Turn local defects into a global convergence statement

Try

A small one-step error does not directly give the error after thousands of steps. Suppose an algorithm’s error satisfies Eₙ₊₁≤(1+Lh)Eₙ+Ch^(p+1). The first term propagates existing error; the second injects a new defect. This recurrence makes two requirements visible: local consistency of order p and a controlled propagation factor. Both are needed for a meaningful global convergence statement.

Predict

Predict why multiplying one local error by the number of steps can be incomplete. Earlier defects can be amplified or damped before the final time. The recurrence weights them by successive powers of 1+Lh. On a fixed time interval these weights remain bounded by an exponential, giving a useful order result. If the interval length grows with refinement or the Lipschitz constant is not uniform, the same conclusion cannot be copied unchanged.

Experiment

Unroll three steps from E₀=0.

Step	Error bound divided by Ch^(p+1)
1	1
2	1+(1+Lh)
3	1+(1+Lh)+(1+Lh)²

The geometric pattern identifies the full sum. Check separately what happens for L=0, where every weight is one and division by L must be avoided.

Understand

After N steps with Nh=T, the bound is E_N≤Ch^(p+1)Σⱼ₌₀ᴺ⁻¹(1+Lh)^j. For L>0 this equals Ch^p[(1+Lh)^N−1]/L and is at most Ch^p(e^(LT)−1)/L. For L=0 it is CTh^p. A nonzero initial error contributes an additional factor at most e^(LT)E₀. The constants must hold on a region containing the exact and numerical paths. This is a one-step convergence proof under explicit stability and local-defect assumptions, not a theorem that any visually smooth trace converges. Rounding errors add further perturbation terms and can eventually dominate refinement.

Apply

Adaptive methods estimate a local error e and choose a new step using h_new≈s h(tol/e)^(1/(p+1)) when that estimator scales like h^(p+1), with safety factor 0<s<1 and bounded growth/shrink ratios. Embedded pairs may have an estimator order different from the accepted solution’s order; use the estimator’s actual exponent. Components need appropriate absolute and relative scales before forming an error norm. A rejected trial must not advance the accepted state. Cache reusable stage data only when the method permits it, count rejected work, and stop with an explicit unmet-tolerance status if minimum step or work limits are reached.

A mixed absolute-relative component scale can be written sᵢ=atolᵢ+rtol·max(|yᵢ|,|yᵢ,new|), with positive absolute tolerances. Divide estimated component errors by these scales before taking a chosen norm. This is a controller convention, not a theorem that each component’s global error meets that scale. State it in the report so a change of units or a nearly zero component does not silently change the meaning of acceptance.

Continue

Local tolerance is not automatically a final-time guarantee. Validate adaptive implementations with smooth known solutions, stiffness, near-zero components, events and discontinuities. Compare accepted mesh points and total function evaluations, not only final output. Dense output and event location require their own formulas and error analysis. The final lesson assembles a verification report that separates mathematical guarantees, finite-precision diagnostics, empirical convergence and modeling uncertainty, so a numerical result can be reviewed rather than merely admired.

Review calculus and linear algebra when a prerequisite needs practice. Continue to Build a numerical result another person can audit.