the code. I post it again now after correcting it.
Solitary-wave interaction: (using matlab)
Use the code section5.m, to study the interaction of solitary
waves in the Whitham equation. Show that a dispersive tail appears
after the interaction, and provide evidence that these oscillations
are not due to numerical instability.
- Hi I Posted This Question Before With A Slight Type Error In The Code I Post It Again Now After Correcting It Solitar 1 (7.57 KiB) Viewed 28 times
%isospectral integration of u_t + uu_x + K*u = 0
% K = fourier transform of dis(k)
N = 512; %input('N=')
L = 8;
a = L/(2*pi);
T = 1.1; %input('T=')
h = 2*pi/N;
dt = .005;
y = a*(h:h:2*pi)';
c1 = 2;
u = exp(-c1*a*(y-pi).^2)/a;
k = [(1:N/2)'; (1-N/2:-1)'];
dis = [1; sqrt(tanh(k)./k)];
Dv = [0; k];
m = a^(-3)*.5*dt*dis.^3;
d1 = (1+i*m)./(1-i*m);
d2 = -.5*i*dt*Dv./(1-i*m);
d3 = .5*d2;
sol = plot(y, a*u, 'r', 'Erasemode', 'background');
box off;
axis([ 0 L -.2 1.75]);
title('Whitham Equation, weak dispersion');
text(1, 1.6, 'dispersion relation = (tanh(k)/k)^{1/2}')
text(1, 1.4, 'u_t + uu_x + K*u = 0, u(x,0) = 2\pi
exp(-2(x-4)^2)/8')
t=0;
while t<T
fftu = fft(u);
fftuu = fft(u.^2);
v = real(ifft(d1.*fftu + 0.5*d2.*fftuu));
w = real(ifft(d1.*fftu + d2.*fftuu));
for n = 1:3
w = v + real(ifft(d3.*fft(w.^2)));
end
u = w;
t = t + dt;
set(sol, 'ydata', a*u);
drawnow;
end
The full Whitham equation is written as nt + mz+K*nz = 0. (2)