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With s unknown stages and two-derivative Hermite data, one can hit order 2s and exact L-stability for every s.

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2026-07-31 11:02 UTC pith:T7XEWPAE

load-bearing objection Solid constructive NA paper: s active stages, order 2s, and exact second-subdiagonal Padé L-stability for every node set, with the algebra holding up. the 3 major comments →

arxiv 2607.24592 v1 pith:T7XEWPAE submitted 2026-07-27 math.NA cs.NA

Arbitrary-Order Pad\'e-Closed Anchored Two-Derivative Time Discretizations: s Active Stages, Order 2s, and L-Stability

Zhixin Huo This is my paper
classification math.NA cs.NA MSC 65L0565L0465L0665L20
keywords multiderivative time integrationtwo-derivative methodHermite anchorPadé approximationarbitrary orderL-stabilityanchored stagesstiff ODEs
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Classical collocation forces a tradeoff: Gauss methods reach order 2s but do not damp infinitely stiff modes to zero, while Radau IIA damps them but only reaches order 2s−1. This paper builds an implicit one-step family that keeps exactly s unknown stage states, feeds each information node both the vector field and its first total time derivative, and still obtains global order 2s. The two free coefficients left in each stage row after the Hermite moment conditions are fixed by the second-subdiagonal Padé approximant to the exponential. A basis theorem shows the closure is unique for any ordered real nodes, preserves the moments, makes the stage determinant exactly the Padé denominator, and forces the accepted step map to be that Padé function—hence L-stable for every positive integer s. Equal-stage numerical comparisons on stiff test problems show the combination of high order and strong damping over wide step-size ranges.

Core claim

For every s≥1 and every ordered real node set 0=c0<c1<⋯<cs=1, the Padé-closed anchored two-derivative method with exactly s active stages satisfies all 2s Hermite moment conditions (global order 2s), realizes the stability function Rs(z)=Ps(z)/Qs(z)=[s−1/s+1]ez exactly, and has det(I−zA−z²Â)=Qs(z), so the accepted one-step map is L-stable.

What carries the argument

The Padé–Hermite basis theorem: the 2s+2 polynomials Πj(z)=T2s(Qs(z)ecj z) and zΠj form a basis of polynomials of degree ≤2s+1, uniquely fixing the two free coefficients per stage row while preserving moments and forcing the stage determinant and endpoint map to be exactly the second-subdiagonal Padé pair.

Load-bearing premise

Global order 2s is proved only for smooth solutions and sufficiently small steps; the practical stiff advantage at large steps rests on linear stability plus experiments, not a stiffness-uniform nonlinear error theorem.

What would settle it

For some s and node set, either the constructed coefficients fail the 2s moment identities or det(I−zA−z²Â)≠Qs(z), or high-precision integration of a smooth nonlinear ODE fails to show observed order approaching 2s.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Any ordered real nodes may be chosen for conditioning or solver behavior without changing the L-stable endpoint map.
  • At fixed active-stage count the family supplies both order 2s and stiff damping, the corner missing from classical Gauss and Radau collocation.
  • The common-denominator identity gives an explicit rational stage solution with no hidden poles, simplifying analysis and block Newton design.
  • When total-derivative information is cheap (Jacobian–vector products, automatic differentiation, or Lax–Wendroff formulations), order 2s and L-stability become available with only s unknown states.

Where Pith is reading between the lines

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

  • Stage-amplification functions Gj=Πj/Qs are explicit, so internal-node placement can be optimized for stiffness-uniform stage bounds without touching endpoint stability.
  • The same anchored Hermite-plus-Padé pattern may extend to three-derivative or IMEX settings if a suitable higher-subdiagonal target remains A-stable.
  • For very large s the monomial basis becomes ill-conditioned, so practical codes will need scaled orthogonal bases or multiprecision coefficient generation even though the theory is node-flexible.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 7 minor

Summary. The paper constructs an arbitrary-order family of implicit one-step time integrators in an "anchored active-stage" two-derivative formulation: the known value Y_0 = u_n serves as a Hermite anchor, s unknown stages are solved for, and each node contributes both L and its total time derivative \dot L. Imposing 2s Hermite moment conditions per row leaves two free coefficients per row, which are uniquely closed by matching the second-subdiagonal Padé approximant [s−1/s+1]_{e^z}. The key algebraic results are: a Padé–Hermite basis theorem (Thm 5.2) showing that the truncated products T_{2s}(Q_s e^{c_j z}) together with their z-multiples span P_{2s+1} for every ordered node set; moment preservation of the closure (Thm 5.3); the exact identity det(I − zA − z²Â) = Q_s(z) (Thm 5.4), excluding hidden stage poles; and the endpoint identity R_s(z) = [s−1/s+1]_{e^z}, which by classical Padé theory (Ehle, order stars) is A- and L-stable for every s (Thm 5.5). Classical order 2s in the smooth small-step regime is proved via the implicit-function theorem and discrete Gronwall (Thm 3.2). Exact symbolic verification is reported through s=6, high-precision tests confirm orders 2–8, and fixed-stage comparisons on the Kaps problem, a two-mode decay system, and a semidiscrete heat equation show favorable accuracy/damping relative to equal-stage Gauss–Legendre and Radau IIA.

Significance. If the results hold — and the proofs and symbolic verification indicate they do — the paper fills a genuine gap in the standard comparison table: Gauss–Legendre gives order 2s without L-stability, Radau IIA gives L-stability at order 2s−1, and this family gives both at s active stages by exploiting second-derivative information. Particularly valuable features are: (i) the construction is valid for *every* ordered node set, decoupling node choice from stability and enabling conditioning-driven node design (illustrated in Table 1); (ii) the exact determinant identity det(I−zA−z²Â)=Q_s is stronger than an endpoint-only statement and rules out hidden poles in the coupled stage solve; (iii) exact symbolic verification through s=6 and 100-digit convergence tests provide concrete reproducibility; (iv) the internal stage transfer functions are available in closed form (G_j=Π_j/Q_s), which is the right starting point for future stage-amplification optimization. The work is a solid contribution to the multiderivative time-integration literature. Its practical-superiority narrative, however, currently rests on experiments that do not fully account for the cost of the total-derivative channel,

major comments (3)
  1. [§9.1, Tables 8–9, Figure 4] §9 and Abstract: the equal-active-stage comparisons with Gauss–Legendre and Radau IIA do not account for the cost of evaluating \dot L = L_u L. Each stage of the proposed method requires a Jacobian–vector product (or equivalent AD/Lax–Wendroff machinery) in addition to L, and the nonlinear solve involves a more complex Jacobian (I - zA - z²\hat A with \hat A carrying derivative-of-L information). Tables 8–9 and Figure 4 report total Newton iterations, which is informative, but iteration counts do not capture per-iteration cost differences; it is possible that at equal wall-clock cost the advantage over Radau IIA narrows considerably. Since the abstract states that the comparisons 'demonstrate the combined high-order accuracy and strong stiff damping... over broad step-size ranges,' the paper should either include a cost-per-step analysis or explicitly qualify the comparison as equal-stag
  2. [§5.5, Theorem 5.5] §5.5, Theorem 5.5 and §10: the internal stage transfer functions G_j(z) = Π_j(z)/Q_s(z) have deg Π_j up to 2s versus deg Q_s = s+1, so for internal nodes the stage ratios can grow like |z|^{s-1} along the negative real axis even though the endpoint map decays like z^{-2}. For stiff problems this means the internal stage values may be enormously amplified relative to u_n (e.g., for the two-mode problem with z = -1000h and s = 4, |G_j| can be of order 10^9 in the relevant regime), which bears directly on roundoff contamination of the endpoint through the coupled solve and on nonlinear-solver starting values. The paper acknowledges this only as future work ('stage-amplification control', §10–11). Given that L-stability is used to argue stiff superiority, at minimum a quantitative statement (e.g., a bound or tabulated values of max_j |G_j(-x)| for the equispaced members in §8) should be adde
  3. [§3, Theorem 3.2] §3, Theorem 3.2: the convergence proof is the standard implicit-function-theorem plus discrete-Gronwall argument and is correct as stated, but its constants (K_J in (14), C in (16)) depend on ||L_u|| and ||\dot L_u|| in a neighborhood of the solution, hence potentially on stiffness in a severe way, and the hypothesis 'sufficiently small Δt' may exclude exactly the large stiff steps showcased in §9.5–9.7. The manuscript is commendably careful elsewhere (e.g., §4.1 explicitly notes that moment matching does not imply A-stability); a parallel one-paragraph caveat here — stating that order 2s is a classical smooth-regime result, that no B-convergence or stiffness-uniform error bound is claimed, and that the stiff experiments are evidence rather than theorem — would align the presentation with the actual strength of the result and preempt misreading of the abstract's framing.
minor comments (7)
  1. [Notation] Throughout, the second-derivative coefficient matrix is typeset in the source as \widehat{A} but renders in several places (abstract, (17), (48)) as what reads as 'bA' or 'b\hat A'; please check the macro and ensure consistent notation between the abstract, §4, and §11.
  2. [Table 2] Table 2: the 'stiff limit' column is typeset ambiguously (e.g., '1/z²−2z+2 2' and 'Rs(z)=O(z−2)' on separate lines within the same row); a cleaner layout with the rational function P_s/Q_s in one column and the O(z^{-2}) decay statement in another would avoid confusion.
  3. [§5.2, Theorem 5.2] §5.2, proof of Theorem 5.2: the s=1 case is handled inline ('π_{0,2}=1/2 and π_{1,2}=0'). It would help the reader to state explicitly why the general sign argument fails for s=1 (c^{s-1}=1 makes the anchor coefficient nonzero), since this is the only point where the proof splits by cases.
  4. [§6.4, Table 1] Table 1 is a useful diagnostic; since κ_2(B_s) is basis- and scaling-dependent (as the text itself notes), consider also reporting the condition number for the scaled/orthogonal basis recommended in §6.3, so the recommendation is actionable rather than only cautionary.
  5. [§7] §7: the range (25∓√105)/780 ≤ C for the sequential L-stable family is given without derivation or citation; since this section is a comparison against prior two-stage fourth-order schemes, a reference (e.g., to [16] or the Li–Du/Tang literature) for this specific range would be appropriate.
  6. [Typos] Several compound words appear fused in the text ('stiffdamping', 'stiffstability', 'stiffintegration', 'stiffproblems'); these are presumably hyphenation artifacts but should be cleaned up for the journal version.
  7. [§1, Table 3] The claim of occupying 'the missing corner' (order 2s with L-stability at s stages, §1 and Table 3) is correct but may surprise readers familiar with order barriers for L-only methods; one sentence noting that the second-derivative channel is what circumvents the classical Gauss/Radau tradeoff would make the positioning clearer.

Circularity Check

0 steps flagged

No significant circularity: order from Hermite moments, Padé closure uniquely proved in-paper, L-stability from classical external Padé theory.

full rationale

The derivation chain is constructive and self-contained. Hermite moment conditions (12) are obtained by Taylor matching of the stage formula to the exact solution and independently enforce local defect O(Δt^{2s+1}) and global order 2s (Theorems 3.1–3.2). The two free coefficients per row are then fixed by requiring the stage polynomials to realize the classically known second-subdiagonal Padé approximant [s−1/s+1]e^z; uniqueness and moment preservation are proved inside the paper via the Padé–Hermite basis theorem (Theorem 5.2) and the truncation identity (Theorem 5.3), not assumed. The stage-determinant identity det(I−zA−z²Â)=Q_s and the endpoint map R_s=P_s/Q_s follow from a maximal-minor argument (Theorems 5.4–5.5). A/L-stability of that rational function is cited from external classical sources (Ehle; order stars), not from a self-fit or self-uniqueness theorem. The author’s related arXiv note [16] appears only as background on sequential two-stage schemes (Section 7) and is not load-bearing for the all-s algebraic claims. No parameter is fitted to data and re-labeled a prediction; no central uniqueness result is imported from overlapping authors. Score 0 is therefore appropriate.

Axiom & Free-Parameter Ledger

1 free parameters · 6 axioms · 1 invented entities

Load-bearing content is mostly standard ODE/Padé mathematics plus the modeling choice that ˙L is available. No data-fitted constants enter the central identities. Node locations are free design choices that the theory proves do not change Rs. The new mathematical objects are the anchored family and the closure basis, not physical entities.

free parameters (1)
  • active node locations c1,...,c_{s-1} = free design choice; examples use equispaced cj=j/s
    Any ordered real nodes with c0=0, cs=1 are allowed; they affect coefficient magnitudes and conditioning (Table 1) but not the endpoint stability function once Padé closure is imposed. Chosen by hand (equispaced or Chebyshev–Lobatto in examples), not fitted to PDE data.
axioms (6)
  • standard math Second-subdiagonal exponential Padé approximants [s−1/s+1]ez are analytic and |R|≤1 in the closed left half-plane, hence A-stable, and tend to 0 at infinity (L-stable).
    Invoked in §4.2 and Theorem 5.5; cited to Ehle (1973) and order-star literature rather than re-proved.
  • domain assumption L and ˙L=Lu L are sufficiently smooth near the exact solution; the exact solution has 2s+1 bounded time derivatives on the step.
    Used for stage consistency and global order in Theorems 3.1–3.2.
  • standard math Implicit-function theorem local solvability of the stage map for small h, with discrete Gronwall giving O(h^{2s}) global error from O(h^{2s+1}) local error.
    Proof of Theorem 3.2; classical one-step convergence framework.
  • standard math Exponential Padé table is normal: Ps and Qs attain stated degrees and are coprime.
    Used in §4.2 and in the determinant argument (Theorem 5.4) via Baker–Graves-Morris.
  • domain assumption Total time derivative information ˙L is computationally available (analytic Jacobian–vector product, AD, or Lax–Wendroff form).
    Practical premise stated in introduction and §9.8; without it the method’s per-stage cost advantage disappears.
  • standard math Hermite value/derivative functionals at distinct nodes are linearly independent; cardinal Hermite derivative polynomials Kj exist with Kj'(ck)=δjk.
    Used inside the Padé–Hermite basis proof (Theorem 5.2).
invented entities (1)
  • Padé-closed anchored two-derivative family (active-stage Hermite formulation with second-subdiagonal closure) independent evidence
    purpose: Realize s unknown stages, order 2s, and exact [s−1/s+1] L-stable endpoint map simultaneously.
    The method class and the specific two-coefficient Padé closure per row are the paper’s constructed objects; they are defined by equations (5)–(6) and (44)–(47), not postulated as physical mediators.

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read the original abstract

An arbitrary-order family of implicit two-derivative one-step methods is constructed in an anchored active-stage formulation. At each information node the method uses both the vector field and its first total time derivative, enriching the local Hermite data without increasing the number of unknown stage states. With the known initial value retained as an anchor and $s$ unknown active stages, $2s$ Hermite moment conditions yield global order $2s$. The two remaining coefficients in each stage row are fixed by the second-subdiagonal Pad\'{e} approximant $[s-1/s+1]_{e^z}$. For every ordered real node set, a Pad\'{e}--Hermite basis theorem proves that the closure is unique, preserves all moment conditions, and gives $\det(I-zA-z^2\widehat A)=Q_s(z)$ and $R_s(z)=P_s(z)/Q_s(z)$. Hence the coupled stage system has no hidden poles and the accepted one-step map is $L$-stable (and therefore $A$-stable) for every positive integer $s$. Exact symbolic verification is reported through $s=6$, and high-precision computations confirm orders $2,4,6,$ and $8$ for the first four members. At equal active-stage count, comparisons with Gauss--Legendre and Radau IIA methods demonstrate the combined high-order accuracy and strong stiff damping of the construction over broad step-size ranges.

Figures

Figures reproduced from arXiv: 2607.24592 by Zhixin Huo.

Figure 1
Figure 1. Figure 1: High-precision convergence for (65). The asymptotic slopes approach 2s for s = 1, 2, 3, 4. s min ℜ{Qs = 0} sampled max |Rs(iy)| excess over one lim x 2 |Rs(−x)| 1 1.00000000 1.0000000000000000 0.00e+00 2 2 1.68709159 1.0000000000000000 0.00e+00 6 3 2.22098003 1.0000000000000002 2.22e-16 12 4 2.66473152 1.0000000000000002 2.22e-16 20 5 3.04820604 1.0000000000000002 2.22e-16 30 6 3.38811794 1.000000000000000… view at source ↗
Figure 2
Figure 2. Figure 2: Endpoint stability diagnostics for the first four members. 0 5 10 15 20 25 30 number of uniform steps 10 −66 10 −57 10 −48 10 −39 10 −30 10 −21 10 −12 10 −3 fast-component magnitude at T =1 Proposed Gauss Radau IIA [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Fast-component damping for the s = 3 methods applied to (66). 9.5. A two-mode stiff decay problem Consider the diagonal system y ′ (t) = [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Error versus total Newton iterations for the Kaps problem. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Fixed-stage comparisons for the semidiscrete heat equation. 9.8. Interpretation of the comparisons The experiments illustrate the principal contribution of the family. At a fixed active-stage count, Gauss supplies order 2s without stiff decay, whereas Radau IIA supplies stiff decay with order 2s − 1. The proposed anchored two-derivative construction supplies both order 2s and endpoint L-stability. On the d… view at source ↗

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Reference graph

Works this paper leans on

26 extracted references · 20 canonical work pages · 1 internal anchor

  1. [1]

    Maximal order for second derivative general linear methods with Runge–Kutta stability,

    A. Abdi and G. Hojjati, “Maximal order for second derivative general linear methods with Runge–Kutta stability,”Applied Numerical Mathematics, vol. 61, no. 10, pp. 1046–1058, 2011. doi:10.1016/j.apnum.2011.06.004

    work page doi:10.1016/j.apnum.2011.06.004 2011

  2. [2]

    Algebraic stability and irreducibility of second derivative methods,

    R. Akbari, G. Hojjati, and A. Abdi, “Algebraic stability and irreducibility of second derivative methods,”Applied Numerical Mathematics, vol. 191, pp. 45–54, 2023. doi:10.1016/j.apnum.2023.05.013

    work page doi:10.1016/j.apnum.2023.05.013 2023

  3. [3]

    G. A. Baker, Jr. and P. Graves-Morris,Padé Approximants, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 59. Cambridge: Cambridge University Press, 1996. doi:10.1017/CBO9780511530074

    doi:10.1017/cbo9780511530074 1996

  4. [4]

    The A-stability of methods with Padé and generalized Padé stability functions,

    J. C. Butcher, “The A-stability of methods with Padé and generalized Padé stability functions,”Numerical Algorithms, vol. 31, nos. 1–4, pp. 47–58, 2002

    2002

  5. [5]

    J. C. Butcher,Numerical Methods for Ordinary Differential Equations, 3rd ed. Chichester, U.K.: Wiley, 2016. doi:10.1002/9781119121534

    doi:10.1002/9781119121534 2016

  6. [6]

    Second derivative extended backward differentiation formulas for the numerical integration of stiffsystems,

    J. R. Cash, “Second derivative extended backward differentiation formulas for the numerical integration of stiffsystems,” SIAM Journal on Numerical Analysis, vol. 18, no. 1, pp. 21–36, 1981. doi:10.1137/0718003

    work page doi:10.1137/0718003 1981

  7. [7]

    On explicit two-derivative Runge–Kutta methods,

    R. P. K. Chan and A. Y . J. Tsai, “On explicit two-derivative Runge–Kutta methods,”Numerical Algorithms, vol. 53, nos. 2–3, pp. 171–194, 2010. doi:10.1007/s11075-009-9349-1

    work page doi:10.1007/s11075-009-9349-1 2010

  8. [8]

    Jacobian-free implicit MDRK methods for stiffsystems of ODEs,

    J. Chouchoulis and J. Schütz, “Jacobian-free implicit MDRK methods for stiffsystems of ODEs,”Applied Numerical Mathematics, vol. 196, pp. 45–61, 2024. doi:10.1016/j.apnum.2023.10.007

    work page doi:10.1016/j.apnum.2023.10.007 2024

  9. [9]

    Explicit strong stability preserving multistage two-derivative time-stepping schemes,

    A. J. Christlieb, S. Gottlieb, Z. J. Grant, and D. C. Seal, “Explicit strong stability preserving multistage two-derivative time-stepping schemes,”Journal of Scientific Computing, vol. 68, no. 3, pp. 914–942, 2016. doi:10.1007/s10915-016-0164-2

    work page doi:10.1007/s10915-016-0164-2 2016

  10. [10]

    A-stable methods and Padé approximations to the exponential,

    B. L. Ehle, “A-stable methods and Padé approximations to the exponential,”SIAM Journal on Mathematical Analysis, vol. 4, no. 4, pp. 671–680, 1973. doi:10.1137/0504057. 23

    work page doi:10.1137/0504057 1973

  11. [11]

    Second derivative multistep methods for stiffordinary differential equations,

    W. H. Enright, “Second derivative multistep methods for stiffordinary differential equations,”SIAM Journal on Numerical Analysis, vol. 11, no. 2, pp. 321–331, 1974. doi:10.1137/0711029

    work page doi:10.1137/0711029 1974

  12. [12]

    High order strong stability preserving multiderivative implicit and IMEX Runge– Kutta methods with asymptotic preserving properties,

    S. Gottlieb, Z. J. Grant, J. Hu, and R. Shu, “High order strong stability preserving multiderivative implicit and IMEX Runge– Kutta methods with asymptotic preserving properties,”SIAM Journal on Numerical Analysis, vol. 60, no. 1, pp. 423–449,

  13. [13]

    Hairer, S

    E. Hairer, S. P. Nørsett, and G. Wanner,Solving Ordinary Differential Equations I: NonstiffProblems, 2nd rev. ed., Springer Series in Computational Mathematics, vol. 8. Berlin: Springer, 1993. doi:10.1007/978-3-540-78862-1

    doi:10.1007/978-3-540-78862-1 1993

  14. [14]

    Multistep-multistage-multiderivative methods for ordinary differential equations,

    E. Hairer and G. Wanner, “Multistep-multistage-multiderivative methods for ordinary differential equations,”Computing, vol. 11, no. 3, pp. 287–303, 1973. doi:10.1007/BF02252917

    work page doi:10.1007/bf02252917 1973

  15. [15]

    Hairer and G

    E. Hairer and G. Wanner,Solving Ordinary Differential Equations II: Stiffand Differential-Algebraic Problems, 2nd rev. ed., Springer Series in Computational Mathematics, vol. 14. Berlin: Springer, 1996. doi:10.1007/978-3-642-05221-7

    doi:10.1007/978-3-642-05221-7 1996

  16. [16]

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

    Z. Huo, “An L-stable implicit two-stage fourth-order temporal discretization scheme for Lax–Wendroff-type solvers applied to stiffproblems,”arXiv preprint arXiv:2512.01628, 2025. doi:10.48550/arXiv.2512.01628

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2512.01628 2025

  17. [17]

    Jackiewicz,General Linear Methods for Ordinary Differential Equations

    Z. Jackiewicz,General Linear Methods for Ordinary Differential Equations. Hoboken, NJ: Wiley, 2009. doi:10.1002/9780470522165

    work page doi:10.1002/9780470522165 2009

  18. [18]

    Construction of two-derivative Runge–Kutta methods of order six,

    Z. Kalogiratou and T. Monovasilis, “Construction of two-derivative Runge–Kutta methods of order six,”Algorithms, vol. 16, no. 12, Art. 558, 2023. doi:10.3390/a16120558

    work page doi:10.3390/a16120558 2023

  19. [19]

    Generalized Runge–Kutta methods of order four with stepsize control for stiffordinary differential equations,

    P. Kaps and P. Rentrop, “Generalized Runge–Kutta methods of order four with stepsize control for stiffordinary differential equations,”Numerische Mathematik, vol. 33, no. 1, pp. 55–68, 1979. doi:10.1007/BF01396495

    work page doi:10.1007/bf01396495 1979

  20. [20]

    Runge–Kutta processes with multiple nodes,

    K. H. Kastlunger and G. Wanner, “Runge–Kutta processes with multiple nodes,”Computing, vol. 9, no. 1, pp. 9–24, 1972. doi:10.1007/BF02236372

    work page doi:10.1007/bf02236372 1972

  21. [21]

    A two-stage fourth order time-accurate discretization for Lax–Wendrofftype flow solvers I: Hyperbolic conservation laws,

    J. Li and Z. Du, “A two-stage fourth order time-accurate discretization for Lax–Wendrofftype flow solvers I: Hyperbolic conservation laws,”SIAM Journal on Scientific Computing, vol. 38, no. 5, pp. A3046–A3069, 2016. doi:10.1137/15M1052512

    work page doi:10.1137/15m1052512 2016

  22. [22]

    High-order multiderivative time integrators for hyperbolic conservation laws,

    D. C. Seal, Y . Güçlü, and A. J. Christlieb, “High-order multiderivative time integrators for hyperbolic conservation laws,” Journal of Scientific Computing, vol. 60, no. 1, pp. 101–140, 2014. doi:10.1007/s10915-013-9787-8

    work page doi:10.1007/s10915-013-9787-8 2014

  23. [23]

    On implicit second derivative two-step peer methods with RK stability for ODEs,

    M. Sharifi, A. Abdi, M. Bra ´s, and G. Hojjati, “On implicit second derivative two-step peer methods with RK stability for ODEs,”Applied Numerical Mathematics, vol. 220, pp. 329–345, 2026. doi:10.1016/j.apnum.2025.10.019

    work page doi:10.1016/j.apnum.2025.10.019 2026

  24. [24]

    Order stars and stability theorems,

    G. Wanner, E. Hairer, and S. P. Nørsett, “Order stars and stability theorems,”BIT Numerical Mathematics, vol. 18, no. 4, pp. 475–489, 1978. doi:10.1007/BF01932026

    work page doi:10.1007/bf01932026 1978

  25. [25]

    On the explicit two-stage fourth-order accurate time discretizations,

    Y . Yuan and H. Tang, “On the explicit two-stage fourth-order accurate time discretizations,”Journal of Computational Mathematics, vol. 41, no. 2, pp. 287–304, 2023. doi:10.4208/jcm.2201-m2020-0288. 24

    work page doi:10.4208/jcm.2201-m2020-0288 2023

  26. [2022]

    doi:10.1137/21M1403175

    work page doi:10.1137/21m1403175