Spatial Discretization

Spatial Discretization#

Motivation#

Ordinary Differential Equations#

We can often think about “the simplest thing we haven’t tried yet”. For spatial data, before we try to express everything with derivatives, we can think about summarizing the activity with a single parameter. This is the basis of Ordinary Differential Equations (ODEs).

\[ \begin{aligned} \frac{d}{dt}\boldsymbol{x}(t) &= \boldsymbol{f}(\boldsymbol{x}(t), t; \boldsymbol{\theta}) && &&\boldsymbol{x}:\mathbb{R}\rightarrow\mathbb{R}^D && \boldsymbol{f}:\mathbb{R}^D\times\mathbb{R}\times\mathbb{R}^{D_\theta}\rightarrow\mathbb{R}^D \end{aligned} \]

Partial Differential Equations#

In many cases, a simple parameter representation of an entire space is impossible. In this case, we have to resort to PDEs whereby we need a spatial discretization.

Difference#

\[ \partial_x u = \lim_{h \rightarrow 0}\frac{u(x + h) + 2 u(x) - u(x-h)}{2h} \]

Spatial Gradients - dealing with the derivatives

u: Field = ...

du_dx = difference(u, axis, step_size, accuracy, order, method)

N[u](x,t)

Analytical (Symbolic)
Finite difference (Slicing, Convolutions)
Finite Volume
Finite Element
Spectral

Other Gradients#

So there are other methods which have been defined within the physics community which describe all of the motion in space. These are:

\[\begin{split} \begin{aligned} \text{grad}(\boldsymbol{f}) &:= \boldsymbol{\nabla}\boldsymbol{f} : \mathbb{R}\rightarrow\mathbb{R}^D\\ \text{div}(\vec{\boldsymbol{f}}) &:= \boldsymbol{\nabla}\cdot\vec{\boldsymbol{f}} : \mathbb{R}^D\rightarrow\mathbb{R}\\ \text{curl}(\vec{\boldsymbol{f}}) &:= \boldsymbol{\nabla}\times\vec{\boldsymbol{f}} : \mathbb{R}^D\rightarrow\mathbb{R}^D \end{aligned} \end{split}\]

And of course we can derive higher order methods from these, e.g. \(\text{Lap}\) and \(\det \boldsymbol{J}(\cdot,\cdot)\). See the differential operators page for a more in-depth walkthrough of some of the most common differential operators.