In addition to the distinction between linear and
nonlinear PDEs, it is important for the computational
scientist to know that there are different classes of PDEs.
Just as different solution techniques are called for in the
linear versus the nonlinear case, different numerical
methods are required for the different classes of PDEs,
whether the PDE is linear or nonlinear. The need for this
specialization in numerical approach is rooted in the physics
from which the different classes of PDEs arise. By analogy
with the conic sections (* ellipse, parabola* and * hyperbola*)
partial differential equations have been classified as
elliptic, parabolic and hyperbolic. Just as an ellipse is a
smooth, rounded object, solutions to elliptic equations tend
to be quite smooth. Elliptic equations generally arise
from a physical problem that involves a diffusion process
that has reached equilibrium, a steady state temperature
distribution, for example. The hyperbola is the
disconnected conic section. By analogy, hyperbolic equations
are able to support solutions with discontinuities, for
example a shock wave. Hyperbolic PDEs usually arise in
connection with mechanical oscillators, such as a vibrating
string, or in convection driven transport problems.
Mathematically, parabolic PDEs serve as a transition from
the hyperbolic PDEs to the elliptic PDEs. Physically,
parabolic PDEs tend to arise in time dependent diffusion
problems, such as the transient flow of heat in accordance
with Fourier's law of heat conduction.