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If a scaled local error estimator behaves as $E\propto h^r$ with approximately unchanged error scale, the displayed update targets a fixed local level. A safety factor $0<\eta <1$ aims below the acceptance boundary. For an embedded Runge-Kutta pair with order of a numerical method $p$ for the accepted output and $p-1$ for the companion formula, the difference generally has $r=p$; for the actual unscaled local truncation error of an order-$p$ formula, $r=p+1$. The exponent belongs to the quantity being controlled. Dividing the defect by $h$ changes that exponent. This follows by solving $E(h_{\mathrm{new}})\approx E(h)(h_{\mathrm{new}}/h{)}^r$ for the next step.^[ref]

Rejecting a trial means retrying from the same accepted time and state, with a smaller step. Advancing time or retaining the rejected output would change the initial value problem being approximated. Minimum and maximum change factors prevent extreme updates; a zero estimate requires finite capped growth. After acceptance, cap the next step at the remaining interval length. Detect a step that cannot advance time in floating-point arithmetic, and report failure rather than an endless retry. An adaptive step-size controller can use earlier estimates to smooth step sequences, but the memoryless rule here is sufficient to explain acceptance and rejection.^[ref]

A Butcher tableau records a Runge-Kutta method's stage matrix $A$, time nodes $c$, and output weights $b$ in the block arrayRead row $i$ above the line as the recipe for Runge-Kutta stage $i$: the left entry gives $t_n+c_{i}h$, and the right entries give the coefficients in $Y_i=y_n+h{\sum }_j{a}_{ij}{k}_j$. Below the line, $b_i$ weights slope $k_i$ in the final update. Zeros on and above the diagonal identify an Explicit Runge-Kutta method. For the internally consistent convention, row sums of $A$ equal $c$; first-order consistency of a numerical method also requires ${\sum }_i{b}_i=1$. An embedded Runge-Kutta pair adds another weight row while retaining $A,c$.

Two Runge-Kutta methods share a stage matrix and nodes but use different output weights. An order-$p$ and order-$q$ pair with $q<p$ gives an output difference generally of order $h^{q+1}$, estimating the lower-order defect. Shared stage evaluations reduce the cost. The adaptive step-size controller exponent must reflect this estimator order, rather than blindly the higher reported solution order.

Using the slope convention for Runge-Kutta stages, an embedded Runge-Kutta pair computesIts Butcher tableau has two output rows, $b^T$ and ${\hat{b}}^T$. Evaluating $f$ is often the expensive part of solving an ordinary differential equation; after the shared slopes have been computed, the extra row needs only another linear combination.^[ref]

Write $\Phi$ for the exact endpoint from the current numerical state, and define signed local defects ${\ell }_q=\Phi -Y^{[q]}$ and ${\ell }_p=\Phi -Y^{[p]}$. Then $d={\ell }_q-{\ell }_p$. If ${\ell }_q=C_q{h}^{q+1}+O(h^{q+2})$ and ${\ell }_p=O(h^{p+1})$, with $C_q\ne 0$, the difference reproduces the leading lower-order defect. It does not reproduce the leading higher-order defect. Here the powers use big O notation and a smooth Taylor expansion; a vanishing leading coefficient or accidental cancellation can make the estimate misleading.

A common choice accepts $Y^{[p]}$ after testing $d$ with absolute and relative error tolerances. With the scale approximately unchanged on a retry, the adaptive step-size controller uses exponent $1/(q+1)$. Accepting a better endpoint does not change the power in the quantity being tested. The Euler-Heun embedded pair makes this visible without a large coefficient table.^[ref]

Error control selects discretization parameters and checks approximations against requested accuracy. Local error estimators, absolute and relative error tolerances and an adaptive step-size controller provide practical local control. Propagation of defects determines the global error, while stiffness, roundoff error and algebraic-solve accuracy impose additional limits.

For an ordinary differential equation, the practical loop is to propose a step, form a local error estimator, scale it by absolute and relative error tolerances, and accept or retry before choosing the next step. An embedded Runge-Kutta pair supplies two endpoints from shared Runge-Kutta stages, making this extra accuracy check inexpensive. The Euler-Heun embedded pair below exposes the distinction between the accepted formula's order of a numerical method and the estimator's step exponent; its Adaptive Euler-Heun integration of exponential growth shows accepted local tests alongside accumulated global error.

A local error estimator approximates a numerical step's defect from the exact solution starting at the same input. Differences between formulas estimate error only after their asymptotic orders and constants are accounted for. Step-doubling error estimation, an embedded Runge-Kutta pair and predictor-corrector error estimation are common constructions. They do not by themselves bound accumulated global error.

Let ${\Phi }_h(t_n,y_n)$ be the exact endpoint of the ordinary differential equation started at the numerical value $y_n$, and let $Y_{n+1}$ be the candidate numerical endpoint. The signed local defect being estimated is ${\ell }_{n+1}={\Phi }_h(t_n,y_n)-Y_{n+1}$. In contrast, actual global error compares $Y_{n+1}$ with the solution of the original initial value problem, including all previous steps. The local truncation error convention inserts original exact data instead. These starting-value distinctions matter even when the same formula is used for every step. A practical local error estimator also relies on a smooth Taylor expansion; a small difference can reflect accidental cancellation rather than a rigorous upper bound.

A Runge-Kutta method advances an initial value problem $y^{\prime }=f(t,y)$, $y(t_0)=y_0$, by sampling its vector field at several trial states within one time step. These samples are Runge-Kutta stages; a weighted combination of their slopes produces the next state. The construction improves on the single tangent line used by the Euler method while requiring evaluations of $f$, rather than explicit higher derivatives of the solution. It applies to scalar equations and systems of ordinary differential equations alike. The coefficient choices distinguish methods such as Heun's method, the classical fourth-order Runge-Kutta method and implicit Runge-Kutta methods.

For $s$ stages, write $A=(a_{ij})$ for the stage matrix, $c_i$ for relative time nodes and $b_i$ for output weights. A time step of length $h$ computesHere $k_i$ is a slope, with units of state per time; $h{k}_i$ is a state increment. Another common convention writes $K_i=h{k}_i$, so its stage states are $Y_i=y_n+{\sum }_j{a}_{ij}{K}_j$ and its output is $y_n+{\sum }_i{b}_i{K}_i$. These are equivalent conventions: including $h$ both inside $K_i$ and again in the output would count the time step twice. The textbook formulation uses the increment convention.^[ref]

Usually the Butcher tableau obeys $c_i={\sum }_j{a}_{ij}$, called internal consistency. Stage $i$ then combines slopes over an effective time $c_{i}h$. The second-order conditions with independent Runge-Kutta abscissae explain what changes when time nodes are specified independently. A Runge-Kutta stage is generally an auxiliary approximation, not a separate accepted solution value at its node. Several stages can even have the same time node but different state arguments, as in the classical fourth-order Runge-Kutta method.

The useful idea is to approximate the integral $y(t_n+h)-y(t_n)={\int }_{t_n}^{t_n+h}f(t,y(t))dt$ without knowing the intermediate exact trajectory. Trial states provide places to sample $f$, and Butcher order conditions make the resulting combination agree with the exact Taylor expansion. More stages alone do not guarantee a higher order of a numerical method: their coefficients must satisfy the appropriate conditions. Embedded Runge-Kutta pairs reuse these same samples for a second output and a practical local error estimator.