Limitations with dynamical systems vs. PDEs?

Question

As a computational scientist, are there limitations to studying dynamical systems — systems of odes in which each state variable evolves with time — compared to studying PDEs?

For instance, it seems that modeling blood flow in the heart is done with PDEs, but in some research papers for fluid-structure interactions for a quasi-2d problem, the modeling is done with a set of ODEs. Which l like; it's easy to learn, it makes sense, and it's accurate and matches physical experiments well.

Could I specialize in dynamical systems modeling and theory and computation (numerical methods) as a computational scientist without being severely limited? Is it, for practical purposes, necessary to learn PDEs and numerical methods for solving PDEs?

I would especially welcome an answer in the context of an example!

Answer 1

PDEs are a form of dynamical system where there is another continuous variable. Usually this is space, so you're looking at how things over time and space instead of just over time.

Here's an example of generalizing an ODE to a PDE. Take your ODE model of chemical reactions which models the concentration of certain chemicals over time. Now generalize the model so that there is a whole tub of these chemicals, but it's not evenly mixed. Things are diffusing around, maybe there's some advective (mean) flow. Now look at how those chemical concentrations are evolving over time at each point in space: this is the PDE generalization (and in this case, it's a PDE u_t = N(u) + f(u) where the f is just the ODE!).

How many foxes and rabbits are there? -> How many foxes and rabbits are there in a a given location?

You can keep going. Conclusion, it's still just a dynamical system, though it has another element to it above just an ODE. Is it necessary to learn PDEs? Depends on if it's necessary for you to use them. Is it necessary for you to use PDEs? That depends on the scientific question you are trying to answer.

If you're a student, I will say that there are enough questions that require a PDE that it can be very useful to learn some of the methods of PDEs. Mathematically, they are a bit more involved than ODEs, but the interesting thing is that the most common way to handle PDEs is... to turn them into ODEs. The finite difference method turns a PDE into a system of ODEs at given points in space. Galerkin or finite element methods pick a basis and expand the infinite dimensional continuous differential operators along the basis to arise at a finite dimensional discrete differential operator... and a discrete differential operator is either just a (non)linear system of equations or an ODE. There are more advanced ways to do expansions as well. The classic Lorenz equation is actually a simplified form of Navier-Stokes, pulling out the interesting dynamics of turbulence in fluids.

So in conclusion, don't avoid PDEs: they are natural and show up in models as they get progressively more detailed. And handling PDEs is "just" a generalization of ODEs. I say "just" since there are some deep mathematical details that need to be addressed when moving to infinite dimensional continuous operators, but that's where non-trivial behavior may not be captured by ODEs!