The following is a timeline of scientific computing, also known as computational science.

Before modern computers

18th century Simpson rediscovers Simpson's rule, a century after Johannes Kepler (who derived it in 1615 after seeing it used for wine barrels). 1733 – The French naturalist Comte de Buffon poses his needle problem.Buffon, G. Editor's note concerning a lecture given 1733 by Mr. Le Clerc de Buffon to the Royal Academy of Sciences in Paris. Histoire de l'Acad. Roy. des Sci., pp. 43-45, 1733; according to Weisstein, Eric W. "Buffon's Needle Problem." From MathWorld--A Wolfram Web Resource. 20 Dec 2012 20 Dec 2012.Buffon, G. "Essai d'arithmétique morale." Histoire naturelle, générale er particulière, Supplément 4, 46-123, 1777; according to Weisstein, Eric W. "Buffon's Needle Problem." From MathWorld--A Wolfram Web Resource. 20 Dec 2012 Euler comes up with a simple numerical method for integrands.Euler, L. Institutionum calculi integralis. Impensis Academiae Imperialis Scientiarum, 1768.Butcher, John C. (2003), Numerical Methods for Ordinary Differential Equations, New York: John Wiley & Sons, .Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: Springer-Verlag, . c. 500 BCE - Urdhva Tiryakbhyam algorithm, a Vedic method for fast integer multiplication; foundational for Indian mathematics. 300 BCE - Babylonian root extraction method, Earliest documented numerical algorithm for square roots. c. 250 BCE - Chinese Remainder Theorem    Systematic solution to simultaneous congruences; used in cryptography.

19th century First formulation of Gram-Schmidt orthogonalisation by Laplace,Laplace, PS. (1816). Théorie Analytique des Probabilités :First Supplement, p. 497ff. to be further improved decades later.Earliest Known Uses of Some of the Words of Mathematics (G). As of Aug 2017. Babbage in 1822, began work on a machine made to compute/calculate values of polynomial functions automatically by using the method of finite differences. This was eventually called the Difference engine. Lovelace's note G on the Analytical Engine (1842) describes an algorithm for generating Bernoulli numbers. It is considered the first algorithm ever specifically tailored for implementation on a computer, and thus the first-ever computer programme.Tom Stoppard's “Arcadia,” at Twenty. By Brad Leithauser. The New Yorker, August 8, 2013. The engine was never completed, however, so her code was never tested. Adams-Bashforth method published.Bashforth, Francis (1883), An Attempt to test the Theories of Capillary Action by comparing the theoretical and measured forms of drops of fluid. With an explanation of the method of integration employed in constructing the tables which give the theoretical forms of such drops, by J. C. Adams, Cambridge. In applied mathematics, Jacobi develops technique for solving numerical equations.Jacobi's Ideas on Eigenvalue Computation in a modern context, Henk van der Vorst.Jacobi method, Encyclopedia of Mathematics.The Early History of Matrix Iterations: With a Focus on the Italian Contribution, Michele Benzi, 26 October 2009. SIAM Conference on Applied Linear Algebra, Monterey Bay – Seaside, California. Gauss Seidel first published. To help with computing tides, Harmonic Analyser is built in 1886. 850 CE: Al-Kindi's frequency analysis – First systematic cryptanalysis technique for breaking substitution ciphers. 1206: Al-Jazari's programmable orchestra – Mechanical automata using pegged cylinders for sequence control (early program storage). 1676: Leibniz's chain rule – Foundation for calculus-based optimization later used in backpropagation. 1738/1763: Bernoulli's utility theory & Bayes' theorem – Probabilistic frameworks for decision-making algorithms.

1900s (decade) 1900 – Runge's work followed by Martin Kutta to invent the Runge-Kutta method for approximating integration for differential equations.MW Kutta. "Beiträge zur näherungsweisen Integration totaler Differentialgleichungen" [Contributions to the approximate integration of total differential equations] (in German). Thesis, Ludwig-Maximilians-Universität München. 1901 – and in B.G Teubner, 1901.Runge, C., "Über die numerische Auflösung von Differentialgleichungen" [About the numerical solution of differential equations](in German), Math. Ann. 46 (1895) 167-178.

1910s (decade) 1910 – A-M Cholesky creates a matrix decomposition scheme. Richardson extrapolation introduced.

1920s 1922 – Lewis Fry Richardson introduces numerical weather forecasting by manual calculation, using methods originally developed by Vilhelm Bjerknes as early as 1895.L F Richardson, Weather Prediction by Numerical Process. Cambridge University Press (1922). 1926 – Grete Hermann publishes foundational paper for computer algebra, which established the existence of algorithms (including complexity bounds) for many of the basic problems of abstract algebra, such as ideal membership for polynomial rings. 1926 Adams-Moulton method. 1927 – Douglas Hartree creates what is later known as the Hartree–Fock method, the first ab initio quantum chemistry methods. However, manual solutions of the Hartree–Fock equations for a medium-sized atom were laborious and small molecules required computational resources far beyond what was available before 1950. 1928 – Leslie Comrie proposes using commercial tabulating machines to perform scientific calculations, and himself uses them to expand Ernest William Brown's lunar calculations.

1930s This decade marks the first major strides to a modern computer, and hence the start of the modern era. Fermi's Rome physics research group (informal name I ragazzi di Via Panisperna) develop statistical algorithms based on Comte de Buffon's work, that would later become the foundation of the Monte Carlo method. See also FERMIAC. Shannon explains how to use electric circuits to do Boolean algebra in "A Symbolic Analysis of Relay and Switching Circuits" John Vincent Atanasoff and Clifford Berry create the first electronic non-programmable, digital computing device, the Atanasoff–Berry Computer, from 1937-42. Complex number calculator created by Stibitz. At Columbia University's Rutherford Laboratory Wallace J. Eckert uses commercial tabulating machinery from IBM, some of it specially modified, for scientific computation.

1940s 1947 – Metropolis algorithm for Monte Carlo simulation (named one of the top-10 algorithms of the 20th century) invented at Los Alamos by von Neumann, Ulam and Metropolis.. Accessed 5 May 2012.S. Ulam, R. D. Richtmyer, and J. von Neumann(1947). Statistical methods in neutron diffusion. Los Alamos Scientific Laboratory report LAMS–551. George Dantzig introduces the simplex method (named one of the top 10 algorithms of the 20th century) in 1947. Systems Optimization Laboratory, Stanford University Huang Engineering Center (site host/mirror). Ulam and von Neumann introduce the notion of cellular automata.Von Neumann, J., Theory of Self-Reproducing Automata, Univ. of Illinois Press, Urbana, 1966. Turing formulated the LU decomposition method.A. M. Turing, Rounding-off errors in matrix processes. Quart. J Mech. Appl. Math. 1 (1948), 287–308 (according to Poole, David (2006), Linear Algebra: A Modern Introduction (2nd ed.), Canada: Thomson Brooks/Cole, .) . A. W. H. Phillips invents the MONIAC hydraulic computer at LSE, better known as "Phillips Hydraulic Computer".The computer model that once explained the British economy. Larry Elliott, The Guardian, Thursday 8 May 2008.Phillip's Economic Computer, 1949. Exhibit at London Science Museum. First hydro simulations occurred at Los Alamos.Richtmyer, R. D. (1948). Proposed Numerical Method for Calculation of Shocks. Los Alamos, NM: Los Alamos Scientific Laboratory LA-671.

1950s First successful weather predictions on a computer occurred.See the review article:- 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.Magnus R. Hestenes and Eduard Stiefel, Methods of Conjugate Gradients for Solving Linear Systems, J. Res. Natl. Bur. Stand. 49, 409-436 (1952).Eduard Stiefel, U¨ ber einige Methoden der Relaxationsrechnung (in German), Z. Angew. Math. Phys. 3, 1-33 (1952).Cornelius Lanczos, Solution of Systems of Linear Equations by Minimized Iterations, J. Res. Natl. Bur. Stand. 49, 33-53 (1952).Cornelius Lanczos, An Iteration Method for the Solution of the Eigenvalue Problem of Linear Differential and Integral Operators, J. Res. Natl. Bur. Stand. 45, 255-282 (1950). Named one of the top 10 algorithms of the 20th century. Equations of State Calculations by Fast Computing Machines introduces the Metropolis–Hastings algorithm. Molecular dynamics invented by Bernie Alder and Wainwright A S Householder invents his eponymous matrices and transformation method (voted one of the top 10 algorithms of the 20th century). 1953 – Enrico Fermi, John Pasta, Stanislaw Ulam, and Mary Tsingou discover the Fermi–Pasta–Ulam–Tsingou problem through computer simulations of a vibrating string.Fermi, E. (posthumously); Pasta, J.; Ulam, S. (1955) : Studies of Nonlinear Problems (accessed 25 Sep 2012). Los Alamos Laboratory Document LA-1940. Also appeared in 'Collected Works of Enrico Fermi', E. Segre ed., University of Chicago Press, Vol.II,978–988,1965. Recovered 21 Dec 2012 A team led by John Backus develops the FORTRAN compiler and programming language at IBM's research centre in San Jose, California. This sped the adoption of scientific programming,W.W. McDowell Award citation: National Medal of Science citation: and is one of the oldest extant programming languages, as well as one of the most popular in science and engineering.

1960s 1960 – First recorded use of the term "finite element method" by Ray Clough to describe the earlier methods of Richard Courant, Alexander Hrennikoff and Olgierd Zienkiewicz in structural analysis.RW Clough, "The Finite Element Method in Plane Stress Analysis," Proceedings of 2nd ASCE Conference on Electronic Computation, Pittsburgh, PA, Sept. 8, 9, 1960. 1961 – John G.F. Francis and Vera Kublanovskaya Also published in: Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki [Journal of Computational Mathematics and Mathematical Physics], 1(4), pages 555–570 (1961). invent QR factorization (voted one of the top 10 algorithms of the 20th century). 1963 – Edward Lorenz discovers the butterfly effect on a computer, attracting interest in chaos theory. 1961 – Using computational investigations of the 3-body problem, Michael Minovitch formulates the gravity assist method.Minovitch, Michael: "A method for determining interplanetary free-fall reconnaissance trajectories," Jet Propulsion Laboratory Technical Memo TM-312-130, pages 38-44 (23 August 1961).Christopher Riley and Dallas Campbell, Oct 22, 2012. "The maths that made Voyager possible". BBC News Science and Environment. Recovered 16 Jun 2013. 1964 – Molecular dynamics invented independently by Aneesur Rahman. 1965 – fast Fourier transform developed by James W. Cooley and John W. Tukey. 1964 – Walter Kohn, with Lu Jeu Sham and Pierre Hohenberg, instigates the development of density functional theory, for which he shares the 1998 Nobel Chemistry Prize with John Pople. This contribution is arguably the earliest work to which Nobels were given for a computer program or computational technique. First regression calculations in economics.

1970s