Point Vortices: Convergence of the Gauss-Legendre VPRK Methods

Each method is integrated over the same time interval with time steps Δt / 2ⁱ, i = 0, …, 7, plus two further halvings that serve as the reference solution. Plotted are the error of the final state against that reference, the order observed from successive halvings, and the span of the relative energy and angular-momentum error over each run.

The dashed reference line of the expected order is drawn only where the method reports an integer order: it is missing for the projected methods, and for a partitioned method it is the minimum over both halves of the tableau rather than the order the method attains here. The measured order is the one to read.

Point Vortices: Convergence of the Gauss-Legendre VPRK Methods

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

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

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

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

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

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