# Title: Convergence of the Godunov advection scheme
#
# Description:
#
# A non-trivial initial tracer distribution is advected by a constant
# velocity field corresponding to a solid rotation around the center
# of the domain. The tracer field after one revolution is compared to the
# initial tracer field to compute the error norms.
#
# Figure \ref{error} and \ref{order} illustrate the convergence of the
# solution with increased resolution. Close to second-order
# convergence is obtained.
#
# \begin{figure}[htbp]
# \caption{\label{error}Evolution of the error as a function of resolution.}
# \begin{center}
# \includegraphics[width=0.8\hsize]{error.eps}
# \end{center}
# \end{figure}
#
# \begin{figure}[htbp]
# \caption{\label{order}Corresponding convergence order.}
# \begin{center}
# \includegraphics[width=0.8\hsize]{order.eps}
# \end{center}
# \end{figure}
#
# Author: St\'ephane Popinet
# Command: sh advection.sh advection.gfs
# Version: 0.8.0
# Required files: advection.sh error.ref order.ref
# Generated files: error.eps order.eps
#
1 0 GfsAdvection GfsBox GfsGEdge {} {
  Time { end = 0.785398 }
  Refine LEVEL
  VariableTracer {} T { gradient = gfs_center_gradient }
  AdvectionParams { cfl = 0.5 }
  Init {} {
    U = -8.*y
    V = 8.*x
    T = {
      double r2 = x*x + y*y; 
      double coeff = 20. + 20000.*r2*r2*r2*r2;
      return (1. + cos(20.*x)*cos(20.*y))*exp(-coeff*r2)/2.;
    }
  }
  OutputErrorNorm { start = end } { awk '{ print LEVEL " " $5 " " $7 " " $9}' } { v = T } {
    s = {
      double r2 = x*x + y*y; 
      double coeff = 20. + 20000.*r2*r2*r2*r2;
      return (1. + cos(20.*x)*cos(20.*y))*exp(-coeff*r2)/2.;
    }
  }
}
GfsBox {}