Guiding Center 4d (Symmetric Field): 2nd Poincaré Integral Invariant with Gauss-Legendre VPRK Methods and Projections
The second Poincaré integral invariant
\[I_{2} = \int_{S} \omega_{ij} (q) \, dq^{i} \wedge dq^{j}\]
of the four-dimensional guiding centre dynamics in the axisymmetric quadratic field $B (x, y, z) = B_{0} (1 + x^{2} + y^{2}) \, e_{z}$, whose coordinates are cartesian, so that the figures show the advected surface in the coordinates it is defined in. The surface is
\[q (\sigma, \tau) = \big( 0.5 (\sigma - 1/2) , \; 0.5 (\tau - 1/2) , \; 0.1 \cos 2 \pi \sigma \cos 2 \pi \tau , \; 0.5 + 0.01 \sin 2 \pi \sigma \sin 2 \pi \tau \big)\]
at magnetic moment $\mu = 10^{-2}$: unlike the flat tokamak surface it is curved in the third component and varies in the parallel velocity as well.
The dynamics is integrated in its variational formulation, with the Gauss-Legendre variational partitioned Runge-Kutta methods and each of the projections onto the constraint submanifold. This is the geometry the pre-0.2 gallery ran with the projected methods only, which is why it has no Gauss-Legendre Runge-Kutta page.
Each method is run over the same time interval at four time steps, $\Delta t \in \{10, 5, 2, 1\}$, whose relative errors are drawn as four curves in one figure. As for the first invariant, this equilibrium is the more benign of the two, and it is the methods rather than the time steps that the figures separate. The time interval and the number of sample points are reduced from the published ones, for the reasons given at the top of src/guiding-center-4d-poincare.jl.
VPRKGauss(1)
SymplecticGauss(1) with standard projection
SymplecticGauss(1) with symmetric projection
SymplecticGauss(1) with symplectic projection
SymplecticGauss(1) with midpoint projection
SymplecticGauss(1) with internal stage projection
Δt = 10.0 failed: MethodError.
Δt = 5.0 failed: MethodError.
Δt = 2.0 failed: MethodError.
Δt = 1.0 failed: MethodError.
No time step completed; no figures for this run.
VPRKGauss(2)
SymplecticGauss(2) with standard projection
SymplecticGauss(2) with symmetric projection
SymplecticGauss(2) with symplectic projection
SymplecticGauss(2) with midpoint projection
SymplecticGauss(2) with internal stage projection
Δt = 10.0 failed: MethodError.
Δt = 5.0 failed: MethodError.
Δt = 2.0 failed: MethodError.
Δt = 1.0 failed: MethodError.
No time step completed; no figures for this run.
VPRKGauss(3)
SymplecticGauss(3) with standard projection
SymplecticGauss(3) with symmetric projection
SymplecticGauss(3) with symplectic projection
SymplecticGauss(3) with midpoint projection
SymplecticGauss(3) with internal stage projection
Δt = 10.0 failed: MethodError.
Δt = 5.0 failed: MethodError.
Δt = 2.0 failed: MethodError.
Δt = 1.0 failed: MethodError.
No time step completed; no figures for this run.
VPRKGauss(4)
SymplecticGauss(4) with standard projection
SymplecticGauss(4) with symmetric projection
SymplecticGauss(4) with symplectic projection
SymplecticGauss(4) with midpoint projection
SymplecticGauss(4) with internal stage projection
Δt = 10.0 failed: MethodError.
Δt = 5.0 failed: MethodError.
Δt = 2.0 failed: MethodError.
Δt = 1.0 failed: MethodError.
No time step completed; no figures for this run.
VPRKGauss(5)
SymplecticGauss(5) with standard projection
SymplecticGauss(5) with symmetric projection
SymplecticGauss(5) with symplectic projection
SymplecticGauss(5) with midpoint projection
SymplecticGauss(5) with internal stage projection
Δt = 10.0 failed: MethodError.
Δt = 5.0 failed: MethodError.
Δt = 2.0 failed: MethodError.
Δt = 1.0 failed: MethodError.
No time step completed; no figures for this run.
VPRKGauss(6)
SymplecticGauss(6) with standard projection
SymplecticGauss(6) with symmetric projection
SymplecticGauss(6) with symplectic projection
SymplecticGauss(6) with midpoint projection
SymplecticGauss(6) with internal stage projection
Δt = 10.0 failed: MethodError.
Δt = 5.0 failed: MethodError.
Δt = 2.0 failed: MethodError.
Δt = 1.0 failed: MethodError.
No time step completed; no figures for this run.