Stability of a Method

A numerical method is stable if, when applied to a stable ODE, the difference between solutions computed from slightly different initial conditions remains bounded.

The standard test case is $\dot{y}=-ay$ with $a>0$, which is trivially stable. The amplification factor $A$ is defined by $y_{i+1}=A\cdot y_i$. Stability requires $|A|\le 1$.

Stability of Common Methods

Explicit Euler

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

Stable when $|1-a\cdot dt|\le 1$, i.e. $a\cdot dt\le 2$. Conditionally stable.

Implicit Euler

$$y_{i+1}=\frac{y_i}{1+a\cdot dt},A=\frac{1}{1+a\cdot dt}$$

Since $a,dt>0$, we always have $|A|<1$. Unconditionally stable.

Explicit Runge-Kutta (general)

All explicit Runge-Kutta methods (including Midpoint method, Ralston method, Fourth order Runge-Kutta) are conditionally stable - there is a maximum step size, though the stability region grows with the order of the method.

Implicit Trapezoid Method

$$A=\frac{1-a\cdot dt/2}{1+a\cdot dt/2}$$

Since $a,dt>0$, $|A|<1$ always. Unconditionally stable.

Richardson Method

The Richardson method uses a centred difference in time:

$$y_{i+1}-y_{i-1}=2dt\cdot g(t_i,y_i)$$

For $\dot{y}=-y$, analysis shows this method is unconditionally unstable: solutions from nearby initial conditions always diverge, regardless of step size. This makes Richardson’s method unsuitable in practice.

Stability of an ODE | Explicit Euler method | Implicit Euler method | Implicit Trapezoid Method | Convergence | Lax Equivalence Theorem