Quasi-random splitting method for accurate and efficient multiphysics simulation

Abstract

We propose a quasi-random operator splitting method for evolution equations driven by multiple mechanisms. The method uses a low-discrepancy sequence to generate the ordering of the subflows, while requiring only one application of each subflow per time step. In particular, for a decomposition into \(p\) operators, the classical multi-operator Strang splitting requires essentially \(2p-2\) subflow evaluations per step, whereas the present method uses only \(p\). In contrast to randomized splitting, the quasi-random scheme is deterministic once the underlying sequence is fixed, so its improved accuracy is achieved in a single run rather than through averaging over many independent realizations. To analyze this method, we develop a convergence framework that exploits the discrepancy structure of the induced ordering sequence and translates it into cancellation in the accumulated local errors. For two operators, this yields an essentially second-order global error bound of order \(O(\tau^{2}|\log \tau|)\) for bounded linear problems. We further extend the analysis to the Allen--Cahn equation and present numerical experiments, including bounded linear systems and the Allen--Cahn equation, which confirm the predicted convergence behavior and demonstrate that the proposed method achieves near-Strang accuracy at a substantially lower computational cost.