計算方法實驗（三）：四階龍格-庫塔方法

四階Runge-Kutta數學原理

給定常微分方程初值問題

$\left\{ \begin{matrix} \frac{\text{dy}}{\text{dx}} = f(x,y),a \leq x \leq b \\ \ y(a) = \alpha\text{\;\;\quad\;h} = \frac{b - a}{N} \\ \end{matrix} \right.\$

記$x_{n} = a + n \cdot h$，$n = 0,1,\cdots,N$，利用四階龍格—庫塔方法

$K_{1} = hf\left( x_{n},y_{n} \right)$

$K_{2} = hf\left( x_{n} + \frac{h}{2},y_{n} + \frac{K_{1}}{2} \right)$

$K_{3} = hf\left( x_{n} + \frac{h}{2},y_{n} + \frac{K_{2}}{2} \right)$

$K_{4} = hf\left( x_{n} + h,y_{n} + K_{3} \right)$

$y_{n + 1} = y_{n} + \frac{1}{6}(K_{1} + 2K_{2} + 2K_{3} + K_{4})$

$n = 0,1,\cdots,N - 1$

可逐次求出微分方程初值問題的數值解$y_{n}$，$n = 1,2,\cdots,N$。

程序流程

核心代碼

#include <cmath>
#include <cstdio>
#include <iostream>
using namespace std;

int n;
double a, b, fa;

double f(double x, double y) { return -y * y; }
double f_(double x) { return 1.0 / (x + 1.0); }

int main() {
    scanf("%lf%lf%lf%d", &a, &b, &fa, &n);
    double x = a, y = fa, h = (b - a) / n;
    for (int i = 1; i <= n; i++) {
        double k1 = h * f(x, y);
        double k2 = h * f(x + h / 2, y + k1 / 2);
        double k3 = h * f(x + h / 2, y + k2 / 2);
        double k4 = h * f(x + h, y + k3);
        x += h;
        y += 1.0 / 6.0 * (k1 + 2 * k2 + 2 * k3 + k4);
        printf("%.2lf\t%lf\t%.2lf\n", x, y, fabs(f_(x) - y));
    }
    return 0;
}

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