Time stepping, stiffness, and stability regions

Forward Euler evaluates the derivative at the current state and follows its tangent for one step. If the exact solution is smooth, the one-step defect in the state is of order h squared; under appropriate Lipschitz and finite-time assumptions, accumulated global error is first order. Some texts divide the local defect by h when defining local truncation error, changing the stated order by one. Always check the convention instead of comparing order labels without their definitions.

For the linear test equation y′=λy, Euler multiplies the current state by one plus hλ. Absolute stability asks when repeated updates do not amplify a decaying mode without bound. Strict damping requires that multiplier to have magnitude below one. This is a statement about fixed step size and repeated steps, distinct from convergence as h approaches zero at a fixed final time. Even inside the stable region, the method may approximate decay rates or phase very poorly.

Backward Euler evaluates the derivative at the unknown next state. For the linear test equation, rearrangement gives an amplification factor equal to one divided by one minus hλ. Every finite mode with negative real λ is strictly damped for positive h. This method is A-stable and also strongly damps modes far into the left half-plane. Such damping is useful for stiff decay but can distort oscillatory or conservative dynamics, where preserving phase or energy structure may be important.

For a nonlinear system, an implicit step requires solving nonlinear equations, often with Newton iteration and an inner linear solver. Their tolerances contribute to the time integration error and should be coordinated with the requested accuracy. A-stability does not make a nonlinear solve infallible, remove discretization error, or permit arbitrary large steps for a faithful trajectory. Other methods trade off order, stability regions, conservation, and cost; the best choice depends on what the solution must preserve.

A system is often called stiff when stability forces an explicit method to resolve fast decaying scales much more finely than needed to follow the slowly changing quantity of interest. The fast mode may already be nearly absent physically, yet an unstable update can resurrect numerical perturbations in that direction. Refining a spatial mesh for a diffusion problem creates increasingly fast modes, tightening the explicit step restriction. The source is the combined spatial and temporal discretization, not merely a badly chosen initial condition.

Adaptive methods estimate local error and adjust step size, but a trustworthy calculation also monitors rejected steps, solver failures, and physical diagnostics. Compare solutions at matched times under refinement, not at equal step indices. For conservation laws or Hamiltonian motion, a method's long-time structural behavior can matter more than a small short-time error estimate. State the norm, time interval, and observable used to judge convergence. A smooth animated curve is a visualization of output, not proof that either the equation or its numerical approximation is stable.