# Title: Convergence for the three-way vortex merging problem
#
# Description:
#
# Another of the test cases presented in Popinet \cite{popinet2003},
# initially used by Almgren et al. \cite{almgren98}, this convergence
# test illustrates the second-order accuracy of Gerris when refinement
# is placed appropriately, either through static refinement or dynamic
# adaptive refinement.
#
# Four vortices are placed in the unit-square, centred at $(0,0)$,
# $(0.09,0)$, $(-0.045,0.045\sqrt{3})$ and $(-0.045,$ $-0.045\sqrt{3})$
# and of strengths $-150$, $50$, $50$, $50$ respectively. The profile of
# each vortex centred around $(x_i,y_i)$ is
# $$
# {1+\tanh(100(0.03-r_i))\over 2},
# $$
# where $r_i=\sqrt{(x-x_i)^2+(y-y_i)^2}$. To initialise the velocity
# field, we use this vorticity as the source term in the Poisson
# equation for the streamfunction $\psi$
# $$
# \nabla^2\psi=\|\nabla\times{\bf U}\|.
# $$
# Each component of the velocity field is then calculated from the
# streamfunction. No-flow boundary conditions are used on the four sides
# of the domain and the simulations are ran to $t=0.25$ using a CFL of
# 0.9.
#
# Two different discretisations are used, each time with up to $L$
# levels of refinement: a grid using static refinement in concentric
# circles of decreasing radius and a grid using dynamic adaptive
# refinement. The ``circle'' grid is constructed by starting from a
# uniform grid with four levels of refinement and by successively adding
# one level to all the cells contained within circles centred on the
# origin and of radii:
# \begin{itemize}
# \item $L=6$: 0.25, 0.15
# \item $L=7$: 0.25, 0.2, 0.15
# \item $L=8$: 0.25, 0.2, 0.175, 0.15
# \item $L=9$: 0.25, 0.2, 0.175, 0.1625, 0.15
# \end{itemize}
# For the dynamically refined grid, the vorticity-based criterion is
# applied at every timestep with a threshold $\tau=4\times10^{-3}$. As
# we do not have an analytical solution for this problem, Richardson
# extrapolation is used.
#
# Figure \ref{vorticity} illustrates the evolution of the vorticity
# and of the adaptively refined grid for $L=8$. The most refined level
# closely follows the three outer vortices as they orbit the central
# one. Far from the vortices, a very coarse mesh is used ($l=3$). One
# may note a few isolated patches of refinement scattered at the
# periphery of the outer vortices. They are due to the numerical noise
# added to the vorticity by the interpolation procedure necessary to
# fill in velocity values for newly created cells. This could be
# improved by using higher-order interpolants.
#
# Table \ref{convergence} summarises the results obtained for the
# first twelve calculations. For fine enough grids close to
# second-order convergence is obtained for both norms and for the two
# discretisations used. The norms of the error on the various grids
# are also comparable for a given resolution.
#
# \begin{table}
# \caption{\label{convergence}Errors and convergence orders in the $x$-component of the
# velocity for the four-way vortex merging problem. The reference
# solution values are given in blue.}
# \begin{center}
# \input{convergence.tex}
# \end{center}
# \end{table}
#
# \begin{figure}
# \caption{\label{vorticity}Contour plots of vorticity (left) and adaptive grids used
# (right) for the four-way vortex merging calculation.}
# \begin{center}
# \begin{tabular}{cc}
# \includegraphics*[width=0.3\hsize]{tv_0_05.eps} &
# \includegraphics*[width=0.3\hsize]{tm_0_05.eps} \\
# \multicolumn{2}{c}{$t=0.05$} \\
# \\
# \includegraphics*[width=0.3\hsize]{tv_0_15.eps} &
# \includegraphics*[width=0.3\hsize]{tm_0_15.eps} \\
# \multicolumn{2}{c}{$t=0.15$} \\
# \\
# \includegraphics*[width=0.3\hsize]{tv_0_25.eps} &
# \includegraphics*[width=0.3\hsize]{tm_0_25.eps} \\
# \multicolumn{2}{c}{$t=0.25$}
# \end{tabular}
# \end{center}
# \end{figure}
#
# Author: St\'ephane Popinet
# Command: sh merging.sh merging.gfs
# Version: 0.6.4
# Required files: merging.sh levels.gfv vorticity.gfv sim.err.ref simc.err.ref
# Running time: 3 minutes
# Generated files: convergence.tex tv_0_05.eps tm_0_05.eps tv_0_15.eps tm_0_15.eps tv_0_25.eps tm_0_25.eps
#
1 0 GfsSimulation GfsBox GfsGEdge {} {
  Time { end = 0.25 }
  AdvectionParams { cfl = 0.9 }
  ApproxProjectionParams { tolerance = 1e-5 }
  ProjectionParams { tolerance = 1e-5 }
  Refine {
    double r = sqrt(x*x + y*y);
    switch (LEVEL) {
    case 6: return r > 0.25 ? 4 : r > 0.15 ? 5 : 6;
    case 7: return r > 0.25 ? 4 : r > 0.2 ? 5 : r > 0.15 ? 6 : 7;
    case 8: return r > 0.25 ? 4 : r > 0.2 ? 5 : r > 0.175 ? 6 : r > 0.15 ? 7 : 8;
    case 9: return r > 0.25 ? 4 : r > 0.2 ? 5 : r > 0.175 ? 6 : r > 0.1625 ? 7 : r > 0.15 ? 8 : 9;
    }
  }
  InitVorticity {} {
    double vortex (double xo, double yo, double s) {
      double r = sqrt ((x - xo)*(x - xo) + (y - yo)*(y - yo));
      return s*(1. + tanh (100.*(0.03 - r)))/2.;
    }
    return vortex (0., 0., -150.) + 
           vortex (0.09, 0., 50.) + 
           vortex (-0.045, 0.0779422863406, 50.) +
           vortex (-0.045, -0.0779422863406, 50.);
  }
  AdaptVorticity { istep = 1 } { maxlevel = LEVEL cmax = 4e-3 }
  OutputSimulation { start = 0.05 } stdout
  EventScript { start = 0.05 } {
    echo Clear
    cat levels.gfv
    echo Save tm_0_05.eps { format = EPS line_width = 0.1 }
    echo Clear
    cat vorticity.gfv
    echo Save tv_0_05.eps { format = EPS line_width = 0.1 }
  }
  OutputSimulation { start = 0.15 } stdout
  EventScript { start = 0.15 } {
    echo Clear
    cat levels.gfv
    echo Save tm_0_15.eps { format = EPS line_width = 0.1 }
    echo Clear
    cat vorticity.gfv
    echo Save tv_0_15.eps { format = EPS line_width = 0.1 }
  }
  OutputSimulation { start = 0.25 } stdout
  EventScript { start = 0.25 } {
    echo Clear
    cat levels.gfv
    echo Save tm_0_25.eps { format = EPS line_width = 0.1 }
    echo Clear
    cat vorticity.gfv
    echo Save tv_0_25.eps { format = EPS line_width = 0.1 }
  }
  OutputSimulation { start = 0.25 } SIM
}
GfsBox {}