Newton-Krylov Methods
Newton-Krylov methods solve large nonlinear systems by combining Newton linearization with Krylov iterative solution of each linear step.
Nesting two iterations
Implicit time integration and steady-state solves require solving large nonlinear algebraic systems F(u) = 0. Newton's method linearizes: given a current guess, it solves the Jacobian system J*delta = -F for a correction and updates the guess, repeating until the residual is small. Each Newton step demands the solution of a large sparse linear system. Newton-Krylov methods use a Krylov iterative solver (typically GMRES) for that inner solve, giving a nested outer-Newton, inner-Krylov structure.
This pairing is powerful because Krylov solvers scale well to very large sparse systems and, crucially, need only the action of the Jacobian on vectors, not the full matrix, which opens the door to matrix-free implementation.
Inexact Newton
Solving each Krylov system to full precision is wasteful when the Newton iterate is still far from the solution. Inexact Newton methods solve the linear system only approximately, tightening the tolerance as the outer iteration converges. Well-chosen forcing terms preserve fast (superlinear or quadratic) Newton convergence while minimizing inner iterations, a major efficiency gain.
Globalization and preconditioning
Pure Newton converges quadratically only near the solution; far away it can diverge. Line searches and trust-region strategies globalize the method, guaranteeing progress from poor initial guesses. Because Krylov convergence depends on the Jacobian's spectrum, a good preconditioner is essential and often the dominant factor in overall performance.
- Outer Newton linearization, inner Krylov linear solve
- Krylov solvers need only Jacobian-vector products
- Inexact Newton adapts inner tolerance to outer progress
- Line search or trust region for robust global convergence
Newton-Krylov solvers, especially in Jacobian-free form, are the standard engine for implicit transport, extended-MHD, and coupled multi-physics fusion simulations.