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arxiv: 1907.08872 · v1 · pith:SOSDYDMDnew · submitted 2019-07-20 · 🧮 math.NA · cs.NA

A simplified Cauchy-Kowalewskaya procedure for the implicit solution of generalized Riemann problems of hyperbolic balance laws

Gino I. Montecinos , Dinshaw S. Balsara This is my paper

Pith reviewed 2026-05-24 18:30 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Cauchy-Kowalewskaya proceduregeneralized Riemann problemhyperbolic balance lawsimplicit Taylor seriesADER schemenumerical methods for PDEs
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The pith

A recursive formula simplifies the Cauchy-Kowalewskaya procedure for implicit GRP solvers in hyperbolic balance laws.

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

The paper develops a recursive formula that replaces the standard Cauchy-Kowalewskaya procedure when constructing implicit Taylor series expansions for generalized Riemann problems. This change lets time derivatives be obtained from space derivatives through direct code implementation rather than symbolic manipulation. The resulting GRP solver is embedded in an ADER framework for hyperbolic balance laws and tested on one-dimensional problems. The approach reduces computational effort while the numerical solutions attain the design orders of accuracy.

Core claim

A recursive formula for the Cauchy-Kowalewskaya procedure is derived within the implicit Taylor series GRP framework for hyperbolic balance laws; the formula is straightforward to implement in computational codes and produces an efficiency gain while preserving the expected theoretical orders of accuracy.

What carries the argument

The recursive formula that replaces the conventional Cauchy-Kowalewskaya procedure inside the implicit Taylor series expansion for GRP.

If this is right

  • The GRP solver integrates directly into ADER schemes for hyperbolic balance laws.
  • One-dimensional numerical tests confirm that theoretical accuracy orders are attained.
  • Direct code implementation removes the need for external symbolic software during the CK step.

Where Pith is reading between the lines

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

  • The recursion may extend to multi-dimensional balance laws if the underlying differential relations remain structurally similar.
  • Wider use could lower the barrier to high-order implicit GRP methods by eliminating dependence on computer algebra tools.

Load-bearing premise

The recursive simplification leaves the truncation errors and stability properties of the original implicit Taylor series GRP framework unchanged.

What would settle it

A convergence test on a smooth exact solution in which the observed order falls below the design order for the chosen polynomial degree would show that the recursion introduces accuracy loss.

Figures

Figures reproduced from arXiv: 1907.08872 by Dinshaw S. Balsara, Gino I. Montecinos.

Figure 1
Figure 1. Figure 1: Sketch of the space-time node distribution. [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Leveque and Yee test. We have used 300 cells, [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The Shu-Osher test. Otput time tout = 0.47, 300 cells and CF L = 0.5. 5. Conclusions In this paper, a simplified Cauchy-Kowalewkaya procedure has been pro￾posed. The strategy uses not only the spatial derivatives of the data but also de derivatives of the Jacobian matrices in both space and time. The simplification allows us to propose a recursive formula which requires the ability of obtaining time and sp… view at source ↗
read the original abstract

The Cauchy-Kowalewskaya (CK) procedure is a key building block in the design of solvers for the Generalised Rieman Problem (GRP) based on Taylor series expansions in time. The CK procedure allows us to express time derivatives in terms of purely space derivatives. This is a very cumbersome procedure, which often requires the use of software manipulators. In this paper, a simplification of the CK procedure is proposed in the context of implicit Taylor series expansion for GRP, for hyperbolic balance laws in the framework of [Journal of Computational Physics 303 (2015) 146-172]. A recursive formula for the CK procedure, which is straightforwardly implemented in computational codes, is obtained. The proposed GRP solver is used in the context of the ADER approach and several one-dimensional problems are solved to demonstrate the applicability and efficiency of the present scheme. An enhancement in terms of efficiency, is obtained. Furthermore, the expected theoretical orders of accuracy are achieved, conciliating accuracy and stability.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a recursive simplification of the Cauchy-Kowalewskaya (CK) procedure for implicit Taylor-series expansions in generalized Riemann problem (GRP) solvers for hyperbolic balance laws, extending the 2015 framework. The recursion is claimed to be straightforward to implement in codes, yielding efficiency gains in ADER schemes while preserving the expected theoretical orders of accuracy, as verified on several one-dimensional test problems.

Significance. If the recursive formula is algebraically equivalent to the original implicit CK procedure (including source terms), the work would provide a practical efficiency improvement for high-order implicit GRP/ADER solvers without altering accuracy or stability properties. The numerical tests recovering design orders are a positive indicator, but the absence of a formal equivalence proof limits the strength of the central claim.

major comments (2)
  1. [derivation of recursive formula] The derivation of the recursive CK formula (main result section following the 2015 reference) does not include a formal inductive proof or algebraic verification that the recursion produces identical time-derivative expressions to the original non-recursive implicit CK procedure, especially when source terms are present and the implicit treatment is active. This equivalence is load-bearing for the claim that accuracy and stability are preserved.
  2. [numerical experiments] Numerical experiments (test problems section) demonstrate recovery of expected orders but do not isolate or test the implicit source-term contributions separately; any hidden truncation introduced by the recursion would not necessarily be detected by standard order-of-accuracy checks on smooth problems.
minor comments (2)
  1. [abstract] The abstract and introduction should explicitly state the precise 2015 reference (Journal of Computational Physics 303 (2015) 146-172) when describing the baseline framework.
  2. [method section] Notation for the recursive coefficients and the implicit time-stepping operator should be introduced with a clear table or diagram to aid implementation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the positive assessment of the practical benefits of the recursive formula. We address each major comment below.

read point-by-point responses

  1. Referee: [derivation of recursive formula] The derivation of the recursive CK formula (main result section following the 2015 reference) does not include a formal inductive proof or algebraic verification that the recursion produces identical time-derivative expressions to the original non-recursive implicit CK procedure, especially when source terms are present and the implicit treatment is active. This equivalence is load-bearing for the claim that accuracy and stability are preserved.

    Authors: We agree that an explicit inductive proof would strengthen the central claim. The recursion is obtained by differentiating the balance law, solving for the highest time derivative, and substituting the lower-order terms recursively, which by construction mirrors the original implicit CK steps. In the revised manuscript we will insert a dedicated subsection containing a formal inductive proof of equivalence that explicitly covers the implicit source-term contributions. revision: yes

  2. Referee: [numerical experiments] Numerical experiments (test problems section) demonstrate recovery of expected orders but do not isolate or test the implicit source-term contributions separately; any hidden truncation introduced by the recursion would not necessarily be detected by standard order-of-accuracy checks on smooth problems.

    Authors: The reported tests already include hyperbolic balance laws with active source terms, and the observed convergence rates match the theoretical orders. Nevertheless, to isolate the implicit source treatment we will add one additional manufactured-solution test focused on a linear source term in the revised version, allowing direct verification that the recursion introduces no extra truncation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; recursive CK formula is algebraic simplification on 2015 base

full rationale

The paper derives a recursive formula for the CK procedure as an algebraic simplification within the implicit Taylor-series GRP framework of the 2015 JCP reference. Validation occurs via 1D numerical tests recovering expected orders of accuracy. The 2015 citation supplies the original implicit procedure but does not bear the load of the recursion itself; the new formula is presented as an independent implementation aid rather than a redefinition or fitted input. No self-definitional reduction, fitted prediction, or ansatz smuggling is exhibited. The absence of a formal equivalence proof between recursive and non-recursive forms is a completeness issue, not a circularity reduction. This is a normal non-finding for a methods paper whose central claim is the simplification and its efficiency gain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the algebraic validity of the recursive CK transformation and on the stability properties inherited from the 2015 implicit framework. No free parameters, invented entities, or ad-hoc axioms are mentioned in the abstract.

axioms (1)
  • domain assumption The implicit Taylor series expansion framework of the 2015 JCP paper remains valid when the CK procedure is replaced by the proposed recursion.
    The abstract positions the new procedure inside that prior framework without re-deriving its stability or consistency.

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Reference graph

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