Newton-Krylov Methods
Solving large nonlinear systems by combining Newton's method with a Krylov solver for the linear step, avoiding explicit Jacobians.
Nonlinear meets iterative
Newton's method solves a nonlinear system by repeatedly linearizing: at each iterate it forms the Jacobian (the matrix of partial derivatives), solves a linear system for the update, and steps. For large problems, two costs dominate: forming the Jacobian and solving the linear system. Newton-Krylov methods address both by solving the Newton linear system with a Krylov method and, crucially, never forming the Jacobian explicitly.
Jacobian-free Krylov
Krylov solvers need only the product of the Jacobian with a vector, not the Jacobian itself. That product is exactly a directional derivative of the residual, which can be approximated by a finite difference of the nonlinear function, or computed exactly by forward-mode automatic differentiation. This Jacobian-free Newton-Krylov (JFNK) approach means the entire nonlinear solve requires only evaluations of the residual function, sidestepping the storage and assembly of a large Jacobian matrix.
- Outer loop: Newton iterations reduce the nonlinear residual
- Inner loop: a Krylov method (usually GMRES) solves each linear Newton step
- Jacobian-vector products come from finite differences or automatic differentiation
- Inexact Newton: solve the inner system only loosely when far from the solution
The role of preconditioning
The inner Krylov solve still needs a good preconditioner to converge quickly, and this is where problem knowledge enters. Physics-based preconditioners approximate the dominant linear couplings cheaply, and are what make JFNK practical for stiff multiphysics. Choosing how loosely to solve the inner system (the forcing term) balances wasted inner iterations against extra outer Newton steps.
Where it is used
Newton-Krylov methods are the standard for fully implicit solution of large nonlinear PDE systems: implicit MHD, reacting flows, and coupled transport. They allow large time steps free of the CFL limit while handling strong nonlinearity, at the price of a well-engineered preconditioner. They are a cornerstone of modern implicit fusion plasma simulation.