def solver_memsave(I, a, T, dt, theta, filename='sol.dat'):
"""
Solve u'=-a*u, u(0)=I, for t in (0,T] with steps of dt.
Minimum use of memory. The solution is stored in a file
(with name filename) for later plotting.
"""
dt = float(dt) # avoid integer division
Nt = int(round(T/dt)) # no of intervals
outfile = open(filename, 'w')
# u: time level n+1, u_1: time level n
t = 0
u_1 = I
outfile.write('%.16E %.16E\n' % (t, u_1))
for n in range(1, Nt+1):
u = (1 - (1-theta)*a*dt)/(1 + theta*dt*a)*u_1
u_1 = u
t += dt
outfile.write('%.16E %.16E\n' % (t, u))
outfile.close()
return u, t