Meaning of "Gradient with respect to coordinates of particle" in SPH

Question

I'm currently trying to implement a simple SPH simulation based on a variety of papers. However as I'm not a trained physicist nor mathematician I have a small issue with the following notation and formulation repeated in a number of different papers and internet sites:

and $\nabla_a$ denotes the gradient taken with respect to the coordinates of particle $a$.

As far as I can tell this is the first paper using this description.

So what is meant by this notation? Is it simply a vector with a partial derivative for each dimension? Or is it something like a directional derivative?

Answer 1

The explict notation is the folowing.

you have i,j particle indexes, and you have the following definitions:

$\vec r_{ij} = \vec r_j - \vec r_i $ , where $\vec r_i$ is the position of the $i$-th particle.

$W(\vec r_{ij},h_i)$ is the smoothing-kernel evaluated between the 2 SPH particles, using the smoothing parameter of the j-th particle.

The notation he uses is the following:

$\nabla_i W_{ij}(h_i) = \frac{\partial W(\vec r_i - \vec r_j,h_i)}{\partial \vec r_i}$

In (most) hidrodynamic aplications of SPH, the usual convention is that for all i, $h_i=h$ is fixed, and $W(\vec r_{ij},h) = w(\frac{|\vec r_i-\vec r_j|}{h})$, and so, you can think that you have an $W(\vec r)$ and you have $\nabla W = \nabla w(r) = w'(r)\hat r$, ando so, in this case, we write:

$\nabla_i W_{ij}(h_i) = w'(r_{ij},h)\frac{\vec r_i - \vec r_j}{|\vec r_i - \vec r_j|}$

and so, you have an explicit formula for this gradient with respect to the i-th particle

It's possible to formulate an SPH formula with $h_i$ depending on the particle, and on the near particles (through the, implicit, dependence on $\rho_i$), and when you do that, you need to introduce 'correcting schemes', which involve calculating things like $\Omega_i = 1-\frac{\partial h_i}{\partial \rho_i}\sum_k m_k\frac{\partial W}{\partial h_i}(r_{ij},h_i)$

I personally don't like this kind of 'correction scheme' which involves changing the smoothing parameter(or using constant, but different parameters for each particle), but, it's a legit try, and in many occasions it works. Still, it makes difficult to enforce symmetries and conservations laws on the system based only on the equations of movement.

Answer 2

Suppose you have a quantity depending on multiple coordinate sets. $ f(x_i,y_i,z_i,x_j,y_j,z_j)$ the gradient $\nabla_i$ indicates that you must take the gradient with respect to $x_i,y_i,z_i$. A 1 dimensional example to make it explicit. Suppose we have the following $$ f(x_1,x_2,x_3) = \exp(-x_1)*x_2 - x_3 $$ (where $x_i$ denotes the coordinate of the $i$'th particle), then $$\nabla_1 f \equiv \frac{\partial}{\partial x_1} f = -\exp(-x_1)*x_2.$$

Answer 3

Suppose a $n$ dimensional vector space.

Each vector $\vec x$ belonging to this vector space has $n$ coordinates $(x)_{1}, (x)_{2}, ...(x)_{n}$

Suppose now that you have a function $\Phi$ of several vectors : $$\Phi = \Phi (\vec x_a, \vec x_b,..... \vec x_l)$$

Then $\vec \nabla_a \Phi$, for instance, is the vector of coordinates: $$\frac{\partial \Phi}{\partial (x_a)_{1}}, \frac{\partial \Phi}{\partial (x_a)_{2}},....,\frac{\partial \Phi}{\partial (x_a)_{n}}$$.