[petsc-users] help me to solve 6-dim PDE with PETSc with advice

[Oleksandr Koshkarov]

Dear All,

I am a new PETSc user and I am still in the middle of the manual (I have 
finally settled to choose PETSc as my main numerical library) so I am 
sorry beforehand if my questions would be naive.

I am trying to solve 6+1 dimensional Vlasov equation with spectral 
methods. More precisely, I will try to solve half-discretized equations 
of the form (approximate form) with pseudospectral Fourier method:

(Equations are in latex format, the nice website to see them is 
https://www.codecogs.com/latex/eqneditor.php)

\frac{dC_{m,m,p}}{dt} =\\
  \partial_x \left ( a_n C_{n+1,m,p} +b_n C_{n,m,p} +c_n C_{n-1,m,p} 
\right ) \\
+ \partial_y \left ( a_m C_{n,m+1,p} +b_m C_{n,m,p} +c_m C_{n,m-1,p}  
\right ) \\
+ \partial_z \left ( a_p C_{n,m,p+1} +b_p C_{n,m,p} +c_p C_{n,m,p-1}  
\right ) \\
+ d_n E_x C_{n-1,m,p} + d_m E_x C_{n,m-1,p} + d_p E_x C_{n,m,p-1} \\
+ B_x (e_{m,p} C_{n,m-1,p-1} + f_{m,p}C_{n,m-1,p+1} + \dots) + B_y 
(\dots) + B_z (\dots)


where a,b,c,d,e,f are some constants which can depend on n,m,p,

n,m,p = 0...N,

C_{n,m,p} = C_{n,m,p}(x,y,z),

E_x = E_x(x,y,z), (same for E_y,B_x,...)

and fields are described with equation of the sort (linear pdes with 
source terms dependent on C):

\frac{dE_x}{dt} = \partial_y B_z - \partial_z B_x + (AC_{1,0,0} + 
BC_{0,0,0}) \\
\frac{dB_x}{dt} = -\partial_y E_z + \partial_z E_x

with A,B being some constants.

I will need fully implicit Crank–Nicolson method, so my plan is to use 
SNES or TS where

I will use one MPI PETSc vector which describes the state of the system 
(all, C, E, B),  then I can evolve linear part with just matrix 
multiplication.

The first question is, should I use something like DMDA? if so, it is 
only 3dimensional but I need 6 dimensional vectots? Will it be faster 
then matrix multiplication?

Then to do nonlinear part I will need 3d mpi-fftw to compute 
convolutions. The problem is, how do I extract subvectors from full big 
vector state? (what is the best approach?)

If you have some other suggestions, please feel free to share

Thanks and best regards,

Oleksandr Koshkarov.