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REVIEW 3 major objections 5 minor 28 references

High-order and Mass-conservative Regularized Implicit-explicit relaxation Runge-Kutta methods for the logarithmic Schr\"{o}dinger equation

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Pith's one-line read The paper develops energy-regularized implicit-explicit relaxation Runge-Kutta methods for the logarithmic Schrödinger equation that conserve mass exactly and keep the full temporal order of the underlying Runge-Kutta pair.

desk verdict A solid, useful combination of existing regularization and IMEX relaxation RK ideas for LogSE, with a fixable gap in the order proof's regularity assumptions. read the letter →

arxiv 2411.08271 v1 pith:CNM57TOQ submitted 2024-11-13 math.NA cs.NA

Jingye Yan , Hong Zhang , Yabing Wei , Xu Qian This is my paper · ORCID
classification math.NAcs.NA MSC 35Q4165L0665M1265M70
keywords logarithmicSchrödingerequationenergyregularizationimplicit-explicitrelaxationRunge-KuttamassconservationFourierpseudo-spectralmethodhigh-ordertimeintegrationnon-differentiablenonlinearity
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 logarithmic Schrödinger equation (LogSE) has a nonlinearity $\ln|u|^2$ that is not differentiable at $u=0$, which blocks standard high-order numerical schemes. The paper removes the singularity by replacing the energy density with a $C^k$ polynomial near zero, then applies implicit-explicit relaxation Runge-Kutta (IMEX RRK) time stepping with Fourier pseudo-spectral discretization in space. Its central claim is that the relaxation coefficient can be chosen to make the semi-discrete scheme conserve mass exactly, while the local truncation error remains $O(\tau^{p+1})$ for a base IMEX Runge-Kutta method of order $p$. If correct, this yields a practical class of linearly implicit, high-order, mass-conserving integrators for a singular nonlinear Schrödinger model, and the numerical experiments confirm orders 2 through 5.

What carries the argument

The load-bearing object is the relaxation coefficient $\gamma_n$ in (3.4), appended to an IMEX Runge-Kutta pair. It is computed from the discrete inner product and the stage derivatives after each step, and by construction it cancels the leading term in the mass variation, forcing exact conservation of the discrete $L^2$ norm. Because the base method is $p$-th order, Lemma 3.1 gives $\gamma_n = 1 + O(\tau^{p-1})$, so the correction is small for small steps, and the Taylor argument in Theorem 3.2 shows the truncation error remains $O(\tau^{p+1})$. The second essential piece is the energy regularization (2.1), which replaces the singular energy density by a $C^k$ polynomial near zero, making the nonlinearity smooth enough for the expansion and for the Runge-Kutta analysis.

What would settle it

Run the IMEX RRK(2,3) scheme on the ERLogSE with an initial datum that has a zero at some point, keep $\varepsilon$ fixed, and measure the $L^2$ temporal error against a fine reference as $\tau$ decreases. If the observed rate falls below 3, or if $\max_n |\gamma_n - 1|$ does not decay like $\tau^2$, the smoothness premise behind Theorem 3.2 is contradicted; mass conservation would still hold algebraically.

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Extended reading notes

Core claim

The paper's central discovery is that any $p$-th order IMEX Runge-Kutta pair can be adapted to the energy-regularized logarithmic Schrödinger equation (ERLogSE) without losing order, by appending a relaxation factor $\gamma_n$ that enforces exact mass conservation. The factor is chosen as in (3.4): it equals 1 when the stage update has zero mass derivative, and otherwise $\gamma_n = -2\,\Re(\sum_j (b^I_j g^I_j + b^E_j g^E_j), u^{\varepsilon,n}_\gamma) / (\tau \|\sum_j (b^I_j g^I_j + b^E_j g^E_j)\|^2)$. Theorem 3.1 shows this choice gives $\|u^{\varepsilon,n+1}_\gamma\|^2 = \|u^{\varepsilon,n}_\gamma\|^2$, and Theorem 3.2 shows the truncation error is $O(\tau^{p+1})$, using the fact that $\gamma_n = 1 + O(\tau^{p-1})$ for a $p$-th order base method. The regularity that makes the analysis possible comes from the energy regularization (2.1), which replaces the non-differentiable $F(\rho) = \rho\ln\rho - \rho$ by a $C^k$ polynomial on $0 \le \rho < \varepsilon^2$.

Load-bearing premise

The load-bearing premise is that the exact ERLogSE solution is smooth enough in time for a Taylor expansion of order $p+1$, and that the base IMEX Runge-Kutta method is at least second-order accurate; the paper takes this smoothness as given rather than deriving it from the $C^k$ energy regularization.

Editorial extensions

If this is right

  • Any $p$-th order IMEX Runge-Kutta pair becomes a mass-conserving scheme for the ERLogSE with the same temporal order; the paper demonstrates this for schemes of orders 2 through 5.
  • The regularization parameter $\varepsilon$ controls the modeling error: the ERLogSE solution and its density converge linearly in $\varepsilon$ to the LogSE solution, while the regularized energy converges quadratically.
  • Mass conservation holds exactly at the semi-discrete level for the chosen relaxation coefficient, independently of the time-step size.
  • Combined with Fourier pseudo-spectral spatial discretization, the method gives spectral accuracy in space, so the overall solver can be high order in both time and space.

Reading between the lines

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

  • The paper leaves implicit that the optimal balance between regularization and time step is $\varepsilon \sim \tau^p$, since the ERLogSE-to-LogSE error is $O(\varepsilon)$ while the temporal error is $O(\tau^p)$; choosing $\varepsilon$ much smaller wastes resolution, and choosing it larger makes the modeling error dominate.
  • The mass-conservation argument uses only the discrete inner product and the relaxation formula, so it should transfer to other spatial discretizations such as finite element or finite difference methods that preserve a compatible inner-product structure; the paper tests only Fourier pseudo-spectral, so this is an untested extension.
  • The same construction likely applies to the regularized sine-Gordon-Schrödinger equation (ERSSE) mentioned in the paper, whose results are stated to be similar but not shown; applying the scheme there would test the generality of the mechanism.
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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

3 major / 5 minor

Summary. The manuscript proposes an energy-regularized logarithmic Schrödinger equation (ERLogSE) and applies implicit-explicit relaxation Runge-Kutta (IMEX RRK) methods in time with Fourier pseudo-spectral discretization in space. The main theoretical claims are Theorem 3.1 (exact mass conservation of the semi-discrete IMEX RRK scheme for a specific relaxation coefficient) and Theorem 3.2 (temporal truncation error O(τ^{p+1}) for a p-th order base IMEX RK method). Numerical experiments report temporal and spatial convergence rates for four IMEX RK pairs and long-time dynamics for two colliding Gaussons.

Significance. If correct, the paper provides a practical linearly implicit, high-order time integrator family for the LogSE that conserves mass exactly, which is valuable because the logarithmic nonlinearity is singular and many existing schemes are low order or non-conservative. The algebraic mass-conservation mechanism is clean and the numerical experiments support the advertised orders in the tested cases. However, the order proof is semi-discrete and rests on unstated regularity assumptions tied to the polynomial degree k of the energy regularization; the fully discrete spatial error is not analyzed. The paper's main contribution is the method construction plus numerical validation, and that contribution is plausible but not yet fully rigorous.

major comments (3)
  1. [Section 3.1, Eq. (3.4)] Equation (3.4) as printed is not a well-defined definition: in the first branch the condition "∑ (b^I_j g^I_j + τ γ_n ∑ b^E_j g^E_j) = 0" contains γ_n on both sides, while the proof of Theorem 3.1 and the subsequent Lemma 3.1 require the simpler condition ∑ (b^I_j g^I_j + b^E_j g^E_j) = 0 with no γ_n inside. This display must be corrected before Theorem 3.1 is a precise statement.
  2. [Section 3, Theorem 3.2] The proof expands the exact ERLogSE solution to order p+1, which requires the solution to have bounded time derivatives through order p+1. The regularized nonlinearity f^k_ε in (2.2) is only C^k in ρ, and when the density crosses ρ = ε^2 the (k+1)-st derivative jumps; for k < p the required Taylor expansion is not justified. The manuscript never states a condition such as k ≥ p, nor any regularity hypothesis on u0 that would imply C^{p+1} regularity of the ERLogSE solution. The rate tests in Section 4.1 use the Gaussian (4.1) with ς = 0, whose density is time-independent at each point, so they do not exercise the switching surface; the two-Gaussian experiment in Section 4.2 is not used for rate measurement. Please state the missing regularity assumption or prove the order result under weaker conditions.
  3. [Section 3, proof of Theorem 3.2] The proof substitutes the exact solution into (3.3) and then invokes Lemma 3.1 for γ_n, but γ_n is defined through stage values of the numerical solution; it is not shown that the γ_n obtained from exact-solution substitutions has the same expansion. The displayed cancellation also drops the O((γ_n−1)^2 τ^2) term by using γ_n − 1 = O(τ^{p−1}), which is valid for p ≥ 2 but should be stated explicitly. As written, the argument is closer to a heuristic order estimate than to a self-contained local truncation error proof of the actual discrete map.
minor comments (5)
  1. [Introduction] The introduction refers to a Section 6, but the paper has only five sections; the conclusion is Section 5.
  2. [Section 4.1.1] The polynomial degree k is denoted by n in the text and figure captions (for example, "larger values of n" and Fig. 4.3 captions with n = 2, n = 4); please use one symbol consistently.
  3. [Section 3 and 4] Theorems 3.1 and 3.2 are stated only for the semi-discrete system, while the abstract and Section 4 describe the fully discrete Fourier pseudo-spectral method; please state explicitly that the same algebraic mass-conservation identity holds for the spectral discretization, or add a remark on the fully discrete setting.
  4. [Table 4.1] The "order" column in Table 4.1 appears to be computed from consecutive values of N; please state the formula used (for example, log(‖e_{N_j}‖/‖e_{N_{j+1}}‖)/log(N_{j+1}/N_j)) so the reader can interpret the very high orders at N = 20.
  5. [Section 4.1] For the exact Gaussian solution (4.1), the paper should state explicitly that the computational domain is chosen large enough so that the boundary truncation error is negligible, and should specify the boundary condition used in the finite-domain computation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: mass conservation is an enforced algebraic identity and the temporal order proof invokes an independent external lemma.

full rationale

The central load-bearing claims are Theorem 3.1 (mass conservation) and Theorem 3.2 (order preservation). Theorem 3.1 is not a prediction: substituting the relaxation coefficient (3.4) into the mass-increment identity (3.6) forces the increment to vanish exactly, so mass conservation is a designed algebraic invariant of the method. Theorem 3.2 derives the truncation error from the known p-th order of the base IMEX RK scheme and from Lemma 3.1, which is quoted from the independent paper Li, Li & Zhang [10]; that lemma's assumptions (base order p >= 2, sufficiently small tau) do not include the target result, and it is not authored by the present authors. No parameter is fitted to data and no 'prediction' is manufactured from a subset of the same data. The only self-citation is [11] (Wang, Yan & Zhang), an earlier first-order IMEX scheme mentioned in the introduction; it is not used in the proofs, in the relaxation construction, or in the numerical validation, so it is not load-bearing. The printed gamma_n formula (3.4) has an apparent typo in its first branch, with gamma_n appearing inside its own condition, and the paper leaves the regularity of u0 to be 'specified later' without doing so; these are correctness or technical defects rather than circular reductions, because neither is used to justify the paper's claims via itself. The numerical convergence tests are self-consistency checks against a fine reference solution, but the order statement stands independently on the cited external lemma and the construction is parameter-free, so the overall circularity score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No new physical entities are introduced. The paper's contribution is a numerical scheme built from two existing components, Bao et al.'s energy regularization and Li et al.'s IMEX relaxation Runge-Kutta framework; the free parameters are method parameters chosen manually.

free parameters (2)
  • epsilon (regularization parameter) = 10^-5 to 10^-10 in experiments
    Chosen by hand to control the smoothing of the logarithmic nonlinearity; the ERLogSE-to-LogSE error is O(epsilon) and the numerical tests vary epsilon.
  • k (polynomial degree in energy regularization) = n = 2, 4, 10 in convergence tests
    Chosen by hand; higher k gives higher regularity and better approximation near rho = 0, but no optimality criterion is given.
assumptions (4)
  • standard math IMEX RK order conditions from [7,8] and Lemma 3.1 from [10] hold for the ERLogSE semi-discretization
    The paper invokes Lemma 3.1 to prove gamma_n = 1 + O(tau^(p-1)) and uses standard IMEX RK convergence order p; these are external results assumed without reproof.
  • domain assumption The exact solution of ERLogSE is sufficiently smooth on [0,T] for the Taylor expansion used in Theorem 3.2
    The proof of Theorem 3.2 expands u^epsilon around t_n + tau and requires higher derivatives; the paper does not state regularity conditions on u0 or the solution.
  • standard math F^epsilon_k defined in (2.1) is in C^k and approximates F to second order, as established by Bao et al. [3]
    The regularization construction and its approximation properties are imported from Bao et al. without reproof.
  • domain assumption The Gaussian solution (4.1) is exact for the LogSE as given in Carles and Gallagher [5]
    Used as a benchmark in the numerical experiments and taken from the cited literature.

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Pith. "Pith review of High-order and Mass-conservative Regularized Implicit-explicit relaxation Runge-Kutta methods for the logarithmic Schr\"{o}dinger equation." pith.science (2026). https://pith.science/paper/CNM57TOQ

@misc{pith2026241108271,
  author       = {Pith},
  title        = {Pith review of: High-order and Mass-conservative Regularized Implicit-explicit relaxation Runge-Kutta methods for the logarithmic Schr\"odinger equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CNM57TOQ}},
  note         = {Machine review of arXiv:2411.08271}
}
abstract

The non-differentiability of the singular nonlinearity (such as $f=\ln|u|^2$) at $u=0$ presents significant challenges in devising accurate and efficient numerical schemes for the logarithmic Schr\"{o}dinger equation (LogSE). To address this singularity, we propose an energy regularization technique for the LogSE. For the regularized model, we utilize Implicit-Explicit Relaxation Runge-Kutta methods, which are linearly implicit, high-order, and mass-conserving for temporal discretization, in conjunction with the Fourier pseudo-spectral method in space. Ultimately, numerical results are presented to validate the efficiency of the proposed methods.

Figures

Figures reproduced from arXiv: 2411.08271 by the authors.

Figure 4.1
Figure 4.1. Convergence order ERLogSE to LogSE, i.e. keˆ ε (t = 1)k, keˆ ε ρ (t = 1)k, e ε E (t = 1). 4.1.2. Convergence rate of the numerical scheme Firstly, we investigate the relaxation coefficient γn at t = 1 for various values of n, ε and the results shown in Fig.4.2, substantiate Lemma 3.1. Next, to assess the temporal convergence rate, we set the mesh size to h = he, and varied the time step as τ = 2 −j × 10−1 for j = 1,… view at source ↗
Figure 4.2
Figure 4.2. Convergence order of max(|γn − 1|) for different IMEX RRK schemes. -2.5 -2 -1.5 -1 -5 -4 -3 -2 (a) RRK(1,2) -2.5 -2 -1.5 -8 -7 -6 -5 -4 -3 (b) RRK(2,3) -2.5 -2 -1.5 -1 -10 -8 -6 -4 (c) RRK(6,4) -2.5 -2 -1.5 -1 -12 -10 -8 -6 (d) RRK(8,5) [PITH_FULL_IMAGE:figures/full_fig_p006_4_2.png] view at source ↗
Figure 4.3
Figure 4.3. Convergence order in time of RRK with different ε. 4.2. Dynamics In this section, we investigate long time dynamics of LogSE with Gaussian-type initial datum in 1D. To this end, we fix λ = −1, x ∈ [−40, 40], τ = 2 × 10−3 , h = 5 16 . The initial datum is chosen as u0(x) = X 2 k=1 bke − ak 2 (x−xk ) 2+ivk x , x ∈ R, (4.3) where bk, ak, xk and vk are real constants, i.e, the initial data is the sum of 2 Gaussons with … view at source ↗
Figures from the paper (1 more)
Figure 4.4
Figure 4.4. Figure 4.4: Plots of |u ε (x, t)| (first column); |u ε (x, t)| at different time (second column) and evolution of mass error (third column) (Case for RRK(2,3)). with the Fourier spectral method, to solve the ERLogSE (2.3). Our analysis demonstrated that these IMEX RRK methods no…

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