How to prove convergence order for exponential splitting (exponential Runge-Kutta) time integrators applied to a semilinear abstract Cauchy problem without assuming the linear part generates an analytic semigroup. The sectoriality condition is replaced by a smoothing hypothesis on an interpolation couple of Banach spaces: each semigroup operator maps the base space boundedly into a second space with a norm controlled by an integrable, non-increasing function whose integral is the growth function. A student learns how Banach's fixed point theorem gives unique internal stage values, how Lagrange-interpolation identities cancel the low-order Taylor terms, and how the discrete Gronwall inequality closes the error recursion to yield an error bounded by the step size to the power of the stage count times the growth function.
A semilinear abstract Cauchy problem is u'(t) = Au(t) + g(t, u(t)), u(0) = u0 on a Banach space X, where A generates a C0-semigroup e^{tA} and g is a locally Lipschitz nonlinearity; its mild solution…