Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am asking only out of curiosity, I did sufficient fluid mechanics to be familiar with shockwaves and the such, but I'm wondering how it relates to the numerical qualities of hyperbolic pdes in general. Wikipedia said the Advection Equation is also difficult to solve numerically, as it is hyperbolic, though it seems 'nicer' than say the Riemann Equation on casual inspection.

share|improve this question

1 Answer 1

Some hyperbolic PDE's are inherently ill-posed because the physical conditions which the equations are modeling are actually elliptic equations. For example, when a hyperbolic equation is modeling a wave equation in a medium of high losses, say, a medium with a complex index of refraction $n$ such that $\Re{n} < \Im{n}$, the wave equation is actually elliptic and imposing Cauchy boundary conditions results in an ill-posed problem.

Fortunately, there are numerous methods of transforming such wave equations, either using differential methods, or finite methods, so that such problems result in a stable solution. Such methods are usually designed for a particular application (e.g., wave propagation through optical materials).

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.