Math Conjectures

Twelve-stage ninth-order explicit Runge–Kutta method

OPENMajorCanonical finite caseProposed c. 1964 · Canonical special case

Canonical statement

Determine whether there exist real coefficients aija_{ij} (1≤j<i≤121\le j<i\le12) and bib_i (1≤i≤121\le i\le12), with ci=∑j<iaijc_i=\sum_{j<i}a_{ij}, such that the explicit Runge--Kutta one-step map
ki=f ⁣(t+cih, y+h∑j<iaijkj),y+=y+h∑i=112biki k_i=f\!\left(t+c_i h,\ y+h\sum_{j<i}a_{ij}k_j\right),\qquad y^+=y+h\sum_{i=1}^{12}b_i k_i
has classical order 99: for every integer q≥1q\ge1, every sufficiently smooth f:R×Rq→Rqf:\mathbb R\times\mathbb R^q\to\mathbb R^q, and every exact solution of y′=f(t,y)y'=f(t,y), one step initialized at y(t)y(t) satisfies y+−y(t+h)=O(h10)y^+-y(t+h)=O(h^{10}) as h→0h\to0.
View source LaTeX
Determine whether there exist real coefficients \(a_{ij}\) (\(1\le j<i\le12\)) and \(b_i\) (\(1\le i\le12\)), with \(c_i=\sum_{j<i}a_{ij}\), such that the explicit Runge--Kutta one-step map
\[
k_i=f\!\left(t+c_i h,\ y+h\sum_{j<i}a_{ij}k_j\right),\qquad y^+=y+h\sum_{i=1}^{12}b_i k_i
\]
has classical order \(9\): for every integer \(q\ge1\), every sufficiently smooth \(f:\mathbb R\times\mathbb R^q\to\mathbb R^q\), and every exact solution of \(y'=f(t,y)\), one step initialized at \(y(t)\) satisfies \(y^+-y(t+h)=O(h^{10})\) as \(h\to0\).
NUM-001:[Butcher1964Processes][Khashin2010Symbolic][Verner2014LowerStage][Butcher2016NumericalMethods]
Numerical analysisAlgebraic geometry

Notes

Explicit Runge–Kutta methods advance the solution of an ordinary differential equation one step using ss stage evaluations of the right-hand side, and a classical question, going back to Butcher's systematic study of high-order processes in the 1960s, asks for the minimal number of stages needed to attain each order [Butcher1964Processes]. This problem is the first unresolved finite case: do real coefficients exist making a twelve-stage explicit method of classical order nine?

The order conditions form a large structured system of polynomial equations indexed by rooted trees — several hundred conditions through order nine — so the question amounts to real solvability of that system [Butcher2016NumericalMethods]. Minimal stage counts are settled through order eight, where eleven stages are necessary and sufficient. For order nine the rigorous lower bound does not exclude twelve stages; a thirteen-stage candidate has been reported via symbolic-numeric solution of the Butcher equations [Khashin2010Symbolic], and exact order-nine methods with fifteen stages are known [Verner2014LowerStage].

The twelve-stage case is undecided in either direction: no solution of the order-nine conditions with twelve stages has been produced, and no proof of nonexistence is known.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.