Backward Euler method

Derivation

Integrating the differential equation ${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}=f(t,y)}$ from ${\displaystyle t_n}$ to ${\displaystyle t_{n+1}=t_n+h}$ yields $${\displaystyle y(t_{n+1})-y(t_n)={\int }_{t_n}^{t_{n+1}}f(t,y(t))\mathrm{d}t.}$$ Now approximate the integral on the right by the right-hand rectangle method (with one rectangle): $${\displaystyle y(t_{n+1})-y(t_n)\approx hf(t_{n+1},y(t_{n+1})).}$$ Finally, use that ${\displaystyle y_n}$ is supposed to approximate ${\displaystyle y(t_n)}$ and the formula for the backward Euler method follows.

The same reasoning leads to the (standard) Euler method if the left-hand rectangle rule is used instead of the right-hand one.

Analysis

The local truncation error (defined as the error made in one step) of the backward Euler Method is ${\displaystyle O(h^2)}$, using the big O notation. The error at a specific time ${\displaystyle t}$ is ${\displaystyle O(h^2)}$. It means that this method has order one. In general, a method with ${\displaystyle O(h^{k+1})}$ LTE (local truncation error) is said to be of kth order.

The region of absolute stability for the backward Euler method is the complement in the complex plane of the disk with radius 1 centered at 1, depicted in the figure. This includes the whole left half of the complex plane, making it suitable for the solution of stiff equations. In fact, the backward Euler method is even L-stable.

The region for a discrete stable system by Backward Euler Method is a circle with radius 0.5 which is located at (0.5, 0) in the z-plane.

Extensions and modifications

The backward Euler method is a variant of the (forward) Euler method. Other variants are the semi-implicit Euler method and the exponential Euler method.

The backward Euler method can be seen as a Runge–Kutta method with one stage, described by the Butcher tableau: $${\displaystyle \begin{array}{cc}1& 1\\ & 1\end{array}}$$

The method can also be seen as a linear multistep method with one step. It is the first method of the family of Adams–Moulton methods, and also of the family of backward differentiation formulas.