Integration Accuracy and Step Size

lob takes a fixed step through the trajectory and lets the caller choose how big it is. This page is about what that knob actually buys, what it costs, and where the library's guidance stops being true.

The knob

Accuracy is not a guarantee lob makes; it is a parameter the caller controls. The step is expressed in down-range inches, and the default is a deliberately coarse grid that is fast and, over the range lob is validated against, good enough. A caller who needs more can ask for a finer step, and lob will spend the time.

The step is not a suggestion the library second-guesses. It is never widened automatically, in either direction. If a caller asks for a coarser step, that is their call, and it is a call they can get wrong.

Two details make the stepping behave. The final step is shortened to land exactly on the requested range instead of overshooting it and reporting a position past the target. A shot that stops making down-range progress before reaching the requested range can leave the solver in a repeating state rather than terminating cleanly.

What a smaller step buys

For smooth trajectories, Heun's global integration error is typically proportional to the square of the step size, while work is roughly inversely proportional to step size. Halving the step typically reduces truncation error by about four times and doubles work, until an error floor or knot effects dominate.

Returns diminish. Past some point the error is no longer dominated by the integration and shrinking the step buys nothing but time, and where that point falls depends on the trajectory rather than on the solver.

Why fixed rather than adaptive

An adaptive integrator estimates its own error and grows or shrinks the step to hit a target. lob does not, and the reason is that adaptivity has a cost that is not obvious from the outside.

Every adaptive step carries an error estimate, which means extra work per step, and every step carries a decision, which means a branch that is taken differently each time. Both are exactly the kind of cost that the drag spline representation was chosen to avoid. It also makes the cost of a solve less predictable, since the number of steps becomes a property of the trajectory rather than of the caller's setting.

What is given up is honest and worth naming: lob cannot tighten the step automatically where a trajectory happens to be hard and loosen it where the trajectory is easy. It spends the same effort on a flat subsonic descent as on a transonic climb. A caller who wants that behavior has to ask for it by choosing the step.

Where the guidance stops

The default step is validated over a range of trajectories, and the library documents which. It also documents a coarser step that a caller may opt into, and the range of trajectories over which that coarser step is defensible. Past that envelope the coarser step stops being a recommendation and becomes a guess, and the library says so rather than quietly continuing to offer it.

The current values are in the API reference rather than here. That is deliberate: a documented default that can drift out of date in prose is worse than a pointer to the constant that defines it.

What backs it

The accuracy claims are measured against registered reference trajectories rather than asserted. Those checks confirm that the default step reproduces the reference trajectories within reporting granularity for the six registered cases and their listed ranges. The convergence checks require successive step-size differences to decrease or reach their numerical floor for the tested cases and channels; they do not measure error against the reference trajectories.

The step-size question is settled by paired experiments rather than argument: the same trajectory solved at two step sizes, compared, so that the measured sensitivity is a property of the trajectory rather than an assumption about it. Where the comparison shows no meaningful change, that is recorded as a limit of what the step size can be held responsible for, which is a more useful statement than a single number.

The method choice is benchmarked separately, at matched accuracy: each candidate method is run at the step size that reaches a common error target, so the comparison is between methods rather than between arbitrary settings. That is what establishes the choice of a second-order step over a higher-order one, which Heun Integration argues from the literature.

Limitations

Fixed step, no embedded error estimate, no adaptive stepping.
The per-step air-density variant is gated by the inverse solver rather than always on. Most shots do not climb far enough for it to matter and it is not free, so the cheap path is the default and the accurate path is paid for per range where it is needed. See Point-Mass Model.