Runge–Kutta methods and renormalization

Rooted trees have been used to calculate the solution of nonlinear flow equations and Runge–Kutta methods. More recently, rooted trees have helped systematizing the algebra underlying renormalization in quantum field theories. The Butcher group and B-series establish a link between these two approaches to rooted trees. On the one hand, this link allows for an alternative representation of the algebra of renormalization, leading to nonperturbative results. On the other hand, it helps to renormalize singular flow equations. The usual approach is extended here to nonlinear partial differential equations. A nonlinear Born expansion is given, and renormalization is used to partly remove the secular terms of the perturbative expansion.