Parallel in time partially explicit splitting scheme for high contrast linear multiscale diffusion problems

Solving multiscale diffusion problems is often computationally expensive due to the spatial and temporal discretization challenges arising from high-contrast coefficients. To address this issue, a partially explicit temporal splitting scheme is proposed. By appropriately constructing multiscale spaces, the spatial multiscale property is effectively captured, and it has been demonstrated that the temporal step size is independent of the contrast. To enhance simulation speed, we propose a parallel algorithm for the multiscale flow problem that leverages the partially explicit temporal splitting scheme. The idea is first to evolve the partially explicit system using a coarse time step size, then correct the solution on each coarse time interval with a fine propagator, for which we consider the all-at-once solver. This procedure is then performed iteratively till convergence. We analyze the stability and convergence of the proposed algorithm. The numerical experiments demonstrate that the proposed algorithm achieves high numerical accuracy for high-contrast problems and converges in a relatively small number of iterations. The number of iterations stays stable as the number of coarse intervals increases, thus significantly improving computational efficiency through parallel processing.

💡 Research Summary

The paper addresses the computational challenge of solving high‑contrast linear multiscale diffusion equations, where traditional explicit time integration requires prohibitively small time steps due to stiffness introduced by large variations in the diffusion coefficient. The authors propose a three‑fold strategy: (1) a partially explicit temporal splitting scheme that decomposes the multiscale solution space into a dominant subspace and a complementary subspace; (2) a construction of multiscale spatial bases using constrained energy minimizing generalized multiscale finite element methods (CEM‑GMsFEM) together with the non‑local multicontinuum (NLMC) approach; and (3) a parallel‑in‑time algorithm based on the parareal framework, where the fine propagator is realized by an all‑at‑once Waveform Relaxation (WR) method that exploits diagonalization of an α‑circulant time‑stepping matrix.

Spatial discretization. The domain Ω is partitioned into coarse cells of size H. For each cell, an auxiliary eigenvalue problem is solved to obtain a set of low‑frequency eigenfunctions. These are then enriched via oversampling and constrained energy minimization to produce multiscale basis functions φ_{p_i}^j. The resulting space V_cem captures the essential multiscale features and yields contrast‑independent convergence. To separate high‑contrast channels (e.g., fractures) from the background matrix, the NLMC technique constructs two auxiliary spaces: V_{H,1} containing basis functions that are non‑zero only on the high‑contrast channels, and V_{H,2} containing matrix‑related bases and a constant mode. This decomposition satisfies the stability condition for the partially explicit scheme, namely Δt ≤ C·γ^{-1}, where γ measures the energy contribution of V_{H,2}.

Partially explicit time splitting. The semi‑discrete system (mass matrix M and stiffness matrix A) is projected onto V_{H,1} and V_{H,2}, yielding two coupled equations. The dominant subspace V_{H,1} is advanced explicitly, while the complementary subspace V_{H,2} is treated implicitly. Because the implicit part only involves the low‑energy component, the resulting time step restriction is independent of the contrast magnitude.

Parareal parallel‑in‑time algorithm. The time interval