Runge–Kutta order conditions

An s-stage method is specified by a matrix A, weights b, and time coordinates c. Its stage derivatives evaluate f at a shifted time and a state assembled from other stage derivatives. A strictly lower-triangular A makes the method explicit, because stage i uses only earlier stages. An implicit tableau couples unknown stages and requires an algebraic solve. Here we assume the usual internal consistency condition $c=A\mathbf1$, so the time coordinates agree with the stage row sums.

The final weights must sum to one to reproduce a constant vector field. Matching the next derivative introduces the weighted average of the stage times. This is already enough to rule out many plausible-looking formulas. A two-stage predictor-corrector that averages the starting slope and an Euler-predicted endpoint slope satisfies the first two conditions. Its order is a statement about smooth general differential equations with those coefficients, not merely about the number of samples in the average.

For an autonomous equation, Taylor expansion first produces the vector field, then its Jacobian applied to itself, then two distinct third-order expressions: a second derivative acting on two copies of the field, and a Jacobian applied twice in a chain. A method must reproduce both third-order coefficients. The first is governed by squared stage row sums, while the second depends on the matrix A acting on c. Treating all third-order information as a single scalar coefficient misses that distinction.

With $c=A\mathbf1$, the conditions displayed below are necessary and sufficient for classical order at least three on sufficiently smooth general systems, including nonautonomous systems represented by adjoining the time variable. The square $c^{\circ2}$ is componentwise. These are not the complete conditions for fourth or higher order, where additional derivative patterns occur. Rooted trees organize those patterns systematically, but the low-order equations can be checked directly by expanding the stages and matching derivatives.

The scalar linear equation reproduces the chain-shaped derivative terms but has no second derivative of the vector field. A method whose stability polynomial agrees with the exponential through a certain power can therefore still fail a branching nonlinear order condition at that order. Conversely a time-only equation tests moment conditions on the quadrature weights but does not test the full stage coupling. A meaningful verification suite needs problems that expose both time dependence and nonlinear state dependence.

Order conditions are algebraic, while usable accuracy also depends on error constants, stage cost, stability, and the smoothness of the actual trajectory. Two second-order methods can agree on every scalar linear problem and differ on a nonlinear problem. This is a reason to inspect a method's full tableau rather than identify it only by its scalar amplification factor. At discontinuities or near singularities, even a correctly designed high-order tableau may not exhibit its nominal convergence rate until the integration is appropriately segmented.