radau5 等求解算法
    English Answer:
    The Radau5 method is a fifth-order implicit Runge-Kutta method for solving ordinary differential equations. It is a member of the Radau family of methods, which are known for their high order of accuracy and efficiency. For a system of equations.
    y' = f(t, y)。
    with initial condition.
    y(t0) = y0。
    the Radau5 method takes the form.
numpy库功能
    y_n+1 = y_n + h  (1/36)  (5k_1 + 9k_2 + 24k_3 + 9k_4 + 5k_5)。
    where.
    k_1 = f(t_n, y_n)。
    k_2 = f(t_n + (1/3)h, y_n + (1/36)h  k_1)。
    k_3 = f(t_n + (1/2)h, y_n + (1/24)h  (2k_2 + k_3))。
    k_4 = f(t_n + (3/4)h, y_n + (1/12)h  (3k_3 + k_4))。
    k_5 = f(t_n + h, y_n + (1/2)h  (k_1 + 4k_4 + k_5))。
    The Radau5 method is a popular choice for solving stiff ordinary differential equations, which are equations that exhibit a wide range of time scales. This is because the method is implicit, which means that it does not require a small time step to maintain stability. However, the method can be more computationally expensive than explicit methods, such as the Runge-Kutta-Fehlberg method.
    Here is a Python implementation of the Radau5 method:
    python.
    import numpy as np.
    def radau5(f, t0, y0, h, nsteps):
        """
        Solve an ordinary differential equation using the Radau5 method.
        Args:
            f: The right-hand side of the ODE.
            t0: The initial time.
            y0: The initial condition.
            h: The step size.
            nsteps: The number of steps to take.
        Returns:
            A numpy array of the solution at each time step.
        """
        t = np.linspace(t0, t0 + h  nsteps, nsteps + 1)。
        y = np.zeros((nsteps + 1, len(y0)))。
        y[0] = y0。
        for i in range(1, nsteps + 1):
            k1 = f(t[i-1], y[i-1])。
            k2 = f(t[i-1] + (1/3)  h, y[i-1] + (1/36)  h  k1)。
            k3 = f(t[i-1] + (1/2)  h, y[i-1] + (1/24)  h  (2  k2 + k3))。

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