Introduction to numerical methods
Numerical methods are at the heart of the CFD process. Researchers dedicate their attention to two fundamental aspects in CFD; i.e. physical modeling and numerics.
In physical modeling, we seek a set of equations or mathematical relations that allow us to close the governing equations. In turbulence modeling for example, one is interested in devising new equations for the extra unknowns that resulted from the averaging process.
On the other hand, the focus in numerics is to devise efficient, robust, and reliable algorithms for the solution of PDEs. PDEs are a combination of differential terms (rates of change) that describe a conservation principle. Without loss in generality, all physical processes can be described by PDEs. Now, the CFD process requires the discretization of the governing PDEs, i.e. the derivation of equivalent algebraic relations that should faithfully represent the original PDEs. This is done by transforming each differential term into an approximate algebraic relation (see Generic scalar transport equation).
Deriving an original numerical algorithm is not only a mathematical challenge. The investigator should also bear in mind the physics behind the term that is being discretized. For example, there are various discretization schemes for the convection term (upwind, QUICK, SOU etc...) because of the special behavior of the convection process. Similarly, the diffusion, convection, and source terms have very specialized treatments.