Schrittwechselmethoden

Schrittwechselmethoden, also known as step-changing methods or adaptive step-size methods, are a class of numerical algorithms used to solve differential equations, particularly initial value problems. The core idea behind these methods is to dynamically adjust the step size during the integration process. This adjustment is based on an estimation of the local error at each step. If the estimated error is too large, the step size is reduced to improve accuracy. Conversely, if the estimated error is sufficiently small, the step size can be increased to speed up the computation.

The primary advantage of Schrittwechselmethoden is their efficiency. By taking smaller steps only when necessary, they can achieve a desired level of accuracy with fewer computational operations compared to methods that use a fixed step size. This is particularly beneficial when solving problems with rapidly changing solutions or regions where high accuracy is required.

A common way to implement Schrittwechselmethoden is by using embedded pairs of Runge-Kutta formulas, such as the Fehlberg or Dormand-Prince pairs. These pairs compute two approximations of the solution at each step using formulas of different orders. The difference between these two approximations provides an estimate of the local truncation error. This error estimate is then compared against a user-defined tolerance to determine whether to accept the step and adjust the step size for the next iteration.

Schrittwechselmethoden are widely employed in various scientific and engineering disciplines, including physics, chemistry, and control theory, where accurate and efficient numerical solutions to differential equations are crucial.