finiteDifferenzenMethoden

FiniteDifferenzenMethoden, often abbreviated as FDM, represent a numerical technique used to approximate solutions to differential equations. The core idea is to replace continuous derivatives with discrete differences, effectively transforming an infinite number of unknown values into a finite number. This is achieved by discretizing the domain of the differential equation into a grid of points. At each grid point, the derivative is approximated by a formula involving the function's values at neighboring points. Common approximations include forward difference, backward difference, and central difference formulas. The choice of approximation impacts the accuracy and stability of the method. Once the differential equation is converted into a system of algebraic equations, these can be solved using standard numerical linear algebra techniques. FDM is widely applied in various fields, including fluid dynamics, heat transfer, and electromagnetics, due to its conceptual simplicity and ease of implementation for many problems. However, its application can become complex for irregular geometries or when high-order accuracy is required, where alternative methods like the Finite-ElementenMethoden might be preferred.