龙格库塔法是一种求解高阶常微分方程的常用方法，在工程当中应用广泛，例如求解物体的运动方程等。
这里我们通过matlab程序编写龙格库塔算法求解二元常微分方程组，假设有常微分方程组：
$\{\begin{array}{l}\ddot{x}-\dot{x}+2\ddot{y}+\dot{y}=-2\mathrm{s}\mathrm{i}\mathrm{n}t-3\mathrm{c}\mathrm{o}\mathrm{s}t\\ \ddot{x}+\ddot{y}=-\mathrm{s}\mathrm{i}\mathrm{n}t-\mathrm{c}\mathrm{o}\mathrm{s}t\\ x(0)=0\\ y(0)=1\\ \dot{x}(0)=1\\ \dot{y}(0)=0\end{array}$
该方程有精确解：
$\{\begin{array}{l}x=\mathrm{s}\mathrm{i}\mathrm{n}t\\ y=\mathrm{c}\mathrm{o}\mathrm{s}t\end{array}$
令$u_1=x$,$u_2=\dot{x}$,$w_1=y$,$w_2=\dot{y}$,则原式可写为
$\{\begin{array}{l}f1(t)=u_2=\dot{x}\\ f2(t)=u_2^{\prime }=\ddot{x}=\mathrm{c}\mathrm{o}\mathrm{s}t-\dot{x}+\dot{y}\\ f3(t)=w_2=\dot{y}\\ f4(t)=w_2^{\prime }=\ddot{y}=-\mathrm{s}\mathrm{i}\mathrm{n}t-2\mathrm{c}\mathrm{o}\mathrm{s}t+\dot{x}-\dot{y}\end{array}$
可以编写四个子函数如下：

function output = f1(x,u1,u2,w1,w2)
output = u2;

function output = f2(x,u1,u2,w1,w2)
output = cos(x)-u2+w2;

function output = f3(x,u1,u2,w1,w2)
output = w2;

function output = f4(x,u1,u2,w1,w2)
output = -sin(x)-2*cos(x)-w2+u2;

四阶龙格库塔函数程序

function [u1,u2,w1,w2] = RK4_2variable(u1,u2,w1,w2,h,a,b)x = a:h:b;for i = 1:length(x)-1k11 = f1(x(i) , u1(i) , u2(i) , w1(i) , w2(i));
k21 = f2(x(i) , u1(i) , u2(i) , w1(i) , w2(i));
L11 = f3(x(i) , u1(i) , u2(i) , w1(i) , w2(i));
L21 = f4(x(i) , u1(i) , u2(i) , w1(i) , w2(i));k12 = f1(x(i)+h/2 , u1(i)+h*k11/2 , u2(i)+h*k21/2 , w1(i)+h*L11/2, w2(i)+h*L21/2);
k22 = f2(x(i)+h/2 , u1(i)+h*k11/2 , u2(i)+h*k21/2 , w1(i)+h*L11/2, w2(i)+h*L21/2);
L12 = f3(x(i)+h/2 , u1(i)+h*k11/2 , u2(i)+h*k21/2 , w1(i)+h*L11/2, w2(i)+h*L21/2);
L22 = f4(x(i)+h/2 , u1(i)+h*k11/2 , u2(i)+h*k21/2 , w1(i)+h*L11/2, w2(i)+h*L21/2);k13 = f1(x(i)+h/2 , u1(i)+h*k12/2 , u2(i)+h*k22/2 , w1(i)+h*L12/2 , w2(i)+h*L22/2);
k23 = f2(x(i)+h/2 , u1(i)+h*k12/2 , u2(i)+h*k22/2 , w1(i)+h*L12/2 , w2(i)+h*L22/2);
L13 = f3(x(i)+h/2 , u1(i)+h*k12/2 , u2(i)+h*k22/2 , w1(i)+h*L12/2 , w2(i)+h*L22/2);
L23 = f4(x(i)+h/2 , u1(i)+h*k12/2 , u2(i)+h*k22/2 , w1(i)+h*L12/2 , w2(i)+h*L22/2);k14 = f1(x(i)+h , u1(i)+h*k13 , u2(i)+h*k23 , w1(i)+h*L13 , w2(i)+h*L23);
k24 = f2(x(i)+h , u1(i)+h*k13 , u2(i)+h*k23 , w1(i)+h*L13 , w2(i)+h*L23);
L14 = f3(x(i)+h , u1(i)+h*k13 , u2(i)+h*k23 , w1(i)+h*L13 , w2(i)+h*L23);
L24 = f4(x(i)+h , u1(i)+h*k13 , u2(i)+h*k23 , w1(i)+h*L13 , w2(i)+h*L23);u1(i+1) = u1(i) + h/6 * (k11 + 2*k12 + 2*k13 + k14);
u2(i+1) = u2(i) + h/6 * (k21 + 2*k22 + 2*k23 + k24);
w1(i+1) = w1(i) + h/6 * (L11 + 2*L12 + 2*L13 + L14);
w2(i+1) = w2(i) + h/6 * (L21 + 2*L22 + 2*L23 + L24);end
end

计算主程序

% main func
clear;clc;
u1(1) = 0;
u2(1) = 1;
w1(1) = 1;
w2(1) = 0;
h=0.01;
a = 0;b=20;
[u1,u2,w1,w2] = RK4_2variable(u1,u2,w1,w2,h,a,b);
figure 
plot(a:h:b,u1,'r-');
hold on
plot(a:h:b,sin(a:h:b),'b-.');
xlabel('time');
ylabel('x');
legend('计算值','精确值');

计算结果如图