pyGP_fields

ImEx implementation for Gross–Pitaevskii like fields. Here we define few classes for implementation of ImEx methods for Gross–Pitaevskii like equations. By Gross–Pitaevskii like we mean partial differential equations of following type: $$\frac{\partial \psi}{\partial t} = i \alpha(\nabla^n\psi) + f(\psi,t,\vec{x}) $$ where $\alpha$ is not dependent on space-variables($\vec{x}$), but it can depend on $t$.

We use (diagonally) Implicit-Explicit(ImEx) Runge-Kutta(RK) time stepping schemes with pseudospectral space discretization. For a quick introduction to both of these topics(and further resources), please see David Ketcheson's short course. We borrow few examples from David's repo and the code here draws inspirations from his code.

Below we outline basic steps for implementation of an ImEx scheme(in context of our notation here):

ImEx notation: we denote the RK coefficients with letters $A$, $B$ and $C$ with following correspondence with Butcher Tableau:

  |
C |     A
  |
------------
  |    B

For ImEx, we use superscript to denote if this set of coefficients belongs to implicit or explicit table of ImEx method. For example $A_{ij}^{ex}$ is $(i,j)th$ component of $A$ of explicit tableau. A ImEx class object holds these coefficients in the variables im_A,ex_A,im_B,ex_B,im_C,ex_C. Please see its read section on how to use that class.

If at current time($t$), solution is $\psi_t$ and we want to find solution $\psi_{t+dt}$ at time $t+dt$, we do:

For an ImEx scheme with $s$ stages:

At each stage i:

$$\psi_k = \psi + (dt)\sum_{j=1}^{i-1}(A_{ij}^{im}K_j^{im} + A_{ij}^{ex}K_j^{ex}) + i(dt)A_{ii}^{im}\alpha\nabla^n\psi_k$$

Defining, $$f_r = \psi + (dt)\sum_{j=1}^{i-1}(A_{ij}^{im}K_j^{im} + A_{ij}^{ex}K_j^{ex})$$ we get $$\psi_k = f_r + i(dt)A_{ii}^{im}\alpha\nabla^n\psi_k$$ Going to Fourier space, denoting corresponding quantities in Fourier space with hat($\hat{()}$): $$\hat{\psi_k} = \frac{\hat{f_r}}{\left[1+i(dt)(A_{ii}^{im})\Lambda\right]}$$ where $$\Lambda\equiv -\alpha (ik)^n$$

We find $\psi_k$ by doing inverse FFT. Then we update $K_i^{im}$ and $K_i^{ex}$, by evaluating $i \alpha(\nabla^n\psi_k)$ and $f(\psi_k,t,\vec{x})$ i.e. given RHS of given PDE with $\psi_k$.

After $s$ stages, the contribution($K$) from all stages is summed over with respective weights($B$) of Butcher Tableau: $$\psi_{t+dt} = \psi_{t} +(dt)\sum_{j=1}^{s}(B_{j}^{im}K_j^{im} + B_{j}^{ex}K_j^{ex})$$

The main class is GPE_scalar_field in GPE. It implements the above mentioned steps in a minimalistic way. Please see the associated information there to see how above steps are executed using 3 different functions in that class and how to use that class.

Here we provide examples from 1-d,2-d and 3-d cases on how to use these classes. All the example files in this repo have phrase "example" in their names along with dimensions. For example RSOC_2d_example is an example of 2-d Rasbha-Spin Orbit coupling case.