我不确定你是否从你的帖子中指出了这一点，但我无法让你的代码按原样运行。然而，我认为主要问题与你提到的第一点有关：“* 在每个动画步骤之后，旧的点应该被删除。*”在绘制动画时，您确实需要明确这一点。目前，您的代码正在为同一个Axes重复创建scatter。就像在动画之外执行此操作一样，这将导致多个数据集相互重叠绘制。
我看到人们这样做的两种主要方式：或者使用绘图的一些set_...方法来 * 更新 * 数据（参见here for scatter plots或here in general）或每次迭代清除Axes或Figure以便绘制新数据。我发现后者更容易/更普遍（如果更懒惰）。下面是一种方法，可以让你的例子做到这一点（我已经编辑了这段代码，删除了对plt.grid和plt.label的调用，因为它们不起作用）：

import matplotlib.pyplot as plt
import matplotlib.animation as animation
import numpy as np

time_steps = 50
N_nodes = 100

positions = []
solutions = []
for i in range(time_steps):
    positions.append(np.random.rand(2, N_nodes))
    solutions.append(np.random.random(N_nodes))

fig, ax = plt.subplots()
marker_size = 5 #upped this to make points more visible

def animate(i):
    """ Perform animation step. """
    #important - the figure is cleared and new axes are added
    fig.clear()
    ax = fig.add_subplot(111, aspect='equal', autoscale_on=False, xlim=(0, 1), ylim=(0, 1))
    #the new axes must be re-formatted
    ax.set_xlim(0,1)
    ax.set_ylim(0,1)
    ax.grid(b=None)
    ax.set_xlabel('x [m]')
    ax.set_ylabel('y [m]')
    # and the elements for this frame are added
    ax.text(0.02, 0.95, 'Time step = %d' % i, transform=ax.transAxes)
    s = ax.scatter(positions[i][0], positions[i][1], s = marker_size, c = solutions[i], cmap = "RdBu_r", marker = ".", edgecolor = None)
    fig.colorbar(s)

ani = animation.FuncAnimation(fig, animate, interval=100, frames=range(time_steps))

ani.save('animation.gif', writer='pillow')

生成以下GIF：

这里，我使用fig.clear()来清除每帧的colorbar;否则，他们中的许多人将被抽走。这意味着每次都必须重新添加Axes和格式。在其他情况下，使用ax.clear()可以很好，并保存add_subplot的步骤。
但是，还有另一种方法可以做到这一点，遵循here。如果您有颜色条Axes的句柄，您可以只清除它们（而不是清除整个Figure），类似于散点图轴：

import matplotlib.pyplot as plt
import matplotlib.animation as animation
import numpy as np

time_steps = 50
N_nodes = 100

positions = []
solutions = []
for i in range(time_steps):
    positions.append(np.random.rand(2, N_nodes))
    solutions.append(np.random.random(N_nodes))

# init the figure, so the colorbar can be initially placed somewhere
marker_size = 5
fig = plt.figure()
ax = fig.add_subplot(111, aspect='equal', autoscale_on=False, xlim=(0, 1), ylim=(0, 1))
s = ax.scatter(positions[0][0], positions[0][1], s = marker_size, c = solutions[0], cmap = "RdBu_r", marker = ".", edgecolor = None)
cb = fig.colorbar(s)

# get the axis for the colobar
cax = cb.ax

def animate(i):
    """ Perform animation step. """
    # clear both plotting axis and colorbar axis
    ax.clear()
    cax.cla()
    #the new axes must be re-formatted
    ax.set_xlim(0,1)
    ax.set_ylim(0,1)
    ax.grid(b=None)
    ax.set_xlabel('x [m]')
    ax.set_ylabel('y [m]')
    # and the elements for this frame are added
    ax.text(0.02, 0.95, 'Time step = %d' % i, transform=ax.transAxes)
    s = ax.scatter(positions[i][0], positions[i][1], s = marker_size, c = solutions[i], cmap = "RdBu_r", marker = ".", edgecolor = None)
    fig.colorbar(s, cax=cax)

ani = animation.FuncAnimation(fig, animate, interval=100, frames=range(time_steps))

ani.save('animation2.gif', writer='pillow')

产生相同的图形。