Definition of convection in high dimensional burgers equation

[qiauil]

Hi! Felix,

I think there might be an issue with the burgers equation.

The convection term of the burgers equation is defined as

$$ \frac{1}{2} \nabla \cdot \left( u \otimes u \right) $$

However, according to Wikipedia, the definition should be

$$ u \cdot \nabla u $$

Note that $$u \cdot \nabla u = \frac{1}{2} \nabla \cdot \left( u \otimes u \right) $$ only hold for 1d case. For high-dimensional cases,

$$ \nabla \cdot \left( u \otimes u \right) = u \cdot \nabla u + u\nabla \cdot u $$

For example, in 3d case the x component of $\nabla \cdot \left( u \otimes u \right)$ is:

$$ u_x\frac{\partial u_x}{\partial x} +u_y\frac{\partial u_x}{\partial y} +u_z\frac{\partial u_x}{\partial z} +u_x(\frac{\partial u_x}{\partial x} +\frac{\partial u_y}{\partial y} +\frac{\partial u_z}{\partial z}) $$

while the x component of $u \cdot \nabla u$ is only the first part, i.e.,

$$ u_x\frac{\partial u_x}{\partial x} +u_y\frac{\partial u_x}{\partial y} +u_z\frac{\partial u_x}{\partial z} $$

In a word, $$u \cdot \nabla u = \nabla \cdot \left( u \otimes u \right) $$ if u is divergence-free, i.e., $\nabla \cdot u =0$. Otherwise $u \cdot \nabla u=\nabla \cdot \left( u \otimes u \right) - u\nabla \cdot u \neq \frac{1}{2}\nabla \cdot \left( u \otimes u \right)$

[Ceyron]

Hi Qiang,

That's a great observation. Since as you write

$$ \nabla \cdot (\mathbf{u} \otimes \mathbf{u}) = (\mathbf{u} \cdot \nabla) \mathbf{u} + \mathbf{u} (\nabla \cdot \mathbf{u}) $$

the way the Burgers PDE is currently defined clashes with the definition on Wikipedia because there is no additional incompressibility constraint $\nabla \cdot \mathbf{u} \neq 0$.

I think this should definitely be addressed. What do you think about changing the current flag of single_channel in

exponax/exponax/nonlin_fun/_convection.py

Line 20 in beea37e

single_channel: bool = False,

to a mode argument, with mode being one of:

1. "conservative": $b_1 \frac{1}{2} \nabla \cdot (\mathbf{u} \otimes \mathbf{u})$
2. "non_conservative": $b_1 (\mathbf{u} \cdot \nabla) \mathbf{u}$
3. "single_channel": $b_1 \frac{1}{2} (1 \cdot \nabla) (u^2) = b_1 u (1 \cdot \nabla u)$ (similar to the the first equation on Wikipedia)

Then, one could forward those settings to all steppers that build upon the convection nonlinearity (should only be Burgers & KdV if I am not mistaken).

[qiauil]

Hi, Felix.

Nice😼! that could be a good solution! But I am still not sure about the single_channel case. Does that mean we only have one channel for the field, i.e., a scalar field, or does it mean a scalar field on 1d mesh🤔? (I would suppose the latter one, as you can't connect a scalar by itself in high-dimensional space) In the torch version, I implemented conservative and nonconservative forms separately, and they both gave the same result in the 1d case. Could you explain the single_cannel's role here a bit more?

Cheers,
Qiang

[Ceyron]

Re mode switch: Cool 👍. Would you like to create a PR for this feature?

Re single_channel: this is a scalar field in 1D, 2D or 3D. In 1D, all three convection forms should give the same results; it's just how they generalize into higher-dimensional spaces.

[qiauil]

Hi, Felix,

I have added a new pull request, #45, where the non-conservative form of convection is added.

I didn't change the single_channel option since I find it is highly used in the docs and other operators. Changing its name to mode may result in unexpected issues for another part of the code. Also, I noticed that the current single_channel model of the convection operator seems to be the conservative form times 0.5, which might be a little bit confusing when users choose conservative and non-conservative models🤔.

Cheers,
Qiang

[Ceyron]

Hi Qiang,

Thanks a lot for the PR!

I didn't change the single_channel option since I find it is highly used in the docs and other operators. Changing its name to mode may result in unexpected issues for another part of the code.

That's a fair point. We could also keep the two boolean flags.

Also, I noticed that the current single_channel model of the convection operator seems to be the conservative form times 0.5, which might be a little bit confusing when users choose conservative and non-conservative models🤔.

Are you suggesting that the single_channel hack needs to be different for both the conservative and neoconservative form?

$$ b_1 \frac{1}{2} (\vec{1} \cdot \nabla) (u^2) = b_1 u (\vec{1} \cdot \nabla u) $$

I would say that there are no symbolic differences as between the conservative and neoconservative forms in higher dimensions.

[qiauil]

Hi, Felix,

I see the points. It seems that you treat $\frac{1}{2} \nabla(\mathbf{u}\mathbf{u})$ as the conservative form according to your reply and comment in #45 , while I believe $\nabla(\mathbf{u}\mathbf{u})$ is the conservative form. I think $\nabla(\mathbf{u}\mathbf{u})$ is the correct form as $\frac{1}{2} $ is only used to calculate the non-conservative form with conservative form in 1d. I will first fix the conflict issue, and we can modify this point with more discussions.

Cheers,
Qiang

[Ceyron]

Hi Qiang,

I am not 100% sure, but keeping the 1/2 factor seems correct because there shouldn't be a difference between 1D and higher dimensions. I will add some more arguments in favor of this later.

[qiauil]

Hi, Felix.

I just checked some material; I would argue that the burgers equation 1D should also be in non-conservative form. You are right, we can discuss it later.

[Ceyron]

I would argue that the burgers equation 1D should also be in non-conservative form.

In order be consistent with the higher dimensional versions, I agree. How about we keep it as an option, effectively allowing four different permutations: multi-channel conservative, multi-channel nonconservative, single-channel conservative and single-channel nonconservative.

For backward compatibility and since it will then also just a be a nice feature of Exponax I suggest we keep the factor of 1/2 for both single-channel conservative, multi-channel conservative in 1D, and multi-channel conservative in 2D/3D.

This basically means we have the following combinations

setup	1D	2D/3D
non-conservative multi-channel	$b_1 u \partial_x u$	$b_1 (\mathbf{u} \cdot \nabla) \mathbf{u}$
non-conservative single-channel	$b_1 u\partial_x u$	$b_1 u(\vec{1} \cdot \nabla) u$
conservative multi-channel	$b_1 \frac{1}{2} \partial_x (u^2)$	$b_1 \frac{1}{2} \nabla \cdot (\mathbf{u} \otimes \mathbf{u})$
conservative single-channel	$b_1 \frac{1}{2} \partial_x (u^2)$	$b_1 \frac{1}{2} (\vec{1} \cdot \nabla) (u^2)$

This would require having two separate single-channel eval methods for the convection nonlinearity.

The default option should be non-conservative and multi-channel.

What do you think?

[qiauil]

Hi, Felix,

I think it should work; we need to make it clear in the docstring, and then It should be all fine for the users to choose!

[Ceyron]

Amazing,
do you want to make the corresponding changes in #45 ?

[qiauil]

Hi, Felix.

I have added the new feature in #45; please have a check on it.

[qiauil]

Hi, Felix,

The new PR is added as #46, please have a check.

[Ceyron]

PR is merged.