Simplify calculation of coefficients in moment-kinetic equation

[johnomotani]

Saving the time derivatives of the moments as they are calculated for the time-advance of the moments, and then using those time derivatives for the coefficients in the moment-kinetic equation, simplifies the code, making fewer places that need to be updated if new
terms are added (e.g. if new terms are added to the RHS of the drift-kinetic equation, only the moment equations and the RHS of the moment-kinetic equation change, not the $\dot z$, $\dot w_\parallel$, $\dot F$ coefficients).

Updates expected values in some tests: discretization errors are changed slightly by the rearrangement of the moment kinetic advection coefficients, and the tests are generally poorly resolved (in order to be fast), so are sensitive to this.

There is a question whether making this change is a good idea, with the concern that there may be some desirable cancellations between terms that would take place analytically but might not happen exactly numerically. Possibly the most concerning is the cancellation of the parallel electric field, but I think that one should be exact up to machine precision, as

dupar_dt = dnupar_dt / 	n + ...
                 = (0.5 * Ez * n + ...) / n + ...
wpa_speed = (0.5 * Ez - dupar_dt + ...) / vth

So that we end up evaluating numerically 0.5 * Ez - 0.5 * Ez * n / n - there will be some rounding error, but no discretization error (for other terms there might be a discretization error due to something like d(n*upar)/dz - n*dupar/dz - upar*dn/dz). For the corresponding lines of code, see:

moment_kinetics/moment_kinetics/src/velocity_moments.jl

Line 962 in e8d95c5

dupar_dt[iz,ir,is] = (dnupar_dt[iz,ir,is] - upar[iz,ir,is] * dn_dt[iz,ir,is]) / n[iz,ir,is]

moment_kinetics/moment_kinetics/src/force_balance.jl

Line 31 in e8d95c5

0.5*geometry.bzed[iz,ir]*fields.Ez[iz,ir]*density[iz,ir,is])

moment_kinetics/moment_kinetics/src/vpa_advection.jl

Lines 410 to 413 in e8d95c5

	(0.5 * Ez[iz,ir]
	- (dupar_dt[iz,ir,is] + (vth[iz,ir,is] * wpa + upar[iz,ir,is]) * dupar_dz[iz,ir,is])
	- wpa * (dvth_dt[iz,ir,is] + (vth[iz,ir,is] * wpa + upar[iz,ir,is]) * dvth_dz[iz,ir,is])
	) / vth[iz,ir,is]

[mrhardman]

I am in favour of anything which simplifies the upgrading of the model. Thanks for looking at the possibility that this might work!

Looking in the code it seems that you have $\partial n / \partial t$, $\partial u / \partial t$, and $\partial p / \partial t$ left explicitly in place everywhere, so the equations are now written as the would be if you normalise them in the moment kinetic way but don't substitute for $\partial n / \partial t$, $\partial u / \partial t$, and $\partial p / \partial t$ -- Is this what you mean when you say we no longer have to change $\dot{z}$, $\dot{w_\|}$, and $\dot{F}$ when we update the DKE and the moment eqns?

Is there a way to make things even more general, so that the function which computes $\dot{z}$, $\dot{w_\|}$, and $\dot{F}$ for the case where no moments are split can be used as a starting point to add the moment kinetic corrections to? This way, we would not have to implement the $E \times B$ and magnetic drifts, e.g., in multiple places.

EDIT: I suppose here the challenge is that $w_\|$, $w_\perp$ have different meanings in the moment-split and standard implementations -- but we could probably write a function to compute the drifts that took as input the "true" velocity $v_\|$ and $v_\perp$, which can easily be computed from the normalised values. These functions can be called in each of the $\dot{z}$, $\dot{w_\|}$, and $\dot{F}$ implementations, so at least the drift implementation is in a single place.

[johnomotani]

@mrhardman ah, my comment was not correct (I'll edit to try and clarify). I was only thinking of adding new terms to the RHS of the kinetic equation, not potentially adding drift terms or other modifications to the free-streaming or acceleration. Potentially we could do something along the lines you're suggesting for drift terms - I think it's something to come back to when we're implementing the 2D version of moment-kinetics.

Yes, using arrays with dn/dt, etc. instead of substituting by hand using the moment equations, then evaluating the resulting expressions is what this PR does.

[mrhardman]

@mrhardman ah, my comment was not correct (I'll edit to try and clarify). I was only thinking of adding new terms to the RHS of the kinetic equation, not potentially adding drift terms or other modifications to the free-streaming or acceleration. Potentially we could do something along the lines you're suggesting for drift terms - I think it's something to come back to when we're implementing the 2D version of moment-kinetics.

Yes, using arrays with dn/dt, etc. instead of substituting by hand using the moment equations, then evaluating the resulting expressions is what this PR does.

Yes, in the best case scenario when modifying the advection velocities and accelerations we would have to modify both $\dot{z}$, $\dot{w}_\|$, $\dot{w}_\perp$, and $\partial n/ \partial t$, $\partial u_\|/ \partial t$, and $\partial p/ \partial t$. I think my suggestion was aimed at keeping the effort in those modifications to the theoretical minimum, happy for you to consider this at a later date if that is not the point of this PR.