[SCM] Gerris Flow Solver branch, upstream, updated. b3aa46814a06c9cb2912790b23916ffb44f1f203
Stephane Popinet
s.popinet at niwa.co.nz
Fri May 15 02:52:44 UTC 2009
The following commit has been merged in the upstream branch:
commit 2fc6ec3cb9c27f990749644b7803f2116400bbf7
Author: Stephane Popinet <s.popinet at niwa.co.nz>
Date: Tue Oct 4 13:19:00 2005 +1000
Geostrophic adjustment test cases
darcs-hash:20051004031900-fbd8f-63b6a367134abbd84e807443b4e173388f6abac5.gz
diff --git a/test/euler/Makefile.am b/test/euler/Makefile.am
index c9984bf..784cf4a 100644
--- a/test/euler/Makefile.am
+++ b/test/euler/Makefile.am
@@ -10,7 +10,8 @@ TESTDIRS = \
channel \
spurious \
capwave \
- waves
+ waves \
+ geo
EXTRA_DIST = \
template.tex \
diff --git a/test/euler/geo/beta/beta.gfs b/test/euler/geo/beta/beta.gfs
new file mode 100644
index 0000000..92059a3
--- /dev/null
+++ b/test/euler/geo/beta/beta.gfs
@@ -0,0 +1,67 @@
+# Title: Geostrophic adjustment on a beta-plane
+#
+# Description:
+#
+# Same as before but a beta-plane, $f = f_0 + \beta y$ is used and
+# the advection terms are included in the momentum equation. No
+# explicit dissipation is added. As in {\cite{dupont}} we chose $\beta
+# = 1.607 \times 10^{- 11}$ m$^{- 1}$s$^{- 1}$. The geostrophic eddy
+# moves slowly westward through the emission of Rossby waves and
+# southward due to the non-linear advection term. The resulting
+# evolution of the total energy is shown on figure \ref{energy}. For
+# our method, the slow decrease in the total energy is due both to the
+# dissipation of potential energy induced by the approximate
+# projection operator and to the dissipative properties of the BCG
+# upwind advection scheme. Another run with the advection terms
+# switched off (figure \ref{energy}, green curve) confirms that the
+# dissipation induced by the approximate projection operator dominates
+# the total dissipation. The results however compare favourably with
+# the finite-element formulations tested by Dupont which all show
+# significantly larger energy dissipation.
+#
+# \begin{figure}[htbp]
+# \begin{center}
+# \includegraphics[width=\hsize]{energy.eps}
+# \end{center}
+# \caption{Evolution
+# of the total energy for the non-linear geostrophic adjustment problem. The
+# C-grid model is based on Sadourny {\cite{}} and implemented by Dupont
+# {\cite{dupont}}. The finite-element formulations are those studied by Dupont. LW:
+# Lynch and Werner {\cite{lynch87}}, LLS: Le Roux et al {\cite{leroux98}}, PZM: Peraire et al
+# {\cite{peraire86}}.}
+# \label{energy}
+# \end{figure}
+#
+# Author: St\'ephane Popinet
+# Command: sh beta.sh beta.gfs
+# Version: 0.8.0
+# Required files: beta.sh c dlw lls pzm llw energy.ref energy-nonlinear.ref
+# Running time: 3 minutes
+# Generated files: energy.eps
+#
+1 0 GfsOcean GfsBox GfsGEdge {} {
+ # dt = 1000 s
+ Time { iend = 1580 dtmax = 0.10285 }
+ Refine 6
+ # Lx = Ly = 1000 km
+ # H0 = 1000 m
+ # g = 0.01 m/s^2
+ PhysicalParams { g = 9.4534734306584e-4 }
+ ApproxProjectionParams { tolerance = 1e-6 }
+ Init {} {
+ # e-folding radius = 100 km
+ # umax = 0.5 m/s = sqrt(200)*exp(-1/2)
+ U = (5.667583815e-4*200.*y*exp (-100.*(x*x + y*y)))
+ V = (- 5.667583815e-4*200.*x*exp (-100.*(x*x + y*y)))
+ P = (5.667583815e-4*exp (-100.*(x*x + y*y)))
+ H = 1
+ }
+ # f0 = 1.0285e-4 s-1
+ # beta = 1.607e-11 m-1s-1
+ SourceCoriolis {} (1. + 0.156246961595*(y + 0.5))
+
+ OutputScalarSum { istep = 150 } { awk '{print $3/1.0285e-4/3600./24. " " $5/6.34646e-06}' > energy } { v = (Velocity2 + P*P/9.4534734306584e-4) }
+}
+GfsBox {
+ front = Boundary
+}
diff --git a/test/euler/geo/beta/beta.sh b/test/euler/geo/beta/beta.sh
new file mode 100644
index 0000000..1d6d291
--- /dev/null
+++ b/test/euler/geo/beta/beta.sh
@@ -0,0 +1,35 @@
+if ! $donotrun; then
+ if gerris2D3 $1; then :
+ else
+ exit 1
+ fi
+ mv -f energy energy-nonlinear
+ if sed 's/Refine 6/Refine 6\nAdvectionParams {scheme = none}/' < $1 |\
+ gerris2D3 -; then :
+ else
+ exit 1
+ fi
+fi
+
+if cat <<EOF | gnuplot; then :
+ set term postscript eps lw 3 color solid 20
+ set output 'energy.eps'
+ set xlabel 'Time (days)'
+ set ylabel 'Normalised total energy'
+ set key bottom left
+ plot 'c' t 'C-grid' w lp, 'energy' t 'Gerris (linear)' w lp, 'energy-nonlinear' t 'Gerris' w lp, 'dlw' t 'Delumped LW' w lp, 'lls' t 'LLS' w lp, 'pzm' t 'PZM' w lp, 'llw' t 'Lumped LW' w lp
+EOF
+else
+ exit 1
+fi
+
+if cat <<EOF | python ; then :
+from check import *
+from sys import *
+if (Curve('energy.ref',1,2) - Curve('energy',1,2)).max() > 1e-2 or \
+ (Curve('energy-nonlinear.ref',1,2) - Curve('energy-nonlinear',1,2)).max() > 1e-2:
+ exit(1)
+EOF
+else
+ exit 1
+fi
diff --git a/test/euler/geo/beta/c b/test/euler/geo/beta/c
new file mode 100644
index 0000000..e497c33
--- /dev/null
+++ b/test/euler/geo/beta/c
@@ -0,0 +1,12 @@
+0 1
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+5.491 0.99960331
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+18.298 0.99952338
+
diff --git a/test/euler/geo/beta/dlw b/test/euler/geo/beta/dlw
new file mode 100644
index 0000000..4841961
--- /dev/null
+++ b/test/euler/geo/beta/dlw
@@ -0,0 +1,12 @@
+0 1
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+17.361 0.94626438
+
diff --git a/test/euler/geo/beta/energy-nonlinear.ref b/test/euler/geo/beta/energy-nonlinear.ref
new file mode 100644
index 0000000..c115e40
--- /dev/null
+++ b/test/euler/geo/beta/energy-nonlinear.ref
@@ -0,0 +1,11 @@
+0 1
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diff --git a/test/euler/geo/beta/energy.ref b/test/euler/geo/beta/energy.ref
new file mode 100644
index 0000000..3de376a
--- /dev/null
+++ b/test/euler/geo/beta/energy.ref
@@ -0,0 +1,11 @@
+0 1
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diff --git a/test/euler/geo/beta/lls b/test/euler/geo/beta/lls
new file mode 100644
index 0000000..23cd688
--- /dev/null
+++ b/test/euler/geo/beta/lls
@@ -0,0 +1,12 @@
+0 0.9999599
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+
diff --git a/test/euler/geo/beta/llw b/test/euler/geo/beta/llw
new file mode 100644
index 0000000..699373c
--- /dev/null
+++ b/test/euler/geo/beta/llw
@@ -0,0 +1,12 @@
+0 1
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+
diff --git a/test/euler/geo/beta/pzm b/test/euler/geo/beta/pzm
new file mode 100644
index 0000000..e4ea0f9
--- /dev/null
+++ b/test/euler/geo/beta/pzm
@@ -0,0 +1,12 @@
+0 1
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+
diff --git a/test/euler/geo/e.ref b/test/euler/geo/e.ref
new file mode 100644
index 0000000..0b52019
--- /dev/null
+++ b/test/euler/geo/e.ref
@@ -0,0 +1,1581 @@
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diff --git a/test/euler/geo/geo.gfs b/test/euler/geo/geo.gfs
new file mode 100644
index 0000000..c8d7dab
--- /dev/null
+++ b/test/euler/geo/geo.gfs
@@ -0,0 +1,119 @@
+# Title: Geostrophic adjustment
+#
+# Description:
+#
+# We consider the geostrophic adjustment problem studied by
+# Dupont {\cite{dupont}} and Le Roux et al {\cite{leroux98}}. A Gaussian bump
+# \[ \eta ( x, y ) = \eta_0 e^{^{- \frac{x^2 + y^2}{R^2}}} \]
+# is initialised in a 1000$\times$1000 km, 1000 m deep square basin. A reduced
+# gravity $g = 0.01$ m/s is used to approximate a 10 m-thick stratified surface
+# layer. On an $f$-plane the corresponding geostrophic velocities are given by
+# \begin{eqnarray*}
+# u ( x, y ) & = & \frac{2 g \eta_0 y}{f_0 R^2} e^{- \frac{x^2 + y^2}{R^2}},\\
+# v ( x, y ) & = & - \frac{2 g \eta_0 x}{f_0 R^2} e^{- \frac{x^2 + y^2}{R^2}},
+# \end{eqnarray*}
+# where $f_0$ is the Coriolis parameter. Following Dupont we set $f_0 = 1.0285
+# \times 10^{- 4}$ s$^{- 1}$, $R = 100$ km, $\eta_0 = 599.5$ m which gives a
+# maximum geostrophic velocity of 0.5 m/s.
+#
+# In the context of the linearised shallow-water equations, the geostrophic
+# balance is an exact solution which should be preserved by the numerical
+# method. In practice, this would require an exact numerical balance between
+# terms computed very differently: the pressure gradient and the Coriolis terms
+# in the momentum equation. If this numerical balance is not exact, the
+# numerical solution will adjust toward numerical equilibrium through the
+# emission of gravity-wave noise which should not affect the stability of the
+# solution. This problem is thus a good test of both the overall accuracy of the
+# numerical scheme and its stability properties when dealing with
+# inertia--gravity waves. We note in particular that a standard A-grid
+# discretisation would develop a strong computational-mode instability in this
+# case. Also, as studied by Leroux et al, an inappropriate choice of
+# finite-element basis functions will result in growing gravity-wave noise.
+#
+# \begin{figure}[htbp]
+# \begin{center}
+# \includegraphics[width=\hsize]{geo_error.eps}
+# \end{center}
+# \caption{Evolution of the maximum error on the surface height for the
+# geostrophic adjustment problem.}
+# \label{geo-error}
+# \end{figure}
+#
+# \begin{figure}[htbp]
+# \begin{center}
+# \begin{tabular}{ccccc}
+# \includegraphics[width=0.18\hsize]{error-100.eps} &
+# \includegraphics[width=0.18\hsize]{error-200.eps} &
+# \includegraphics[width=0.18\hsize]{error-300.eps} &
+# \includegraphics[width=0.18\hsize]{error-400.eps} &
+# \includegraphics[width=0.18\hsize]{error-1500.eps} \\
+# (a) & (b) & (c) & (d) & (e)
+# \end{tabular}
+# \end{center}
+# \caption{Evolution of the surface-height error field. (a) $t =$1.157
+# days, (b) $t = 2.315$ days, (c) $t =$3.472 days, (d) $t =$4.630 days, (e) $t
+# =$17.361 days.}
+# \label{geo-error1}
+# \end{figure}
+#
+# Figures \ref{geo-error} and \ref{geo-error1} summarise the results obtained
+# when running the geostrophic adjustment problem on a $64 \times 64$ uniform
+# grid with a timestep $\Delta t = 1000$ s. The maximum error on the height
+# field (Figure \ref{geo-error}) is small even after 18 days. After a strong
+# initial transient corresponding to the emission of gravity waves, the error
+# reaches a minimum at day 3 and then slowly grows with time with modulations
+# due to the reflexions of the initial gravity waves on the domain boundaries.
+# As illustrated on figure \ref{geo-error1}, this growth is not due to any
+# instability of the solution but to the slow decrease of the maximum amplitude
+# of the Gaussian bump due to numerical energy dissipation.
+#
+# Author: St\'ephane Popinet
+# Command: sh geo.sh geo.gfs
+# Version: 0.8.0
+# Required files: geo.sh geo.gfv e.ref
+# Running time: 3 minutes
+# Generated files: geo_error.eps error-100.eps error-200.eps error-300.eps error-400.eps error-1500.eps
+#
+1 0 GfsOcean GfsBox GfsGEdge {} {
+ # dt = 1000 s
+ Time { iend = 1580 dtmax = 0.10285 }
+ Refine 6
+ # Lx = Ly = 1000 km
+ # H0 = 1000 m
+ # g = 0.01 m/s^2
+ PhysicalParams { g = 9.4534734306584e-4 }
+ AdvectionParams { scheme = none }
+ ApproxProjectionParams { tolerance = 1e-6 }
+ Init {} {
+ # e-folding radius = 100 km
+ # umax = 0.5 m/s = sqrt(200)*exp(-1/2)
+ U = (5.667583815e-4*200.*y*exp (-100.*(x*x + y*y)))
+ V = (- 5.667583815e-4*200.*x*exp (-100.*(x*x + y*y)))
+ P = (5.667583815e-4*exp (-100.*(x*x + y*y)))
+ H = 1
+ }
+ # f0 = 1.0285e-4 s-1
+ SourceCoriolis {} 1
+
+ OutputErrorNorm { istep = 1 } { awk '{print $3/1.0285e-4/3600./24. " " $9*1000e6*1.0285e-4*1.0285e-4/0.01;}' > e } { v = P } {
+ s = (5.667583815e-4*exp (-100.*(x*x + y*y)))
+ unbiased = 1
+ v = E
+ }
+ OutputSimulation { istart = 100 iend = 500 istep = 100 } stdout {}
+ EventScript { istart = 100 iend = 500 istep = 100 } { echo "Save error-$GfsIter.eps { format = EPS }"}
+ OutputSimulation { istart = 1500 } stdout {}
+ EventScript { istart = 1500 } { echo "Save error-$GfsIter.eps { format = EPS }"}
+ EventScript { start = end } {
+ cat <<EOF | gnuplot
+ set term postscript eps lw 3 color solid 20
+ set output 'geo_error.eps'
+ set xlabel 'Time (days)'
+ set ylabel 'Maximum error on surface height (cm)'
+ plot 'e.ref' u 1:(\$2*100.) t '' w l, 'e' u 1:(\$2*100.) t 'ref' w l
+EOF
+ }
+}
+GfsBox {
+ front = Boundary
+}
diff --git a/test/euler/merging/levels.gfv b/test/euler/geo/geo.gfv
similarity index 84%
copy from test/euler/merging/levels.gfv
copy to test/euler/geo/geo.gfv
index 52abaa9..d61f0af 100644
--- a/test/euler/merging/levels.gfv
+++ b/test/euler/geo/geo.gfv
@@ -1,4 +1,4 @@
-# GfsView 2D
+# GfsView 3D
View {
tx = 0 ty = 0
q0 = 0 q1 = 0 q2 = 0 q3 = 1
@@ -11,11 +11,11 @@ View {
Squares {
r = 0 g = 0 b = 0
shading = Constant
- maxlevel = 8
+ maxlevel = -1
} {
n.x = 0 n.y = 0 n.z = 1
- pos = 0
-} Level {
+ pos = -0.5
+} E {
amin = 1
amax = 1
cmap = Jet
diff --git a/test/euler/geo/geo.sh b/test/euler/geo/geo.sh
new file mode 100644
index 0000000..7596ee8
--- /dev/null
+++ b/test/euler/geo/geo.sh
@@ -0,0 +1,16 @@
+if ! $donotrun; then
+ if gerris2D3 $1 | gfsview-batch2D3 geo.gfv; then :
+ else
+ exit 1
+ fi
+fi
+
+if cat <<EOF | python ; then :
+from check import *
+from sys import *
+if (Curve('e',1,2) - Curve('e.ref',1,2)).max() > 1e-3:
+ exit(1)
+EOF
+else
+ exit 1
+fi
diff --git a/test/euler/template.tex b/test/euler/template.tex
index 46e246a..9fa75c5 100644
--- a/test/euler/template.tex
+++ b/test/euler/template.tex
@@ -54,6 +54,8 @@ current stable branch of the version-controlled source code.
\section{Shallow-water}
+\input{geo/geo.tex}
+\input{geo/beta/beta.tex}
\input{waves/waves.tex}
\bibliographystyle{plain}
diff --git a/test/euler/tests.bib b/test/euler/tests.bib
index 0eb7951..ddc0561 100644
--- a/test/euler/tests.bib
+++ b/test/euler/tests.bib
@@ -21,6 +21,14 @@
local_url = {almgren1.ps.gz}
}
+ at PhdThesis{dupont,
+ 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{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},
@@ -37,6 +45,24 @@
year = 1932
}
+ at 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,
+ volume = 126,
+ pages = {1931-1951}
+}
+
+ at 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,
+ number = 7,
+ pages = {871-909}
+}
+
@Article{minion96,
author = {M. L. Minion},
title = {A Projection Method for Locally Refined Grids},
@@ -48,6 +74,15 @@
local_url = {minion96.ps.gz}
}
+ at 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,
+ 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},
--
Gerris Flow Solver
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