📚 Numerical Methods for Phase Paths | 相路径的数值计算方法
Phase paths provide a geometric view of the solutions to systems of differential equations. In the IB Mathematics Analysis & Approaches HL course, students study systems of first-order differential equations and their phase portraits. When exact solutions are difficult or impossible to obtain, numerical methods — such as Euler’s method, the improved Euler method, and Runge-Kutta methods — offer powerful tools for approximating phase paths. This article explores these techniques, their accuracy, and their application to typical IB problems.
相路径为微分方程组的解提供了几何视角。在IB数学分析与方法高级水平课程中，学生学习一阶微分方程组及其相图。当精确解难以或无法求得时，数值方法——如欧拉法、改进欧拉法和龙格-库塔法——为近似计算相路径提供了强有力的工具。本文将探讨这些数值技术、它们的精度以及它们在典型IB问题中的应用。
1. The Phase Plane | 相平面基础
Consider the autonomous system
dx/dt = f(x, y), dy/dt = g(x, y)
The phase plane is the x-y plane, in which every point (x, y) represents a possible state of the system. A phase path (or phase trajectory) is the curve traced by a solution (x(t), y(t)) as the parameter t varies. The collection of all phase paths for different initial conditions is called the phase portrait.
相平面即 x-y 平面，其中每个点 (x, y) 代表系统的一个可能状态。相路径（或相轨迹）是解 (x(t), y(t)) 随参数 t 变化所描绘出的曲线。不同初始条件对应的所有相路径的集合称为相图。
- The velocity vector (f(x, y), g(x, y)) is tangent to the phase path at every point. | 速度向量 (f(x, y), g(x, y)) 在每个点均与相路径相切。
- For an autonomous system, phase paths never cross (uniqueness theorem). | 对于自治系统，相路径永不相交（唯一性定理）。
- The direction field helps sketch approximate phase paths by eye. | 方向场有助于徒手勾画近似的相路径。
2. Euler’s Method for Systems | 欧拉方法求解系统
Euler’s method is the simplest numerical scheme for approximating a solution curve. Given an initial point (x₀, y₀) and a step size h, we generate successive points using the recurrence
xₙ₊₁ = xₙ + h·f(xₙ, yₙ) yₙ₊₁ = yₙ + h·g(xₙ, yₙ)
Each step advances the solution by h in time, using the slope at the current point. The local truncation error is O(h²), and the global error after many steps is O(h).
欧拉法是最简单的数值求解方案。给定初始点 (x₀, y₀) 和步长 h，利用递推关系逐点生成后续的点。每一步按当前点的斜率将解推进时间 h。局部截断误差为 O(h²)，多步之后的全局误差为 O(h)。
Example. For the simple harmonic oscillator system x’ = y, y’ = −x with (x₀, y₀) = (1, 0) and h = 0.1, the first step gives x₁ = 1 + 0.1·(0) = 1.000 and y₁ = 0 + 0.1·(−1) = −0.100.
示例。对于简谐振子系统 x’ = y, y’ = −x，初值 (x₀, y₀) = (1, 0)，步长 h = 0.1，第一步得到 x₁ = 1 + 0.1·(0) = 1.000，y₁ = 0 + 0.1·(−1) = −0.100。
3. Improved Euler Method (Heun’s Method) | 改进欧拉法（亨氏法）
The improved Euler method is a predictor-corrector scheme. We first compute a tentative (predictor) point, then use the average of the slopes at the original and predictor points for the actual step:
x* = xₙ + h·f(xₙ, yₙ) y* = yₙ + h·g(xₙ, yₙ)
xₙ₊₁ = xₙ + (h/2)·[f(xₙ, yₙ) + f(x*, y*)]
yₙ₊₁ = yₙ + (h/2)·[g(xₙ, yₙ) + g(x*, y*)]
The local truncation error improves to O(h³), and the global error is O(h²) — one order better than Euler’s method.
改进欧拉法是一种预估-校正格式。首先计算一个试探性的（预估）点，然后取原点和预估点处斜率的平均值来完成实际步进。局部截断误差改善为 O(h³)，全局误差为 O(h²)——比欧拉法高一阶。
Example. For the oscillator above, the predictor gives x* = 1.000, y* = −0.100. The corrector then yields x₁ = 1 + 0.05·[0 + (−0.1)] = 0.995, y₁ = 0 + 0.05·[−1 + (−1)] = −0.100.
示例。对于上述振子系统，预估得到 x* = 1.000，y* = −0.100。校正后得到 x₁ = 1 + 0.05·[0 + (−0.1)] = 0.995，y₁ = 0 + 0.05·[−1 + (−1)] = −0.100。
4. Fourth-Order Runge-Kutta Method (RK4) | 四阶龙格-库塔法
RK4 is the most widely used classical scheme. It evaluates four slopes per step and combines them with weighted averages. For the system x’ = f(x, y), y’ = g(x, y):
k₁ = h·f(xₙ, yₙ) ℓ₁ = h·g(xₙ, yₙ)
k₂ = h·f(xₙ + k₁/2, yₙ + ℓ₁/2)
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