtime differences depend strongly on these choices.
  4. [References] Reference [21] is cited as "in preparation" and "under review"; if the manuscript is being finalized, this reference should either be updated to a published version or removed as a support for the claim about dynamic versus static moduli.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the scheme, stability analysis, and error analysis are self-contained; the C-hat=1 boundary-condition extension is explicitly a rule-of-thumb, not a circular derivation.

full rationale

The derivation chain is self-contained. The finite-difference scheme is constructed directly from the PTZ equations (1)-(3) by placing v half-shifted in space and time relative to sigma, and epsilon half-shifted relative to v, with the rheological equation discretized by the alpha-weighted formula (32); this is an explicit construction, not an output of the stability analysis. The stability conditions are obtained by computing the transfer matrix (51), its characteristic polynomial (52), and applying Jury/Routh-Hurwitz criteria, yielding (58)-(60). No parameter is fitted to a subset of data and then renamed as a prediction: the only settings (alpha=1/2, C-hat=1) are chosen by the analysis itself. The self-citations to refs [1], [17], and [18] provide the staggered-placement idea and the thermodynamic PTZ model as inputs; they are not used to define away any target claim. The equality extension (67) of the strict condition (66) is explicitly labelled by the authors as a rule-of-thumb (Section 4, paragraph after Eq. (43)), i.e. an unproven assumption about boundary conditions; this is a correctness or assumption limitation, not a circular reduction. The COMSOL comparison is an external benchmark, and the energy conservation check is a non-built-in test. Hence score 0.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The scheme introduces one free parameter, alpha, which tunes the time interpolation of the relaxation equation. The central stability and accuracy claims rest on the PTZ model as given, the sufficiency of von Neumann analysis for the boundary-value problem, and standard Taylor expansion arguments. No new physical entities are introduced.

free parameters (1)
  • alpha = 1/2 (optimal for second-order accuracy)
    Weight parameter for time interpolation in the discretized rheological equation (32); alpha=1/2 chosen by hand for second-order accuracy, other values alpha=0,1 discussed.
assumptions (3)
  • domain assumption The PTZ model (Eqs. 1-3) is a valid continuum thermodynamic model for the solid.
    The entire numerical study is built on this model; it is taken from prior internal variable theory (ref 18), not derived here.
  • domain assumption The von Neumann stability criterion (|xi|<=1 for all k) is sufficient for stability of the initial-boundary value problem.
    Section 4 states this as a rule-of-thumb and notes boundary conditions may allow only certain eigenmodes; no proof is given for the specific boundary conditions used in Section 5.
  • standard math The Taylor expansion and standard finite difference error analysis are valid.
    Used for second-order accuracy derivations in Section 3.

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Cite this review

Pith. "Pith review of Thermodynamical extension of a symplectic numerical scheme with half space and time shift demonstrated on rheological waves in solids." pith.science (2026). https://pith.science/paper/4SM3PCRI

@misc{pith2026190807975,
  author       = {Pith},
  title        = {Pith review of: Thermodynamical extension of a symplectic numerical scheme with half space and time shift demonstrated on rheological waves in solids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4SM3PCRI}},
  note         = {Machine review of arXiv:1908.07975}
}
read the original abstract

On the example of the Poynting-Thomson-Zener rheological model for solids, which exhibits both dissipation and wave propagation - with nonlinear dispersion relation -, we introduce and investigate a finite difference numerical scheme. Our goal is to demonstrate its properties and to ease the computations in later applications for continuum thermodynamical problems. The key element is the positioning of the discretized quantities with shifts by half space and time steps with respect to each other. The arrangement is chosen according to the spacetime properties of the quantities and of the equations governing them. Numerical stability, dissipative error and dispersive error are analysed in detail. With the best settings found, the scheme is capable of making precise and fast predictions. Finally, the proposed scheme is compared to a commercial finite element software, COMSOL, which demonstrates essential differences even on the simplest - elastic - level of modelling.

Figures

Figures reproduced from arXiv: 1908.07975 by the authors.

Figure 1
Figure 1. Visualization of the finite difference numerical scheme. Velocity values stay at triangles, strain and stress values at rhombuses, filled symbols denote values calculated via the scheme, while empty ones represent initial and boundary conditions. First, new velocities are determined from (23), then new strains according to (25), and finally new stress values are obtained from (26) or (32), respectively. Grey indicat… view at source ↗
Figure 2
Figure 2. Snapshot of the shape of the fully born stress pulse near the left end of the sample. 5.1. Hookean wave propagation For the Hooke system, our scheme is symplectic, with very reliable long-time behaviour. This is well visible in [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Left: snapshot of the stress pulse right before its 15th bouncing back from the boundary. Middle: spacetime picture of the wave propagation. Bouncing back from free ends makes stress change sign. Right: elastic energy, kinetic energy, and their sum as functions of time. Calculation done with N = 200 space cells and C = 1. 5.2. Poynting–Thomson–Zener wave propagation For the PTZ system, we find that the principally o… view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: Upper row: α = 1/2, Lower row: α = 0 calculation of the stress signal when it starts its 7th bouncing, with Cˆ = 1. From left to right: N = 400, 800, 1600 space cells. Actually, α = 1/2 offers that realibility already at N = 50, and even N = 25 ‘does a decent job’, as …
Figure 5
Figure 5. Figure 5: The same α = 1/2 prediction with N = 25, 50, 100 space cells, from left to right, respectively. space cells along the sample t˜ ˜e t˜ elastic kinetic elastic+kinetic+rheological thermal total 0 0.01 0.02 0.03 0.04 0.05 0 1 2 3 4 5 6 7 [PITH_FULL_IMAGE:figures/full_fig…
Figure 6
Figure 6. Figure 6: α = 1/2, Cˆ = 1 spacetime picture and energy conservation, N = 200. With α = 1/2, the spacetime picture and total energy conservation are not less satisfactory, as visible in [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Wavy dispersion error and decrease by dissipative error for the Hooke system when C = 1/2, with N = 100. -1.0 -0.5 0.5 1.0 Re ξ -1.0 -0.5 0.5 1.0 Im ξ 0.5 1.0 1.5 2.0 2.5 3.0 k  Δx  -3 -2 -1 1 2 3 arg ξ -1.0 -0.5 0.5 1.0 Re ξ -1.0 -0.5 0.5 1.0 Im ξ 0.5 1.0 1.5 2.0 2.…
Figure 8
Figure 8. Figure 8: Upper row: case of C = 1, lower row: case of C = 1/2. Left: the two roots ξ± in the complex plane, right: k dependence of the argument of ξ±. 6.2. Poynting–Thomson–Zener case In case of a dissipative system like the PTZ one, it is hard to detect the dissipative error, …
Figure 9
Figure 9. Figure 9: The stress signal provided by the scheme with Cˆ = 1/2, N = 200, for comparison with [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Visualization of the three branches ξ0(k), ξ+(k), ξ−(k) for Cˆ = 1. Upper row: α = 1/2, lower row: α = 0 [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: Same as [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: In Figures 10–11, the roots are not exactly on the unit circle – here, ∆t dependence of |ξ0| and |ξ±| is displayed, at a neutral value k∆x = π/4, for Cˆ = 1 and α = 1/2. 7. Solutions using the finite element software COMSOL Finally, for comparative reasons, we present…
Figure 13
Figure 13. Figure 13: Applying the proposed scheme in case of two different pulse lengths; τ˜b = 0.2 (left) and τ˜b = 0.04 (right). The dimensionless space and time steps are ∆x˜ = ∆˜t = 0.01. This time step is actually not much smaller than the shorter pulse length so, for example, the ti…
Figure 14
Figure 14. Figure 14: Rear-side velocity history in time for τ˜b = 0.2, with maximum BDF order being 2 (left) and 5 (right), respectively. 7.2. Runge–Kutta-based schemes: Cash–Karp 5 This scheme results in unstable solutions, independently of the corresponding settings (initial time step, …
Figure 15
Figure 15. Figure 15: Rear-side velocity history in time, using the Dormand–Prince time stepping method (left: τ˜b = 0.2, right: τ˜b = 0.04). 7.4. Runge–Kutta-based schemes: RK34 Using stiffness detection, this scheme solves the problem in the fastest and most efficient way. However, when …
Figure 16
Figure 16. Figure 16: Rear-side velocity history in time, using the RK34 time stepping method (left: τ˜b = 0.2, right: τ˜b = 0.04). Since this COMSOL option proved the best, in order to test the mesh dependence of its solution, we have examined the τ˜b = 0.04 case with 300 space cells (∆x˜…
Figure 17
Figure 17. Figure 17: Rear-side velocity history in time, for pulse length τ˜b = 0.04, with 300 nodes. Left: solution by our scheme, right: COMSOL RK34 result. To summarize, compared to our scheme realized in Matlab, COMSOL run times are 100–1000–10000 times larger, with large memory deman…

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