Discontinuities

A discontinuity of the discrete trajectory, together with the means of integrating a Lagrangian across it.

Discontinuous Galerkin variational integrators approximate the trajectory by a piecewise polynomial that need not be continuous at the interval boundaries. The resulting non-conservative product of the one-form and the jump is given a meaning by integrating along a path $\Phi$ connecting the two one-sided limits $q^-$ and $q^+$,

\[\int \limits_0^1 \vartheta \big( \Phi(\tau; q^-, q^+) \big) \cdot \frac{d \Phi}{d\tau} (\tau; q^-, q^+) \, d\tau \approx \sum \limits_{i=1}^{\sigma} \beta_i \, \vartheta \big( \Phi (\gamma_i) \big) \cdot \Phi' (\gamma_i) ,\]

so a Discontinuity pairs

• path: the connecting path, a PathIntegral such as PathIntegralLinear or PathIntegralTrigonometric, and
• quadrature: the quadrature rule with $\sigma$ nodes $\gamma_i$ and weights $\beta_i$ used to evaluate that integral.

Used by DGVIPI.

Constructor

Discontinuity(path::PathIntegral, quadrature::QuadratureRule)

Example

Discontinuity(PathIntegralLinear(), LobattoLegendreQuadrature(2))

which integrates along a straight line between the one-sided limits with the two-point Lobatto rule, i.e. a trapezoidal flux.

Abstract supertype of the paths used to integrate a Lagrangian across a jump of a discontinuous discrete trajectory.

A concrete PathIntegral provides the two functions $\varphi^{-}$ and $\varphi^{+}$ that build the connecting path from the one-sided limits,

\[\Phi (\tau; q^-, q^+) = q^- \, \varphi^{-} (\tau) + q^+ \, \varphi^{+} (\tau) , \qquad \varphi^{-} (0) = \varphi^{+} (1) = 1 , \qquad \varphi^{-} (1) = \varphi^{+} (0) = 0 ,\]

through the interface

evaluate_l(path, τ)     # φ⁻(τ)
evaluate_r(path, τ)     # φ⁺(τ)
derivative_l(path, τ)   # dφ⁻/dτ (τ)
derivative_r(path, τ)   # dφ⁺/dτ (τ)

Implementations: PathIntegralLinear, PathIntegralTrigonometric. Paired with a quadrature rule in a Discontinuity.

PathIntegralLinear is a path integral along a linear path

\[\phi (\tau; q^-, q^+) = (1-\tau) q^- + \tau q^+ .\]

PathIntegralTrigonometric is a path integral along a cos^2/sin^2 path

\[\phi (\tau; q^-, q^+) = \cos^2 (\pi \tau / 2) q^- + \sin^2 (\pi \tau / 2) q^+ .\]