## Abstract

Excessive spatial parallelization can introduce a performance bottleneck due to the communication overhead. While time-parallel method multigrid-reduction-in-time (MGRIT) provides an alternative to enhance concurrency, it generally requires large numbers of iterations to converge or even fails when applied to advection-dominated problems. To enhance the convergence of MGRIT, we propose the use of consecutive-step coarse-grid operators in MGRIT, rather than the standard rediscretized coarse-grid operators. The consecutive-step coarse-grid operator is defined as the multiplication of several fine-grid operators, which is able to track the advective characteristic more accurately than the standard rediscretized one. Numerical results show that multilevel MGRIT using the proposed operator is more efficient than the one using the standard rediscretized operator when applied to linear advection problems. Moreover, we perform time-parallel computing of the Euler equations and the Navier–Stokes equations by using the proposed method. Spatial coarsening is also considered. Compared with the MGRIT using the standard rediscretization approach, the developed method demonstrates enhanced robustness and efficiency in handling complex flow problems, including cases involving multidimensional shock waves and contact discontinuities.

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## Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

The authors acknowledge the reviewers for their valuable comments, which have significantly improved the quality of the manuscript.

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The work is supported by the National Natural Science Foundation of China (Grant Nos. 91952203 and 11902271), and 111 project on Aircraft Complex Flows and the Control (Grant No. B17037).

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## Appendix

### Appendix

The difference between the blocking MGRIT and the proposed non-blocking MGRIT is elaborated with the help of Fig. 9. The figure shows two iterations of blocking and non-blocking versions of MGRIT. For simplicity, V-cycle two-level MGRIT with F-relaxation is used for illustration. Four coarse time points are distributed on four processors. The coarse level is solved sequentially so there exists step-like feature in the figure. Each iteration is composed of F-relaxations, restrictions, coarse-level solving and finally prolongations. For the blocking MGRIT, the first processor completes the first V-cycle iteration and then waits until the last process finishes it. After that, processors will perform an MPI_Allreduce operation to obtain the current error, which is used for stopping the iteration. If not converged, the second iteration proceeds in the same way as the first one, as is shown in Fig. 9a. By contrast, if the blocking communication is removed, the second iteration on any processor starts immediately after the first iteration finishes, as is shown in Fig. 9b. It reduces considerable communication overhead especially when the number of processors is large.

We give quantitative analysis of the speedup *S* achieved by the non-blocking version over the blocking one. Let \(T_F\) and \(T_C\) denote the time consumed by a relaxation on the fine level and by a solving step on the coarse level, respectively. Likewise, let \(T_R\) and \(T_P\) denote the time cost of a restriction and a prolongation, respectively. Define \(n_{iter}\) to be the number of iterations, and \(N_C\) to be the number of time points on the coarse level, which is the same as the number of processors in Fig. 9. Then, the total time cost of blocking MGRIT is

The total time consumed by non-blocking MGRIT is

The speedup *S* defined by \(T_{Block}/T_{NBlock}\) can be rearranged as

Algebra analysis indicates that the speedup increases as \(N_C\) or \(n_{iter}\) increase. Therefore, the benefit of non-blocking MGRIT is exploited when \(N_C\) and \(n_{iter}\) are large, which is often the case for hyperbolic equations. To stably advance hyperbolic equations, CFL number on the coarsest level cannot be intensively large, which means \(N_C\) is generally not very small.

One issue of the non-blocking version of MGRIT is that we still need a guideline to stop the iteration. To address this issue, one can perform a blocking communication every other certain number of iterations to obtain the error norm. The time cost of possible extra non-blocking iterations is generally much less than the blocking overhead. To add the non-blocking strategy in XBraid library, one has to modify two parts. The first part is in function *braid_DriveCheckConvergence*, where the convergence flag is modified to ensure that iterations do not stop too early. The second part is in function *braid_FRestrict*, where the global error norm is computed. A conditional statement is added to ensure that the *MPI_Allreduce* operation is performed only when needed.

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Zhen, M., Ding, X., Qu, K. *et al.* Enhancing the Convergence of the Multigrid-Reduction-in-Time Method for the Euler and Navier–Stokes Equations.
*J Sci Comput* **100**, 40 (2024). https://doi.org/10.1007/s10915-024-02596-0

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DOI: https://doi.org/10.1007/s10915-024-02596-0