HeLaZ (Hermite-Laguerre Z-pinch solver, 2020)

![] (phi_demo.gif)

Roadmap : (Current version 1.4)

0. Write MOLI matlab solver in Fortran using Monli1D as starting point

   0.0 go from 1D space to 2D fourier and from Hermite basis to Hermite-Laguerre basis

   0.1 implement linear Poisson equation in fourier space

   0.2 implement moment hierarchy linear terms

   0.3 RK4 time solver

   0.4 Benchmark with MOLI matlab results for Z-pinch (cf. kz_linear script) Note : benchmark_*.m compares MOLI and HeLaZ linear results

   0.5 Load COSOlver matrices

   0.6 Benchmarks now include Dougherty, Lenard-Bernstein and Full Coulomb collision operators Note : for full Coulomb, one must store a precomputed matrix from COSOlver in the iCa folder

1. Implementation of the non linear Poisson brackets term

   1.0 FFTW3 has been used to treat the convolution as a product and discrete fourier transform

   1.1 Methods in fourier_mod.f90 have been validated by tests on Hasegawa Wakatani system

   1.1 Qualitative test : find similar turbulences as Hasegawa Wakatani system with few moments

   1.2 Zonal flows are observed in a similar way to Ricci Rogers 2006 with GS2

   1.3 Linear analysis showed that a certain amount of PJ are recquired to trigger mode

    1.3.1 The \eta_B = 0.5 case is easier since it converged better in linear analysis than \eta_B = 1.0
   
    1.3.2 Collisionality helps

   1.4 Quantitative study with stationary average particle flux \Gamma_\infty

    1.4.1 Convergence study of \Gamma_\infty w.r.t. P and J
   
    1.4.2 Direct comparison with GS2 results of Ricci,Rogers 2006