Predictorcorrectorvarianten

Predictor-corrector variants are numerical methods used to solve ordinary differential equations by combining a predictor step that estimates the solution at a new time point with a corrector step that refines that estimate. They are commonly applied to initial value problems and can be formulated as explicit, implicit, or semi-implicit schemes. A typical multi-step predictor-corrector pair uses information from previous steps to predict y_{n+1}, then solves an implicit or semi-implicit equation to obtain a more accurate y_{n+1}. In variable-step or adaptive schemes, the process may include several evaluations of the right-hand side and multiple corrective iterations.

One prominent family of predictor-corrector methods is the Adams-Bashforth–Moulton (AB–M) pair. Adams-Bashforth provides an explicit predictor, while Adams-Moulton provides an implicit corrector. By combining these two, one can achieve higher-order accuracy while preserving relatively lower computational cost per step. Other variants pair a simple explicit predictor, such as Euler or Runge-Kutta-inspired predictions, with a trapezoidal or higher-order implicit corrector. In practice, several variants are described by the PECE (Predict-Evaluate-Correct-Evaluate) or PEC (Predict-Evaluate-Correct) patterns, which emphasize the sequence of prediction, derivative evaluation, correction, and possible re-evaluation.

Variants differ in order of accuracy, stability properties, and computational cost. Implicit correctors are favored for stiff problems, often requiring a Newton iteration to solve the implicit equation at each step. Predictor accuracy, the number of corrector iterations, and step-size control all influence performance and stability. Applications span a wide range of ODEs and DAEs in science and engineering, where flexible accuracy and adaptivity are advantageous.