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Convergence & Error Analysis

Intuition

How do you trust a simulation? You measure its error scaling. Run the same problem with \(N\), \(2N\), \(4N\) steps; if error falls by the factor your method's order promises, the implementation is correct and you know exactly what resolution buys. A straight line on a log–log plot is the closest thing numerical analysis has to a signature.

The procedure

  1. Choose a benchmark with a known answer — here, the cross-field cycloid.
  2. Define the error at the final time: \(\text{Error}(N) = |\mathbf{x}_\text{numerical}(T) - \mathbf{x}_\text{exact}(T)|\)
  3. Run each method with \(N = 10, 20, 50, 100, \dots\) steps.
  4. Plot \(\log(\text{Error})\) vs \(\log(N)\): the slope is \(-p\), the method's order.

Expected results

Method Slope Doubling \(N\) divides error by
Forward/Backward Euler −1 2
Heun / Leapfrog −2 4
RK3 −3 8
RK4 −4 16

Reading the plot like a pro

  • Slope shallower than expected → a bug (often an \(\mathcal{O}(\Delta t)\) inconsistency in initialization — e.g. leapfrog's half-step start).
  • Curve flattening at small \(\Delta t\) → floating-point round-off has taken over; you've hit the accuracy floor (~\(10^{-15}\) relative for float64).
  • Error growing with \(N\) → instability, not truncation error; no amount of resolution fixes a scheme used outside its stability region.
  • Order ≠ long-time fidelity: RK4's beautiful slope says nothing about its energy drift on Hamiltonian systems — see leapfrog.

Local vs global error

Local truncation error (one step): \(\mathcal{O}(\Delta t^{p+1})\). Number of steps to fixed time \(T\): \(N = T/\Delta t\). Global error: \(N \times \mathcal{O}(\Delta t^{p+1}) = \mathcal{O}(\Delta t^p)\). The lost power of \(\Delta t\) is the price of accumulation.

Common mistakes

  • Benchmarking against another simulation instead of an exact solution — you may be measuring the other code's error.
  • Testing convergence at one \(N\). Order is a slope; it needs several points.
  • Comparing methods at equal step count rather than equal cost (RK4 does 4 evaluations per step).

Knowledge graph position

Prerequisites: the integrator pages. Leads to: trustworthy field solvers and PIC diagnostics.

Quiz

Q1 (computational). A mystery integrator gives errors \(8\times10^{-3}\) at \(N = 100\) and \(10^{-3}\) at \(N = 200\). What's its order?

Answer

Error ratio 8 for a factor 2 in \(N\) ⇒ \(2^p = 8 \Rightarrow p = 3\). It's RK3-class.

Q2 (conceptual). Your leapfrog code shows slope −1 instead of −2. Most likely cause?

Answer

A first-order inconsistency somewhere — classically, initializing the staggered velocity at \(t = 0\) instead of \(t = \Delta t/2\) (missing initial half-kick), or a non-centered force evaluation. One \(\mathcal{O}(\Delta t)\) ingredient poisons the whole scheme's order.

Q3 (MCQ). At very small \(\Delta t\), the measured error stops decreasing because:

  • (a) the method's order saturates (b) round-off error dominates
  • (c) the exact solution is unknown (d) Python loops are slow
Answer

(b). Truncation error shrinks below the float64 round-off floor; further refinement only accumulates more rounding.