我正在实现RKF4(5)集成器,但是我无法弄清楚我的代码是否有效,并且我不了解本地截断错误,或者我的代码是否无效。
我为代码块的大小表示歉意,但在这种情况下,最小的可重现示例相当大。
import numpy as np
def RKF45(state, derivative_function, h):
"""
Calculate the next state with the 4th-order calculation, along with a 5th-order error
check.
Inputs:
state: the current value of the function, float
derivative_function: A function which takes a state (given as a float)
and returns the derivative of a function at that point
h: step size, float
"""
k1 = h * derivative_function(state)
k2 = h * derivative_function(state + (k1 / 4))
k3 = h * derivative_function(state + (k1 * (3/32)) + (k2 * (9/32)))
k4 = h * derivative_function(state + (k1 * (1932/2197)) + (k2 * (-7200/2197)) + (k3 * (7296/2197)))
k5 = h * derivative_function(state + (k1 * (439/216)) + (k2 * (-8)) + (k3 * (3680/513)) + (k4 * (-845/4104)))
k6 = h * derivative_function(state + (k1 * (-8/27)) + (k2 * (2)) + (k3 * (-3544/2565)) + (k4 * (1859/4104)) + (k5 * (-11/40)))
y1 = state + ((25/216) * k1) + ((1408/2565) * k3) + ((2197/4101) * k4) - ((1/5)*k5)
y2 = state + ((16/135) * k1) + ((6656/12825) * k3) + ((28561/56430) * k4) - ((9/50) * k5) + ((2/55) * k6)
return(y1, y2)
def integrate_RKF45(t0, tmax, tol, h_init, x_0, df, verbose = False):
"""
integrate a function whose derivative is df from t0 to tmax
t0: starting time
tmax: end time
h_init: initial timestep
x_0: starting position
df: a function which takes x and returns the derivative of a function at x
"""
h = h_init
x_i = x_0
t = t0
while t < tmax:
h = min(h, tmax - t)
y1, y2 = RKF45(x_i, df, h)
err_i = np.abs(y1 - y2)
R = 2 * err_i / h
delta = (tol/R)**(1/4)
if verbose:
print(f"t: {t:0.2e}, dt: {h:0.2e}, x: {x_i:0.2e}, err: {err_i:0.2e}")
if err_i < tol:
t += h
x_i = y1
elif err_i > tol:
h *= delta
return(x_i)
def exponential(x_0, k=1):
"""
A simple test function, this returns the input, so it'll integrate to e^x.
"""
return(k * x_0)
if __name__ == "__main__":
integrate_RKF45(t0 = 0.,
tmax = 0.15,
tol = 1e-4,
h_init = 1e-2,
x_0 = 1.,
df = exponential,
verbose=True)
因此,此代码works在一定程度上返回了我提供的任何函数的积分的近似值。但是,本地截断错误似乎太大。运行上面的代码将输出:
t: 0.00e+00, dt: 1.00e-02, x: 1.00e+00, err: 3.95e-06
t: 1.00e-02, dt: 1.00e-02, x: 1.01e+00, err: 3.99e-06
t: 2.00e-02, dt: 1.00e-02, x: 1.02e+00, err: 4.03e-06
t: 3.00e-02, dt: 1.00e-02, x: 1.03e+00, err: 4.07e-06
t: 4.00e-02, dt: 1.00e-02, x: 1.04e+00, err: 4.11e-06
t: 5.00e-02, dt: 1.00e-02, x: 1.05e+00, err: 4.16e-06
t: 6.00e-02, dt: 1.00e-02, x: 1.06e+00, err: 4.20e-06
t: 7.00e-02, dt: 1.00e-02, x: 1.07e+00, err: 4.24e-06
t: 8.00e-02, dt: 1.00e-02, x: 1.08e+00, err: 4.28e-06
t: 9.00e-02, dt: 1.00e-02, x: 1.09e+00, err: 4.32e-06
t: 1.00e-01, dt: 1.00e-02, x: 1.11e+00, err: 4.37e-06
t: 1.10e-01, dt: 1.00e-02, x: 1.12e+00, err: 4.41e-06
t: 1.20e-01, dt: 1.00e-02, x: 1.13e+00, err: 4.46e-06
t: 1.30e-01, dt: 1.00e-02, x: 1.14e+00, err: 4.50e-06
t: 1.40e-01, dt: 1.00e-02, x: 1.15e+00, err: 4.55e-06
err
值是4阶和5阶方法之间的差。我的印象是,n^th
阶方法的局部截断误差为O(dt^(n+1))
阶,这意味着上述积分的误差应该在1e-9
左右,而不是1e-6
。
所以我的代码错误还是我的理解错误?谢谢!
请参见https://math.stackexchange.com/questions/2701385/adaptive-step-size-in-rk45-for-second-order-ode/2701678#2701678,您似乎对方法系数使用了相同的损坏源。
y1中的分母4101是错误的,必须为4104。
增量因子应该稍微减轻一点,delta = (tol/R)**(1/5)
或delta = (tol/R)**(1/6)
,并在每个步骤中都应用,也要在成功的步骤中应用。
本地错误err_i
的参考错误为tol*h
,这就是为什么在R
中用h
除以的原因。
这将导致迭代步骤
t: 0.000000e+00, dt: 1.00e-02, x: 1.000000e+00, err: 1.28e-13
t: 1.000000e-02, dt: 1.40e-01, x: 1.010050e+00, err: 6.60e-08
t: 1.500000e-01, dt: 3.88e-01, x: 1.161834e+00
或间隔较长时间
t: 0.000000e+00, dt: 1.00e-02, x: 1.000000e+00, err: 1.28e-13
t: 1.000000e-02, dt: 2.27e-01, x: 1.010050e+00, err: 7.18e-07
t: 2.372490e-01, dt: 4.31e-01, x: 1.267757e+00, err: 2.02e-05
t: 6.680839e-01, dt: 4.76e-01, x: 1.950509e+00, err: 5.03e-05
t: 6.680839e-01, dt: 4.47e-01, x: 1.950509e+00, err: 3.73e-05
t: 1.115525e+00, dt: 3.84e-01, x: 3.051213e+00, err: 2.81e-05
t: 1.500000e+00, dt: 3.89e-01, x: 4.481769e+00
所有提到的更正均在RKF45中给出了新的循环
while t < tmax:
h = min(h, tmax - t)
y1, y2 = RKF45(x_i, df, h)
err_i = np.abs(y1 - y2)
R = err_i / h
delta = 0.95*(tol/R)**(1/5)
if verbose:
print(f"t: {t:0.6e}, dt: {h:0.2e}, x: {x_i:0.6e}, err: {err_i:0.2e}")
if R < tol:
t += h
x_i = y1
h *= delta
if verbose:
print(f"t: {t:0.6e}, dt: {h:0.2e}, x: {x_i:0.6e}")