The main result of this paper is the proof that the so-called Kahan’s method is actually equivalent to a Runge-Kutta method. Let us recall the notion of Kahan’s method. Consider a system of differential equations given by a vector field \[ \dot{x} = f(x), \] where \(f(x) = Q(x) + B x + c\), \(x \in \mathbb R^n\), \(Q\) being a quadratic form, \(B \in \mathbb R^{n \times n}\), and \(c \in \mathbb R^n\). Consider the numerical integration method \(x \mapsto x'\) with step \(h\) given by \[ \frac{x-x'}{h} = Q(x, x') + \frac{1}{2} B(x+x') + c, \] where \[ Q(x, x') = \frac{1}{2} ( Q(x+x') - Q(x) - Q(x') ) \] is the symmetric bilinear form obtained from \(Q\) by polarization.
Due to the fact that this method is a Runga-Kutta method, it shares a number of properties with the Runga-Kutta methods. This result permits to explain some of the properties of the Kahan’s method, and brings new open questions.