Numerical Differential Equations

Numerical methods for Differential Equations.

Euler’s Method

$$y^{\prime }=f(t,y),y(t_0)=y_0$$ $$t_n,y_n\to t_{n+1},y_{n+1}$$

• $t_{n+1}=t_n+h$
• $y_{n+1}=y_n+h{k}_n$
• where $k_n=f(t_n,y_n)$
• Local Truncation Error: $O(h^2)$
• Global Truncation Error: $O(h)$

Heun’s Method

$$y^{\prime }=f(t,y),y(t_0)=y_0$$ $$t_n,y_n\to t_{n+1},y_{n+1}$$

• $t_{n+1}=t_n+h$
• $y_{n+1}=y_n+h\cdot \text{avg}(k_{n,1},k_{n,2})$
• where $k_{n,1}=f(t_n,y_n)$
• where $k_{n,2}=f(t_n+h,y_n+h{k}_{n,1})$
• Local Truncation Error: $O(h^3)$
• Global Truncation Error: $O(h^2)$

Runge-Kutta Method

$$y^{\prime }=f(t,y),y(t_0)=y_0$$ $$t_n,y_n\to t_{n+1},y_{n+1}$$

• $t_{n+1}=t_n+h$
• $y_{n+1}=y_n+h\cdot \text{avg}(k_{n,1},k_{n,2},k_{n,2},k_{n,3},k_{n,3},k_{n,4})$
• where $k_{n,1}=f(t_n,y_n)$
• where $k_{n,2}=f\left(t_n+\frac{1}{2}h,y_n+\frac{h}{2}{k}_{n,1}\right)$
• where $k_{n,3}=f\left(t_n+\frac{1}{2}h,y_n+\frac{h}{2}{k}_{n,2}\right)$
• where $k_{n,4}=f\left(t_n+h,y_n+\frac{h}{2}{k}_{n,3}\right)$
• Local Truncation Error: $O(h^5)$
• Global Truncation Error: $O(h^4)$