Differenzenapproximationen

Differenzenapproximationen, also known as finite difference approximations, are a fundamental technique in numerical analysis used to approximate derivatives of a function. These methods replace the continuous derivative with a difference quotient, which can be computed from function values at discrete points. The most basic forms are the forward difference, backward difference, and central difference.

The forward difference approximation for the first derivative of a function f(x) at a point x is given by (f(x+h) - f(x)) / h, where h is a small step size. The backward difference is (f(x) - f(x-h)) / h. The central difference, often more accurate, is (f(x+h) - f(x-h)) / (2h). These approximations are derived from Taylor series expansions of the function.

Higher-order derivatives can also be approximated using combinations of function values at different points. The accuracy of these approximations depends on the step size h and the order of the approximation. As h approaches zero, the difference approximation converges to the true derivative. However, choosing a very small h can lead to round-off errors due to the limited precision of computer arithmetic. Differenzenapproximationen are widely used in solving differential equations numerically, such as in computational fluid dynamics and heat transfer simulations.