The governing equations, that is, the Navier – Stokes equations in continuum mechanics are a set of coupled non–linear partial differential equations derived from the conservation laws for mass, momentum and energy.

For a numerical solution of a mathematical continuum model, the focus is to devise efficient, robust, and reliable algorithms for the solution of the partial differential equations. To do this the partial differential equations are converted to a discrete system of algebraic equation using a discretization procedure. In case of a CFD simulation, the discretization of the governing equations, that is, the derivation of equivalent algebraic relations should accurately represent the equations and the physics modeled.

There are a number of discretization techniques that have been developed to solve different kinds of problems in CFD. These methods can be broadly classified into two categories:
  • Mesh based methods – These methods require division of the problem domain into a grid or mesh over which the governing equations are discretized. The points are positioned according to a topological connectivity which ensures the compatibility of the numerical technique used. Examples of these methods include Finite Difference (FD), Finite Volume (FV) and Finite Element (FE) methods.
  • Mesh free methods – These methods use a collection of nodes in the domain which do not have any apparent connectivity. These methods are particularly important in simulations where the nodes are created or destroyed, simulations where the deformations are so large that the connectivity might introduce distortion and in cases where the domain possesses discontinuities or singularities. A few examples of these methods are: Smoothed Particle Hydrodynamics (SPH), Finite Pointset Method (FPM), Meshless Local Petrov Galerkin (MLPG) and Lattice Boltzman methods.

Of the methods discussed above, mesh based methods are more popular and are among the most widely used in CFD codes. The following sections describe the mesh based discretization methods (FD, FV and FE) and the time discretization techniques used to solve the partial differential equations governing a fluid flow.