Implicit Euler Method

The implicit Euler method (also called backward Euler) uses the backward difference approximation for the time derivative.

$$y_{i+1}=y_i+dt\cdot g(t_{i+1},y_{i+1})$$

Unlike the Explicit Euler method, $y_{i+1}$ appears on both sides, so the equation must be solved (e.g. algebraically or iteratively) to isolate $y_{i+1}$ at each step.

Example: Linear Test ODE

For $\dot{y}=-ay$, the implicit Euler method gives

$$y_{i+1}=y_i+dt\cdot (-a\cdot y_{i+1})$$ $$y_{i+1}(1+a\cdot dt)=y_i$$ $$y_{i+1}=\frac{y_i}{1+a\cdot dt}$$

Stability

The amplification factor is $\frac{1}{1+a\cdot dt}$. Since $a>0$ and $dt>0$, this is always less than 1 in magnitude, so the implicit Euler method is unconditionally stable - it remains stable for any step size.

This comes at the cost of needing to solve for $y_{i+1}$ at each step, which for nonlinear $g$ requires a root-finding method.

See Stability of a method for the general framework.

Explicit Euler method | Finite differences | Stability of a method | BTCS scheme