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Computational physics uses numerical algorithms to obtain and test predictions of physical models. It includes solving differential equations, simulating stochastic processes and extracting physical quantities from data. The mathematical model, its numerical approximation and its agreement with an experiment are separate questions: an accurately executed algorithm can still solve an unsuitable model.

A simple example is the harmonic oscillator $\ddot{x}=-{\omega }^{2}x$. The explicit Euler method advances it byFor a unit mass, define $E_n=(v_n^2+{\omega }^2{x}_n^2)/2$. Direct substitution givesThe physical oscillator conserves energy, but this algorithm creates it on every step. The discrepancy is a numerical effect, not a discovery about the oscillator. Changing the time step and comparing with a known solution exposes the problem; for long integrations, the choice of numerical method matters as well as the size of the step.

Richard Feynman's lecture on planetary motion gives a concrete historical teaching example: forces determine accelerations, which update velocities and positions through successive arithmetic steps. A computer extends the same approach to systems whose coupled motion has no convenient closed expression.^[ref] Conservation checks, comparison against solvable limits, and convergence under finer numerical resolution help distinguish a reliable calculation from an attractive plot. Agreement with measurements then tests the physical model itself.

Robert H. Swendsen co-developed the Swendsen-Wang algorithm with Jian-Sheng Wang. Their 1987 paper introduced collective updates for spin-system Monte Carlo simulations, reducing the difficulty of sampling large correlated regions near a phase transition.^[ref]

The key distinction is between physical dynamics and a sampling algorithm's dynamics. Replacing many single-spin changes by a collective update can preserve the intended equilibrium probability distribution while changing how quickly samples decorrelate. An improvement in simulation speed is therefore not a claim that a real magnet changes by that same update rule. Swendsen's Carnegie Mellon profile identifies his field as computational physics.^[ref]