Predictor-corrector methods

Predictor-corrector based on RK2

The PredCorrRK2XY defines a predictor-corrector based on the RK2 time integration method. The RK2 method is implemented in the ITimeStepper method (see rk2).

This time integration method is applied to a a guiding-centre equations system documented in guiding_centre.

\[ \partial_t f(t, x, y) + (E(t, x, y)\wedge e_z) \cdot \nabla f(t, x, y) = 0, \\ - \Delta \phi(t, x, y) = -\partial_x^2 \phi(t, x, y) -\partial_y^2 \phi(t, x, y) = f(t, x, y), \\ E(t, x, y) = - \nabla \phi(t, x, y). \]

Applied to this equations system, we have, writing \(\{t^n\}_n \text{ the time discretisation, and } f^n = f(t^n, \cdot, \cdot)\):

For \(n\geq 0\),

Advect on a half time step:
From \(f^n\), we compute \(\phi^n\) with the Poisson-like equation (FFTPoissonSolver);
From \(\phi^n\), we compute \(E^n\) by deriving (FFTPoissonSolver);
From \(f^n \text{ and } E^n\), we compute \(f^{n+1/2}\) by advecting (BslAdvection1D) on \(\frac{dt}{2}\);
Advect on a full time step:
From \(f^{n+1/2}\), we compute \(\phi^{n+1/2}\) with the Poisson-like equation (FFTPoissonSolver);
From \(\phi^{n+1/2}\), we compute \(E^{n+1/2}\) by deriving (FFTPoissonSolver);
From \(f^n \text{ and } E^{n+1/2}\), we compute \(f^{n+1}\) by advecting (BslAdvection1D) on \(dt\).