# Numerical methods

Numerical method is the field of science which relies on the approximations (within provided error limits) of mathematical formulations or relationships. There are many examples such as- approximating the circle with a polygon and approximating an infinite series. These methods got popularized mainly after the invention of computers. Since the advent of high computational facilities, scientists are interested in large-scale simulations of complex phenomenon.

Some of the basic numerical methods required in mathematics are:

- Solving Linear Algebraic Equations of the form Ax=b
- Interpolation and extrapolation of data
- Evaluation of functions (e.g. Lagendre, Bessels, etc.)
- Generation of pseudo-random number
- Maximization and minimizing the function
- Transormations (Fourier, Laplace)
- Solution of differential equations (ODE and PDE)

Some of the high end in simulations required in Earth science are:

- Atmospheric Science

- Weather prediction,
- Simulating ocean atmosphere interaction for CO2 modelling

- Electro-magnetics

- wave propagation in materials

- Geophysics

- Seismics:
- wave propagation in earth
- Migration
- Full Waveform inversion

- Magnetotelluric: Simulating earth's magnetic field
- Reservoir: Reservoir modelling for time lapse changes.