Computational Fluid Dynamics - Elementary (e)

Partial differential equations - Classification

In fluid dynamics, the governing equations are partial differential equations (PDE) containing 1st- and 2nd-order derivatives in space and 1st-order derivatives in time. Therefore, in order to solve the fluid dynamics problems it is necessary to know how to solve the PDEs.

In this chapter, a simple classification of 2nd-orser, linear PDEs are introduced. The general form of the PDE would be:

\begin{equation} A\frac{\partial^2 u}{\partial x^2} + B\frac{\partial^2 u}{\partial x \partial y} + C\frac{\partial^2 u}{\partial y^2} + D\frac{\partial u}{\partial x} + E\frac{\partial u}{\partial y} + Fu + G = 0; \end{equation}
in which the capital letters A-G are constants. This PDE can be classified as elliptic, parabolic or hyperbolic considering its discriminant, B² - 4AC. It means that the lower-order derivatives are not important in the classification.

If the discriminant is negative, the PDE is Elliptic and if it is positive, the PDE is Hyperbolic. Otherwise, it is Parabolic.

More information regarding the classification of PDEs can be found in the following document:

classification of PDEs.pdf