# Title: Potential flow around a sphere
#
# Description:
#
# The axisymmetric potential flow around a sphere is computed (Figure
# \ref{isolines}) and compared to the theoretical solution
# \cite{lamb}. A large domain is used together with variable spatial
# resolution to minimise the influence of the finite domain size.
#
# Figure \ref{error} and \ref{order} illustrate the convergence of the
# solution for the horizontal component of velocity with increased
# resolution.
#
# \begin{figure}[htbp]
# \caption{\label{isolines}Isolines of the velocity components ($x$ in red, $y$ in blue).}
# \begin{center}
# \includegraphics[width=0.8\hsize]{isolines.eps}
# \end{center}
# \end{figure}
#
# \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 axi.sh axi.gfs
# Version: 1.3.0
# Required files: axi.sh error.ref order.ref isolines.gfv
# Generated files: error.eps order.eps isolines.eps
#
1 0 GfsAxi GfsBox GfsGEdge {} {
    Time { end = 0 }
    PhysicalParams { L = 50 }
    AdvectionParams { scheme = none }
    ApproxProjectionParams { tolerance = 1e-10 }
    Refine 4
    Refine (LEVEL + 1./50.*(x*x + y*y)*(4. - LEVEL))
    Global {
	#define A0 0.5
	#define U0 1.
    }
    Solid (ellipse (0., 0., A0, A0))
    Init {} {
	U = U0
	Phi = {
	    double r = sqrt (cx*cx + cy*cy);
	    return U0*A0*A0*A0*cx/(2.*r*r*r);
	}
    }
    OutputErrorNorm { start = end } { awk '{ print LEVEL " " $7 " " $9}' } { v = U } {
 	s = (dx("Phi") + 1.)
    }
    OutputSimulation { start = end } sim-LEVEL.gfs
}
GfsBox {
    left = Boundary { BcDirichlet U U0 }
    right = Boundary { BcDirichlet U U0 }
}