Monte Carlo simulation (voted one of the top 10 algorithms of the 20th century) invented at Los Alamos by von Neumann, Ulam and Metropolis.
First hydro simulations at Los Alamos occurred.
Ulam and von Neumann introduce the notion of cellular automata.
A routine for the Manchester Baby written to factor a large number (2^18), one of the first in computational number theory. The Manchester group would make several other breakthroughs in this area.
Hestenes, Stiefel, and Lanczos, all from the Institute for Numerical Analysis at the National Bureau of Standards, initiate the development of Krylov subspace iteration methods. Voted one of the top 10 algorithms of the 20th century.
Equations of State Calculations by Fast Computing Machines introduces the Metropolis–Hastings algorithm. Also, important earlier independent work by Alder and S. Frankel.
Fermi, Ulam and Pasta with help from Mary Tsingou, discover the Fermi–Pasta–Ulam problem.
Molecular dynamics invented by Alder and Wainwright
Householder invents his eponymous matrices and transformation method (voted one of the top 10 algorithms of the 20th century).
John G.F. Francis and Vera Kublanovskaya invent QR factorization (voted one of the top 10 algorithms of the 20th century).
First recorded use of the term "finite element method" by Ray Clough, to describe the methods of Courant, Hrenikoff and Zienkiewicz, among others. See also here.
Using computational investigations of the 3 body problem, Minovitch formulates the gravity assist method.
Molecular dynamics was invented independently by Aneesur Rahman.
Kruskal and Zabusky follow up the Fermi–Pasta–Ulam problem with further numerical experiments, and coin the term "soliton".
Edward Lorenz discovers the butterfly effect on a computer, attracting interest in chaos theory.
Frenchman Verlet (re)discovers a numerical integration algorithm, (first used in 1791 by Delambre, by Cowell and Crommelin in 1909, and by Carl Fredrik Störmer in 1907, hence the alternative names Störmer's method or the Verlet-Störmer method) for dynamics, and the Verlet list.
Mandelbrot, from studies of the Fatou, Julia and Mandelbrot sets, coined and popularized the term 'fractal' to describe these structures' self-similarity.
Kenneth Appel and Wolfgang Haken prove the four colour theorem, the first theorem to be proved by computer.
Fast multipole method invented by Rokhlin and Greengard (voted one of the top 10 algorithms of the 20th century).
The appearance of the first research grids using volunteer computing – GIMPS (1996), distributed.net (1997) and Seti@Home (1999).
Kepler conjecture is almost all but certainly proved algorithmically by Thomas Hales in 1998.
In computational group theory, God's number is shown to be 20.
Mathematicians completely map the E8-group.
Hales completes the proof of Kepler's conjecture.
Timeline of computational mathematics Wikipedia (Text) CC BY-SA
