Separate numerical accuracy from physical completeness

A physical model simplifies reality by choosing which effects to include. A numerical method approximates the solution of that model. Making the numerical solution more accurate does not automatically add an omitted force or correct a wrong initial condition — open definition. These are different sources of discrepancy.

A toy central-gravity orbit can converge toward its own ideal solution while still disagreeing with a real spacecraft affected by thrust or drag. State the intended question before demanding more detail: a simple model can be useful when its limits match the learning or analysis purpose.

In the velocity-first exercise, reducing the step from 0.1 to 0.05 changes the first updated velocity to (−0.05, 1) and position to (0.9975, 0.05). Those states occur at different times, so their coordinates should not be compared as if they were simultaneous. Compare runs at shared output times instead.

For a fictional convergence — open definition study, suppose an output at the same final time is 1.20, 1.10 and 1.05 as steps are halved. The differences are shrinking, which is useful evidence about this calculation. It does not alone prove a particular exact answer or guarantee that every other output has converged.

Computers often represent real numbers with finite binary precision. Some simple decimal fractions are therefore stored approximately. Python’s documentation explains why arithmetic such as adding decimal-looking values can differ slightly from exact decimal expectations. More printed digits do not create more physical information.

Use tolerances tied to the problem and inspect sensitivity to numerical choices. Very small steps also increase computation and can expose accumulated rounding effects; smaller is not a universal guarantee of a better result. Preserve the model, units, parameters, method and versions so another person can reproduce the calculation and judge the same limits.