[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
}
@Article{ghia82,
- 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}
}
@Article{gueyffier98,
- 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,
+ 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},
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+ title = {Finite elements for shallow-water equations ocean
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pages = {1931-1951}
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+ wet--dry fronts over complex topography},
+ author = {Q. Liang and A.G.L. Borthwick},
+ journal = {Computers and Fluids},
+ year = 2008,
+ publisher = {Elsevier}
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+ title = {A wave propagation algorithm for hyperbolic systems
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+ Fluids},
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}
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+ journal = {ANZIAM J},
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+ year = 2006
}
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@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,
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+ author = {N. Zhang and Z.C. Zheng},
+ title = {An improved direct-forcing immersed-boundary method
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+ journal = {Journal of Computational Physics},
+ year = 2007,
+ volume = 221,
+ pages = {250-268}
}
--
Gerris Flow Solver
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