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:

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

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

Atmospheric Science

1. Weather prediction,
2. Simulating ocean atmosphere interaction for CO2 modelling

Electro-magnetics

1. wave propagation in materials

Geophysics

1. Seismics:
   1. wave propagation in earth
   2. Migration
   3. Full Waveform inversion
2. Magnetotelluric: Simulating earth's magnetic field
3. Reservoir: Reservoir modelling for time lapse changes.