= Error control for ODE solvers
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>.
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