# Green-Naghdi soliton

The Green-Naghdi system of equations admits solitary wave solutions of the form \displaystyle \eta(\psi) = ah_0\mathrm{sech}^2\left(\psi\frac{\sqrt{3ah_0}} {2h_0\sqrt{h_0(1+a)}}\right) \displaystyle u(\psi) = \frac{c\eta}{h_0+\eta} with \psi = x - ct and the soliton velocity c^2=g(1+a)h_0.

#include "grid/multigrid1D.h"
#include "green-naghdi.h"

The domain is 700 metres long, the acceleration of gravity is 10 m/s2. We need to set the dispersion parameter \alpha_d to one. We compute the solution in one dimension for a number of grid points varying between 128 and 1024.

int main()
{
X0 = -200.;
L0 = 700.;
G = 10.;
alpha_d = 1.;
for (N = 128; N <= 1024; N *= 2)
run();
}

We follow Le Métayer et al, 2010 (section 6.1.2) for the values of h_0 and a.

double h0 = 10, a = 0.21;

double sech2 (double x) {
double a = 2./(exp(x) + exp(-x));
return a*a;
}

double soliton (double x, double t)
{
double c = sqrt(G*(1. + a)*h0), psi = x - c*t;
double k = sqrt(3.*a*h0)/(2.*h0*sqrt(h0*(1. + a)));
return a*h0*sech2 (k*psi);
}

event init (i = 0)
{
double c = sqrt(G*(1. + a)*h0);
foreach() {
double eta = soliton (x, t);
h[] = h0 + eta;
u.x[] = c*eta/(h0 + eta);
}
}

We output the profiles and reference solution at regular intervals.

event output (t = {0,7.3,14.6,21.9,29.2}) {
if (N == 256) {
foreach()
fprintf (stdout, "%g %g %g %g\n", x, h[] - h0, u.x[], soliton (x, t));
fprintf (stdout, "\n");
}
}

We compute the error between the theoretical and numerical solutions at t = 29.2.

event error (t = end) {
scalar e[];
foreach()
e[] = h[] - h0 - soliton (x, t);
fprintf (stderr, "%d %g\n", N, normf(e).max/(a*h0));
}
set grid
set xlabel 'x'
set ylabel 'z'
plot 'out' u 1:4 w l t 'exact', 'out' w l t 'numerical' Depth profiles for N = 256. (script)

The method has a second-order rate of convergence as expected.

set logscale
set xlabel 'N'
set ylabel 'max|e|/a'
set xtics 128,2,1024
set cbrange [1:2]
set grid
fit [5:] a*x+b 'log' u (log($1)):(log($2)) via a,b
plot [100:1250]'log' u 1:2 pt 7 t '', \
exp(b)*x**a t sprintf("%.0f/N^{%4.2f}", exp(b), -a) Relative error as a function of resolution. (script)