Coordinate-wise splitting algorithms for ODE simulation via Koopman-Lie product formulas
For researchers in numerical ODE simulation, this work offers an alternative splitting framework with theoretical justification, though it is incremental as it applies known product formulas to coordinate-wise decompositions.
The paper presents coordinate-wise splitting algorithms for ODE simulation using Koopman-Lie product formulas, achieving comparable accuracy to RK45 with reduced computational cost in some cases. Numerical experiments on Lotka-Volterra, Van der Pol, and Lorenz systems show trade-offs between splitting order, time steps, and runtime.
We present a computational framework for simulating finite-dimensional ordinary differential equations by combining classical Koopman-Lie product formulas with coordinate-wise frozen subflows. The setting is model-known, since the vector field is assumed to be available, and no data-driven approximation of the Koopman operator is attempted. Under standard assumptions, the Koopman-Lie generator associated with the flow admits a coordinate decomposition into partial generators. This decomposition leads to elementary updates in which all but one state variable are frozen, and the resulting frozen scalar subproblems are evaluated either in closed form or by one-dimensional solves. Lie-Trotter, Strang, and higher-order exponential compositions are then converted into state-update algorithms for two- and three-dimensional systems, with the semigroup and product-formula theory used as background justification for the constructions. We also record the exponential-term counts produced by the recursive constructions used in the implementation. These counts are presented as implementation costs. Numerical experiments on the Lotka-Volterra, Van der Pol, and Lorenz systems compare the coordinate-wise splitting algorithms with high-accuracy RK45 reference solutions using root-mean-square errors and work-precision curves. The results illustrate the practical trade-off between splitting order, number of time steps, number of exponential factors, and runtime.