{
  "abstract": "We develop controlled B-series and controlled Runge-Kutta methods as extensions of classical B-series and Runge-Kutta theory to controlled-driven rough differential equations. The driving signal is a path controlled by an underlying $α$-Hölder step-$2$ rough path, with $α\\in(1/3,1/2]$, as arises when the output of one rough system drives another. Controlled B-series retain the classical rooted-tree and elementary-differential structure, while incorporating driver-dependent coefficients defined recursively through controlled rough integration. We establish finite-order expansions of the exact solution and its controlled Runge-Kutta approximation, and derive corresponding tree-based order conditions. Under suitable smoothness, solvability, stability, and boundedness assumptions, matching through tree order $p$ yields local error $O(|t-s|^{(p+1)α})$ and global error $O(h^{(p+1)α-1})$, provided $(p+1)α>1$. The standard rough differential equation formulation is recovered when the controlled driver is the reference path itself, and the classical time-driven case recovers the usual B-series and Runge-Kutta methods. We further construct a simplified method using only increments of the controlled driver. Segmentwise canonical lifting of its piecewise linear interpolation gives explicit tree coefficients and reduces the third-order conditions to the classical ones. Assuming a Wong-Zakai solution-approximation rate $r_0>0$, the simplified method has global error $O(h^{\\min\\{r_0,(p+1)α-1\\}})$. Numerical experiments with fractional Brownian controlled drivers illustrate the resulting convergence behaviour.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14196",
  "author_total": 3,
  "authors": [
    {
      "name": "Xinyuan An"
    },
    {
      "name": "Xingya Fan"
    },
    {
      "name": "Xing Gao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b9bgyknp0bkjx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14196",
  "fulltext_url": null,
  "id": "b9bgyknp0bkjx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14196",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2612|Numerical methods for differential equations"
    ]
  ],
  "openalex": "W7213290394",
  "pdf_url": "https://arxiv.org/pdf/2609.14196",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b9bgyknp0bkjx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14196",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14196"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Controlled B-series, controlled Runge-Kutta methods and controlled rough differential equations",
  "title_text": "Controlled B-series, controlled Runge-Kutta methods and controlled rough differential equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/b9bgyknp0bkjx",
  "venue": "arXiv",
  "versions": []
}