Chapter II
Identifying the type of the PDE
When a physician sees a sick patient, he does not immediately offer the patient the medicine, or
give a shot. He or she spends a lot of time taking the patient's vital signs, and observing the symptoms
before offering a cure. A little time spent diagnosing the problem, and identifying the problem, goes a long
way towards the successful resolution of the problem.
The same is true of CFD. Before we select an algorithm for solving a PDE, we should identify the
nature of the PDE – is it elliptic, parabolic, or hyperbolic. Incorrect diagnosis, or jumping into
programming without identifying the equation type will cause needless frustration on the part of the CFD
engineer.
Many of the PDEs we encounter in CFD are second order PDEs, and may be linear or nonlinear.
In any first course in partial differential equations, the following three types of PDEs are often discussed:
Elliptic PDE:
template.
A commonly encountered PDE is called the Poisson's equation and has the following
2F  q
Or ,
2F 2F
 2  q ( x, y )
x 2
y
(1)
If the right hand side is zero, then the Poisson's equation is called the Laplace's equation. In this
PDE, F is the unknown, while (x,y) are independent variables. This type of PDE occur in many physical
applications – stress analysis, temperature distribution on plates and shells, ground water pressure field,
electrical fields generated by charges, magnetic fields, and of course, in fluid dynamics.
A point P in a field governed by an elliptic PDE is influenced by the entire region surrounding it.
In other words, if there are two points P and Q in a field, the point P affects the point Q, and vice versa. The
domain of dependence of a point P governed by an elliptic PDE is shown in the figure below. The edge of
the domain is called the boundary. On this boundary, one should prescribe the property being solved for
(the function F in equation 1) or its derivative normal to the boundary, F/n. Sepcifying F is called the
Dirichlet boundary condition. Specifying the normal derivative is called the Dirichlet boundary condition.
One should be specified, or the other, but not both. A common mistake in solving PDEs analytically or
numerically is the over-specification of boundary conditions.
y
P
x
Parabolic PDE:
template:
A commonly encountered PDE is called the diffusion equation has the following
F
2F
 2
x
y
(2)
As before, F is the unknown, while x and y are any two independent variables. This type of PDE
occurs in the variation and decay of temperature in rods, in boundary layer equations, diffusion of
pollutants or chemicals in still water or air, in neutron transport equations, and many more.
A typical point P governed by such a PDE has the following domain of dependence (from
mathematical and physical considerations):
Y
P
X
Notice that the domain of dependence of the point P is the vertical line passing through Y, and all the
domain to the left of it. One can solve this type of PDE by marching from the left to the right. We need not
know, a priori, any information to the right of the point P in order to solve for F at the point P. We do need
to specify F (Dirichlet) or F/n (Neumann) along some initial line (the left edge of the hatched region
shown), and at the top and bottom of the domain.
Hyperbolic PDE: A commonly encountered PDE is called the wave equation, and has the following
template.
2
2F
2  F

c
x 2
y 2
(3)
Here c is a function of (x,y). It occurs in the propagation of sound waves through 2-D domains, vibration of
guitar strings and telephone wires, in electromagnetics, and in fluid mechanics. From mathematics (and
physics) this equation has the following domain of dependence:
y
P
x
Here the domain of dependence is the triangular shape region to the left of point P. The slope of
the top and bottom boundaries is c(x,y). One can solve this equation also by marching from the left to
right. At the left most edge of the domain, one should prescribe both F and its derivative normal to the
leftmost edge. This is called a Cauchy boundary condition. Nothing else need to be (or should be)
specified.
Some commonly encountered Fluid Dynamics PDEs and their type:
Now that we have looked at the templates, we can look at the PDEs in fluid mechanics and see which
template do they best fit. Of course the PDEs may be nonlinear, may contain mixed derivatives (i.e. first
and second derivatives, both), and may contain more than two independent variables – e.g. (x,y,z) or (x,y,t).
In such cases, do the following:



Forget about non-linearity. It has no bearing on the equation type.
Keep only the highest derivatives in each independent directions on the left, move lower order
derivatives to the right.
Consider two independent directions at a time.
While mathematicians will turn in their graves, these ad hoc rules often work well. Here are some
examples.
Incompressible Potential Flow equation:
This equation in 2-D is given by:
 2  0
Or ,
 2  2
 2 0
x 2
y
(4)
Comparing this equation to the three templates given earlier, we immediately conclude that template (1) fits
our equation best. Thus incompressible potential flow equation is elliptic. A point P in the incompressible
flow is governed by all its surroundings. The tail of an airfoil can affect what is going on at the nose, and
vice versa.
The three-dimensional incompressible flow is governed by:
 2  0
Or ,
 2  2  2
 2  2 0
x 2
y
z
(5)
This PDE contains three independent directions (x,y,z). Using our thumb rules above, consider
two directions at a time. Take the derivative in the third direction to the right, if it makes you feel better.
It will be easy to see that the equation remains elliptic in any two directions we choose. The tail of a Boeing
747 can affect what is going on at the nose, and vice versa.
Linearized Compressible Potential Flow equation:
In AE 4001, we derived: