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 one-dimensional low-discrepancy sequence to determine the ordering of the subflows through a direct point-to-permutation construction, while applying each subflow exactly once per time step. Thus, for a decomposition into \(p\) operators, the proposed method requires only \(p\) subflow evaluations per step, compared with essentially \(2p-2\) evaluations for the classical multi-operator Strang splitting when \(p\geq 3\). In contrast to randomized splitting, the quasi-random ordering is deterministic once the underlying sequence is fixed and avoids the run-to-run fluctuations caused by random permutations. To analyze the resulting scheme, we develop a convergence framework that connects the discrepancy of the induced operator orderings with cancellation of the leading commutator errors accumulated over time. For an arbitrary fixed number of bounded linear operators, we establish a global error of order \(O(τ^2\log N)\), equivalently \(O(τ^2|\logτ|)\) for fixed terminal time, which is essentially second order. We further extend the same cancellation mechanism to the nonlinear Allen--Cahn equation and obtain the same essentially second-order convergence behavior under suitable Sobolev regularity assumptions. Numerical experiments for bounded linear systems and the Allen--Cahn equation confirm the predicted convergence rates and show that the proposed quasi-random splitting strategy retains the low per-step computational cost of a Lie-type splitting method while achieving accuracy close to second order.

arXiv comment32 pages, 6 figures