The scientific foundation

Physical modelling

Identify the governing principles (conservation of energy, momentum and mass; thermodynamics; transport) and the approximations that make a problem tractable. A good model states its assumptions so that its range of validity is known.

Example: a pendulum obeys

The familiar period $T_0=2\pi \sqrt{L/g}$ follows only after replacing $\sin \theta$ by $\theta$. The exact period is $T=4\sqrt{L/g}K\left({\sin }^2({\theta }_0/2)\right)$, with $K$ the complete elliptic integral of the first kind. At a release angle of 60° the small-angle formula is already 6.8 % short ($T/T_0=1.073$). Knowing where an approximation fails is part of the model.

Mathematical modelling

Translate the physics into equations: ordinary and partial differential equations, conservation laws, stochastic processes and statistical models. Non-dimensionalisation exposes which parameters matter and which can be neglected.

Numerical methods

Most realistic models have no closed-form solution. Discretisation, numerical integration, linear and nonlinear solvers and optimisation turn them into computations whose accuracy can be measured and controlled. See Scientific Computing.

Uncertainty

Inputs are measured imperfectly, parameters are inferred from limited data and models are approximations. Each of these is a source of uncertainty that propagates to the result. Reporting the result with an interval, and with the input that dominates it, is what makes it usable. See NOTE / 001.

Quantitative reasoning

Order-of-magnitude estimates, dimensional analysis, limiting cases and sanity checks catch errors before they reach a conclusion. A result that cannot survive a back-of-the-envelope check is not ready to be interpreted.