Generic PDE Terms

[jejjohnson]

This will feature some fairly generic PDE terms that keep cropping up that we may want to solve.

Diffusion

Some generic Diffusion equation ν (∂²η/∂x² + ∂²η/∂y²) that we may want to solve

$$ \begin{aligned} &=\nu \left( \frac{\partial^2 \eta}{\partial x^2} + \frac{\partial^2 \eta}{\partial y^2} + \frac{\partial^2 \eta}{\partial z^2}\right) \end{aligned} $$

• Naive
  • 1D
  • 2D

Advection

Some generic advection terms that we may want to solve, i.e. u ∂η/∂x + v ∂η/∂y + w ∂η/∂z

$$ \begin{aligned} &=u \frac{\partial \eta}{\partial x} + v \frac{\partial \eta}{\partial y} + w \frac{\partial \eta}{\partial z} \end{aligned} $$

• Naive
  • 1D
  • 2D
• Upwind
  • 1D
  • 2D
• #21

Note: I've also seen instantiations of this with the name "determinant Jacobian".

[roxyboy]

I think the generic advection term should read u da/dx + v db/dy (+ w dc/dz) as advection always entails a gradient of 'something' being transported by u and v (and w) and not the other way around.

[jejjohnson]

Makes sense. But wouldn’t the a b (and c) be the same variable? Mainly the advection term from the definition of the Jacobian/advection term part?

I think I was a naive with the “generic advection term” label because I did it thinking about the tutorial not the actual PDEs, e.g. shallow water, QG, Material derivative, etc.

[roxyboy]

I guess it depends on the level of "generic" but yes, I'd only envision a use case with a=b=c.

[jejjohnson]

Closing due to #22 and #20