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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.

The Euler-Heun 1(2) pair combines the explicit Euler method and Heun's method. Here 1(2) names their respective order of a numerical method; the accepted endpoint in this article is the second-order one. Its shared Runge-Kutta stages and endpoints areThe first output row in this Butcher tableau is Heun's method; the second is the explicit Euler method. Both outputs cost two evaluations of $f$ together, rather than one for Euler method plus two for a separately computed Heun's method.

For the initial value problem $y^{\prime }=y$, $y(0)=1$, take $h=0.2$. The first slope is $k_1=1$; the stage state is $1+0.2k_1=1.2$, giving $k_2=1.2$. ThereforeThe exact exponential function gives signed local defects approximately $0.02140275816$ for the explicit Euler method and $0.00140275816$ for Heun's method. Their difference is exactly $0.02$. Thus the pair difference estimates the larger Euler defect; calling it the actual Heun defect would miss by a factor about fourteen here.

More generally, a step from $y_n$ on $y^{\prime }=y$ hasThese original Taylor expansions separate three quantities: the accepted Heun's method has local truncation error $O(h^3)$ and global error $O(h^2)$ under suitable numerical stability, while the tested local error estimator is $O(h^2)$. For $E=|d|/s$ with $s$ held approximately fixed, use $h_{\mathrm{new}}=\eta h{E}^{-1/2}$. A cube-root update would model a different quantity, the actual higher-order local defect.

An explicit Runge-Kutta method has $a_{ij}=0$ whenever $j\ge i$. Its stage matrix is strictly lower triangular: compute each Runge-Kutta stage using already known slopes, starting with $Y_1=y_n$. In contrast, an implicit Runge-Kutta method requires solving coupled stage equations, whose existence is addressed by stage solvability of an implicit Runge-Kutta method.

Here Heun's method means the explicit endpoint-averaging Runge-Kutta method: take an Euler predictor $Y_2=y_n+hf(t_n,y_n)$, sample $k_2=f(t_n+h,Y_2)$ there, and average it with $k_1=f(t_n,y_n)$ for the output. The second Runge-Kutta stage is evaluated at the predictor, not at the corrected endpoint. In particular it requires no implicit solve as the ODE trapezoidal rule does. Naming conventions vary between books; the formulas specify the method unambiguously.^[ref]

For smooth $f$, a Taylor expansion about $(t_n,y_n)$ gives $k_2=f+h(f_t+f_{y}f)+O(h^2)$, where $f_y$ is the state Jacobian matrix. Consequently the output is $y_n+hf+h^2(f_t+f_{y}f)/2+O(h^3)$, matching the exact solution through degree two by the chain rule. This gives local truncation error $O(h^3)$ and, with the usual bounded smoothness and Lipschitz continuity hypotheses on a fixed finite interval, global error $O(h^2)$. The Euler-Heun embedded pair compares this output with its Euler predictor.

For $y^{\prime }=y$, $y_n=1$ and $h=0.2$, the sampled slopes are $1$ and $1.2$. Transporting the second slope to the starting point would end at $1.24$; the average slope $1.1$ instead ends at $1.22$. The exact exponential function reaches $e^{0.2}\approx 1.22140275816$. The magnified endpoint comparison separates the pair difference $0.02$ from the actual Heun local defect $0.00140275816$.

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.