Everything must be made as simple as possible. But not simpler.
― Albert Einstein
A few weeks ago, I wrote about the variety of partial differential equations with their hyperbolic, parabolic and elliptic character. The point of the essay was noting that parabolic and elliptic equations have some intrinsically unphysical aspects. While this is true to an extent, I failed to note the linearity of these concepts. Some of my wise readers called me on this oversight in particular with respect to nonlinear parabolic equations that exhibit clear wave like behavior. Where linear parabolic equations produce infinite speeds that are clearly unphysical, the nonlinearity of the diffusion produces finite wave speeds and fronts usually associated with compressible flows. The nonlinear diffusion equations are used to model porous media and radiative transfer. The higher the degree of nonlinearity, the sharper and more shock-like the fronts behave. Furthermore, this sort of diffusion can be extremely useful for numerically stabilizing the solution of hyperbolic PDEs. My failure to note this texture was a significant oversight on my part.
One of the points of looking at linear equations is related to the nature of our knowledge of analytical solutions. Our ability to solve problems where the equations are linear is far greater than for nonlinear equations. The most general approach to understanding equations uses the technique of linearization to help understand the nature of solutions. For nonlinear hyperbolic equations this approach has been quite powerful for providing some systematic explanations of the basic character of solutions. As the nonlinearity of the equations grows, the limitations of this common approach grow. Eventually the linearized equations are of little use for practical understanding. Nonlinear diffusion is a powerful example of this. Eventually compressible fluid dynamics also departs from any strong connection to the linear analysis. In particular the nature of turbulence is definitely nonlinear and poorly understood. Moreover, the nonlinearity that produces turbulence is only quadratic, yet eludes any real deep understanding analytically. One can only imagine the mysteries that surround greater nonlinearity.
Other aspects of modeling with PDEs are problematic. One issue that comes up frequently with hyperbolic heat conduction is relativity. Generally, models are not Lorenz invariant. Almost every equation can be recast in a relativistic form to fix this issue. For many cases this has no significant influence on the solutions to the model. In hyperbolic heat conduction the changes return the equations to satisfying the second law of thermodynamics. The speeds in the equations are generally not relativistic, which makes this an extremely curious result. In greater depth, the entire notion that grounds the second law is equilibrium thermodynamics, and the processes in hyperbolic heat conduction are out of equilibrium. We need to carefully apply principles and assumptions when their foundations are being shaken.
Simplicity is a great virtue but it requires hard work to achieve it and education to appreciate it. And to make matters worse: complexity sells better.
― Edsger W. Dijkstra