Discrete electric work
The nonrelativistic electric operator corrects a known second-order radial truncation error in its isotropic energy moment. The correction acts on the distribution-function derivative at each electric push. It does not adjust a completed step’s total energy or add an energy-accounting ledger.
Centered-stencil identity
Let \(h=\Delta v\), \(v_j=(j+\tfrac12)h\), \(j=0,\ldots,N-1\), and \(v_g=v_{N-1}+h\). The current code uses odd parity for the lower \(f_1\) ghost cell and zero for the upper ghost cell. Define the real contraction
The uncorrected isotropic electric rate is
Summation by parts on this finite centered grid gives the exact discrete identities
The terms containing \(s_{N-1}\) are the upper-tail contributions of this stencil. They vanish for a resolved distribution whose tail is negligible. The middle energy term is an interior discretization defect and persists even with a vanishing tail. Tests with all electric-field components and both vanishing and finite tails verify these identities directly, including independent harmonic truncation.
Local correction and supported scope
The added energy rate is the exact opposite of the interior defect:
A local density-neutral basis \(\psi_j=(v_j^2-\langle v^2\rangle)f_{0,j}\) realizes it through
This is a distribution-dependent rank-one, \(O(h^2)\) correction for a resolved smooth distribution. It leaves the density rate, higher harmonics, and explicit upper-tail terms unchanged. It uses the same density-neutral moment helper as the ion-frame remap and local exchange. A second centering pass suppresses density cancellation, and a variance-scaled roundoff threshold rejects zero or degenerate radial energy responses. This prevents a single occupied radial node from generating artificial density through division by a rounding-only response. Physical use requires a nonnegative distribution with resolved radial energy spread. Large explicit increments can make \(f_0\) negative; this correction supplies no positivity guarantee. Positivity checks and temporal, radial, spatial, and angular convergence remain necessary.
TzoufrasVlasov(..., conserve_electric_work=True) is the nonrelativistic default.
Setting it to False retains the original operator for controlled comparisons.
The correction requires streaming speed equal to radial momentum, \(v=p\) in the
solver’s nonrelativistic normalization. Other streaming speeds are rejected when
the correction is enabled. BaseVFP2D explicitly disables it in relativistic
mode; a relativistic discrete energy correction is not implemented.
Coupled verification and remaining errors
The finite-field coupling diagnostic is also run to \(t=2\), four times its original interval, on the same \(12\times4\) grid. At \(n_v=32\), \(\Delta t=0.01\), the accounted defect changes from +7.4362% with the original stencil to -0.008376% with the work correction. The percentages use initial transverse magnetic perturbation energy. Halving \(\Delta t\) gives -0.008362%; doubling \(n_v\) to 64 gives -0.002703%. At that finer resolution the raw defect is +1.0518%, with +1.0545% current-projection work recorded separately. Quasineutrality remains below \(3\times10^{-14}\) in these checks.
The declared regression requires less than 0.1% accounted defect at \(t=2\) and more than hundredfold improvement over the uncorrected control. It also bounds the sensitivity to temporal refinement and requires radial refinement to reduce the residual by more than a factor of two, to below 0.004%. These results include the pressure-work fix that removes a duplicate electron-energy source; the old radial-difference bound measured approach to that source’s nonzero error plateau. This interval is about 3.1% of the guide-field Alfvén period, so it is not a full-period kinetic-wave or experiment-duration validation. The ideal magnetic subsystem has a separate full-period test.
The correction does not repair the electron momentum discretization. For an isotropic distribution, for example, its electric-force moment remains
A dedicated test preserves and exposes this residual. The ion magnetic source uses \(c^2(\nabla\times\mathbf B)\times\mathbf B\) directly, so the electron force-moment defect does not enter that source. Electron current projection and its lab-frame work remain separate limitations of the coupled kinetic method.
The earlier short, initially unmagnetized nonlinear check now has an accounted residual near \(9\times10^{-10}\) of its total-energy scale on both coarse and fine grids. Its old radial/spatial error ratios no longer measure convergence of the removed work defect. The updated test tightens its absolute residual bound and limits refinement sensitivity; it does not identify the remaining residual’s cause. Longer flow-time convergence, angular refinement, and projection-free energy closure remain open.