NewtonKrylovVerfahren

NewtonKrylovVerfahren is a numerical method used to solve nonlinear systems of equations. It combines the Newton-Raphson method with Krylov subspace methods to efficiently solve large-scale problems. The Newton-Raphson method is an iterative technique that linearizes the nonlinear system at each iteration, while Krylov subspace methods are used to solve the resulting linear systems.

The Newton-Krylov method is particularly useful for problems where the Jacobian matrix (the matrix of partial derivatives of the nonlinear system) is large and sparse. In such cases, direct methods for solving the linear systems may be computationally expensive or infeasible. Krylov subspace methods, such as the Generalized Minimal Residual (GMRES) method or the Bi-Conjugate Gradient Stabilized (BiCGSTAB) method, can be used to iteratively solve the linear systems without explicitly forming the Jacobian matrix.

The Newton-Krylov method has been successfully applied to a wide range of problems in science and engineering, including fluid dynamics, structural mechanics, and chemical engineering. It is a powerful tool for solving nonlinear systems of equations, especially for large-scale problems where traditional methods may fail. However, it requires careful implementation and tuning to achieve optimal performance.