M. Abdelbagi; A. Laadhari.

Cubic-Convergent Newton-Type Methods for Nonlinear Differential Equations: Applications to Navier–Stokes via Finite Elements. Journal of Physics: Conference Series. IOP Publishing.

Accepted.  1-12 (2025).

DOI 10.100/…

We investigate cubic-convergent Newton-type methods for nonlinear differential problems, focusing on the variants of Kou, Homeier, and Weerakoon and their multidimensional formulations. The schemes are validated on scalar roots, a system of nonlinear equations, an initial value problem, and then applied to the steady incompressible Navier–Stokes problem. Across all tests, cubic methods reach the target tolerance in fewer iterations than classical Newton, with measured orders confirming third- versus second-order behavior. Notably, only Kou’s method reuses a single Jacobian per outer iteration, reducing assembly and linear-solve cost while retaining cubic convergence; the other cubic variants require additional Jacobian evaluations or factorizations. Overall, the results highlight the practical efficiency of high-order Newton-type strategies, with Kou’s method delivering the best performance in our experiments.