Guiding Center 4d (Tokamak): 1st Poincaré Integral Invariant with Gauss-Legendre VPRK Methods and Projections
The first Poincaré integral invariant
\[I_{1} = \oint_{\gamma} \vartheta_{i} (q) \, dq^{i}\]
of the four-dimensional guiding centre dynamics, on the same loop as the Gauss-Legendre Runge-Kutta page: a circle of radius $0.1$ in the poloidal plane of the medium-size axisymmetric tokamak about $(R, Z) = (1.75, 0)$, at $u = 0.5$ and $\mu = 10^{-3}$.
Here 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 — comparing the methods that preserve the structure of the dynamics with those that do not is the point of these pages. The internal stage projection has no integrator in GeometricIntegrators 0.16 and its runs fail here, as they do on the trajectory pages.
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 on the Gauss-Legendre Runge-Kutta page. The projection is what separates the runs: over this interval the unprojected method loses 3·10⁻¹ of the invariant at $\Delta t = 10$ and 3·10⁻³ at $\Delta t = 1$, its symmetric projection 3·10⁻⁵ and 5·10⁻¹¹.
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.