II.1 Properties of 2nd order PDE’s

A 2nd order PDE is a differential equation whose highest derivative is of the form

• elliptic - such as the heat conduction equation
• parabolic - such as the diffusion equation
• hyperbolic - these govern wave propagation problems

Amongst other things, the type of equation we are dealing with will affect the type of boundary conditions that we will apply.

An elliptic problem is a steady-state problem where the conditions at any point are dependent on the conditions at all the other points. The steady-state heat conduction equation is a typical example. The dependency of the solution is shown in figure 1.

Figure 1: Dependence diagram for an elliptic problem

Elliptic problems are boundary value problems - to complete the solution one must specify the value of the variable on the boundaries of the region under consideration.

Parabolic and hyperbolic equations represent time-marching or propagation problems, problems that evolve through time. In a hyperbolic problem, the conditions at a given point are affected by the properties at other points in the problem at past times. In effect, information about the solution propagates around the domain at a finite speed. For example, many flow problems are hyperbolic in nature. Information about a disturbance somewhere in the flow propagates through the flow at the speed of sound, and when this information reaches a point, the flow conditions at that point are affected by the disturbance. Parabolic problems are similar, but the propagation of information is instantaneous. Thus a disturbance anywhere in the domain instantly starts to affect the solution at all other points. Both parabolic and hyperbolic equations are initial value problems. To solve them, the value of the variable must be specified at all points in the domain at the start of the problem. The solution dependency for hyperbolic and elliptic problems looks like figure 2.

Figure 2: Solution dependency for hyperbolic (left) and parabolic (right) problems.

Now, the bad news. Classifying PDE’s involving 1 variable - say a PDE describing the temperature of a plate (elliptic if steady-state, parabolic if not) is fairly easy. In CFD we typically solve at least 4 linked PDE’s, and working out the classification is not easy. However the behaviour of various types of flow is as follows :

In general, time-dependent fluid flow problems are parabolic or hyperbolic, and so we are involved in an initial value problem. We solve this by creating an initial field and then advancing the solution small steps (timesteps) t until we reach the desired solution. Steady state flow problems may be elliptic, but in fact the timestepping method is so useful that such problems are often solved in the same way, by starting from an initial state and marching forward in time until the solution ceases to change.