Order of Consistency for the Euler-Heun Method

Euler-Heun Method

$$y(t+h)=y+\frac{h}{2}\left[f(t,y)+f(\underline{t+h},\underline{y+hf(t,y)})\right]$$

The Euler-Heun method’s increment function is $\Phi (t,y,h)=\frac{1}{2}[f(t,y)+f(t+h,y+hf(t,y))]$.

1. Set up the truncation error formula:

   $$\tau (t,h)=\frac{y(t+h)-y(t)}{h}-\frac{1}{2}[f(t,y(t))+f(t+h,y(t)+hf(t,y(t)))]$$

2. Use Taylor Series for all terms:

   • Left Part: We expand $y(t+h)$ to a higher order: $y(t+h)=y(t)+h{y}^{\prime }(t)+\frac{h^2}{2}{y}^{\prime \prime }(t)+\mathcal{O}(h^3)$. This gives:

     $$\frac{y(t+h)-y(t)}{h}=y^{\prime }(t)+\frac{h}{2}{y}^{\prime \prime }(t)+\mathcal{O}(h^2)$$

   • Right Part: We use a multivariate Taylor expansion for the second $f$ term:

     $$f(t+h,y+hf)=f(t,y)+h\frac{\partial f}{\partial t}+(hf)\frac{\partial f}{\partial y}+\mathcal{O}(h^2)$$

     So the full increment function is:

     $$\Phi (t,y,h)=\frac{1}{2}[f+(f+h\frac{\partial f}{\partial t}+hf\frac{\partial f}{\partial y}+\mathcal{O}(h^2))]=f+\frac{h}{2}\left(\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial y}\right)+\mathcal{O}(h^2)$$

3. Substitute and Simplify: Using $y^{\prime }=f$ and $y^{\prime \prime }=\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial y}$, the increment function becomes:

   $$\Phi (t,y(t),h)=y^{\prime }(t)+\frac{h}{2}{y}^{\prime \prime }(t)+\mathcal{O}(h^2)$$

   Now, we substitute everything back into the truncation error formula:

   $$\tau (t,h)=\left(y^{\prime }(t)+\frac{h}{2}{y}^{\prime \prime }(t)+\mathcal{O}(h^2)\right)-\left(y^{\prime }(t)+\frac{h}{2}{y}^{\prime \prime }(t)+\mathcal{O}(h^2)\right)=\mathcal{O}(h^2)$$

The local truncation error is of the second order in $h$. Therefore, the Euler-Heun method has an order of consistency of 2.