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==Discretization==
The central process in CFD is the process of [[discretization]], i.e. the process of taking differential equations with an infinite number of [[degrees of freedom]], and reducing it to a system of finite degrees of freedom. Hence, instead of determining the solution everywhere and for all times, we will be satisfied with its calculation at a finite number of locations and at specified time intervals. The [[partial differential equations]] are then reduced to a system of algebraic equations that can be solved on a computer. Errors creep in during the discretization process. The nature and characteristics of the errors must be controlled in order to ensure that:
1) we are solving the correct equations (consistency property)
2) that the error can be decreased as we increase the number of degrees of freedom (stability and
Once these two criteria are established, the power of computing machines can be leveraged to solve the problem in a numerically reliable fashion. Various discretization schemes have been developed to cope with a variety of issues. The most notable for our purposes are: [[finite difference methods]], finite volume methods, [[finite element methods]], and [[spectral methods]].
==Finite Difference Method: ==
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