Implicit solvers and Newton tolerances

Backward Euler seeks a value Z satisfying $G(Z)=Z-Y_n-hf(t_{n+1},Z)=0$. Newton's method linearizes G at a current guess, solves for a correction, and updates that guess. The exact Newton matrix is identity minus h times the state Jacobian of f. A small h often makes this matrix close to identity, but stiff rates, couplings, and scaling can change its conditioning. The mathematical implicit formula does not specify how many iterations are sufficient.

A predictor from previous accepted values can provide a good initial guess, but Newton convergence remains local without additional safeguards. A line search or damping can prevent a bad full correction from leaving the useful region. Reusing a factorization gives a modified Newton method, which can save work when the Jacobian changes slowly but can converge more slowly. Failure to reduce corrections or residuals may require refreshing the Jacobian, improving the linear solve, rejecting the step, or reducing h.

Near a regular root, first-order perturbation analysis relates the state error to the inverse residual Jacobian applied to the residual. A small raw residual can correspond to a large state error if that inverse is large. Multiplying the entire equation by a small constant also makes its residual small without changing the root. Residual tolerances must therefore use a meaningful scaling and an estimate of how residual error affects the state, rather than an arbitrary numerical threshold.

Correction-based stopping can estimate the remaining error when the iterations have entered a predictable contraction regime. But a small step in an iteration is not always proof of a small error: damping, stagnation, or a poor approximate Jacobian can make corrections tiny. Use both progress and a residual or correction criterion tied to the integration scales. In a scalar problem a derivative bounded away from zero can give a rigorous local error bound by the mean-value theorem; in a system one needs an appropriate inverse or nonlinear regularity bound.

An order-p one-step formula has an ideal local defect of order h to the p plus one. A sufficient fixed-interval strategy is to keep each algebraic state error at that scale or below, so its propagated contribution does not spoil the intended global order under the same stability assumptions. A fixed absolute nonlinear tolerance may dominate as h shrinks, causing a refinement plateau. Adaptive codes usually tie nonlinear error to the current componentwise integration weights and reserve a fraction of the allowed local estimate.

There is another nested solve inside Newton when a Krylov method approximates the correction. Its linear residual should be small enough that the resulting correction error does not obstruct nonlinear convergence, but high accuracy in early nonlinear iterations may be unnecessary. Preconditioners affect that trade and must account for the step-dependent matrix. Track nonlinear iterations, linear iterations, rejected trials, and Jacobian updates together; a solver that takes few accepted steps can still be expensive if each step hides a difficult algebraic problem.