electron_kinetic_equation

add_wall_boundary_condition_to_Jacobian!(jacobian, phi, pdf, ppar, vthe, upar, z,
                                         vperp, vpa, vperp_spectral, vpa_spectral,
                                         vpa_adv, moments, vpa_diffusion, me_over_mi,
                                         ir)

All the contributions that we add in this function have to be added with a -'ve sign so that they combine with the 1 on the diagonal of the preconditioner matrix to make rows corresponding to the boundary points which define constraint equations imposing the boundary condition on those entries of δg (when the right-hand-side is set to zero).

calculates the contribution to the residual of the electron kinetic equation from the z advection term: residual = zdot * d(pdf)/dz + wpadot * d(pdf)/dwpa - pdf * prefactor INPUTS: zadvectionterm = dummy array to be filled with the contribution to the residual from the z advection term pdf = modified electron pdf used in the kinetic equation = (true electron pdf / dense) * vthe vthe = electron thermal speed z = z grid vpa = vparallel grid zspectral = spectral representation of the z grid scratchdummy = dummy array to be used for temporary storage OUTPUT: zadvectionterm = updated contribution to the residual from the z advection term

electron_adaptive_timestep_update!(scratch, t, t_params, moments, phi, z_advect,
                                   vpa_advect, composition, r, z, vperp, vpa,
                                   vperp_spectral, vpa_spectral,
                                   external_source_settings, num_diss_params;
                                   evolve_ppar=false)

Check the error estimate for the embedded RK method and adjust the timestep if appropriate.

electron_backward_euler!(old_scratch, new_scratch, moments, phi,
    collisions, composition, r, z, vperp, vpa, z_spectral, vperp_spectral,
    vpa_spectral, z_advect, vpa_advect, scratch_dummy, t_params,
    external_source_settings, num_diss_params, nl_solver_params, ir;
    evolve_ppar=false, ion_dt=nothing) = begin

Take a single backward euler timestep for the electron shape function $g_e$ and parallel pressure $p_{e∥}$.

Update the electron distribution function using backward-Euler for an artifical time advance of the electron kinetic equation until a steady-state solution is reached.

Note that this function does not use the runge_kutta timestep functionality. t_params.previous_dt[] is used to store the (adaptively updated) initial timestep of the pseudotimestepping loop (initial value of t_params.dt[] within electron_backward_euler_pseudotimestepping!()). t_params.dt[] is adapted according to the iteration counts of the Newton solver.

electron_kinetic_equation_euler_update!(f_out, ppar_out, f_in, ppar_in, moments,
                                        z, vperp, vpa, z_spectral, vpa_spectral,
                                        z_advect, vpa_advect, scratch_dummy,
                                        collisions, composition,
                                        external_source_settings,
                                        num_diss_params, t_params, ir;
                                        evolve_ppar=false, ion_dt=nothing,
                                        soft_force_constraints=false)

Do a forward-Euler update of the electron kinetic equation.

When evolve_ppar=true is passed, also updates the electron parallel pressure.

Note that this function operates on a single point in r, given by ir, and f_out, ppar_out, f_in, and ppar_in should have no r-dimension.

fill_electron_kinetic_equation_Jacobian!(jacobian_matrix, f, ppar, moments,
                                         collisions, composition, z, vperp, vpa,
                                         z_spectral, vperp_specral,
                                         vpa_spectral, z_advect, vpa_advect,
                                         scratch_dummy, external_source_settings,
                                         num_diss_params, t_params, ion_dt,
                                         ir, evolve_ppar, include=:all)

Fill a pre-allocated matrix with the Jacobian matrix for electron kinetic equation and (if evolve_ppar=true) the electron energy equation.

add_identity=false can be passed to skip adding 1's on the diagonal. Doing this would produce the Jacobian for a steady-state solve, rather than a backward-Euler timestep solve. [Note that rows representing boundary points still need the 1 added to their diagonal, as that 1 is used to impose the boundary condition, not to represent the 'new f' in the time derivative term as it is for the non-boundary points.]

fill_electron_kinetic_equation_v_only_Jacobian!()
    jacobian_matrix, f, ppar, dpdf_dz, dpdf_dvpa, z_speed, moments, zeroth_moment,
    first_moment, second_moment, third_moment, dthird_moment_dz, phi, collisions,
    composition, z, vperp, vpa, z_spectral, vperp_spectral, vpa_spectral, z_advect,
    vpa_advect, scratch_dummy, external_source_settings, num_diss_params, t_params,
    ion_dt, ir, iz, evolve_ppar)

Fill a pre-allocated matrix with the Jacobian matrix for a velocity-space solve part of the ADI method for electron kinetic equation and (if evolve_ppar=true) the electron energy equation.

fill_electron_kinetic_equation_z_only_Jacobian_f!(
    jacobian_matrix, f, ppar, dpdf_dz, dpdf_dvpa, z_speed, moments, zeroth_moment,
    first_moment, second_moment, third_moment, dthird_moment_dz, collisions,
    composition, z, vperp, vpa, z_spectral, vperp_spectral, vpa_spectral, z_advect,
    vpa_advect, scratch_dummy, external_source_settings, num_diss_params, t_params,
    ion_dt, ir, ivperp, ivpa, evolve_ppar)

Fill a pre-allocated matrix with the Jacobian matrix for a z-direction solve part of the ADI method for electron kinetic equation and (if evolve_ppar=true) the electron energy equation.

fill_electron_kinetic_equation_z_only_Jacobian_ppar!(
    jacobian_matrix, ppar, moments, zeroth_moment, first_moment, second_moment,
    third_moment, dthird_moment_dz, collisions, composition, z, vperp, vpa,
    z_spectral, vperp_spectral, vpa_spectral, z_advect, vpa_advect, scratch_dummy,
    external_source_settings, num_diss_params, t_params, ion_dt, ir, evolve_ppar)

Fill a pre-allocated matrix with the Jacobian matrix for a z-direction solve part of the ADI method for electron kinetic equation and (if evolve_ppar=true) the electron energy equation.

get_electron_split_Jacobians!(ivperp, ivpa, ppar, moments, collisions, composition,
                              z, vperp, vpa, z_spectral, vperp_spectral, vpa_spectral,
                              z_advect, vpa_advect, scratch_dummy,
                              external_source_settings, num_diss_params, t_params,
                              ion_dt, ir, evolve_ppar

Fill a pre-allocated matrix with the Jacobian matrix for electron kinetic equation and (if evolve_ppar=true) the electron energy equation.

implicit_electron_advance!()

Do an implicit solve which finds: the steady-state electron shape function $g_e$; the backward-Euler advanced electron pressure which is updated using $g_e$ at the new time-level.

The r-dimension is not parallelised. For 1D runs this makes no difference. In 2D it might or might not be necessary. If r-dimension parallelisation is needed, it would need some work. The simplest option would be a non-parallelised outer loop over r, with each nonlinear solve being parallelised over {z,vperp,vpa}. More efficient might be to add an equivalent to the 'anyv' parallelisation used for the collision operator (e.g. 'anyzv'?) to allow the outer r-loop to be parallelised.

updateelectronpdf is a function that uses the electron kinetic equation to solve for the updated electron pdf

The electron kinetic equation is: zdot * d(pdf)/dz + wpadot * d(pdf)/dwpa = pdf * pre_factor

INPUTS:
scratch = `scratch_pdf` struct used to store Runge-Kutta stages
pdf = modified electron pdf @ previous time level = (true electron pdf / dens_e) * vth_e
dens = electron density
vthe = electron thermal speed
ppar = electron parallel pressure
ddens_dz = z-derivative of the electron density
dppar_dz = z-derivative of the electron parallel pressure
dqpar_dz = z-derivative of the electron parallel heat flux
dvth_dz = z-derivative of the electron thermal speed
z = struct containing z-coordinate information
vpa = struct containing vpa-coordinate information
z_spectral = struct containing spectral information for the z-coordinate
vpa_spectral = struct containing spectral information for the vpa-coordinate
scratch_dummy = dummy arrays to be used for temporary storage
dt = time step size
max_electron_pdf_iterations = maximum number of iterations to use in the solution of the electron kinetic equation
ion_dt = if this is passed, the electron pressure is evolved in a form that results in
         a backward-Euler update on the ion timestep (ion_dt) once the electron
         pseudo-timestepping reaches steady state.

OUTPUT: pdf = updated (modified) electron pdf

use Picard iteration to solve the electron kinetic equation

The electron kinetic equation is: zdot * d(pdf)/dz + wpadot * d(pdf)/dwpa = pdf * pre_factor Picard iteration uses the previous iteration of the electron pdf to calculate the next iteration: zdot * d(pdf^{i+1})/dz + wpadot^{i} * d(pdf^{i})/dwpa = pdf^{i} * prefactor^{i}

INPUTS: pdf = modified electron pdf @ previous time level = (true electron pdf / dense) * vthe dens = electron density vthe = electron thermal speed ppar = electron parallel pressure ddensdz = z-derivative of the electron density dppardz = z-derivative of the electron parallel pressure dqpardz = z-derivative of the electron parallel heat flux dvthdz = z-derivative of the electron thermal speed z = struct containing z-coordinate information vpa = struct containing vpa-coordinate information zspectral = struct containing spectral information for the z-coordinate vpaspectral = struct containing spectral information for the vpa-coordinate scratchdummy = dummy arrays to be used for temporary storage maxelectronpdfiterations = maximum number of iterations to use in the solution of the electron kinetic equation OUTPUT: pdf = updated (modified) electron pdf

updateelectronpdfwithshooting_method is a function that using a shooting method to solve for the electron pdf

The electron kinetic equation is: zdot * d(pdf)/dz + wpadot * d(pdf)/dwpa = pdf * prefactor The shooting method is 'explicit' in z, solving zdoti * (pdf{i+1} - pdf{i})/dz{i} + wpadot{i} * d(pdf{i})/dwpa = pdf{i} * prefactor{i}

INPUTS:
pdf = modified electron pdf @ previous time level = (true electron pdf / dens_e) * vth_e
dens = electron density
vthe = electron thermal speed
ppar = electron parallel pressure
ddens_dz = z-derivative of the electron density
dppar_dz = z-derivative of the electron parallel pressure
dqpar_dz = z-derivative of the electron parallel heat flux
dvth_dz = z-derivative of the electron thermal speed
z = struct containing z-coordinate information
vpa = struct containing vpa-coordinate information
vpa_spectral = struct containing spectral information for the vpa-coordinate
scratch_dummy = dummy arrays to be used for temporary storage

OUTPUT: pdf = updated (modified) electron pdf

updateelectronpdfwithtime_advance is a function that introduces an artifical time derivative to advance the electron kinetic equation until a steady-state solution is reached.

The electron kinetic equation is: zdot * d(pdf)/dz + wpadot * d(pdf)/dwpa = pdf * pre_factor

INPUTS:
pdf = modified electron pdf @ previous time level = (true electron pdf / dens_e) * vth_e
dens = electron density
vthe = electron thermal speed
ppar = electron parallel pressure
ddens_dz = z-derivative of the electron density
dppar_dz = z-derivative of the electron parallel pressure
dqpar_dz = z-derivative of the electron parallel heat flux
dvth_dz = z-derivative of the electron thermal speed
z = struct containing z-coordinate information
vpa = struct containing vpa-coordinate information
z_spectral = struct containing spectral information for the z-coordinate
vpa_spectral = struct containing spectral information for the vpa-coordinate
scratch_dummy = dummy arrays to be used for temporary storage
max_electron_pdf_iterations = maximum number of iterations to use in the solution of the electron kinetic equation
io_electron = info struct for binary file I/O
initial_time = initial value for the (pseudo-)time
ion_dt = if this is passed, the electron pressure is evolved in a form that results in
         a backward-Euler update on the ion timestep (ion_dt) once the electron
         pseudo-timestepping reaches steady state.

OUTPUT: pdf = updated (modified) electron pdf