用Python实现四阶龙格-库塔（Runge-Kutta）方法求解高阶微分方程
问题
应用四阶龙格-库塔（Runge-Kutta）方法求解如下二阶初值问题:
\begin{equation}
\left\{
\begin{aligned}
t^2x''(t)-2tx'(t)+2x(t) & = t^3\ln t, & t\in [1,5]\\
x(t) & = 1, & t=1 \\
x'(t) & = 0. & t=1
\end{aligned}
\right.
\end{equation}
要求:取步长h=0.01,给出解x(t)的图像和在t=0,0.5,1,1.5,2,2.5,3,3.5,4,4.5,5处的近似解.
求解步骤
-
Step1. 将原问题归结为其等价问题
引进新的变量
y(t)=x'(t)将高阶微分方程的初值问题归结为如下一阶微分方程组的初值问题来求解.\begin{equation} \left\{ \begin{aligned} x'(t) & = y, & t\in [1,5]\\ y'(t) & = \frac{t^3\ln t +2ty -2x}{t^2}, & t\in [1,5]\\ x(t) & = y, & t=1\\ y(t) & = 0. & t=1 \end{aligned} \right. \end{equation} -
Step2. 四阶龙格-库塔方法的离散格式
针对以上一阶微分方程组的初值问题应用四阶龙格-库塔方法构造得到以下离散格式:
\begin{equation} \left\{ \begin{aligned} x_{n+1} & = x_n +\frac{h}{6}(K_1 + 2K_2 + 2K_3 + K_4),\\ y_{n+1} & = y_n +\frac{h}{6}(L_1 + 2L_2 + 2L_3 + L_4). \end{aligned} \right. \end{equation}其中
\begin{equation} \left\{ \begin{aligned} K_1 & = f(t_n,x_n,y_n),\\ K_2 & = f(t_n + \frac{h}{2},x_n + \frac{h}{2}K_1,y_n + \frac{h}{2}L_1),\\ K_3 & = f(t_n + \frac{h}{2},x_n + \frac{h}{2}K_2,y_n + \frac{h}{2}L_2),\\ K_4 & = f(t_n + h,x_n + hK_3,y_n + hL_3),\\ L_1 & = g(t_n,x_n,y_n),\\ L_2 & = g(t_n + \frac{h}{2},x_n + \frac{h}{2}K_1,y_n + \frac{h}{2}L_1),\\ L_3 & = g(t_n + \frac{h}{2},x_n + \frac{h}{2}K_2,y_n + \frac{h}{2}L_2),\\ L_4 & = f(t_n + h,x_n + hK_3,y_n + hL_3). \end{aligned} \right. \end{equation}这是一步法，利用节点
t_n上的值x_n,y_n，由 (4) 式顺序计算K_1,L_1,K_2,L_2,K_3,L_3,K_4,L_4，然后代入(3)式即可求得节点t_{n+1}上的x_{n+1},y_{n+1}.- Step3. 利用龙格-库塔法求解高阶微分方程的
Python代码如下:
# 开发者: Leo 刘 # 开发环境: macOs Big Sur # 开发时间: 2021/9/28 4:31 下午 # 邮箱 : 517093978@qq.com # @Software: PyCharm # ---------------------------------------------------------------------------------------------------------- import math import matplotlib.pyplot as plt def f(t, x, y): a = y return a def g(t, x, y): a = (t ** 3 * math.log(t) + 2 * t * y - 2 * x) / t ** 2 return a def RK4(t, x, y, h): """ :param t: t 的初始值 :param x: x 的初始值 :param y: y 的初始值 :param h: 时间步长 :return: 迭代新解 """ tarray, xarray, yarray = [], [], [] while t <= 5: tarray.append(t) xarray.append(x) yarray.append(y) t += h K_1 = f(t, x, y) L_1 = g(t, x, y) K_2 = f(t + h / 2, x + h / 2 * K_1, y + h / 2 * L_1) L_2 = g(t + h / 2, x + h / 2 * K_1, y + h / 2 * L_1) K_3 = f(t + h / 2, x + h / 2 * K_2, y + h / 2 * L_2) L_3 = g(t + h / 2, x + h / 2 * K_2, y + h / 2 * L_2) K_4 = f(t + h, x + h * K_3, y + h * L_3) L_4 = g(t + h, x + h * K_3, y + h * L_3) x = x + (K_1 + 2 * K_2 + 2 * K_3 + K_4) * h / 6 y = y + (L_1 + 2 * L_2 + 2 * L_3 + L_4) * h / 6 return tarray, xarray, yarray def main(): tarray, xarray, yarray = RK4(1, 1, 0, 0.01) print("龙格-库塔 数值结果".center(168)) print('-' * 178) print("对象\\时刻\t", "t=0\t\t", " t=0.5\t\t", " t=1\t\t\t", " t=1.5\t\t", " t=2\t\t\t", " t=2.5\t\t\t", " t=3\t\t\t", " t=3.5\t\t", " t=4\t\t\t", " t=4.5\t\t\t", " t=5") print('-' * 178) print("x:", end='') for i in range(len(xarray)): if i % 40 == 0: print("\t\t", "%8.4f" % xarray[i], end='') print('\n', '-' * 177) print("y:", end='') for i in range(len(yarray)): if i % 40 == 0: print("\t\t", "%8.4f" % yarray[i], end='') print('\n', '-' * 177) plt.figure('龙格-库塔 数值结果') plt.subplot(221) # plt.plot(tarray, xarray, label='x_runge_kutta') plt.scatter(tarray, xarray, label='x_scatter', s=1, c='#DC143C', alpha=0.6) # plt.xlabel('t') plt.legend() plt.subplot(222) # plt.plot(tarray, yarray, label='y_runge_kutta') plt.scatter(tarray, yarray, label='y_scatter', s=1, c='#DC143C', alpha=0.6) # plt.xlabel('t') plt.legend() plt.subplot(212) # plt.plot(tarray, xarray, label='runge_kutta') plt.scatter(tarray, xarray, label='Numerical solution scatter', s=1, c='#DC143C', alpha=0.6) plt.xlabel('t') plt.legend() plt.show() if __name__ == "__main__": main()- Step4. 代码的运行结果如下：
龙格-库塔 数值结果 ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 对象\时刻 t=0 t=0.5 t=1 t=1.5 t=2 t=2.5 t=3 t=3.5 t=4 t=4.5 t=5 ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- x: 1.0000 0.8579 0.5080 0.1042 -0.1564 -0.0416 0.7105 2.3875 5.2995 9.7769 16.1685 --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- y: 0.0000 -0.6689 -1.0161 -0.9205 -0.2855 0.9689 2.9116 5.6026 9.0952 13.4374 18.6728 --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - Step3. 利用龙格-库塔法求解高阶微分方程的
问题来源: 《微分方程数值解》-M.Ran
