# How to get started with numerically solving a Stochastic Navier Stokes equation

I originally posted the question on the math stackexchange, and was told I should try here.

I’m researching Stochastic PDE, in particular the Navier Stokes Equation, and would like to estimate the probability of a certain event involving the solution to the SNSE. As such, I’d need to first numerically solve the equation.

I’m an absolute beginner at Numerical Analysis and scientific computing, so I need some advice on getting started. I’ve seen some tutorials on implementing finite differences for deterministic PDE, but I’m having some trouble finding some on SPDE. Most searches resulted in highly technical results (which I understand, given the nature of the subject), but perhaps there are more “beginner friendly” sources that I can use to work my way up.

• Learn ODEs, then PDEs, then SDEs, and then try SPDEs. It's a long path but for a reason: there's a lot of information embedded in the equation that is expected to be understood. Commented Jul 2, 2019 at 11:13
• @Chris Rackauckas, fair enough. It’s probably better I can do all of those anyway. Commented Jul 2, 2019 at 15:48
• @ChrisRackauckas, would you mind adding an answer. Or, if you prefer, you could suggest some references and let the post as "community post". Commented Jul 2, 2019 at 16:39
• I second this. You can't solve an SPDE without solving the PDE first. Learn how people solve the Navier-Stokes equation, and then go from there. Commented Jul 2, 2019 at 23:42

Take a classic paper like this one from Davie and Gaines on solving the stochastic heat equation. By equation (2) they say

We consider finite-difference approximations to (1). The simplest such approximation is the explicit scheme

Equation 2 is then really simple: it's just the 3 point [1 -2 1] stencil of the Laplacian for the spatial part with an Euler-Maruyama type time discretization. And that's the point: the expected knowledge in the field includes that you know something about ODEs, SDEs, and PDEs.

So I would say that, until you know why I would choose the stochastic heat equation as a source before looking at stochastic Navier-Stokes, and until you know why that discretization is "obvious", you should be filling in background knowledge.

• ODEs are how things evolve over time deterministically.
• PDEs are a infinite dimensional extension of ODEs (or an infinite dimensional extension of linear algebra, depending on how you think about it).
• SDEs are an extension of ODEs which describe how things evolve over time stochastically with continuous randomness
• SPDEs are an infinite dimensional extension of SDEs, or a stochastic extension of PDEs.
• Navier-Stokes is a nonlinear PDE, which is even more difficult than a linear PDE, so you'll want some knowledge of that as well.
• Stochsatic Navier-Stokes is a nonlinear SPDE, which means that not only is the rigorous discussion deeply caked in functional analysis discussion, a lot is also unknown.

The numerical methods in the field both take inspiration from PDEs and SDEs given these roots, and those fields both look to ODEs as the simplified case. Thus there really isn't a way to make this more "beginner friendly" because when you're this deep in the hierarchy it's expected that the finite dimensional or deterministic case is something you already understand well, and it's pretty hard to understand how to do SPDEs without understanding the simplified forms.