CFD 14 : Mathematical Classification of PDE of Second Order

Partial differential equations of fluid dynamics and heat transfer belong to the class of quasilinear PDE, which means that they are linear in their highest-order derivatives, but perhaps not in other terms.

The quasilinear PDE can be classified into three types according to the existence and form of their characteristics, the special lines in the solution domain. . One aspect is, however, very important for us: The information in the solutions tends to propagate along the characteristics if they exist. This has deep implications not only for the mathematical properties of the solution but also for the choice of numerical methods. To put it simply, different numerical methods must be used for equations of different types.

The classification is applicable to a broad range of systems of quasilinear PDE. For simplicity, we will limit the discussion to a single linear equation of second order for a function of two variables φ(x, y). The most general form of such an equation is

where A, B, C, D, E, F, and G are known coefficients that can be either constants or functions of x and y. Our choice of is not as arbitrary as it may seem. Many equations of fluid dynamics and heat transfer are of the second order, for example, the Navier-Stokes equations or the heat transfer equation. Their highest-order derivatives have the same general form as the highest-order derivatives of above equation. The lower order terms are quite different, but, they are of little importance for the classification.

The characteristics of equation can be defined as curves, on which the second derivatives φxx , φyy, and φyy are not uniquely determined by the other terms of the equation. It can be shown that, if a characteristic curve y = y(x) exists, its slope is given by

The classification is based on this relation, more specifically, on the value of B^2 − 4AC. The characteristics at the point (x, y) can be of three possible forms:

  1. B^2 − 4AC >0. There are two real characteristics intersecting at this point. The equation is called hyperbolic.
  2. B^2 − 4AC = 0. There is one real characteristic. The equation is called parabolic.
  3. B^2 − 4AC < 0. No real characteristics exist at this point. The equation is called elliptic.

If A, B, and C are constants, the classification holds in the entire domain

If A, B, or C are functions of x and y in space, the classification must be done separately for each point (x, y). The equation may be of a different type in different parts of space