我正在尝试计算向量场的散度：

Fx  = np.cos(xx + 2*yy)
Fy  = np.sin(xx - 2*yy)
F = np.array([Fx, Fy])

分析解决方案

这是基于散度的解析计算的散度 (div(F) = dF/dx + dF/dy ) 的样子（请参阅此处的Wolfram Alpha ）：

• dFx/dx = d/dx cos(x+2y) = -sin(x+2y)
• dFy/dy = d/dy sin(x-2y) = -2*cos(x-2y)

分歧：

div_analy = -np.sin(xx + 2*yy) - 2*np.cos(xx - 2*yy)

编码：

# Number of points (NxN)
N = 50
# Boundaries
ymin = -2.; ymax = 2.
xmin = -2.; xmax = 2.


# Create Meshgrid
x = np.linspace(xmin,xmax, N)
y = np.linspace(ymin,ymax, N)
xx, yy = np.meshgrid(x, y)

# Analytic computation of the divergence (EXACT) 
div_analy = -np.sin(xx + 2*yy) - 2*np.cos(xx - 2*yy)

# PLOT
plt.imshow(div_analy , extent=[xmin,xmax,ymin,ymax],  origin="lower", cmap="jet")

数值解

现在，我试图获得相同的数值，所以我使用这个函数来计算散度

def divergence(f,sp):
    """ Computes divergence of vector field 
    f: array -> vector field components [Fx,Fy,Fz,...]
    sp: array -> spacing between points in respecitve directions [spx, spy,spz,...]
    """
    num_dims = len(f)
    return np.ufunc.reduce(np.add, [np.gradient(f[i], sp[i], axis=i) for i in range(num_dims)])

当我使用此函数绘制散度图时：

 # Compute Divergence
 points = [x,y]
 sp = [np.diff(p)[0] for p in points]
 div_num = divergence(F, sp)
    
 # PLOT
 plt.imshow(div_num, extent=[xmin,xmax,ymin,ymax], origin="lower",  cmap="jet")

......我明白了：

问题

数值解不同于解析解！我究竟做错了什么？