Rosenbrock methods

Remarks

Clause 1 holds because the right-hand side of the $i$-th equation involves only $k_1, \ldots, k_{i-1}$, so the stages are computed one after the other. For clause 2, with $f(y) = \lambda y$ and $J = \lambda$, each $k_i$ is $y_n$ times a rational function of $z = h\lambda$ with denominator $(1 - \gamma z)^i$. For clause 3, $(1 - h\lambda) k_1 = h\lambda y_n$ gives $y_{n+1} = y_n / (1 - h\lambda)$, the same as the backward Euler method: $\abs{R(z)} \le 1$ when $\operatorname{Re} z \le 0$, and $R(z) \to 0$ as $\abs{z} \to \infty$. The order conditions of Rosenbrock methods use the exact Jacobian; methods that allow an approximate $J$ are called W-methods.

Going deeper

For a student

Rosenbrock methods are linearly implicit Runge–Kutta methods: the coefficients are chosen so that the order conditions hold when $J$ is the exact Jacobian. One LU factorisation of $I - h\gamma J$ per step serves all stages, which is why all diagonal coefficients are equal. They are popular for moderately stiff problems of medium size, for example in chemical kinetics, and embedded pairs give error estimates for step size control.