ABM.py

""" From "COMPUTATIONAL PHYSICS" & "COMPUTER PROBLEMS in PHYSICS"
    by RH Landau, MJ Paez, and CC Bordeianu (deceased)
    Copyright R Landau, Oregon State Unv, MJ Paez, Univ Antioquia, 
    C Bordeianu, Univ Bucharest, 2017. 
    Please respect copyright & acknowledge our work."""

# ABM.py:   Adams BM method to integrate ODE
# Solves y' = (t - y)/2,    with y[0] = 1 over [0, 3]

from visual.graph import *

numgr = gdisplay(x=0, y=0, width=600, height=300, xmin=0.0, xmax = 3.0,
          title="Numerical Solution", xtitle='t', ytitle='y', ymax=2., ymin=0.9)
numsol = gcurve(color=color.yellow, display = numgr)
exactgr = gdisplay(x=0, y=300, width=600, height=300, title="Exact solution",
            xtitle='t', ytitle='y', xmax=3.0, xmin=0.0, ymax=2.0, ymin=0.9)

exsol =  gcurve(color = color.cyan, display = exactgr)
n = 24                                                 # N steps > 3
A = 0; B = 3.                                           
t =[0]*500;     y =[0]*500;     yy=[0]*4     
                        
def f(t, y):                                      # RHS F function
    return  (t - y)/2.0

def rk4(t, yy, h1):             
    for i in range(0, 3):
        t  = h1 * i
        k0 = h1 * f(t, y[i])
        k1 = h1 * f(t + h1/2., yy[i] + k0/2.)
        k2 = h1 * f(t + h1/2., yy[i] + k1/2.)
        k3 = h1 * f(t + h1, yy[i] + k2 )
        yy[i + 1] = yy[i]  +  (1./6.) * (k0  +  2.*k1  +  2.*k2 + k3)
        print(i,yy[ i])
    return yy[3]

def ABM(a,b,N):
# Compute 3 additional starting values using rk
   h = (b-a) / N                          # step
   t[0] = a;    y[0] = 1.00;    F0  = f(t[0], y[0])
   for k in range(1, 4):                   
      t[k] = a  +  k * h
   y[1]  = rk4(t[1], y, h)                      # 1st step
   y[2]  = rk4(t[2], y, h)                      # 2nd step
   y[3] = rk4(t[3], y, h)                       # 3rd step
   F1 = f(t[1], y[1])        
   F2 = f(t[2], y[2])        
   F3 = f(t[3], y[3])
   h2 = h/24.

   for k in range(3, N):                               # Predictor
      p = y[k]  +  h2*(-9.*F0  +  37.*F1 - 59.*F2 + 55.*F3)
      t[k + 1] = a + h*(k+1)                       # Next abscissa
      F4 = f(t[k+1], p)                        
      y[k+1] = y[k] + h2*(F1-5.*F2 + 19.*F3 + 9.*F4)   # Corrector
      F0 = F1                                     # Update values
      F1 = F2
      F2 = F3
      F3 = f(t[k + 1], y[k + 1])
   return t,y

print("  k     t      Y numerical      Y exact")
t, y = ABM(A,B,n)
for k in range(0, n+1):
    print (" %3d  %5.3f  %12.11f  %12.11f " %(k,t[k],y[k],(3.*exp(-t[k]/2.)-2.+t[k])))
    numsol.plot(pos = (t[k], y[k]) )
    exsol.plot(pos = (t[k], 3.*exp(-t[k]/2.) -2. + t[k]))