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Quasi-random splitting method for accurate and efficient multiphysics simulation

Lei Li, Yunxiao Liu, Chenchen Wan

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 \(O(τ^{2}|\log τ|)\) 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.

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 operators, the classical multi-operator Strang splitting requires essentially subflow evaluations per step, whereas the present method uses only . 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 \(O(τ^{2}|\log τ|)\) 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.

Paper Structure

This paper contains 18 sections, 16 theorems, 217 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setup and Notations
  3. General setup of the quasi-random splitting method
  4. Other notations
  5. The quasi-random sequence and the distribution of induced sign sequence
  6. Preliminary results on the quasi-random sequence
  7. The induced sign sequence and its distribution
  8. Convergence of the quasi-random splitting for two operators
  9. A common error template for the two-operator quasi-random splitting
  10. Application I: bounded linear operators
  11. Application II: the Allen--Cahn equation
  12. Numerical experiments
  13. Experiments for bounded linear systems
  14. Experiments for the Allen--Cahn equation
  15. Two-operator diffusion--reaction test
  16. ...and 3 more sections

Key Result

Theorem 3.1

Let $\{z_n\}_{n\ge1}$ be the radical-inverse sequence defined as above. Then for any $n_0\ge 0$, and $N\ge 2$,

Figures (4)

  • Figure 1: Comparison of the convergence behavior for the bounded linear operators in the discrete $L^2$ norm. The quasi-random splitting method is compared with the randomized splitting method for time steps $\tau=2^{-4},2^{-5},\dots,2^{-8}$.
  • Figure 2: Comparison of the convergence behavior for the Allen--Cahn equation in the discrete $L^2$ norm. The quasi-random splitting method is compared with the randomized splitting method for time steps $\tau=2^{-10},2^{-11},\dots,2^{-15}$.
  • Figure 3: Comparison of the convergence behavior for the Allen--Cahn equation in the discrete $W^{1,2}$ norm. The quasi-random splitting method is compared with the randomized splitting method for time steps $\tau=2^{-10},2^{-11},\dots,2^{-15}$.
  • Figure 4: Three-operator Allen--Cahn equation with background flow: convergence of the quasi-random splitting method in the discrete $L^2$ norm and $W^{1,2}$ norm for $\tau=2^{-10},2^{-11},\dots,2^{-15}$. The reported quantity is the deterministic error $\mathcal{E}^{h,3}_{2,{\rm qr}}(\tau)$ and $\mathcal{E}^{h,3}_{1,2,{\rm qr}}(\tau)$.

Theorems & Definitions (30)

  • Definition 3.1: One-dimensional discrepancy
  • Theorem 3.1
  • Theorem 3.2: Koksma inequality
  • Definition 3.2: Sign sequence
  • Lemma 3.1
  • proof
  • Corollary 3.1
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • ...and 20 more