Scikit-fdiff | 二维浅水方程计算与可视化

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发布于 2025-03-21 17:28:25

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代码可运行

%matplotlib inline

Scikit-fdiff | 二维浅水方程计算与可视化

引言

在科学计算和工程领域，偏微分方程（PDE）的求解一直是一个复杂且耗时的任务。Scikit-fdiff是一个旨在简化这一过程的Python库，它通过有限差分法（Finite Difference Method）来离散化空间导数，并结合方法线（Method of Lines）将PDE转化为高维常微分方程（ODE），从而利用标准的ODE数值积分方法进行求解。

Scikit-fdiff的核心特点

符号化离散化：Scikit-fdiff使用Sympy进行符号化离散化，能够自动生成ODE的精确雅可比矩阵。
多种后端支持：目前支持Numpy和Numba作为后端，能够将符号化的ODE转化为高效的数值计算例程。
灵活的求解器：支持Scipy中的标准ODE求解器，并提供了一些专门优化的求解器，能够利用雅可比矩阵的快速计算。
模拟处理工具：提供实时显示、结果持久化、后处理接口等实用功能。

适用场景

Scikit-fdiff适用于以下场景：

模型原型设计：当你需要快速验证模型时，Scikit-fdiff能够提供非最优但准确的解决方案。
刚性问题的求解：对于刚性系统，Scikit-fdiff提供了高阶隐式求解器和自适应时间步长控制。
简单几何域：由于有限差分法的限制，Scikit-fdiff主要适用于矩形几何域。

安装

Scikit-fdiff可以通过PyPI和Conda安装：

!pip install scikit-fdiff -i https://pypi.mirrors.ustc.edu.cn/simple/

Looking in indexes: https://pypi.mirrors.ustc.edu.cn/simple/
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模型设置

以下是一个使用Scikit-fdiff求解二维浅水方程的完整示例：

from skfdiff import Model, Simulation
import pylab as pl
import numpy as np
from scipy.signal.windows import gaussian

# 定义模型
model = Model(["-(dx((H + h) * u) + dy((H + h) * v))",
               "-(u * dxu + v * dyu) - g * dxh + nu * (dxxu + dyyu)",
               "-(u * dxv + v * dyv) - g * dyh + nu * (dxxv + dyyv)"],
               ["h(x, y)", "u(x, y)", "v(x, y)"],
               parameters=["H(x, y)", "nu", "g"],
               boundary_conditions="periodic")

设置初始条件

L = 10
x = y = np.linspace(-L / 2, L / 2, 56)
xx, yy = np.meshgrid(x, y, indexing="ij")
h = (gaussian(x.size, x.size // 20)[:, None] *
     gaussian(y.size, y.size // 20)[None, :]) + 1
h = np.roll(h, 12, axis=0)
h = np.roll(h, 12, axis=1)
H = np.zeros_like(h)
u = np.zeros_like(h)
v = np.zeros_like(h)
initial_fields = model.Fields(x=x, y=y, h=h, u=u, v=v,
                              H=H, g=9.81, nu=0)

运行模拟

simulation = Simulation(model, initial_fields, dt=.1, tmax=1)
container = simulation.attach_container()
tmax, final_fields = simulation.run()

79f1a7 running: t: 1: : 10it [00:09,  1.09it/s]

可视化结果

import matplotlib.pyplot as plt
# 使用 matplotlib 可视化结果
plt.figure(figsize=(10, 8))

# 绘制高度场 h
plt.subplot(2, 2, 1)
plt.contourf(xx, yy, container.data.h[-1], levels=50, cmap='viridis')
plt.colorbar(label='Height (h)')
plt.title('Height Field (h)')
plt.xlabel('x')
plt.ylabel('y')

# 绘制速度场 u
plt.subplot(2, 2, 2)
plt.contourf(xx, yy, container.data.u[-1], levels=50, cmap='plasma')
plt.colorbar(label='Velocity (u)')
plt.title('Velocity Field (u)')
plt.xlabel('x')
plt.ylabel('y')

# 绘制速度场 v
plt.subplot(2, 2, 3)
plt.contourf(xx, yy, container.data.v[-1], levels=50, cmap='plasma')
plt.colorbar(label='Velocity (v)')
plt.title('Velocity Field (v)')
plt.xlabel('x')
plt.ylabel('y')

# 绘制总速度场 magnitude
speed = np.sqrt(container.data.u[-1]**2 + container.data.v[-1]**2)
plt.subplot(2, 2, 4)
plt.contourf(xx, yy, speed, levels=50, cmap='inferno')
plt.colorbar(label='Speed Magnitude')
plt.title('Speed Magnitude')
plt.xlabel('x')
plt.ylabel('y')

plt.tight_layout()
plt.show()