8. Concepts

8.1 Numerical Path Continuation

Numerical path continuation (also called homotopy continuation) is used to track the solutions $u$ of a nonlinear system $f(u, p) = 0$ as a parameter $p$ is varied.

8.1.1 Pseudo-Arclength Continuation

In pseudo-arclength continuation, an additional arclength constraint is added to the system, allowing for the tracking of solution branches around limit points (folds) where natural continuation fails.

8.2 Discretization Schemes

• Finite Differences: $d^n u / dx^n$ is approximated using grid points and stencils.

• Pseudospectral Methods: (Reserved for future implementation).