Why guiding-centre interface maps fail at SPECTRE sheets¶

Multi-volume SPECTRE tracing requires integmode > 0 and crossing_level = 3: guiding-centre (GC) motion inside each volume, full orbit (FO) near each current sheet (DOC/spectre-interface-crossing.md). This note derives why the alternatives are not valid models of a sharp sheet: fixed-$\mu$ GC resets across the sheet (the former crossing_level = 0 point-fixed reset and crossing_level = 1 tangential reset, also with RK45 GC), a smoothed Boris layer, and unresolved single-branch steps. The algebra is checked exactly by concept/symplectic_sheet/proofs/verify_rejected_models.py (SymPy); analytic estimates and cited theorems are argued here.

Setting¶

SI units: charge $q\neq0$, mass $m$, parallel velocity $u$, $\mu = m v_\perp^2/(2B)$, $B = |\mathbf B| > 0$ (Gaussian: $q \to q/c$). The stationary sheet $\Sigma$ has $\mathbf B_\pm\cdot n = 0$ and a finite tangential jump. $A$ is continuous and piecewise smooth up to $\Sigma$ with $\nabla\times A = \mathbf B_\pm$; only its derivatives jump. Crossings are isolated and transversal, finitely many per bounded time, without grazing.

T1. GC resets¶

T1a (composition criterion). On the fixed-$\mu$ GC phase spaces let $\Gamma_\pm = (qA + m u b_\pm)\cdot dx$, $\omega_\pm = -d\Gamma_\pm$, $H_\pm = m u^2/2 + \mu B_\pm$, with nondegenerate $\omega_\pm$, transversal flow and a smooth invertible reset $R: S_-\to S_+$ between event hypersurfaces. The flow–reset–flow map at fixed physical time is symplectic iff

\[R^*(\omega_+|_{S_+}) = \omega_-|_{S_-} \quad\text{and}\quad d(H_+\circ R - H_-)|_{TS_-} = 0 .\]

Proof. Split a variation $v = t + aX_-$, $t\in TS_-$. The saltation map sends it to $Sv = DR\,t + aX_+$. With $w = s + bX_-$ and $\Delta H = H_+\circ R - H_-$,

\[\omega_+(Sv,Sw) - \omega_-(v,w) = (R^*\omega_+ - \omega_-)(t,s) + a\,d\Delta H(s) - b\,d\Delta H(t),\]

using $\iota_{X_\pm}\omega_\pm = dH_\pm$; $t,s,a,b$ are independent. Within energy-preserving resets, symplecticity is presymplecticity on the event surface. Energy equality itself is not necessary (a constant shift of $H_+$ leaves its flow unchanged).

T1b (point-fixed resets). Let $R$ keep the surface point and $\mu$ and map $u\mapsto f(x,u)$. Presymplecticity forces $f_u b_+ = b_-$, hence $b_+ = \pm b_-$ and $f_u = \pm1$; with energy conservation and $\mu>0$ also $B_+ = B_-$.

Proof. Evaluate the pulled-back two-form on $(\partial_u, t)$ for tangent $t$: $-m f_u b_+\cdot t = -m b_-\cdot t$ for all $t$. Unit norms give $f_u^2 = 1$; differentiating energy equality gives $f f_u = u$, so $f^2 = u^2$ and $\mu(B_+ - B_-) = 0$. The energy root $f^2 = u^2 - 2\mu(B_+ - B_-)/m$ therefore fails at any direction jump.

Tangential resets escape T1b: for the sheet $z=0$ with $b_- = (1,0,0)$, $b_+ = (\cos\phi,\sin\phi,0)$, $A_- = B(0,-z,0)$, $A_+ = B(z\sin\phi,-z\cos\phi,0)$, the in-plane rotation $R(x,y,u) = (Q_\phi(x,y),u)$ is presymplectic and energy preserving. It is an algebraic counterexample, not a physical displacement. The tangential reset with a scalar energy solve still has no general two-form identity: in this uniform slab the energy residual vanishes at zero displacement, the reset is the identity, and its two-form defect is $m(b_+ - b_-)\cdot dx\wedge du \neq 0$.

T1c (loss of GC ordering). For $b(z) = (\cos\alpha(z),\sin\alpha(z),0)$ and constant $B$,

\[b\cdot\nabla\times b = -\alpha', \qquad B^*_\parallel = B + \frac{m u}{q}\,b\cdot\nabla\times b = B(1 - \rho_\parallel\alpha'), \qquad \rho_\parallel = \frac{m u}{qB},\]

and $\det\omega = (mqB^*_\parallel)^2$ in $(x,y,z,u)$. Let $\alpha = \alpha_- + \Delta\alpha\,s(z/L)$ on $[0,L]$ with monotone $C^1$ $s$, $s(0)=0$, $s(1)=1$, $s'(0)=s'(1)=0$. If $\rho_\parallel\Delta\alpha > 0$ and $L < \rho_\parallel\Delta\alpha$, $B^*_\parallel$ has a zero (since $\max s'\ge1$), where the GC two-form degenerates; at fixed scaled position with $s'>0$, $B^*_\parallel/B$ diverges as $L\to0$. For $\rho_\parallel\Delta\alpha < 0$ there is no zero, but the slow-variation conditions $\rho|\nabla b|\ll1$, $\rho|\nabla\ln B|\ll1$ require $\rho \ll L/\max(|\Delta\alpha|, |\ln(B_+/B_-)|)$. A sharp sheet has $L = 0$, so any finite gyroradius violates the GC ordering at any nonzero jump.

T1d (μ is not conserved). $x$ and $v$ of the full orbit are continuous through a bounded magnetic step. In the equal-$B$ slab of T1b with $v = (v_t, 0, v_n)$, $v_n\neq0$,

\[\mu_+ - \mu_- = \frac{m v_t^2\sin^2\phi}{2B},\]

a finite jump independent of any step size. No fixed-$\mu$ reset reproduces this, whatever its two-form properties.

T2. A smoothed Boris layer changes the problem¶

Replace the jump by a layer of width $w$ and compare with the sharp field at matched entry $x, v$, $|v| = v_0$, $|v_n|\ge\eta v_0$. With $\Omega = |q|B_\text{ref}/m$, $\rho = v_0/\Omega$, $\epsilon = w/\rho$, the exit-velocity discrepancy is $O(\epsilon)$. Using $z$ as independent variable, $dv/dz = (q/m)(v\times\mathbf B)/v_n$, so to first order

\[\frac{\Delta I}{m} = \frac{q}{m v_{n0}}\,v_0\times\int(\mathbf B_w - \mathbf B_0)\,dz + O(v_0\epsilon^2).\]

For $\mathbf B_w = \mathbf B_- + \Delta\mathbf B\,s(z/w)$, $s(\xi) = 6\xi^5 - 15\xi^4 + 10\xi^3$, the integral is $-w\Delta\mathbf B/2$: the leading term does not vanish in general. Exact witness: $\mathbf B = B_y(z)e_y$, $q>0$, $B_- = B_\text{ref}$, $B_+ = 2B_\text{ref}$, entry $v = (0,0,v_0)$, $w = \rho/3$: the sharp exit has $v_x = -2v_0/3$, the layer exit $v_x = -v_0/2$. The Hamiltonian change in matched gauges is only $O(\epsilon)$ ($H_w - H_0 = -q\pi\cdot\delta A/m + q^2|\delta A|^2/(2m)$, $\delta A = O(B_\text{ref}w)$), while the force changes at order one.

With step $h$, $\delta = \Omega h$ and $\kappa = v_0h/w$, resolved Boris needs $\kappa\ll1$ and $\delta\ll1$; the per-crossing error relative to the sharp target is $O(\epsilon + \delta^2/\epsilon)$ at cost $O(1/\delta)$ steps per gyration. Width and step must both converge; a fixed width converges to a different field.

T3. A single-branch step across a sheet is first order¶

Let a step $h$ cross at fraction $\theta\in(0,1)$ and use one side’s field throughout. With $[[f]] = f_+ - f_-$ the leading local error is $-(1-\theta)h[[f]]$ (incoming branch) or $\theta h[[f]]$ (outgoing branch). For uniform fields with $G_\pm v = (q/m)v\times\mathbf B_\pm$, the exact map is $e^{(1-\theta)hG_+}e^{\theta hG_-} = I + h(\theta G_- + (1-\theta)G_+)

O(h^2)$, while Boris applies the Cayley rotation $C_h(G) = (I - hG/2)^{-1}(I + hG/2) = I + hG + O(h^2)$ of one side. $C_h^TC_h = I$: exact speed conservation does not remove the error. A midpoint-branch choice has coefficient $\theta$ or $-(1-\theta)$, with mean zero and variance $1/12$ for uniform $\theta$. If such kicks of size $D = |[[f]]|$ were independent, $N$ crossings would accumulate $\sqrt N hD/\sqrt{12}$; coherent accumulation $O(NhD)$ is also possible. Exact event splitting has no such term.

T4. RK45 GC does not repair the reset¶

Decreasing the RK45 tolerance with a fixed-$\mu$ reset converges to the exact GC flows composed with the same reset. T1c and T1d hold for exact per-volume flows, so the model error remains.

T5. Full-orbit alternatives¶

T5a (exact FO flow). $H = |p - qA|^2/(2m)$ has a symplectic, energy-preserving flow across the sheet. In each volume $\dot Z = J\nabla H$. Continuity of $A$ makes $x$, $p$, $H$ continuous at the event; the gradient jump of $H$ is normal to the event surface, so the saltation matrix is $S = \bigl(\begin{smallmatrix}I&0\\ c\,nn^T&I\end{smallmatrix}\bigr)$ with $S^TJS = J$. A cyclic coordinate in a symmetry-compatible matched gauge keeps its canonical momentum; this is why SIMPLE maps $p_\zeta$ across sheets by the $A_\zeta$ jumps in axisymmetric fields.

T5b (resolved Poincaré Gauss 2). For smooth $K = g(x)(H - E)$ with $0 < g_\text{min} \le g$, two-stage Gauss–Legendre ($b_i = 1/2$, $a_{11} = a_{22} = 1/4$, $a_{12} = 1/4 - \sqrt3/6$, $a_{21} = 1/4 + \sqrt3/6$) is symplectic of order four in fictitious time, and under the usual analyticity and compactness assumptions $|K_n - K_0| \le Ch_\tau^4$ over exponentially long times (Hairer, Lubich and Wanner 2006). On a shell $K = k$, $X_K/g = X_{H - k/g}$, so $|H_n - E| \le Ch_\tau^4/g_\text{min}$. The step must resolve every scale of $K$, $h_\tau \ll L/(g|v_n|)$; smooth backward-error analysis does not cover a sharp sheet.

T5c (event-resolved FO RK45). With finitely many uniformly transversal crossings, piecewise smooth Lipschitz fields and exact event landing, RK45 converges as step and tolerance go to zero (induction over events, using continuous dependence of event times on the incoming state). It is not symplectic: explicit RK cannot satisfy $2b_ia_{ii} = b_i^2$ with $a_{ii} = 0$. For the harmonic oscillator, Dormand–Prince ($R(z) = \sum_{j\le5}z^j/j! + z^6/600$) gives $|R(i\eta)|^2 = 1 - \eta^6/1800 + \eta^8/1600 - \eta^{10}/14400 + \eta^{12}/360000$, an energy drift linear in the number of steps for small $N\eta^6$. SIMPLE uses this method for the bounded FO phases near sheets.

References¶

E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration, 2nd ed., Springer (2006), https://doi.org/10.1007/3-540-30666-8
E. Hairer, Variable time step integration with symplectic methods, Appl. Numer. Math. 25, 219 (1997), https://doi.org/10.1016/S0168-9274(97)00061-5