Richardson Method

The Richardson method (also called the symmetric or central-difference method) approximates the time derivative using a centred difference spanning two steps:

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

This is second-order in time ($O(d{t}^2)$ per step) because the centred difference cancels the first-order error term.

Stability Analysis

Despite its higher formal accuracy, the Richardson method is unconditionally unstable. Applying it to the test ODE $\dot{y}=-y$ gives a two-step recurrence. Analysis shows that one of the two independent solutions grows without bound regardless of the step size $dt$. This means the numerical solution always diverges, even for stable ODEs with arbitrarily small $dt$.

This is in sharp contrast to methods like the Implicit Trapezoid Method and Implicit Euler method, which are unconditionally stable. The Richardson method’s instability makes it unsuitable for practical use, despite its apparent accuracy.

Stability of a method | Explicit Euler method | Implicit Trapezoid Method | Stability of an ODE