{"id":"24308b9b-f125-4bd8-95aa-ad2bd9bfd143","arxiv_id":"2507.04991","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Paired Explicit Relaxation Runge-Kutta (P-ERRK) methods give entropy-stable multirate time integration for compressible flows, with measured speedups of 3-4x over standalone relaxed schemes.","lead":"This paper introduces a multirate time-stepping method that keeps the mathematical entropy of compressible-flow simulations conserved or decaying, while running up to four times faster than standard options. It blends two existing tools, paired explicit Runge-Kutta schemes and relaxation scaling, into a single integrator for non-uniform grids.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the central claims are well supported by the theory, validation, and reproducible code; the acknowledged semidiscrete EC/ES precondition is a scope boundary, not a flaw.","rationale":"The central claim is not that P-ERRK makes arbitrary semidiscretizations entropy-stable; it is that the time integrator inherits exact entropy conservation/stability from EC/ES spatial discretizations. Since b_i>=0 and the relaxation equation is solved to round-off accuracy, the fully discrete entropy identity follows directly from the semidiscrete property. The AMR and shock-capturing caveats are explicitly acknowledged and do not affect the conforming-mesh EC/ES validation. The only candidate concern I considered was the fixed-final-time order of accuracy under variable gamma_n: the relaxed step is p-th order at t_n+gamma_n*dt, so comparing at a nominal final time could in principle introduce an O(dt^(p-1)) time-shift error. However, the paper's order studies use smooth periodic/analytic solutions where gamma_n-1 is observed to be far below the worst-case bound (e.g., O(10^-(4+p)) in Section 4.3.1), and the reported convergence rates are consistent with design order; I therefore do not treat this as load-bearing. I also considered the gamma_n>1 stability caveat, but the simulations remain stable and the paper's robustness claims are empirical. The reader's weakest assumption correctly identifies the main scope condition; the paper states it repeatedly, so no verdict change is warranted.","tokens_in":46640,"tokens_out":17500,"duration_ms":213968,"concrete_test":"Run the isentropic-vortex AMR convergence study of Section 4.3.1 with entropy-conservative mortar fluxes (e.g., the flux-correction approach of ref. [99]) and check that the global entropy defect stays at round-off while design order is retained; this would confirm that the AMR caveat is an implementation gap in Trixi.jl rather than a limitation of P-ERRK itself.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no internally inconsistent or unsupported step that would change the verdict. The paper's central claim is scoped correctly: P-ERRK is a time-integration layer that preserves the entropy behavior of the underlying semidiscretization, not a mechanism that creates entropy stability from a non-EC/ES spatial discretization. Given (2.15)/(2.17) and nonnegative weights b_i>=0, the shared-weight relaxation update (2.20) with the direction (3.1) reproduces the semidiscrete entropy production exactly (up to Newton tolerance), so entropy conservation/stability transfers to the fully discrete level. The paper explicitly flags where this precondition fails: L2 projection mortars break exact entropy conservation in Section 4.3.1, and production runs with shock capturing/BR1 (Sections 5.1, 5.2, 5.5) cannot be strictly entropy-stable semidiscretizations. These are honest scope limitations, not defects in the method or its validation. Convergence studies show design order, linear invariants are conserved to round-off, and performance claims are backed by reproducible code with detailed RHS counts and runtimes.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces Paired Explicit Relaxation Runge-Kutta (P-ERRK) methods, which combine paired explicit Runge-Kutta (P-ERK) multirate integrators with the relaxation methodology. The central idea is to preserve, at the fully discrete level, the entropy conservation or entropy stability of the underlying semidiscretization by solving a scalar relaxation equation for a parameter gamma_n that scales the update. Optimized schemes of orders two, three, and four are presented, together with convergence studies, tests of linear invariant conservation, entropy-defect measurements, robustness demonstrations (advection, Kelvin-Helmholtz instability, positivity preservation), and performance comparisons against standalone relaxed Runge-Kutta methods on several compressible flow problems, including Navier-Stokes-Fourier, visco-resistive MHD, and inviscid transonic configurations. The manuscript also provides a reproducibility repository. The theoretical derivation in Sections 2 and 3 is transparent: shared weights across partitions, nonnegative weights b_i, the direction d_n = sum_i b_i K_i, and the relaxation equation enforce consistency between the entropy change of the semidiscretization and that of the relaxed update.","tokens_in":46867,"tokens_out":5300,"duration_ms":62411,"significance":"If the results hold, the paper provides a practical, high-order multirate time integrator that inherits entropy behavior from entropy-conservative or entropy-stable semidiscretizations and substantially reduces RHS evaluations and runtime relative to standalone relaxed Runge-Kutta schemes. The main strengths are the optimized schemes up to fourth order, the reproducible implementation, the convergence studies in Section 4.3 showing designed orders for L1 errors, the round-off-level entropy conservation in Section 4.1 when the semidiscretization is EC, and the round-off-level preservation of linear invariants in Section 4.2. The entropy-conservation results are built into the method by construction, so Figures 4b and 7b are implementation checks rather than independent predictions; this is inherent to relaxation methods and is not a flaw. The paper is also honest about scope: Section 4.3.1 explicitly notes that L2 projection mortars break exact entropy conservation in the AMR convergence study, and the production runs in Sections 5.1, 5.2, and 5.5 use BR1 or shock-capturing semidiscretizations that are not strictly entropy-stable.","major_comments":[],"minor_comments":[{"comment":"The text says 'The values in Table 2 correspond to the run from t = 0 to t = 30 t_c', but the performance comparison appears in Table 3, not Table 2; the cross-reference should be corrected.","section":"Section 5.1.2"},{"comment":"The passage 'we confirmed by performing the same convergence study with the standard P-ERRK schemes' appears to refer to the non-relaxed baseline; it should read 'standard P-ERK schemes'.","section":"Section 4.3.3"},{"comment":"The captions of Figures 10a and 10b refer to 'Fig. 8a' and 'Fig. 8b'; these should refer to Figures 10a and 10b, respectively.","section":"Figure 10"},{"comment":"Several typographical errors should be corrected: 'erronous' in Section 2.4, 'conenctrate' in Section 5.1.2, 'reprensented' in Section 5.5.1, 'svaings' in Section 5.5.2, 'obatined' in the caption of Figure B.19, and 'signficantly'/'constrast' in the caption of Figure 16.","section":"Throughout"},{"comment":"The concluding sentence that P-ERRK methods 'introduce no defects in the global entropy' would be more precise if it explicitly restated the scope condition that this guarantee holds when the underlying semidiscretization is entropy-conservative or entropy-stable; the paper states this condition in the technical sections, but the conclusion is currently slightly broader in wording.","section":"Section 6"}],"recommendation":"minor_revision","confidential_remarks":"The paper is technically sound and the reproducibility repository is a significant strength. The requested changes are limited to editorial corrections and a wording clarification in the conclusion; no technical revisions are needed. The fit with the journal's scope is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid subfield contribution and worth a serious referee. The genuinely new part is pairing the P-ERK multirate framework with relaxation, then optimizing second- through fourth-order P-ERRK families and testing them on DGSEM discretizations of Euler, Navier-Stokes-Fourier, and visco-resistive MHD in 2D and 3D. The closest prior work used a second-order multirate scheme and only Burgers/ODEs; this paper goes well beyond that.\n\nWhat it does well: the convergence studies show designed orders in L1 on non-uniform and randomly partitioned meshes, linear invariants are preserved to round-off, and global entropy stays at machine precision when the spatial discretization is entropy-conservative. The speedup tables count RHS evaluations and wall time, and the reproducibility repository is real. I also believe the time-limiting story: the advection and Kelvin-Helmholtz examples show relaxation suppressing spurious oscillations and even reducing step counts, which makes sense as a nonlinear stabilizer.\n\nSoft spots: the entropy-conservation plots are implementation checks, not independent predictions. Equation (3.3) is solved precisely so the entropy defect vanishes, so the evidentiary weight has to come from the accuracy, invariant, and performance results, and those hold. Second, the fully discrete entropy property is conditional on the semidiscretization being EC/ES. The paper says this plainly, including where it fails (AMR L2 mortars, shock-captured production runs), so it is a scope boundary rather than a hidden flaw. Third, the time-limiting interpretation is empirical; there is no theorem that relaxation will cure partition-interface oscillations. The L∞ degradation in the random-assignment tests is attributed to different error constants across member schemes, and the authors check the same behavior for unrelaxed P-ERK, so that is handled.\n\nBottom line: this is not a paradigm shift, but it is careful, honest, and useful. The methods are implemented, reproducible, and outperform standard relaxed RK by factors of two to four on realistic problems. The citation pattern is fine; self-citations point to the actual P-ERK and relaxation literature. I would cite it, and I would send it to review. The main questions for referees are about the generality of the time-limiting claim and the cost of the relaxation solve in different implementations, not about a load-bearing flaw.","headline":"A careful, reproducible combination of P-ERK with relaxation; the entropy plots are construction checks, but the accuracy, invariant, and speedup claims hold up.","tokens_in":47396,"tokens_out":2863,"would_cite":true,"duration_ms":29392,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65L06","65M20","76-04","70K20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Paired Explicit Relaxation Runge-Kutta methods combine P-ERK multirate pairing with per-step entropy relaxation, yielding explicit high-order integrators that conserve linear invariants and match the semidiscrete entropy without losing…","keywords":["entropy stability","multirate time integration","relaxation Runge-Kutta","paired explicit Runge-Kutta","method of lines","high-order methods","discontinuous Galerkin"],"falsifier":"Run the weak blast wave or isentropic vortex test from the paper with a P-ERRK scheme on an entropy-conservative DGSEM discretization and record $H(U^{n+1}) - H(U^n)$ over many steps; the claim says the defect stays at round-off, so any drift above roughly $10^{-12}$ in double precision would falsify it. Repeating the same run with a non-entropy-stable spatial discretization should show the defect being controlled by the space scheme, not the time integrator.","tokens_in":46459,"feed_emoji":"⚡","tokens_out":4663,"duration_ms":49488,"temperature":0.7,"pith_summary":"This paper develops Paired Explicit Relaxation Runge-Kutta (P-ERRK) methods, which join the P-ERK idea of pairing cheap and expensive Runge-Kutta schemes across mesh partitions with the relaxation trick of rescaling each step to enforce a discrete entropy condition. The paper claims these methods keep the order of accuracy and linear-invariant conservation of the underlying P-ERK schemes, introduce no defect in the global entropy, and outperform standard relaxed Runge-Kutta integration in practice. On compressible Euler, Navier-Stokes, and visco-resistive magnetohydrodynamics test cases, right-hand-side evaluations are reduced by factors up to four and speedups above three are observed. A sympathetic reader cares because this offers a practical way to obtain high-order, entropy-stable, multirate time integration on non-uniform meshes without changing the spatial discretization.","feed_headline":"Multirate Runge-Kutta schemes cut RHS work by up to 4x","feed_subtitle":"High-order entropy-stable multirate methods beat single-rate relaxed Runge-Kutta in runtime by over 3x.","key_machinery":"The central object is the relaxation parameter $\\gamma_n$, a per-step scalar rescaling of the Runge-Kutta update, together with the sparse-weight P-ERK structure that makes computing the entropy change cheap. The relaxed update is $U^{n+1}(\\gamma_n) = U^n + \\gamma_n \\Delta t \\sum_i b_i K_i$, and $\\gamma_n$ is chosen by solving the relaxation equation so that the discrete entropy change matches the semidiscrete one. The P-ERK pairing assigns high-stage schemes with larger stability domains to regions with high characteristic speeds and cheaper schemes to slow regions, while the shared weight vector $b^T$ across partitions guarantees conservation of linear invariants and lets the relaxation parameter act as a time-limiter that improves nonlinear stability.","core_discovery":"On the paper's own terms, the central discovery is that combining Paired Explicit Runge-Kutta methods with relaxation yields a family of explicit multirate integrators that are entropy-conservative when the semidiscretization is entropy-conservative and entropy-stable when it is entropy-dissipative, while preserving design order up to four. The relaxation parameter $\\gamma_n$ solves a scalar nonlinear equation per step so that $H(U^{n+1}(\\gamma_n)) = H(U^n) + \\gamma_n \\Delta H_n$, forcing the fully discrete entropy to follow the semidiscrete entropy tendency. Validation shows entropy defects at round-off for entropy-conservative setups, conserved mass, momentum, and energy, designed convergence orders on non-uniform meshes, and improved robustness in under-resolved and adaptively refined simulations.","pith_inferences":["Editorial extension: the relaxation parameter could serve as a cheap nonlinearity-based indicator for adaptive mesh refinement or shock-capturing activation, since it deviates from unity precisely when the standard scheme's entropy starts changing.","Editorial extension: applying the same relaxation equation to implicit-explicit P-ERK variants, which the paper lists as future work, should yield entropy-stable IMEX methods with the same per-step scalar solve.","Editorial extension: because the cost of computing the entropy correction scales with the number of nonzero entries in the weight vector, future tableau optimization for sparser $b$ would directly reduce overhead in entropy-stable codes."],"forward_implications":["If the central claim is right, entropy-stable multirate time stepping can be dropped into existing entropy-conserving or entropy-stable spatial discretizations without re-deriving the space scheme.","The relaxation mechanism acts as a time-limiter, so under-resolved or AMR-driven simulations run longer before positivity loss, extending the reach of explicit high-order discontinuous Galerkin methods.","Right-hand-side evaluation savings of factors two to four over single-rate relaxed Runge-Kutta methods make multirate integrators attractive for production-scale compressible flow and MHD simulations.","The relaxation parameter can be interpreted as an adaptive timestep controller based on a nonlinear solution functional rather than an embedded error estimate, which suggests a new robustness diagnostic for explicit time integration."],"supporting_citations":[{"why":"Supplies the Paired Explicit Runge-Kutta method framework that pairs schemes with different stability domains and costs.","marker":"[45]"},{"why":"Introduces the relaxation Runge-Kutta mechanism and gives existence, uniqueness, accuracy, and linear-stability results for the relaxation parameter.","marker":"[37]"},{"why":"Extends relaxation to fully discrete explicit entropy-stable schemes for the Euler and Navier-Stokes equations, forming the entropy-stability basis used here.","marker":"[38]"},{"why":"Provides the fourth-order P-ERK tableaus that the fourth-order P-ERRK schemes are built from.","marker":"[47]"},{"why":"Closest prior work extending relaxation to a multirate method; the present paper generalizes it to higher order and quantifies speedups.","marker":"[39]"},{"why":"Supplies the entropy-conserving and kinetic-energy-preserving numerical flux used in the entropy-conservative semidiscretizations.","marker":"[26]"},{"why":"Provides the multirate P-ERK framework with dynamic partitioning on adaptively refined meshes whose robustness the relaxation improves.","marker":"[68]"},{"why":"Supplies the subcell shock-capturing approach used in the production simulations to stabilize under-resolved flows.","marker":"[81]"}],"fun_headline_variants":["Multirate relaxation RK hits 4x speedup with entropy stability","Entropy-stable multirate RK cuts runtime by 4x","High-order entropy-conservative multirate RK up to 4x faster","P-ERRK: entropy-stable multirate up to 4x speedup","Relaxed multirate RK entropy-stable up to 4x faster"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The fully discrete entropy property holds only when the underlying spatial semidiscretization is already entropy-conservative or entropy-stable; the time integrator cannot restore entropy behavior that the space discretization lacks.","fun_headline_variants_meta":{"raw":{"variants":["Multirate relaxation RK hits 4x speedup with entropy stability","Entropy-stable multirate RK cuts runtime by 4x","High-order entropy-conservative multirate RK up to 4x faster","P-ERRK: entropy-stable multirate up to 4x speedup","Relaxed multirate RK entropy-stable up to 4x faster"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001556,"raw_usage":{"total_tokens":6233,"prompt_tokens":976,"completion_tokens":5257,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":592,"completion_tokens_details":{"reasoning_tokens":5153}},"tokens_in":592,"tokens_out":5257,"duration_ms":39134,"temperature":1.0,"reasoning_tokens":5153,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:35:02.665337+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the weak blast wave or isentropic vortex test from the paper with a P-ERRK scheme on an entropy-conservative DGSEM discretization and record $H(U^{n+1}) - H(U^n)$ over many steps; the claim says the defect stays at round-off, so any drift above roughly $10^{-12}$ in double precision would falsify it. Repeating the same run with a non-entropy-stable spatial discretization should show the defect being controlled by the space scheme, not the time integrator.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Paired Explicit Runge-Kutta method framework that pairs schemes with different stability domains and costs."},{"cited_title":"Doehring, L","cited_arxiv_id":null,"evidence_quote":"Provides the fourth-order P-ERK tableaus that the fourth-order P-ERRK schemes are built from."}],"review_version":1}