Breakup of a rectangular perturbation into a train of solitons

We reproduce the study of Madsen et al, 2008. An initial rectangular perturbation of width 2b and height a is superposed on an ocean of constant depth h_0.

We solve the one-dimensional problem with an adaptive “bitree”. We compute both the Saint-Venant solution and the Green-Naghdi solutions.

#include
#if SAINT_VENANT
# include
#else
# include
#endif

We need a very high level of refinement to accurately reproduce the results of Madsen et al. The depth h_0 and gravity are both set to unity (Madsen et al use these non-dimensional units). The amplitude is set to 0.1 and half-width to 12.2 as done in section 3.2.1 of Madsen et al, 2008.

#define MAXLEVEL 15
double a = 0.1, b = 12.2;

int main() {
N = 1 << MAXLEVEL;
L0 = 4000.;
G = 1.;
#if !SAINT_VENANT
#endif
run();
}

The initial conditions are a (half)rectangle sitting on the axis of symmetry at x=0.

event init (i = 0) {
foreach() {
h[] = 1. + a*(x < b);
zb[] = -1.;
}
}

We save the free-surface elevation at times t\sqrt{g/h_0}= 35, 90, 700 and 2000 as in Figure 1 of Madsen et al, 2008. We use a frame of reference travelling with the wave i.e. we use (x-b)/h_0-t\sqrt{g/h_0} as coordinate.

event plot (t = {35, 90, 700, 2000}) {
char name;
sprintf (name, "t-%g", t);
FILE * fp = fopen (name, "w");
foreach()
fprintf (fp, "%g %g\n", x - b - t, eta[]);
fclose (fp);
}

We need a low error threshold for adaptation on \eta.

#if TREE
event adapt (i++) {
astats s = adapt_wavelet ({eta}, (double[]){1e-6}, MAXLEVEL);
fprintf (stderr, "%g refined %d cells, coarsened %d cells\n",
t, s.nf, s.nc);
}
#endif

Finally we compare the two results (for Green-Naghdi and Saint-Venant) with the results of Madsen et al, 2008, Figure 1. The GN solutions reproduces the breakup of the initial perturbation into a train of solitary waves at long times, while the Saint-Venant solution becomes very inaccurate.

The agreement between Madsen et al and the GN solution is quite good given that the model of Madsen et al is a higher-order Boussinesq expansion than the GN model (Madsen et al, 2006).

set term svg enhanced size 640,480 font ",9"
set multiplot layout 4,1 scale 1.,1.
set ytics 0,0.05,0.1
set yrange [-0.02:0.1]
set xtics -50,10,50
plot [-50:50]'../madf1a.plot' w l t 'Madsen et al, 2008', \
'< sort -n -k1,2 t-35' w l t 'SGN', \
'< sort -n -k1,2 ../madsen-sv/t-35' w l t 'Saint-Venant'
set xtics -100,20,100
set xrange [-100:100]
unset key
plot '../madf1b.plot' w l, \
'< sort -n -k1,2 t-90' w l, \
'< sort -n -k1,2 ../madsen-sv/t-90' w l
plot '../madf1c.plot' w l, \
'< sort -n -k1,2 t-700' w l, \
'< sort -n -k1,2 ../madsen-sv/t-700' w l
plot '../madf1d.plot' w l, \
'< sort -n -k1,2 t-2000' w l, \
'< sort -n -k1,2 ../madsen-sv/t-2000' w l
unset multiplot Evolution of surface elevation (script)