非线性微分方程 (Korteweg–De Vries) 随着时间的推移变得不稳定

Question

我想用初始条件 U(x,0) = n(n+1)/cosh^2(x) 求解 Korteweg-De Vries 方程。然而，当 n>1（n 是整数）时，我的系统在数值上变得不稳定，这可能是什么原因？

import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp

# Parameters
L = 40    # Length of the interval
N = 256   # Number of grid points
dx = L / N  # Spatial step size
x = np.linspace(-L/2, L/2, N)  # Discretized spatial domain
dt = 0.1  # Time step size
T = 20    # End time
n = 1     # Parameter n for the initial condition

# Initial condition u(x,0) = n*(n+1)/cosh^2(x)
u0 = (n * (n + 1)) / np.cosh(x)**2

# Function to compute derivatives according to the KdV equation
def derivs(t, u):
    # Periodic boundary conditions are applied using np.roll
    uxx = np.roll(u, -1) - 2 * u + np.roll(u, 1)  # Second derivative
    uxx /= dx**2
    uxxx = np.roll(uxx, -1) - np.roll(uxx, 1)  # Third derivative
    uxxx /= 2 * dx
    # KdV equation: time derivative of u
    return -6 * u * (np.roll(u, -1) - np.roll(u, 1)) / (2 * dx) - uxxx

# Solve the KdV equation using a 4th-order Runge-Kutta method
sol = solve_ivp(derivs, [0, T], u0, method='RK45', t_eval=np.arange(0, T, dt))

# Animation of the solution
plt.figure()
for i in range(len(sol.t)):
    plt.clf()  # Clear the previous plot
    plt.plot(x, sol.y[:, i])  # Plot the solution at time t[i]
    plt.title(f"Solution of the KdV equation at t={sol.t[i]:.1f}")
    plt.xlabel("x")  # Label for the x-axis
    plt.ylabel("u(x,t)")  # Label for the y-axis
    plt.ylim([-0.5, n*(n+1)+0.5])  # Set y-axis limits for better visualization
    plt.pause(0.1)  # Short pause to animate the plot

plt.show()  # Show the final plot window

Answer 1

对于一般的 n 来说，你的初始条件是错误的。应该是

u0 = (n * (n + 1)) / np.cosh( 0.5 * np.sqrt(2*n*(n+1.0)) * x ) ** 2

尽管如此，您仍然会得到少量的色散。