[SCM] Gerris Flow Solver branch, upstream, updated. e8f73a07832050124d2b8bf6c6f35b33180e65a8

Stephane Popinet popinet at users.sf.net
Tue Nov 24 12:25:26 UTC 2009


The following commit has been merged in the upstream branch:
commit 696894947a4bda786d90a4ae6521668e29a39d91
Author: Stephane Popinet <popinet at users.sf.net>
Date:   Thu Oct 29 16:36:51 2009 +1100

    New 'cosine bell' test case
    
    darcs-hash:20091029053651-d4795-8b9d50728e23b9f15a84dccc83d061f2607832a8.gz

diff --git a/test/Makefile.am b/test/Makefile.am
index e5d94c1..94ac6ea 100644
--- a/test/Makefile.am
+++ b/test/Makefile.am
@@ -28,7 +28,8 @@ TESTDIRS = \
 	waves \
 	nz \
 	parabola \
-	lonlat
+	lonlat \
+	cosine
 
 EXTRA_DIST = \
 	template.tex \
diff --git a/test/cosine/cosine.gfs b/test/cosine/cosine.gfs
new file mode 100644
index 0000000..4c22221
--- /dev/null
+++ b/test/cosine/cosine.gfs
@@ -0,0 +1,184 @@
+# Title: Advection of a cosine bell around the sphere
+#
+# Description:
+#
+# This test case was suggested by Williamson et
+# al. \cite{williamson92} (Problem \#1). A "cosine bell" initial
+# concentration is given by
+# $$
+# h(\lambda,\theta)=(h_0/2)(1+\cos(\pi r/R))
+# $$
+# if $r<R$ and 0 otherwise, with $R=1/3$ and
+# $$
+# r=\arccos[\sin\theta_c\sin\theta+\cos\theta_c\cos\theta\cos(\lambda-\lambda_c)]
+# $$
+# the great circle distance between longitude, latitude
+# $(\lambda,\theta)$ and the center initially taken as
+# $(\lambda_c,\theta_c)=(3\pi/2,0)$.
+#
+# The advection velocity field corresponds to solid-body rotation at an
+# angle $\alpha$ to the polar axis of the spherical coordinate
+# system. It is given by the streamfunction
+# $$
+# \psi=-u_0(\sin\theta\cos\alpha-\cos\lambda\cos\theta\sin\alpha)
+# $$
+#
+# The cosine bell field is rotated once around the sphere and should
+# come back exactly to its original position. The difference between
+# the initial and final fields is a measure of the accuracy of the
+# advection scheme coupled with the spherical coordinate mapping (the
+# "conformal expanded spherical cube" metric in our case).
+#
+# For the "spherical cube" metric, two angles are considered: 45
+# degrees which rotates the cosine bell above four of the eight "poles"
+# of the mapping and 90 degrees which avoids the poles entirely. Mass
+# is conserved to within machine accuracy in either case.
+#
+# The mesh is adapted dynamically according to the gradient of tracer
+# concentration.
+#
+# \begin{figure}[htbp] 
+# \caption{\label{solution45}Tracer field after one rotation around the
+# sphere with $\alpha=45^\circ$ (red). Reference solution
+# (green). Zero level contour line (blue). Equivalent static
+# resolutions (a) $16\times 16\times 6$. (b) $32\times 32\times
+# 6$. (c) $64\times 64\times 6$. (d) $128\times 128\times 6$.}
+# \begin{center}
+# \begin{tabular}{cc}
+# (a) \includegraphics[width=0.45\hsize]{isolines-4-45.eps} &
+# (b) \includegraphics[width=0.45\hsize]{isolines-5-45.eps} \\
+# (c) \includegraphics[width=0.45\hsize]{isolines-6-45.eps} &
+# (d) \includegraphics[width=0.45\hsize]{isolines-7-45.eps}
+# \end{tabular}
+# \end{center}
+# \end{figure}
+#
+# \begin{figure}[htbp] 
+# \caption{\label{solution90}Tracer field after one rotation around the
+# sphere with $\alpha=90^\circ$ (red). Reference solution
+# (green). Zero level contour line (blue). Equivalent static
+# resolutions (a) $16\times 16\times 6$. (b) $32\times 32\times
+# 6$. (c) $64\times 64\times 6$. (d) $128\times 128\times 6$.}
+# \begin{center}
+# \begin{tabular}{cc}
+# (a) \includegraphics[width=0.45\hsize]{isolines-4-90.eps} &
+# (b) \includegraphics[width=0.45\hsize]{isolines-5-90.eps} \\
+# (c) \includegraphics[width=0.45\hsize]{isolines-6-90.eps} &
+# (d) \includegraphics[width=0.45\hsize]{isolines-7-90.eps}
+# \end{tabular}
+# \end{center}
+# \end{figure}
+#
+# \begin{figure}[htbp] 
+# \caption{\label{error}Relative error norms (as defined in
+# \cite{williamson92}) as functions of spatial resolution. The results
+# of Rossmanith \cite{rossmanith2006} using a gnomonic spherical cube
+# metric and a different 2nd-order advection scheme are also
+# reproduced for comparison. (a) $\alpha=45^\circ$. (b)
+# $\alpha=90^\circ$.}
+# \begin{center}
+# \begin{tabular}{c}
+# (a) \includegraphics[width=0.7\hsize]{order-45.eps} \\
+# (b) \includegraphics[width=0.7\hsize]{order-90.eps}
+# \end{tabular}
+# \end{center}
+# \end{figure}
+#
+# Author: St\'ephane Popinet
+# Command: sh cosine.sh
+# Version: 091029
+# Required files: cosine.sh isolines.gfv reference.gfv zero.gfv error-45.ref error-90.ref rossmanith45 rossmanith90
+# Running time: 6 minutes
+# Generated files: isolines-4-45.eps  isolines-5-90.eps  isolines-7-45.eps  order-90.eps isolines-4-90.eps  isolines-6-45.eps  isolines-7-90.eps isolines-5-45.eps  isolines-6-90.eps  order-45.eps
+#
+Define U0 (2.*M_PI)
+
+6 12 GfsAdvection GfsBox GfsGEdge {} {
+  PhysicalParams { L = 2.*M_PI/4. }
+  MetricCubed LEVEL
+  Time { end = 1 }
+  Refine LEVEL
+  VariableTracer T { 
+      gradient = gfs_center_gradient 
+      cfl = 1
+  }
+  Global {
+      #define DTR (M_PI/180.)
+      double bell (double x, double y, double t) {
+	  double h0 = 1.;
+	  double R = 1./3.;
+	  double lc = 3.*M_PI/2. + U0*t, tc = 0.;
+	  x *= DTR; y *= DTR;
+	  double r = acos (sin(tc)*sin(y) + cos (tc)*cos (y)*cos (x - lc));
+	  return r >= R ? 0. : (h0/2.)*(1. + cos (M_PI*r/R));
+      }
+  }
+  Init {} { T = bell(x,y,0) }
+  VariableStreamFunction Psi -U0*(sin (y*DTR)*cos (DTR*ALPHA) - cos (x*DTR)*cos (y*DTR)*sin (DTR*ALPHA))
+  AdaptGradient { istep = 1 } { cmax = 1e-4 maxlevel = LEVEL } T
+  OutputTime { istep = 10 } stderr
+#  OutputSimulation { istep = 10 } stdout
+  OutputSimulation { start = end } end-LEVEL-ALPHA.gfs
+  OutputErrorNorm { istep = 1 } { awk '{ print LEVEL,$3,$5,$7,$9}' > error-LEVEL-ALPHA } { v = T } {
+      s = bell(x,y,t)
+      v = E
+      relative = 1
+  }
+  OutputScalarSum { istep = 1 } t-LEVEL-ALPHA { v = T }
+  OutputScalarSum { istep = 1 } area-LEVEL-ALPHA { v = 1 }
+  EventScript { start = end } {
+      ( cat isolines.gfv
+        echo "Save isolines.gnu { format = Gnuplot }"
+        echo "Clear"
+        cat reference.gfv
+        echo "Save reference.gnu { format = Gnuplot }"
+	echo "Clear"
+        cat zero.gfv
+        echo "Save zero.gnu { format = Gnuplot }"
+      ) | gfsview-batch2D end-LEVEL-ALPHA.gfs
+      cat <<EOF | gnuplot
+        set term postscript eps lw 2 18 color
+        set output 'isolines-LEVEL-ALPHA.eps'
+        set size ratio -1
+        set xlabel 'Longitude'
+        set ylabel 'Latitude'
+        unset key
+        plot [-120:-60][-30:30]'isolines.gnu' w l, 'reference.gnu' w l, 'zero.gnu' w l
+EOF
+      fixbb isolines-LEVEL-ALPHA.eps
+      rm -f isolines.gnu reference.gnu zero.gnu
+
+      # check mass conservation
+      if awk '
+        BEGIN { min = 1000.; max = -1000.; }{ 
+          if ($5 < min) min = $5; 
+          if ($5 > max) max = $5; 
+        }
+        END {
+          if (max - min != 0.)
+            exit (1);
+        }' < t-LEVEL-ALPHA; then
+	  exit 0
+      else
+	  exit $GFS_STOP
+      fi
+  }
+}
+GfsBox {}
+GfsBox {}
+GfsBox {}
+GfsBox {}
+GfsBox {}
+GfsBox {}
+1 2 right
+2 3 top
+3 4 right
+4 5 top
+5 6 right
+6 1 top
+1 3 top left
+3 5 top left
+5 1 top left
+2 6 bottom right
+4 2 bottom right
+6 4 bottom right
diff --git a/test/cosine/cosine.sh b/test/cosine/cosine.sh
new file mode 100644
index 0000000..7166c49
--- /dev/null
+++ b/test/cosine/cosine.sh
@@ -0,0 +1,60 @@
+levels="4 5 6 7"
+alphas="45 90"
+
+if ! $donotrun; then
+    for alpha in $alphas; do
+	for i in $levels; do
+	    if gerris2D -DLEVEL=$i -DALPHA=$alpha cosine.gfs 2> log-$i-$alpha; then :
+	    else
+		exit 1
+	    fi
+	done
+    done
+fi
+
+for alpha in $alphas; do
+    rm -f error-$alpha
+    for i in $levels; do
+	tail -n 1 error-$i-$alpha >> error-$alpha
+    done
+
+    if cat <<EOF | gnuplot ; then :
+set term postscript eps color enhanced lw 2 18
+set output 'order-$alpha.eps'
+set logscale
+set xtics 16,2,256
+set key spacing 1.5 bottom left
+ftitle(a,b) = sprintf("%.0f/x^{%4.2f}", exp(a), -b)
+f2(x)=a2+b2*x
+fit [3:]f2(x) 'error-$alpha' u (log(2**\$1)):(log(\$4)) via a2,b2
+fm(x)=am+bm*x
+fit [3:]fm(x) 'error-$alpha' u (log(2**\$1)):(log(\$5)) via am,bm
+set xlabel 'Spatial resolution'
+set ylabel 'Relative error norms'
+plot 'error-$alpha' u (2**\$1):4 t 'L2' w lp ps 2,  exp(f2(log(x))) t ftitle(a2,b2), \
+     'error-$alpha' u (2**\$1):5 t 'Max' w lp ps 2, exp(fm(log(x))) t ftitle(am,bm), \
+     'rossmanith$alpha' u 1:3 t 'L2 (Rossmanith)' w lp ps 2, \
+     'rossmanith$alpha' u 1:4 t 'Max (Rossmanith)' w lp ps 2 lt 1
+EOF
+    else
+	exit 1
+    fi
+done
+
+if cat <<EOF | python ; then :
+from check import *
+from sys import *
+c = Curve()
+print (Curve('error-45',1,4) - Curve('error-45.ref',1,4)).max()
+print (Curve('error-45',1,5) - Curve('error-45.ref',1,5)).max()
+print (Curve('error-90',1,4) - Curve('error-90.ref',1,4)).max()
+print (Curve('error-90',1,5) - Curve('error-90.ref',1,5)).max() 
+if (Curve('error-45',1,4) - Curve('error-45.ref',1,4)).max() > 1e-10 or\
+   (Curve('error-45',1,5) - Curve('error-45.ref',1,5)).max() > 1e-10 or\
+   (Curve('error-90',1,4) - Curve('error-90.ref',1,4)).max() > 1e-10 or\
+   (Curve('error-90',1,5) - Curve('error-90.ref',1,5)).max() > 1e-10:
+    exit(1)
+EOF
+else
+   exit 1
+fi
diff --git a/test/cosine/error-45.ref b/test/cosine/error-45.ref
new file mode 100644
index 0000000..2b246e0
--- /dev/null
+++ b/test/cosine/error-45.ref
@@ -0,0 +1,4 @@
+4 1 1.002e+00 5.316e-01 5.358e-01
+5 1 3.577e-01 2.092e-01 1.909e-01
+6 1 8.026e-02 5.107e-02 4.311e-02
+7 1 1.557e-02 1.174e-02 1.248e-02
diff --git a/test/cosine/error-90.ref b/test/cosine/error-90.ref
new file mode 100644
index 0000000..1d84b60
--- /dev/null
+++ b/test/cosine/error-90.ref
@@ -0,0 +1,4 @@
+4 1 6.565e-01 4.073e-01 3.483e-01
+5 1 2.002e-01 1.266e-01 9.351e-02
+6 1 3.949e-02 2.585e-02 1.578e-02
+7 1 8.840e-03 6.328e-03 5.954e-03
diff --git a/doc/examples/boussinesq/boussinesq.gfv b/test/cosine/isolines.gfv
similarity index 79%
copy from doc/examples/boussinesq/boussinesq.gfv
copy to test/cosine/isolines.gfv
index b55c6a2..25a2b41 100644
--- a/doc/examples/boussinesq/boussinesq.gfv
+++ b/test/cosine/isolines.gfv
@@ -1,18 +1,18 @@
 # GfsView 2D
 View {
-  tx = 0 ty = -0.55
+  tx = -2.01967 ty = -2.03136
   sx = 1 sy = 1 sz = 1
   q0 = 0 q1 = 0 q2 = 0 q3 = 1
-  fov = 18.5
+  fov = 5.09252
   r = 0.3 g = 0.4 b = 0.6
   res = 1
   lc = 0.001
   reactivity = 0.1
 }
-Linear {
+Isoline {
   r = 0 g = 0 b = 0
   shading = Constant
-  maxlevel = 8
+  maxlevel = -1
 } {
   n.x = 0 n.y = 0 n.z = 1
   pos = 0
@@ -23,4 +23,6 @@ Linear {
 } 0 {
   reversed = 0
   use_scalar = 1
+} {
+  n = 9
 }
diff --git a/doc/examples/boussinesq/boussinesq.gfv b/test/cosine/reference.gfv
similarity index 68%
copy from doc/examples/boussinesq/boussinesq.gfv
copy to test/cosine/reference.gfv
index b55c6a2..09882fc 100644
--- a/doc/examples/boussinesq/boussinesq.gfv
+++ b/test/cosine/reference.gfv
@@ -1,26 +1,28 @@
 # GfsView 2D
 View {
-  tx = 0 ty = -0.55
+  tx = -2.04454 ty = -2.04048
   sx = 1 sy = 1 sz = 1
   q0 = 0 q1 = 0 q2 = 0 q3 = 1
-  fov = 18.5
+  fov = 4.18068
   r = 0.3 g = 0.4 b = 0.6
   res = 1
   lc = 0.001
   reactivity = 0.1
 }
-Linear {
-  r = 0 g = 0 b = 0
+Isoline {
+  r = 0 g = 0.733654 b = 0.0148928
   shading = Constant
-  maxlevel = 8
+  maxlevel = -1
 } {
   n.x = 0 n.y = 0 n.z = 1
   pos = 0
-} T {
+} bell(x,y,0) {
   amin = 0 min = 0
   amax = 0 max = 1
   cmap = Jet
 } 0 {
   reversed = 0
   use_scalar = 1
+} {
+  n = 9
 }
diff --git a/test/cosine/rossmanith45 b/test/cosine/rossmanith45
new file mode 100644
index 0000000..a8c0e02
--- /dev/null
+++ b/test/cosine/rossmanith45
@@ -0,0 +1,6 @@
+# see Rossmanith, 2006, JCP 213, Table 3, t = 1
+# resolution L1 L2 Linfty
+20    5.02e-1 	4.01e-1		4.62e-1
+40    1.12e-1 	1.07e-1 	1.76e-1
+80    2.12e-2 	2.23e-2 	5.85e-2
+160   4.77e-3 	5.32e-3 	1.96e-2
diff --git a/test/cosine/rossmanith90 b/test/cosine/rossmanith90
new file mode 100644
index 0000000..8ab0ce7
--- /dev/null
+++ b/test/cosine/rossmanith90
@@ -0,0 +1,6 @@
+# see Rossmanith, 2006, JCP 213, Table 2, t = 1
+# resolution L1 L2 Linfty
+20    2.48e-1 1.96e-1 1.83e-1
+40    5.70e-2 4.62e-2 4.57e-2
+80    1.47e-2 1.20e-2 1.48e-2
+160   3.77e-3 3.22e-3 6.26e-3
diff --git a/doc/examples/boussinesq/boussinesq.gfv b/test/cosine/zero.gfv
similarity index 77%
copy from doc/examples/boussinesq/boussinesq.gfv
copy to test/cosine/zero.gfv
index b55c6a2..8298c7b 100644
--- a/doc/examples/boussinesq/boussinesq.gfv
+++ b/test/cosine/zero.gfv
@@ -1,18 +1,18 @@
 # GfsView 2D
 View {
-  tx = 0 ty = -0.55
+  tx = -2.01967 ty = -2.03136
   sx = 1 sy = 1 sz = 1
   q0 = 0 q1 = 0 q2 = 0 q3 = 1
-  fov = 18.5
+  fov = 5.09252
   r = 0.3 g = 0.4 b = 0.6
   res = 1
   lc = 0.001
   reactivity = 0.1
 }
-Linear {
+Isoline {
   r = 0 g = 0 b = 0
   shading = Constant
-  maxlevel = 8
+  maxlevel = -1
 } {
   n.x = 0 n.y = 0 n.z = 1
   pos = 0
@@ -23,4 +23,6 @@ Linear {
 } 0 {
   reversed = 0
   use_scalar = 1
+} {
+  n = 0 levels = 0
 }
diff --git a/test/template.tex b/test/template.tex
index 9b1e119..db4c5ee 100644
--- a/test/template.tex
+++ b/test/template.tex
@@ -112,6 +112,7 @@ branch only.
 
 \input{lonlat/lonlat.tex}
 \input{lonlat/coriolis/coriolis.tex}
+\input{cosine/cosine.tex}
 
 \bibliographystyle{plain}
 \bibliography{tests}
diff --git a/test/tests.bib b/test/tests.bib
index 26dcf5f..f5bd5eb 100644
--- a/test/tests.bib
+++ b/test/tests.bib
@@ -1,261 +1,321 @@
 @Article{almgren97,
-  author = 	 {A. S. Almgren and J. B. Bell and P. Colella and T. Marthaler},
-  title = 	 {A Cartesian Grid Projection Method for the Incompressible Euler Equations in Complex Geometries},
-  journal = 	 {SIAM J. Sci. Comp.},
-  year = 	 1997,
+  author =	 {A. S. Almgren and J. B. Bell and P. Colella and
+                  T. Marthaler},
+  title =	 {A Cartesian Grid Projection Method for the
+                  Incompressible Euler Equations in Complex
+                  Geometries},
+  journal =	 {SIAM J. Sci. Comp.},
+  year =	 1997,
   volume =	 18,
   number =	 5,
-  pages =        {1289-1309},
-  url =          {http://seesar.lbl.gov/ccse/Publications/almgren/abcm.sisc/paper.ps.gz},
-  local_url =    {almgren.ps.gz}
+  pages =	 {1289-1309},
+  url =
+                  {http://seesar.lbl.gov/ccse/Publications/almgren/abcm.sisc/paper.ps.gz},
+  local_url =	 {almgren.ps.gz}
 }
 
 @Article{almgren98,
-  author = 	 {A. S. Almgren and J. B. Bell and P. Colella and L. H. Howell and M. L. Welcome},
-  title = 	 {A Conservative Adaptive Projection Method for the Variable Density Incompressible Navier-Stokes Equations},
-  journal = 	 {J. Comput. Phys.},
-  year = 	 1998,
+  author =	 {A. S. Almgren and J. B. Bell and P. Colella and
+                  L. H. Howell and M. L. Welcome},
+  title =	 {A Conservative Adaptive Projection Method for the
+                  Variable Density Incompressible Navier-Stokes
+                  Equations},
+  journal =	 {J. Comput. Phys.},
+  year =	 1998,
   volume =	 142,
   pages =	 {1-46},
-  url =          {http://seesar.lbl.gov/ccse/Publications/almgren/abchw96/paper.ps.gz},
-  local_url =    {almgren1.ps.gz}
+  url =
+                  {http://seesar.lbl.gov/ccse/Publications/almgren/abchw96/paper.ps.gz},
+  local_url =	 {almgren1.ps.gz}
 }
 
 @Book{bird87,
   author =	 {R. B. Bird and R. C. Armstrong and O. Hassager},
-  title = 	 {Dynamics of polymeric liquids},
-  publisher = 	 {Wiley-Interscience},
-  year = 	 1987,
+  title =	 {Dynamics of polymeric liquids},
+  publisher =	 {Wiley-Interscience},
+  year =	 1987,
   edition =	 {second edition}
 }
 
 @article{blanco1995,
-  title={The structure of the axisymmetric high-{R}eynolds number flow around an ellipsoidal bubble of fixed shape},
-  author={Blanco, A. and Magnaudet, J.},
-  journal={Physics of Fluids},
-  volume={7},
-  pages={1265},
-  year={1995},
-  publisher={AIP}
+  title =	 {The structure of the axisymmetric high-{R}eynolds
+                  number flow around an ellipsoidal bubble of fixed
+                  shape},
+  author =	 {Blanco, A. and Magnaudet, J.},
+  journal =	 {Physics of Fluids},
+  volume =	 7,
+  pages =	 1265,
+  year =	 1995,
+  publisher =	 {AIP}
 }
 
 @PhdThesis{dupont,
-  author = 	 {F. Dupont},
-  title = 	 {Comparison of numerical methods for modelling ocean circulation in basins with irregular coasts},
-  school = 	 {McGill University},
-  year = 	 2001,
+  author =	 {F. Dupont},
+  title =	 {Comparison of numerical methods for modelling ocean
+                  circulation in basins with irregular coasts},
+  school =	 {McGill University},
+  year =	 2001,
   address =	 {Montreal}
 }
 
 @article{fadlun2000,
-  title={Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations},
-  author={Fadlun, EA and Verzicco, R. and Orlandi, P. and Mohd-Yusof, J.},
-  journal={Journal of Computational Physics},
-  volume={161},
-  number={1},
-  pages={35--60},
-  year={2000}
+  title =	 {Combined immersed-boundary finite-difference methods
+                  for three-dimensional complex flow simulations},
+  author =	 {Fadlun, EA and Verzicco, R. and Orlandi, P. and
+                  Mohd-Yusof, J.},
+  journal =	 {Journal of Computational Physics},
+  volume =	 161,
+  number =	 1,
+  pages =	 {35--60},
+  year =	 2000
 }
 
 @Article{fornberg1988,
-  author = 	 {B. Fornberg},
-  title = 	 {Steady viscous flow past a sphere at high {R}eynolds number},
-  journal = 	 {J. Fluid Mech.},
-  year = 	 1988,
-  volume = 	 190,
-  pages = 	 471
+  author =	 {B. Fornberg},
+  title =	 {Steady viscous flow past a sphere at high {R}eynolds
+                  number},
+  journal =	 {J. Fluid Mech.},
+  year =	 1988,
+  volume =	 190,
+  pages =	 471
 }
 
 @article{gerlach2006,
-  author =      {D. Gerlach and G. Tomar and G. Biswas and F. Durst},
-  title =       {Comparison of surface tension methods for surface tension dominant two-phase flows},
-  journal =     {Int. J.  Heat Mass Transfer},
-  year =        2006,
-  pages=        {740-754},
-  volume=       49
+  author =	 {D. Gerlach and G. Tomar and G. Biswas and F. Durst},
+  title =	 {Comparison of surface tension methods for surface
+                  tension dominant two-phase flows},
+  journal =	 {Int. J.  Heat Mass Transfer},
+  year =	 2006,
+  pages =	 {740-754},
+  volume =	 49
 }
 
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-  author = 	 {U. Ghia, K.N. Ghia, C.T. Shin},
-  title = 	 {High-{R}e solution for incompressible flow using the {N}avier-{S}tokes equations and the multigrid method},
-  journal = 	 {J. Comput. Phys.},
-  year = 	 1982,
+  author =	 {U. Ghia, K.N. Ghia, C.T. Shin},
+  title =	 {High-{R}e solution for incompressible flow using the
+                  {N}avier-{S}tokes equations and the multigrid
+                  method},
+  journal =	 {J. Comput. Phys.},
+  year =	 1982,
   volume =	 48,
   pages =	 {387-411}
 }
 
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-  title = 	 {Volume of fluid interface tracking with smoothed surface stress methods for three-dimensional flows},
-  journal = 	 {J. Comp. Phys.},
-  year = 	 1998,
+  author =	 {D. Gueyffier and A. Nadim and J. Li and
+                  R. Scardovelli and S. Zaleski},
+  title =	 {Volume of fluid interface tracking with smoothed
+                  surface stress methods for three-dimensional flows},
+  journal =	 {J. Comp. Phys.},
+  year =	 1998,
   volume =	 152,
   pages =	 {423-456}
 }
 
 @Book{lamb,
   author =	 {H. Lamb},
-  title = 	 {Hydrodynamics},
-  publisher = 	 {Dover},
-  year = 	 1932
+  title =	 {Hydrodynamics},
+  publisher =	 {Dover},
+  year =	 1932
 }
 
 @Article{leroux98,
-  author = 	 {D. Y. Leroux and C. A. Lin},
-  title = 	 {Finite elements for shallow-water equations ocean models},
-  journal = 	 {Monthly Weather Review},
-  year = 	 1998,
+  author =	 {D. Y. Leroux and C. A. Lin},
+  title =	 {Finite elements for shallow-water equations ocean
+                  models},
+  journal =	 {Monthly Weather Review},
+  year =	 1998,
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   pages =	 {1931-1951}
 }
 
 @Article{liang2008,
-  title  =    {Adaptive quadtree simulation of shallow flows with wet--dry fronts over complex topography},
-  author =    {Q. Liang and A.G.L. Borthwick},
-  journal=    {Computers and Fluids},
-  year =      {2008},
-  publisher = {Elsevier}
+  title =	 {Adaptive quadtree simulation of shallow flows with
+                  wet--dry fronts over complex topography},
+  author =	 {Q. Liang and A.G.L. Borthwick},
+  journal =	 {Computers and Fluids},
+  year =	 2008,
+  publisher =	 {Elsevier}
 }
 
 @Article{lynch87,
-  author = 	 {D. R. Lynch and F. E. Werner},
-  title = 	 {3-{D} hydrodynamics on finite elements. {P}art I: linearized harmonic model},
-  journal = 	 {Int. J. for Num. Methods in Fluids},
-  year = 	 1987,
+  author =	 {D. R. Lynch and F. E. Werner},
+  title =	 {3-{D} hydrodynamics on finite elements. {P}art I:
+                  linearized harmonic model},
+  journal =	 {Int. J. for Num. Methods in Fluids},
+  year =	 1987,
   number =	 7,
   pages =	 {871-909}
 }
 
 @article{masliyah1970,
-  title={Numerical study of steady flow past spheroids},
-  author={Masliyah, J.H. and Epstein, N.},
-  journal={Journal of Fluid Mechanics},
-  volume={44},
-  number={03},
-  pages={493--512},
-  year={1970},
-  publisher={Cambridge Univ Press}
+  title =	 {Numerical study of steady flow past spheroids},
+  author =	 {Masliyah, J.H. and Epstein, N.},
+  journal =	 {Journal of Fluid Mechanics},
+  volume =	 44,
+  number =	 03,
+  pages =	 {493--512},
+  year =	 1970,
+  publisher =	 {Cambridge Univ Press}
 }
 
 @Article{minion96,
-  author = 	 {M. L. Minion},
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-  volume =       127,
-  pages =        {158-177},
-  year = 	 1996,
-  url =          {http://citeseer.nj.nec.com/minion96projection.html},
-  local_url =    {minion96.ps.gz}
+  author =	 {M. L. Minion},
+  title =	 {A Projection Method for Locally Refined Grids},
+  journal =	 {J. Comput. Phys.},
+  volume =	 127,
+  pages =	 {158-177},
+  year =	 1996,
+  url =		 {http://citeseer.nj.nec.com/minion96projection.html},
+  local_url =	 {minion96.ps.gz}
 }
 
 @Article{peraire86,
-  author = 	 {J. Peraire and O. C. Zienkiewicz and K. Morgan},
-  title = 	 {Shallow water problems: a general explicit formulation},
-  journal = 	 {Int. J. for Num. Methods in Eng.},
-  year = 	 1986,
+  author =	 {J. Peraire and O. C. Zienkiewicz and K. Morgan},
+  title =	 {Shallow water problems: a general explicit
+                  formulation},
+  journal =	 {Int. J. for Num. Methods in Eng.},
+  year =	 1986,
   volume =	 22,
   pages =	 {547-574}
 }
 
 @Article{popinet2003,
-  author = 	 {S. Popinet},
-  title = 	 {Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries},
-  journal = 	 {J. Comput. Phys.},
-  year = 	 2003,
+  author =	 {S. Popinet},
+  title =	 {Gerris: a tree-based adaptive solver for the
+                  incompressible Euler equations in complex
+                  geometries},
+  journal =	 {J. Comput. Phys.},
+  year =	 2003,
   volume =	 190,
   number =	 2,
   pages =	 {572-600},
-  url =          {http://gfs.sf.net/gerris.pdf}
+  url =		 {http://gfs.sf.net/gerris.pdf}
 }
 
 @Article{popinet99,
-  author =       {S. Popinet and S. Zaleski},
-  title =        {A front tracking algorithm for the accurate representation of surface tension},
-  journal =      {Int. J. Numer. Meth. Fluids},
-  volume =       30,
-  pages =        {775-793},
-  year =         1999
+  author =	 {S. Popinet and S. Zaleski},
+  title =	 {A front tracking algorithm for the accurate
+                  representation of surface tension},
+  journal =	 {Int. J. Numer. Meth. Fluids},
+  volume =	 30,
+  pages =	 {775-793},
+  year =	 1999
 }
 
 @Article{prosperetti81,
-  author = 	 {A. Prosperetti},
-  title = 	 {Motion of two superposed viscous fluids},
-  journal = 	 {Phys. Fluids},
-  year = 	 1981,
+  author =	 {A. Prosperetti},
+  title =	 {Motion of two superposed viscous fluids},
+  journal =	 {Phys. Fluids},
+  year =	 1981,
   volume =	 24,
   pages =	 {1217-1223}
 }
 
 @TechReport{rider95,
-  author = 	 {W. J. Rider},
-  title = 	 {Approximate projection methods for incompressible flows: Implementation, variants and robustness},
-  institution =  {Los Alamos National Laboratory},
-  year = 	 1995,
+  author =	 {W. J. Rider},
+  title =	 {Approximate projection methods for incompressible
+                  flows: Implementation, variants and robustness},
+  institution =	 {Los Alamos National Laboratory},
+  year =	 1995,
   number =	 {LA-UR-2000},
-  local_url =    {rider95.ps.gz},
-  url =          {http://www-xdiv.lanl.gov/XHM/personnel/wjr/Web_papers/proj/proj.ps.Z},
-  pages=         {81-85}
+  local_url =	 {rider95.ps.gz},
+  url =
+                  {http://www-xdiv.lanl.gov/XHM/personnel/wjr/Web_papers/proj/proj.ps.Z},
+  pages =	 {81-85}
 }
 
 @Article{rossmanith2004,
-  author = 	 {J. A, Rossmanith and D. S. Bale and R. J. LeVeque},
-  title = 	 {A wave propagation algorithm for hyperbolic systems on curved manifolds},
-  journal = 	 {Journal of Computational Physics},
-  year = 	 2004,
-  volume = 	 199,
-  pages = 	 {631-662}
+  author =	 {J. A, Rossmanith and D. S. Bale and R. J. LeVeque},
+  title =	 {A wave propagation algorithm for hyperbolic systems
+                  on curved manifolds},
+  journal =	 {Journal of Computational Physics},
+  year =	 2004,
+  volume =	 199,
+  pages =	 {631-662}
+}
+
+ at article{rossmanith2006,
+  title =	 {A wave propagation method for hyperbolic systems on
+                  the sphere},
+  author =	 {Rossmanith, J.A.},
+  journal =	 {Journal of Computational Physics},
+  volume =	 213,
+  number =	 2,
+  pages =	 {629--658},
+  year =	 2006,
+  publisher =	 {Elsevier}
 }
 
 @article{rudman97,
-  title={Volume-tracking methods for interfacial flow calculations},
-  author={Rudman, M.},
-  journal={International Journal for Numerical Methods in Fluids},
-  volume={24},
-  number={7},
-  pages={671--691},
-  year={1997}
+  title =	 {Volume-tracking methods for interfacial flow
+                  calculations},
+  author =	 {Rudman, M.},
+  journal =	 {International Journal for Numerical Methods in
+                  Fluids},
+  volume =	 24,
+  number =	 7,
+  pages =	 {671--691},
+  year =	 1997
 }
 
 @TechReport{rutgers-waves,
-  author = 	 {E. Curchitser},
-  title = 	 {Waves in a circular basin},
-  institution =  {Rutgers University},
-  year = 	 2005,
-  url =          {http://marine.rutgers.edu/po/index.php?model=test-problems&title=circle}
+  author =	 {E. Curchitser},
+  title =	 {Waves in a circular basin},
+  institution =	 {Rutgers University},
+  year =	 2005,
+  url =
+                  {http://marine.rutgers.edu/po/index.php?model=test-problems&title=circle}
 }
 
 @Article{sampson2006,
-  title =   {Moving boundary shallow water flow above parabolic bottom topography},
-  author =  {J. Sampson and A. Easton and M. Singh},
-  journal = {ANZIAM J},
-  volume =  {47},
-  year   =  {2006}
+  title =	 {Moving boundary shallow water flow above parabolic
+                  bottom topography},
+  author =	 {J. Sampson and A. Easton and M. Singh},
+  journal =	 {ANZIAM J},
+  volume =	 47,
+  year =	 2006
 }
 
 @Article{torres00,
-  author = 	 {D. J. Torres and J. U. Brackbill},
-  title = 	 {The Point-Set method: front-tracking without connectivity},
-  journal = 	 {J. Comput. Phys.},
-  year = 	 2000,
-  volume = 	 165,
-  pages = 	 {620-644}
+  author =	 {D. J. Torres and J. U. Brackbill},
+  title =	 {The Point-Set method: front-tracking without
+                  connectivity},
+  journal =	 {J. Comput. Phys.},
+  year =	 2000,
+  volume =	 165,
+  pages =	 {620-644}
 }
 
 @Article{vola2004,
-  author = 	 {D. Vola and F. Babik and J.-C Latch\'e},
-  title = 	 {On a numerical strategy to compute gravity currents of non-Newtonian fluids},
-  journal = 	 {J. Comput. Phys.},
-  year = 	 2004,
+  author =	 {D. Vola and F. Babik and J.-C Latch\'e},
+  title =	 {On a numerical strategy to compute gravity currents
+                  of non-Newtonian fluids},
+  journal =	 {J. Comput. Phys.},
+  year =	 2004,
   volume =	 201,
   number =	 2,
   pages =	 {397-420}
 }
 
+ at Article{williamson92,
+  author =	 {{Williamson}, D.~L. and {Drake}, J.~B. and {Hack},
+                  J.~J. and {Jakob}, R. and {Swarztrauber}, P.~N.},
+  title =	 {A standard test set for numerical approximations to
+                  the shallow water equations in spherical geometry},
+  journal =	 {Journal of Computational Physics},
+  year =	 1992,
+  volume =	 102,
+  pages =	 {211-224},
+  doi =		 {10.1016/S0021-9991(05)80016-6}
+}
+
 @Article{zhang2007,
-  author = 	 {N. Zhang and Z.C. Zheng},
-  title = 	 {An improved direct-forcing immersed-boundary method for finite difference applications},
-  journal = 	 {Journal of Computational Physics},
-  year = 	 2007,
-  volume = 	 221,
-  pages = 	 {250-268}
+  author =	 {N. Zhang and Z.C. Zheng},
+  title =	 {An improved direct-forcing immersed-boundary method
+                  for finite difference applications},
+  journal =	 {Journal of Computational Physics},
+  year =	 2007,
+  volume =	 221,
+  pages =	 {250-268}
 }

-- 
Gerris Flow Solver



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