Operator Splitting Methods: Numerical Solutions of Ordinary Differential Equations via Separation of Variables

Abstract: This paper applies the concept of linear semigroups induced by nonlinear flows, originally developed by Dorroh and Neuberger in the 1990s, to the approximation of uniquely solvable initial value problems for nonlinear ordinary differential equations. Building on a framework rooted in the earlier works of Lie, Kowalewski, and Groebner, we analyze nonlinear systems through the lens of the Koopman-Lie semigroup exp(tK), where K is the linear Lie generator associated with the flow induced by the nonlinear differential equation. A central feature of this approach is the decomposition K = K1 + ... + KN, which enables the use of operator splitting methods. We revisit the foundational first-order splitting scheme introduced by H. F. Trotter in 1959 and extend it to higher-order schemes with improved error bounds. These theoretical developments are supported by numerical examples that demonstrate the accuracy and efficiency of the proposed methods, which are based entirely on the classical separation of variables technique for solving ordinary differential equations.