Some Questions about Brusselator Example

[yangheart]

Dear Dr. Zwicker,

I am really appreciated that you developed such an amazing package for us to solve PDE. I am really interested in the example, called Brusselator - Using the PDE class. But there are conditions there which I am still confused about. Could you help me to figure out them?

Firstly, does t_range=20 mean t $\in$ [0,20] or mean t$\in$ [0, 20dt]?

Secondly, for bc="natural", does it mean there is no bounday condition?

Thirdly, If I set up bc="neumann", does it means neumann condition is 0?

Thank you so much!

[david-zwicker]

Great to hear that you like our work! To your questions:

1. The former: t_range specifies the time range in time units not in the number of time steps
2. Natural boundary conditions are 'periodic' for periodic grid axes and 'neumann' for non-periodic grid axes
3. Indeed, 'neumann' implies a vanishing derivative at the boundary

You can read more about boundary conditions in the documentation.

[david-zwicker]

You can simply set an interval for the tracker that is smaller than your timestep dt. When doing this, you might want to use the latest master branch of the package, since we fixed a bug that might slow down your simulation in this case.

[yangheart]

Dear Dr. Zwicker,

Sorry for disturbing again. I am still confused about the conditions of this example. From the paper, it said that: "The standard PDE classes support additive Gaussian white noise, but alternatives, including multiplicative noise, can be specified in userdefined classes." Does it mean that the main body of the PDE in the Brusselator Example involves a Gaussian noise term?

Thank you so much!

[david-zwicker]

Which version of the Brusselator Example do you refer to? Could you give a link to the example? Generally, the py-pde package support stochastic partial differential equations, but the actual implementations vary a bit. Also, the paper does not refer to the newest version of the package, so some of its descriptions might be outdated.

[yangheart]

Here is the example of Brusselator that I refer to:
https://py-pde.readthedocs.io/en/latest/examples_gallery/pde_brusselator_expression.html#sphx-glr-examples-gallery-pde-brusselator-expression-py

The equation on this page seems to be deterministic since the white noises are only added to the initial values. I am wondering if any stochastic terms are involved in the main body of the equation besides the initial values?

[david-zwicker]

This example solves the deterministic equations for the Brusselator, starting from random initial conditions, as you suspected. However, if you add the noise argument to the call to PDE, you wind up with the stochastic version, as described in the documentation of PDE. In this case uncorrelated Gaussian white noise is added to both fields and the value supplied by the noise argument sets the noise strength.