What is the possible source of error in Runge-Kutta 4 algorithm in python for solving SHM
Question:
I wrote a simple RK4 program in python to solve SHM equation:
y''[t] = -w^2*y[t]
by writing:
z'[t] = -w^2*y
y'[t] = z
Following is the RK4 code:
def rk4_gen(t, y, z, h):
k1 = h*f1(y, z, t)
l1 = h*f2(y, z, t)
k2 = h*f1(y+k1/2.0, z+l1/2.0, t+h/2.0)
l2 = h*f2(y+k1/2.0, z+l1/2.0, t+h/2.0)
k3 = h*f1(y+k2/2.0, z+l2/2.0, t+h/2.0)
l3 = h*f2(y+k2/2.0, z+l2/2.0, t+h/2.0)
k4 = h*f1(y+k3, z+l3, t+h)
l4 = h*f2(y+k3, z+l3, t+h)
y = y + k1/6.0 + k2/3.0 + k3/3.0 + k4/6.0
z = z + l1/6.0 + l2/3.0 + l3/3.0 + l4/6.0
t = t + h
return t, y, z
def f1(y, z, t):
return z
def f2(y, z, t):
return -y
The plot of the result is coming almost sinusoidal, but the values exceeding beyond -1 and 1 by small amount.
Step 0 : t = 0.000, y = 0.00000000, z = 1.00000000
Step 1 : t = 0.100, y = 0.09987500, z = 0.99583542
Step 2 : t = 0.200, y = 0.19891812, z = 0.98171316
Step 3 : t = 0.300, y = 0.29613832, z = 0.95775779
Step 4 : t = 0.400, y = 0.39056108, z = 0.92419231
Step 5 : t = 0.500, y = 0.48123826, z = 0.88133615
Step 6 : t = 0.600, y = 0.56725756, z = 0.82960208
Step 7 : t = 0.700, y = 0.64775167, z = 0.76949228
Step 8 : t = 0.800, y = 0.72190710, z = 0.70159347
Step 9 : t = 0.900, y = 0.78897230, z = 0.62657115
Step 10 : t = 1.000, y = 0.84826536, z = 0.54516314
Step 11 : t = 1.100, y = 0.89918085, z = 0.45817226
Step 12 : t = 1.200, y = 0.94119609, z = 0.36645847
Step 13 : t = 1.300, y = 0.97387644, z = 0.27093037
Step 14 : t = 1.400, y = 0.99687982, z = 0.17253614
Step 15 : t = 1.500, y = 1.00996028, z = 0.07225423
Step 16 : t = 1.600, y = 1.01297061, z = -0.02891646
Step 17 : t = 1.700, y = 1.00586398, z = -0.12996648
Step 18 : t = 1.800, y = 0.98869457, z = -0.22988588
Step 19 : t = 1.900, y = 0.96161722, z = -0.32767438
Step 20 : t = 2.000, y = 0.92488601, z = -0.42235127
Step 21 : t = 2.100, y = 0.87885191, z = -0.51296534
Step 22 : t = 2.200, y = 0.82395944, z = -0.59860439
Step 23 : t = 2.300, y = 0.76074238, z = -0.67840440
Step 24 : t = 2.400, y = 0.68981857, z = -0.75155827
Step 25 : t = 2.500, y = 0.61188388, z = -0.81732398
Step 26 : t = 2.600, y = 0.52770540, z = -0.87503206
Step 27 : t = 2.700, y = 0.43811390, z = -0.92409250
Step 28 : t = 2.800, y = 0.34399560, z = -0.96400066
Step 29 : t = 2.900, y = 0.24628344, z = -0.99434256
Step 30 : t = 3.000, y = 0.14594781, z = -1.01479910
Step 31 : t = 3.100, y = 0.04398694, z = -1.02514942
Step 32 : t = 3.200, y = -0.05858305, z = -1.02527330
Step 33 : t = 3.300, y = -0.16073825, z = -1.01515248
Step 34 : t = 3.400, y = -0.26145719, z = -0.99487106
Step 35 : t = 3.500, y = -0.35973108, z = -0.96461480
Step 36 : t = 3.600, y = -0.45457385, z = -0.92466944
Step 37 : t = 3.700, y = -0.54503210, z = -0.87541801
Step 38 : t = 3.800, y = -0.63019464, z = -0.81733718
Step 39 : t = 3.900, y = -0.70920170, z = -0.75099262
Step 40 : t = 4.000, y = -0.78125355, z = -0.67703353
Step 41 : t = 4.100, y = -0.84561868, z = -0.59618627
Step 42 : t = 4.200, y = -0.90164114, z = -0.50924723
Step 43 : t = 4.300, y = -0.94874724, z = -0.41707502
Step 44 : t = 4.400, y = -0.98645148, z = -0.32058195
Step 45 : t = 4.500, y = -1.01436144, z = -0.22072502
Step 46 : t = 4.600, y = -1.03218196, z = -0.11849644
Step 47 : t = 4.700, y = -1.03971818, z = -0.01491378
Step 48 : t = 4.800, y = -1.03687770, z = 0.08899018
Step 49 : t = 4.900, y = -1.02367164, z = 0.19217774
Step 50 : t = 5.000, y = -1.00021473, z = 0.29361660
Why are the value of y or z exceeding beyond range [-1,1]? Is there any type-casting error that is propagating??
Answers:
No, it’s either step size or just the way floating point numbers work.
Cut the step size in half and see if the error goes down.
You need to understand that all numerical algorithms like this are approximations. You should not expect to see a perfect 1.0 at the top of the sine wave or a perfect 0.0 when it crosses the x-axis.
You can’t represent 0.1 exactly in binary any more than you can represent 1/3 exactly in decimal. And then there’s the way IEEE floating point numbers work. You need to understand these things.
Here’s another thought: How confident are you that your RK4 implementation is correct? I don’t have the time to pore through it right now, but I’d wonder why you don’t use a library like NumPy instead of your own? Perhaps they understand numerical methods better than we do, and more users means finding and fixing bugs faster.
You wrote TDSE in your comment. Am I correct in assuming that this is the Time Dependent Schrodinger Equation? If yes, are you using complex numbers correctly in your solution? Is the integration scheme treating them properly?
I wrote a simple RK4 program in python to solve SHM equation:
y''[t] = -w^2*y[t]
by writing:
z'[t] = -w^2*y
y'[t] = z
Following is the RK4 code:
def rk4_gen(t, y, z, h):
k1 = h*f1(y, z, t)
l1 = h*f2(y, z, t)
k2 = h*f1(y+k1/2.0, z+l1/2.0, t+h/2.0)
l2 = h*f2(y+k1/2.0, z+l1/2.0, t+h/2.0)
k3 = h*f1(y+k2/2.0, z+l2/2.0, t+h/2.0)
l3 = h*f2(y+k2/2.0, z+l2/2.0, t+h/2.0)
k4 = h*f1(y+k3, z+l3, t+h)
l4 = h*f2(y+k3, z+l3, t+h)
y = y + k1/6.0 + k2/3.0 + k3/3.0 + k4/6.0
z = z + l1/6.0 + l2/3.0 + l3/3.0 + l4/6.0
t = t + h
return t, y, z
def f1(y, z, t):
return z
def f2(y, z, t):
return -y
The plot of the result is coming almost sinusoidal, but the values exceeding beyond -1 and 1 by small amount.
Step 0 : t = 0.000, y = 0.00000000, z = 1.00000000
Step 1 : t = 0.100, y = 0.09987500, z = 0.99583542
Step 2 : t = 0.200, y = 0.19891812, z = 0.98171316
Step 3 : t = 0.300, y = 0.29613832, z = 0.95775779
Step 4 : t = 0.400, y = 0.39056108, z = 0.92419231
Step 5 : t = 0.500, y = 0.48123826, z = 0.88133615
Step 6 : t = 0.600, y = 0.56725756, z = 0.82960208
Step 7 : t = 0.700, y = 0.64775167, z = 0.76949228
Step 8 : t = 0.800, y = 0.72190710, z = 0.70159347
Step 9 : t = 0.900, y = 0.78897230, z = 0.62657115
Step 10 : t = 1.000, y = 0.84826536, z = 0.54516314
Step 11 : t = 1.100, y = 0.89918085, z = 0.45817226
Step 12 : t = 1.200, y = 0.94119609, z = 0.36645847
Step 13 : t = 1.300, y = 0.97387644, z = 0.27093037
Step 14 : t = 1.400, y = 0.99687982, z = 0.17253614
Step 15 : t = 1.500, y = 1.00996028, z = 0.07225423
Step 16 : t = 1.600, y = 1.01297061, z = -0.02891646
Step 17 : t = 1.700, y = 1.00586398, z = -0.12996648
Step 18 : t = 1.800, y = 0.98869457, z = -0.22988588
Step 19 : t = 1.900, y = 0.96161722, z = -0.32767438
Step 20 : t = 2.000, y = 0.92488601, z = -0.42235127
Step 21 : t = 2.100, y = 0.87885191, z = -0.51296534
Step 22 : t = 2.200, y = 0.82395944, z = -0.59860439
Step 23 : t = 2.300, y = 0.76074238, z = -0.67840440
Step 24 : t = 2.400, y = 0.68981857, z = -0.75155827
Step 25 : t = 2.500, y = 0.61188388, z = -0.81732398
Step 26 : t = 2.600, y = 0.52770540, z = -0.87503206
Step 27 : t = 2.700, y = 0.43811390, z = -0.92409250
Step 28 : t = 2.800, y = 0.34399560, z = -0.96400066
Step 29 : t = 2.900, y = 0.24628344, z = -0.99434256
Step 30 : t = 3.000, y = 0.14594781, z = -1.01479910
Step 31 : t = 3.100, y = 0.04398694, z = -1.02514942
Step 32 : t = 3.200, y = -0.05858305, z = -1.02527330
Step 33 : t = 3.300, y = -0.16073825, z = -1.01515248
Step 34 : t = 3.400, y = -0.26145719, z = -0.99487106
Step 35 : t = 3.500, y = -0.35973108, z = -0.96461480
Step 36 : t = 3.600, y = -0.45457385, z = -0.92466944
Step 37 : t = 3.700, y = -0.54503210, z = -0.87541801
Step 38 : t = 3.800, y = -0.63019464, z = -0.81733718
Step 39 : t = 3.900, y = -0.70920170, z = -0.75099262
Step 40 : t = 4.000, y = -0.78125355, z = -0.67703353
Step 41 : t = 4.100, y = -0.84561868, z = -0.59618627
Step 42 : t = 4.200, y = -0.90164114, z = -0.50924723
Step 43 : t = 4.300, y = -0.94874724, z = -0.41707502
Step 44 : t = 4.400, y = -0.98645148, z = -0.32058195
Step 45 : t = 4.500, y = -1.01436144, z = -0.22072502
Step 46 : t = 4.600, y = -1.03218196, z = -0.11849644
Step 47 : t = 4.700, y = -1.03971818, z = -0.01491378
Step 48 : t = 4.800, y = -1.03687770, z = 0.08899018
Step 49 : t = 4.900, y = -1.02367164, z = 0.19217774
Step 50 : t = 5.000, y = -1.00021473, z = 0.29361660
Why are the value of y or z exceeding beyond range [-1,1]? Is there any type-casting error that is propagating??
No, it’s either step size or just the way floating point numbers work.
Cut the step size in half and see if the error goes down.
You need to understand that all numerical algorithms like this are approximations. You should not expect to see a perfect 1.0 at the top of the sine wave or a perfect 0.0 when it crosses the x-axis.
You can’t represent 0.1 exactly in binary any more than you can represent 1/3 exactly in decimal. And then there’s the way IEEE floating point numbers work. You need to understand these things.
Here’s another thought: How confident are you that your RK4 implementation is correct? I don’t have the time to pore through it right now, but I’d wonder why you don’t use a library like NumPy instead of your own? Perhaps they understand numerical methods better than we do, and more users means finding and fixing bugs faster.
You wrote TDSE in your comment. Am I correct in assuming that this is the Time Dependent Schrodinger Equation? If yes, are you using complex numbers correctly in your solution? Is the integration scheme treating them properly?