# Title: Convergence for a simple periodic problem
#
# Description:
#
# This is one of the test cases presented in Popinet \cite{popinet2003}.
# Following Minion \cite{minion96} and Almgren et al. \cite{almgren98},
# this convergence test illustrates the second-order
# accuracy of Gerris for flows without solid boundaries. This
# problem uses a square unit domain with periodic boundary conditions in
# both directions. The initial conditions are taken as
# \begin{eqnarray*}
# u(x,y) &=& 1-2\cos(2\pi x)\sin(2\pi y), \\
# v(x,y) &=& 1+2\sin(2\pi x)\cos(2\pi y). 
# \end{eqnarray*}
# The exact solution of the Euler equations for these initial conditions
# is
# \begin{eqnarray*}
# u(x,y,t) &=& 1-2\cos(2\pi(x-t))\sin(2\pi(y-t)), \\
# v(x,y,t) &=& 1+2\sin(2\pi(x-t))\cos(2\pi(y-t)), \\
# p(x,y,t) &=& -\cos(4\pi(x-t))-\cos(4\pi(y-t)).
# \end{eqnarray*}
# As in \cite{almgren98} nine runs are performed on grids with $L=5,6$
# and $7$ levels of refinement (labelled ``uniform'') and with one
# (labelled $r=1$) or two (labelled $r=2$) additional levels added only
# within the square defined by the points $(-0.25,-0.25)$ and
# $(0.25,0.25)$. The length of the run for each case is 0.5, the CFL number is
# 0.75. For each run both the $L_2$ and $L_\infty$ norms of the error in
# the $x$-component of the velocity is computed. Table \ref{minion1}
# gives the errors and order of convergence obtained.
#
# Close to second-order convergence is obtained (asymptotically in
# $L$) for the $L_2$ and $L_\infty$ norms. The values
# obtained are comparable to that in \cite{minion96,almgren98}.
# \begin{table}
# \caption{\label{minion1}Errors and convergence orders in the $x$-component of the velocity
# for a simple periodic problem. The reference solution values are given in blue.}
# \begin{center}
# \input{minion1.tex}
# \end{center}
# \end{table}
#
# Author: St\'ephane Popinet
# Command: sh periodic.sh periodic.gfs
# Version: 0.6.4
# Required files: periodic.sh r0.ref r1.ref r2.ref
# Running time: 3 minutes
# Generated files: minion1.tex
#
1 2 GfsSimulation GfsBox GfsGEdge {} {
  Time { end = 0.5 }
  AdvectionParams { cfl = 0.75 }
  Refine (x < -0.25 || x > 0.25 || y < -0.25 || y > 0.25 ? LEVEL : LEVEL + BOX)
  Init {} {
    U = (1. - 2.*cos (2.*M_PI*x)*sin (2.*M_PI*y))
    V = (1. + 2.*sin (2.*M_PI*x)*cos (2.*M_PI*y))
  }
  ApproxProjectionParams { tolerance = 1e-6 }
  ProjectionParams { tolerance = 1e-6 }
  OutputErrorNorm { start = end } stdout { v = U } {
    s = (1. - 2.*cos (2.*M_PI*(x - t))*sin (2.*M_PI*(y - t)))
  }
}
GfsBox {}
1 1 right
1 1 top