Two pillars became three
For centuries physics rested on two pillars: theory proposes an equation, experiment tests it. But most equations of real physics have no closed-form solution. Newton's second law for three gravitating bodies, the flow of a turbulent fluid, a folding protein — pen and paper simply run out. In the mid-20th century the computer became a third pillar: a way to extract quantitative predictions from equations we cannot solve by hand.
Computation is not a replacement for thinking; it is an instrument, like a telescope. Numerical physics turns a differential equation into an arithmetic recipe a machine can run; experimental physics turns a detector's clicks into a number with an honest error bar. This track teaches both crafts and the discipline that ties them together — the scientific method made quantitative.
Everything reduces to arithmetic: discretization
The single deepest idea in computational physics is discretization: replace the continuous — space, time, a field — with a finite grid of numbers, and replace derivatives and integrals with differences and sums. A machine cannot handle the infinitesimal dx; it can add and multiply finitely many numbers, very fast.
The Taylor series is the workhorse: every finite-difference formula is a rearranged Taylor expansion, and truncating it — keeping finitely many terms — is where numerical error is born.
From the series you can read off an approximation to a derivative and, crucially, how big its error is. This is the recurring bargain of numerics: a smaller step h means smaller truncation error but more steps (more cost), and eventually more round-off error. Choosing h wisely is an art with rules.
The other half: measurement is a craft
The experimental pillar has its own hard discipline. A raw reading is not a measurement; a measurement is a value with an uncertainty. Reporting the number without the error bar is, to a physicist, close to meaningless — it hides whether a claimed effect is real. See measurement uncertainty and accuracy and precision.
The irreducible shape of a physical result: a best estimate \bar{x} together with the uncertainty \sigma_{\bar{x}} that says how much to trust it.
The two families of experimental error mirror the two numerical ones. A statistical (random) error scatters repeated readings and shrinks as you take more data; a systematic error biases them all the same way and does not shrink with more data — you must hunt it down. Confusing the two is the classic blunder (systematic and random errors).
Closing the loop
Put the pieces in a cycle. A theory gives a model; computation turns the model into a prediction with numbers; an experiment yields data with error bars; statistical inference compares the two and feeds back to refine the theory. This track walks that loop: numerical integration (Guide 2), Monte Carlo and molecular dynamics (Guide 3), the measurement craft of errors, propagation and fitting (Guide 4), and finally signal, noise and discovery (Guide 5).
Two everyday tools deserve a nod up front, because a good physicist reaches for them before touching a computer. Dimensional analysis — checking that units balance — catches most algebra mistakes and can even guess the form of an answer. And order-of-magnitude reasoning, or Fermi estimation, gives a back-of-envelope figure that tells you whether a simulation's output is even plausible.