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

Yating Wang, Zhengya Yang, Wing Tat Leung

TL;DR

The paper addresses the computational challenge of high-contrast multiscale diffusion by introducing a partially explicit temporal splitting that achieves a contrast-independent time step. It couples this with multiscale space construction via CEM-GMsFEM and NLMC, and a parallel-in-time framework using Parareal with a waveform-relaxation-based all-at-once fine solver. The authors prove stability and convergence, derive full-discretization error bounds, and validate the approach through numerical experiments showing robust performance as the number of coarse intervals increases. The resulting method enables efficient, accurate simulations on parallel architectures for linear high-contrast diffusion problems, with clear potential for extensions to nonlinear cases.

Abstract

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 managed, 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 both the sequential solver and 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.

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

TL;DR

The paper addresses the computational challenge of high-contrast multiscale diffusion by introducing a partially explicit temporal splitting that achieves a contrast-independent time step. It couples this with multiscale space construction via CEM-GMsFEM and NLMC, and a parallel-in-time framework using Parareal with a waveform-relaxation-based all-at-once fine solver. The authors prove stability and convergence, derive full-discretization error bounds, and validate the approach through numerical experiments showing robust performance as the number of coarse intervals increases. The resulting method enables efficient, accurate simulations on parallel architectures for linear high-contrast diffusion problems, with clear potential for extensions to nonlinear cases.

Abstract

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 managed, 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 both the sequential solver and 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.

Paper Structure

This paper contains 13 sections, 5 theorems, 77 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partially explicit temporal splitting scheme
  4. Parareal algorithm
  5. Multiscale space construction
  6. CEM-GMsFEM
  7. Construction of two multiscale subspaces
  8. Parallel in time for partially explicit temporal splitting scheme
  9. WR method via diagonalization for all-at-once system
  10. Parareal all-at-once partially explicit temporal splitting algorithm
  11. Convergence of the main algorithm
  12. Numerical experiments
  13. Conclusion

Key Result

Theorem 4.1

Let $u(t)$ be the exact solution of eqnAa and satisfy $u(t)=U(t)+W(t)$, $U^{j}(t)$ and $W^{j}(t)$ are the solution of eqnZz-eqnZx at $j$-th iteration respectively. Then it holds for $j=1,2,\cdots$ and $n=0,1,2,\cdots,N-1$ that where $\gamma$ is defined by

Figures (6)

  • Figure 1: Example 1. Left:$\kappa$; Right:$f$
  • Figure 2: Example 1. Left: reference solution; Right: Algorithm \ref{['main_algorthm']} solution.
  • Figure 3: Example 1. Left: Max differences between the Algorithm \ref{['main_algorthm']} solution. Right: Convergence rate.
  • Figure 4: Example 2. Left:$\kappa$; Right:$f$
  • Figure 5: Example 2. Left: reference solution; Right: Algorithm \ref{['main_algorthm']} solution.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Theorem 4.1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 4.2