Pith Number

pith:C4WFFQB3

pith:2026:C4WFFQB3LA2UGTMRNZ4B6RMMFD

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Low Stage High Order Explicit Runge--Kutta Methods via Q- and D-Conditions: General Theory and Efficient Recursive Construction

Jizu Huang, Junyuan He

A Q/D-space reformulation of order conditions yields explicit Runge-Kutta methods of even order p with stage count (p squared minus 2p plus 8) over 4.

arxiv:2605.16995 v1 · 2026-05-16 · math.NA · cs.NA

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Claims

C1strongest claim

For every even order p ≥ 4 the construction produces explicit Runge-Kutta methods with stage number s(p) = (p² - 2p + 8)/4; the Q/D conditions together with B(p) are sufficient to guarantee order p.

C2weakest assumption

The Q- and D-space residual conditions remain linearly independent and the two structured linear systems at each recursive step remain solvable for the chosen stage count; this is invoked when the authors state that the Butcher coefficients are obtained from the two linear systems (abstract and §3-4).

C3one line summary

A Q/D-space reformulation of Butcher simplifying assumptions yields sufficient order conditions and a recursive linear-system construction for explicit Runge-Kutta methods of even order p with s(p)=(p²-2p+8)/4 stages.

References

38 extracted · 38 resolved · 2 Pith anchors

[1] Tensor networks and hi- erarchical tensors for the solution of high-dimensional partial diﬀerential equations 2016

[2] Counting Rooted Trees: The Universal Law t(n) ∼ Cρ −nn−3/2 2006

[3] A 3(2) pair of Runge-Kutta formulas 1989

[4] Coeﬃcients for the study of Runge-Kutta integration processes 1963

[5] Implicit runge-kutta processes 1964

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:03:35.030479Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

172c52c03b5835434d916e781f458c28c1adfdb856e7e417c9b813b8c21ff9c7

Aliases

arxiv: 2605.16995 · arxiv_version: 2605.16995v1 · doi: 10.48550/arxiv.2605.16995 · pith_short_12: C4WFFQB3LA2U · pith_short_16: C4WFFQB3LA2UGTMR · pith_short_8: C4WFFQB3

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/C4WFFQB3LA2UGTMRNZ4B6RMMFD \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 172c52c03b5835434d916e781f458c28c1adfdb856e7e417c9b813b8c21ff9c7

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "2a84312a414541211905507e3f056935936770ce587ba254f6110f374e3e66a6",
    "cross_cats_sorted": [
      "cs.NA"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.NA",
    "submitted_at": "2026-05-16T13:43:57Z",
    "title_canon_sha256": "3dae47c10513ca7b75122c5fd34c682a5b0473f0f4118990ac3811a7c9175355"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.16995",
    "kind": "arxiv",
    "version": 1
  }
}