{"id":"0913e367-650d-4c20-8440-b2c076af7090","arxiv_id":"2507.14692","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For three high-order compact spatial schemes coupled with SSPRK3 time stepping, the paper identifies stability limits and dispersion-error regions for 1D and 2D convection-dispersion (KdV-type) equations.","lead":"The paper studies three mathematical recipes for simulating waves that both travel and spread out, and maps the parameter settings where each recipe stays accurate or breaks down. This gives researchers and engineers practical guidance for choosing simulation parameters in KdV-type wave models, which appear in coastal, plasma, and optics applications.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"GSA plots contradict the stated stability threshold: Fig. 2 lists max |Gnum|=21 at Dα=0.11, so Dα,cr=0.12 is not established without specifying the Nc window; the central critical-number claim is under-specified.","rationale":"The reader's weakest assumption concerns transfer of the linear thresholds to nonlinear KdV/mKdV experiments. My concern is more direct: the linear threshold itself is not crisply supported by the paper's own figures and captions. The text calls Dα=0.11 stable for CNCS6 while the figure caption reports max |Gnum|=21.09, and similar contradictions appear for CNCS8 and CCS8. This does not necessarily mean the mathematics is false; an independent check of the RK3 stability polynomial suggests a pure-dispersion threshold near Dα≈0.11–0.12 is plausible because |Gnum|≤1 requires |Dα (keq^(2))^3 h^3|≤√3. The issue is that the paper defines Dα,cr as a single number without specifying the Nc window over which the stability comparison is made, and the experimental validation uses a time-step formula that appears unable to reach Dα=0.12 with the stated CFL=0.11. These are fixable presentation and verification gaps rather than proof of a false central claim, so the reader's conditional verdict is retained. The check I propose — reproducing the GSA maxima and the (Nc,Dα,kh) stability region — would settle whether the concern lands. Credit is due for the standard GSA framework and for the apparent consistency between the RK3 imaginary-axis stability limit and the order of magnitude of the reported thresholds.","tokens_in":26225,"tokens_out":25673,"duration_ms":267749,"concrete_test":"Recompute |Gnum| from Eq. (27) and the equivalent-wavenumber formulas B.1–B.4 on the paper's stated grid (kh over [0,π] for CNCS and [0,2π] for CCS; Nc over the figure range) and extract max |Gnum| for Dα=0.11/0.12 (CNCS) and 0.011/0.012 (CCS). If max |Gnum| at the 'stable' Dα is >1 within the plotted Nc range, or if no (kh,Nc) rectangle can be specified in which Dα=0.11 is stable while Dα=0.12 is not, then the headline critical-number claim is not supported as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing weak point is that the linear stability threshold itself — the paper's headline result — is not actually pinned down by the presented GSA plots. Section 3.1.1 states that for Dα=0.11 'SSPRK3-CNCS6 remains stable throughout the Nyquist range (kh=π)', and §3.1.3 makes the analogous claim for CCS8 at Dα=0.011. But the caption of Fig. 2(i) reports max |Gnum|=21.09 at Dα=0.11, and Fig. 4(i) reports max |Gnum|=2.60e+02 at Dα=0.011. A |Gnum| maximum larger than one is precisely the instability indicator used to define Dα,cr, so either the captions are wrong, the relevant Nc window is silently restricted, or the words 'stable throughout' are false. A further inconsistency is that for Example 4.1 with N=100 and h=2π/100, the time-step rule (46) with the stated CFL=0.11 gives Dα≈0.11/(1+2h²)≈0.109<0.12, so the claimed destabilizing Dα=0.12 run cannot be produced with the stated CFL. The critical Dα values are therefore under-specified as functions of Nc and not validated by the numerical experiments as described.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents a global spectral analysis (GSA) of the fully discrete one- and two-dimensional linear convection–dispersion equation, using three high-order compact spatial discretizations (CNCS6, CNCS8, CCS8) combined with SSPRK3 time integration. The authors derive the numerical amplification factor Gnum, normalized phase speed, and normalized group velocity as functions of wavenumber, Courant number Nc, and dispersion number Dα, and use these to identify critical dispersion numbers Dα,cr and Courant limits beyond which |Gnum| exceeds unity, with associated nonphysical features such as q-waves, phase reversal, and reversed energy transport. The theoretical predictions are compared with numerical experiments for linear problems, KdV and mKdV soliton problems, and small-dispersion KdV shock problems.","tokens_in":26509,"tokens_out":4001,"duration_ms":46793,"significance":"If the claimed thresholds are correct, the paper offers practically useful guidance for selecting discretization parameters for convection–dispersion simulations. The derivation of Gnum in Eq. (27) from the SSPRK3 polynomial is standard, the equivalent wavenumber expressions in Appendix B are explicitly given, and the convergence tables in Section 4 are consistent with the nominal orders of the schemes. The paper also makes a useful comparison between node-centered and cell-centered compact schemes. However, the central stability-threshold claims are not actually pinned down by the presented contour plots: the figure captions report maximum |Gnum| values far above unity for parameter values that the text describes as stable. In addition, the normalized phase and group velocity diagnostics divide by quantities that vanish on curves, so contour extrema of order 10^4–10^6 include singular features rather than physically meaningful growth. These issues affect the headline conclusions and require careful revision.","major_comments":[{"comment":"The central stability claim that CNCS6 is 'stable throughout the Nyquist range' at Dα=0.11 is contradicted by the caption of Fig. 2(i), which reports max |Gnum|=21.09 at that parameter value; similarly, Fig. 3(i) reports max |Gnum|=25.16 at Dα=0.11 for CNCS8, and Fig. 4(i) reports max |Gnum|=2.60e+02 at Dα=0.011 for CCS8. Since |Gnum|>1 is precisely the instability criterion used to define Dα,cr, the critical values Dα,cr=0.12 and 0.012 are not established unless the relevant (Nc,kh) window over which the maximum is taken is specified and the text is reconciled with the displayed maxima.","section":"§3.1.1–3.1.3 and Figs. 2–4"},{"comment":"The normalized phase speed cnum/cph and group velocity vg,num/vg divide by cph=c−αk² and vg=c−3αk², respectively, which vanish along curves in the (Nc,kh) plane; in nondimensional form the denominators are Nc−Dα(kh)² and Nc−3Dα(kh)². The contour extrema reported in Figs. 2–7, ranging up to 10^6, therefore include singular values along these curves, and the associated statements about expansion of 'negative q-wave' and 'reversed energy transport' regions are not meaningful until the analysis is restricted to the nonsingular set or a different normalization is used.","section":"Eqs. (31)–(32) and (43)–(45)"},{"comment":"The numerical experiment in Example 4.1 uses N=100, h=2π/100, and the stated rule CFL=0.11 for CNCS; substituting into Eq. (46) with g'(u)=2 and f'(u)=1 gives Dα = CFL/(1+2h²) ≈ 0.109, so the claimed destabilizing run at Dα=0.12 cannot be produced with the stated CFL value. The paper needs to specify either how Dα=0.12 is obtained in practice or correct the time-step prescription, because the validation of Dα,cr depends on actually performing the run at the claimed critical value.","section":"§4.1, Eq. (46), and Fig. 8"},{"comment":"The statement that nonlinear blow-up occurs at Dα,cr=0.12 for CNCS and 0.012 for CCS, with 'these results are not presented here,' is an unsupported assertion. Since the GSA applies to the linear constant-coefficient equation, the transfer of the threshold to the nonlinear KdV/mKdV computations is load-bearing and needs direct evidence; nonlinearity and the filtering procedure used in Example 4.3 could plausibly shift the thresholds.","section":"§4.2, Example 4.2"}],"minor_comments":[{"comment":"The phrase 'Courant Friedrichs Lewy number' should be hyphenated as 'Courant–Friedrichs–Lewy number'.","section":"Abstract"},{"comment":"The text says that for Dα=0.11 the scheme 'remains stable throughout the Nyquist range (kh=π)', but the caption of Fig. 2(i) reports max |Gnum|=21.09; please clarify whether the contour plots are clipped or the stability statement refers to a restricted (Nc,kh) window.","section":"§3.1.1"},{"comment":"The critical value Dα,cr=0.12 in the 2D analysis is stated for a fixed Courant number (Nc=0.9 for CNCS6, Nc=0.7 for CNCS8, Nc=0.45 for CCS8), but the 1D analysis quotes thresholds over a range of Nc; the dependence of Dα,cr on Nc should be stated explicitly to avoid ambiguity.","section":"§3.2.1"},{"comment":"The caption of Fig. 9 does not identify the line styles or colors used for CNCS6, CNCS8, and CCS8; please add a legend or specify the correspondence in the caption.","section":"Fig. 9"},{"comment":"The sentence 'To validate the theoretical findings, we validate our spectral predictions through numerical experiments' is repetitive; consider rewording to avoid the doubled 'validate'.","section":"§1, Introduction"}],"recommendation":"major_revision","confidential_remarks":"The manuscript applies a well-established GSA framework to a new equation and provides useful convergence data, but the stability-threshold claims are internally inconsistent with the displayed maximum amplification factors. I would encourage the editor to require the authors to reconcile the captions with the text, define the critical thresholds as functions of the spectral window, and either provide the nonlinear blow-up evidence or soften the corresponding claims. The paper is within the scope of the journal and the issues appear fixable, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nMy take on arXiv:2507.14692: it's a global spectral analysis of three compact schemes (CNCS6, CNCS8, CCS8) with SSPRK3 time stepping applied to the 1D/2D linear convection-dispersion equation, producing the usual |Gnum|, phase speed, and group velocity maps. The derivation of Gnum is standard and the convergence tables in Section 4 are clean and match the formal orders, so the numerical machinery is credible. The 2D extension with anisotropy and the kh∈[0,2π] treatment for the cell-centered CCS8 are genuinely useful for people simulating dispersive waves.\n\nThe soft spots are real, though. The headline result — the critical dispersion numbers Dα ≈ 0.12 for the node-centered schemes and ≈ 0.012 for CCS8 — is not actually established by the plots as printed. The caption of Fig. 2(i) reports max |Gnum| = 21.09 at Dα = 0.11, while the text says the scheme is stable throughout the Nyquist range; Fig. 4(i) reports max |Gnum| = 2.60e+02 at Dα = 0.011, again described as stable. A |Gnum| maximum above one is exactly the instability indicator used to define the threshold. Either the captions are wrong, the text is silently restricting to a subrange, or the numbers don't mean what the text says. Until that is resolved, Dα,cr is under-specified as a function of Nc and kh.\n\nTwo smaller issues need attention. First, the normalized phase speed and group velocity (Eqs. 31–32) divide by expressions that vanish on curves in the (Nc, kh) plane, so contour extrema up to 1e5 or 1e6 are singular artifacts, not meaningful physical values. That doesn't invalidate the |Gnum| analysis, but it does contaminate the q-wave / reversed-energy interpretation. Second, the nonlinear validation is asserted rather than shown: the paper states that blow-up occurs at the critical Dα for the KdV and mKdV examples but does not present those results. Given that the thresholds come from a linear constant-coefficient analysis, this transferability is load-bearing and should be demonstrated.\n\nThe novelty claim also deserves a careful check against [27] (Ashwin et al., J. Sci. Comput. 2015), which is cited but not compared in scope. The CCS8 scheme itself comes from an unreplicated preprint by the same group, which is not a problem for reproducibility (the formulas are in the appendix) but does mean the scheme's properties are not yet independently vetted.\n\nOverall, this is worth a serious referee: the underlying math is sound, the application is useful, and the data tables are reproducible. But the manuscript needs a round of revision to sort out the plot-vs-text contradiction and to show, not just assert, the nonlinear blow-up runs.\n\nRecommendation: send it to peer review with a request for major revision.","headline":"Useful GSA maps for three compact schemes on the linear KdV model, but the headline stability thresholds are not actually pinned down by the figures as printed.","tokens_in":27062,"tokens_out":6823,"would_cite":false,"duration_ms":71840,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q53","65M06","65M12"],"pacs":[],"model":"deepseek-v4-flash","headline":"Global spectral analysis fixes the critical dispersion numbers of compact convection-dispersion schemes at $D_{\\alpha,\\mathrm{cr}}=0.12$ (node-centered) and $0.012$ (cell-centered), beyond which spurious waves and reversed energy…","keywords":["global spectral analysis","convection-dispersion equation","Korteweg-de Vries equation","compact finite difference schemes","SSPRK3","amplification factor","q-waves","dispersion number"],"falsifier":"Evaluate $|G_{\\rm num}|$ from Eq. (27) using the equivalent-wavenumber formulas of Appendix B at $N_c=1.4$ and $D_\\alpha=0.12$ over $kh\\in[0,\\pi]$: the claimed threshold stands only if some wavenumber gives $|G_{\\rm num}|>1$; if none does, the critical dispersion number for CNCS6/CNCS8 is misidentified. Alternatively, run Example 4.2 at $D_\\alpha=0.115$ and $D_\\alpha=0.125$ and compare the evolution of the maximum amplitude, since the paper states the nonlinear blow-up results themselves are not presented.","tokens_in":26034,"feed_emoji":"🌊","tokens_out":11142,"duration_ms":611696,"temperature":0.7,"pith_summary":"This paper establishes sharp stability limits for three fully discrete compact schemes on the convection-dispersion equation, the linear prototype behind the KdV and mKdV equations. Coupled with SSPRK3 time stepping, the sixth- and eighth-order node-centered schemes CNCS6 and CNCS8 remain stable up to a dispersion number $D_\\alpha\\approx0.11$ over the full Nyquist range, with $D_{\\alpha,\\mathrm{cr}}=0.12$ as the critical value; the eighth-order cell-centered CCS8 has $D_{\\alpha,\\mathrm{cr}}=0.012$, an order of magnitude smaller, in exchange for resolving wavenumbers up to $kh=2\\pi$. Beyond these thresholds the numerical amplification factor $|G_{\\rm num}|$ exceeds unity, and the normalized phase speed and group velocity develop negative regions that the paper identifies as spurious q-waves, phase reversal, and reversed energy transport. Numerical tests on linear 1D/2D problems, KdV single and double solitons, small-dispersion KdV, and mKdV interactions confirm that instability appears at the predicted critical dispersion numbers and that the cell-centered scheme is markedly more accurate but computationally dearer.","feed_headline":"Dispersion number 0.12 marks the stability edge","feed_subtitle":"For compact wave schemes solving KdV-type equations, cell-centered CCS8 needs D≈0.012; past the edge, q-waves and reversed energy appear.","key_machinery":"The load-bearing object is the numerical amplification factor of the fully discrete scheme, $G_{\\rm num}=1-A+\\tfrac{A^2}{2}-\\tfrac{A^3}{6}$, with $A=N_c\\, i k_{\\rm eq}^{[1]}h + D_\\alpha\\big(-i (k_{\\rm eq}^{[2]})^3 h^3\\big)$ in 1D and the obvious four-term sum in 2D. The factors $k_{\\rm eq}^{[1]}$ and $k_{\\rm eq}^{[2]}$ are the equivalent wavenumbers of the first- and third-derivative compact stencils, rational trigonometric functions of $kh$ listed in Appendix B; they convert the physical dispersion relation $\\omega=ck-\\alpha k^3$ into the numerical relation used to plot $c_{\\rm num}/c_{\\rm ph}$ and $v_{g,\\rm num}/v_g$. Stability is decided by whether $|G_{\\rm num}|$ exceeds unity anywhere in the $(N_c,kh)$ plane, and nonphysical behavior is read from sign changes in the two velocity ratios, with the spectral domain extended to $kh\\in[0,2\\pi]$ for CCS8 to exploit its cell-centered degrees of freedom. The 2D analysis folds in the mesh aspect ratio $AR$ and wave angle $\\theta$ through direction-wise Courant and dispersion numbers, fixing $AR=1$ and $\\theta=45^\\circ$ for the reported contours.","core_discovery":"On the paper's own terms, the central discovery is that global spectral analysis of the fully discrete linear equation $u_t+c u_x+\\alpha u_{xxx}=0$ (with the 2D extension $u_t+c_xu_x+c_yu_y+\\alpha(u_{xxx}+u_{yyy})=0$) identifies dispersion numbers $D_\\alpha=\\alpha\\Delta t/h^3$ at which the schemes change character: $D_{\\alpha,\\mathrm{cr}}=0.12$ for both CNCS6 and CNCS8, and $D_{\\alpha,\\mathrm{cr}}=0.012$ for CCS8. Below these values, with Courant numbers $N_c\\lesssim1.3$ for the node-centered schemes and $N_c\\lesssim0.65$ for CCS8, $|G_{\\rm num}|\\le1$ over the full resolved spectrum, and the numerical phase speed and group velocity ratios stay close to unity over a wide wavenumber range. At the critical values, stability is lost at high wavenumbers ($kh\\approx2.5$ for CNCS, $kh\\approx5.35$ for CCS8), and the contour plots of $c_{\\rm num}/c_{\\rm ph}$ and $v_{g,\\rm num}/v_g$ show negative regions where waves travel backward and energy is transported opposite to the physical direction, which the paper calls q-waves. The experiments in Section 4 are reported as validating these thresholds: linear and nonlinear KdV-type simulations blow up at $D_{\\alpha,\\mathrm{cr}}$, and CCS8 systematically produces errors about an order of magnitude smaller than CNCS8 at at least double the computational cost.","pith_inferences":["The transfer of linear periodic-grid thresholds to the nonlinear KdV/mKdV runs is asserted, not demonstrated: the paper says the nonlinear blow-up results are not presented. A direct check is to run the single-soliton example at $D_\\alpha=0.115$ and $0.125$ and record whether the solution remains bounded near the predicted edge.","The same $G_{\\rm num}$ machinery can be applied at other aspect ratios and incidence angles; the reported $AR=1$, $\\theta=45^\\circ$ slice leaves open the question of how directional anisotropy shifts $D_{\\alpha,\\mathrm{cr}}$ in the 2D case.","Because the q-wave and phase-reversal regions appear inside the stable zone, the contours of $c_{\\rm num}/c_{\\rm ph}$ and $v_{g,\\rm num}/v_g$ could be used as local error estimators for adaptive stepping or grid refinement in KdV-type solvers, an extension the paper does not pursue.","The claimed doubled spectral resolution of CCS8 can be tested directly with a single-mode linear problem at $kh\\in(\\pi,2\\pi)$: the observed phase error should match the paper's equivalent-wavenumber curves and stay small where those curves stay near unity."],"forward_implications":["For CNCS6 and CNCS8 with SSPRK3, keeping $D_\\alpha\\le0.11$ and $N_c\\lesssim1.3$ avoids $|G_{\\rm num}|>1$ over the complete $kh\\in[0,\\pi]$ range; at $D_{\\alpha,\\mathrm{cr}}=0.12$ the usable wavenumber range drops to about $kh\\le2.5$.","For CCS8, $D_\\alpha$ must be kept near $0.011$–$0.012$ and $N_c$ below about $0.65$, but the scheme then resolves the doubled range $kh\\in[0,2\\pi]$, making it the choice when high-wavenumber fidelity matters more than large time steps.","Spurious q-waves, phase reversal, and reversed energy transport can occur even where $|G_{\\rm num}|\\le1$, so parameter selection should be guided by the normalized phase speed and group velocity contours, not by stability alone.","The same critical dispersion numbers carry over to the 2D linear tests at $AR=1$, $\\theta=45^\\circ$; CNCS8 shrinks the negative-group-velocity area relative to CNCS6 but does not eliminate it.","CCS8's accuracy advantage is about an order of magnitude in $L_\\infty$ error over CNCS8 at similar resolution, with convergence orders near 8 for CNCS schemes and near 7 for CCS8, at a runtime cost at least double that of CNCS."],"supporting_citations":[{"why":"Supplies the CCS8 third-derivative compact scheme, its equivalent wavenumber, and the filtering procedure used in Example 4.3.","marker":"[30]"},{"why":"Supplies the CNCS6 and CNCS8 compact approximations for first and third derivatives on which the node-centered analysis rests.","marker":"[26]"},{"why":"Source of the CNCS stencil coefficients and the equivalent-wavenumber formulas (B.1)-(B.2).","marker":"[28]"},{"why":"Source of the CCS first-derivative scheme and its equivalent-wavenumber expression (B.3).","marker":"[44]"},{"why":"Establishes the global spectral analysis framework: numerical dispersion relation, DRP concept, and property contours.","marker":"[16]"},{"why":"Introduces the error-dynamics and amplification-factor approach beyond von Neumann analysis that the paper follows for $G_{\\rm num}$.","marker":"[38]"},{"why":"Defines spurious q-waves, the artifact the paper maps to negative phase and group velocity regions.","marker":"[50]"},{"why":"Provides the SSPRK3 time integrator whose amplification factor enters Eqs. (27) and (40).","marker":"[46]"}],"fun_headline_variants":["D=0.12 is the stability cliff for compact wave schemes","Cell-centered CCS8 stable only for D≤0.012","Spurious q-waves and reversed flow appear at critical D","Exact stability bounds pinned for KdV-type compact schemes","Compact schemes: stable below Dcr, blow up above"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the linear, constant-coefficient, periodic-grid thresholds computed from Eq. (13) also govern the nonlinear KdV and mKdV simulations, so that choosing CFL=0.11 for CNCS and 0.011 for CCS and expecting blow-up at $D_{\\alpha,\\mathrm{cr}}=0.12$ and $0.012$ follows from the linear analysis; the paper explicitly notes that the nonlinear blow-up results are not presented.","fun_headline_variants_meta":{"raw":{"variants":["D=0.12 is the stability cliff for compact wave schemes","Cell-centered CCS8 stable only for D≤0.012","Spurious q-waves and reversed flow appear at critical D","Exact stability bounds pinned for KdV-type compact schemes","Compact schemes: stable below Dcr, blow up above"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000965,"raw_usage":{"total_tokens":4239,"prompt_tokens":1208,"completion_tokens":3031,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":824,"completion_tokens_details":{"reasoning_tokens":2946}},"tokens_in":824,"tokens_out":3031,"duration_ms":23139,"temperature":1.0,"reasoning_tokens":2946,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:51:56.720202+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate $|G_{\\rm num}|$ from Eq. (27) using the equivalent-wavenumber formulas of Appendix B at $N_c=1.4$ and $D_\\alpha=0.12$ over $kh\\in[0,\\pi]$: the claimed threshold stands only if some wavenumber gives $|G_{\\rm num}|>1$; if none does, the critical dispersion number for CNCS6/CNCS8 is misidentified. Alternatively, run Example 4.2 at $D_\\alpha=0.115$ and $D_\\alpha=0.125$ and compare the evolution of the maximum amplitude, since the paper states the nonlinear blow-up results themselves are not presented.","supporting_citations":[{"cited_title":"A novel central compact finite-difference scheme for third derivatives with high spectral resolution","cited_arxiv_id":"2405.00569","evidence_quote":"Supplies the CCS8 third-derivative compact scheme, its equivalent wavenumber, and the filtering procedure used in Example 4.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the CNCS6 and CNCS8 compact approximations for first and third derivatives on which the node-centered analysis rests."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of the CNCS stencil coefficients and the equivalent-wavenumber formulas (B.1)-(B.2)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of the CCS first-derivative scheme and its equivalent-wavenumber expression (B.3)."},{"cited_title":"Sengupta, High accuracy computing methods: fluid flows and wave phenomena, Cambridge University Press, 2013","cited_arxiv_id":null,"evidence_quote":"Establishes the global spectral analysis framework: numerical dispersion relation, DRP concept, and property contours."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the error-dynamics and amplification-factor approach beyond von Neumann analysis that the paper follows for $G_{\\rm num}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines spurious q-waves, the artifact the paper maps to negative phase and group velocity regions."},{"cited_title":"Gottlieb, C.-W","cited_arxiv_id":null,"evidence_quote":"Provides the SSPRK3 time integrator whose amplification factor enters Eqs. (27) and (40)."}],"review_version":1}