Numerical and algorithmic methods for simulating physical systems. Molecular dynamics, Monte Carlo, density-functional theory, lattice gauge theory, N-body, computational fluid dynamics, spectral methods.
Computational Physics
Molecular dynamics (MD)
Numerical integration of Newton's equations for a many-atom system under a prescribed force field (Lennard-Jones, AMBER, CHARMM). Produces…
Monte Carlo methods (physics)
Stochastic sampling of statistical ensembles (Metropolis, heat-bath, Wolff cluster). Convergence controlled by detailed balance and…
Lattice gauge theory
Non-perturbative regularisation of gauge theories on a discrete spacetime lattice (Wilson 1974). Primary tool for QCD calculations of…
Computational fluid dynamics (CFD)
Numerical solution of Navier–Stokes and related PDEs via finite-volume, finite-element, or spectral methods. Central to aerospace,…
N-body simulation
Direct or hierarchical (tree/Barnes–Hut, FMM, particle-mesh) integration of gravitational N-body systems. Enables galactic,…
Finite volume method
Discretisation based on integral conservation laws over control volumes; natively handles shocks and conserves fluxes. Backbone of CFD…
Variational Monte Carlo (VMC)
Stochastic minimisation of ⟨ψ_T|H|ψ_T⟩/⟨ψ_T|ψ_T⟩ over a trial wavefunction's parameters using Metropolis sampling. Benchmarks for…
Density matrix renormalisation group (DMRG)
Variational algorithm (White 1992) optimising matrix-product states for 1D quantum lattice systems; extends via PEPS/MERA to higher…
Symplectic integrator
Numerical scheme for Hamiltonian systems that exactly preserves the symplectic 2-form; stable long-time integration (leapfrog, Yoshida,…
Ab initio electronic-structure methods
Hartree–Fock, post-HF (MP2, CC), DFT, CI, MCSCF: hierarchy of first-principles methods for molecular and solid-state electronic structure.
Tensor networks (MPS, PEPS, MERA)
Variational Ansätze efficient for area-law entangled states. DMRG = MPS optimisation; PEPS in 2D; MERA for critical systems.…
Quantum Monte Carlo
VMC, DMC, AFQMC, PIMC methods for many-body quantum systems. Sign problem limits fermionic applications. Benchmarks for homogeneous…
Machine-learning potentials (NNP, GAP)
Neural-network + Gaussian-process surrogates for PES at DFT accuracy, MD-timestep cost. Behler-Parrinello, SchNet, GAP, MACE. Materials +…
Finite volume method
Integrate conservation laws over control volumes; naturally conservative; upwind/Godunov/Riemann for hyperbolic systems.
Spectral methods
Expand in global basis (Fourier, Chebyshev); exponential convergence for smooth; FFT implementation; used in turbulence DNS.
Monte Carlo simulation
Sampling from distributions to compute integrals/averages; Metropolis-Hastings for importance sampling; quantum MC.
Molecular dynamics (MD)
Integrate Newton's equations for N particles with empirical potentials (LJ, EAM, reactive); fs timestep; NVE, NVT, NPT.
Lattice Boltzmann method
Discrete velocity lattice with BGK collision; recovers Navier–Stokes at low Mach; good parallelism; multiphase, porous flows.
FFT-based PDE solvers
Spectral accuracy in periodic domains; diagonalize Laplacian; N log N scaling; used in pseudospectral turbulence codes.
Multigrid methods
Hierarchy of coarse grids accelerates elliptic solvers to O(N); V, W, F cycles; essential for Poisson in simulations.
Parallel HPC scaling
Amdahl & Gustafson laws; strong vs weak scaling; MPI+OpenMP+GPU; exascale systems (Frontier, Aurora 2022+).
Markov chain Monte Carlo (MCMC)
Construct Markov chain with target stationary distribution; Metropolis, Gibbs, Hamiltonian MC; Bayesian inference, lattice QCD.
Tensor networks (MPS, PEPS, MERA)
Efficient parametrization of low-entanglement states; DMRG for 1D; MERA for critical; PEPS for 2D; quantum simulation basis.
Verification, validation, UQ
V&V ensures 'solving equations right' (code) and 'solving right equations' (physics); UQ quantifies input uncertainty propagation.
Undecidability of the spectral gap in 2D quantum lattice models (Computational Physics)
Computational-physics application of L0 halting-problem and Gödel incompleteness. Cubitt, Pérez-García, and Wolf 2015 (Nature 528:207–211)…
Chaitin's halting probability Ω and the incomputability limit of physical simulation (Computational Physics)
Computational-physics application of L0 Chaitin Ω and Kolmogorov complexity. Chaitin's Ω is the probability that a self-delimiting…