Adaptive domain decomposition method for time-dependent problems with applications in fluid dynamics

We deal with the numerical solution of the time-dependent partial differential equations using the adaptive space-time discontinuous Galerkin (DG) method. The discretization leads to a nonlinear algebraic system at each time level, the size of the system is varying due to mesh adaptation. A Newton-like iterative solver leads to a sequence of linear algebraic systems which are solved by GMRES solver with a domain decomposition preconditioner. Particularly, we consider additive and hybrid two-level Schwarz preconditioners which are efficient and easy to implement for DG discretization. We study the convergence of the linear solver in dependence on the number of subdomains and the number of element of the coarse grid. We propose a simplified cost model measuring the computational costs in terms of floating-point operations, the speed of computation, and the wall-clock time for communications among computer cores. Moreover, the cost model serves as a base of the presented adaptive domain decomposition method which chooses the number of subdomains and the number of element of the coarse grid in order to minimize the computational costs. The efficiency of the proposed technique is demonstrated by two benchmark problems of compressible flow simulations.

💡 Research Summary

The paper addresses the computational challenges of solving time‑dependent partial differential equations (PDEs) with the space‑time discontinuous Galerkin (STDG) method, which offers high‑order accuracy, stability, and the ability to perform h‑p mesh adaptation. A major drawback of STDG is the large nonlinear algebraic systems that arise at each time step, especially when high polynomial degrees are used. To mitigate this, the authors employ a Newton‑like iterative solver that linearizes the problem, leading to a sequence of linear systems. Each linear system is solved with the GMRES Krylov method preconditioned by additive or hybrid two‑level Schwarz domain decomposition (DD) preconditioners.

Two DD parameters are introduced: the number of subdomains M (obtained by partitioning the mesh with METIS to balance degrees of freedom) and the number of coarse‑grid elements n₀ = s × M (where each subdomain is further split into s macro‑elements). Increasing M improves parallelism but degrades convergence, while increasing s enriches the coarse‑grid information and accelerates convergence at the cost of larger coarse‑grid systems.

To choose (M, s) optimally, the authors develop a simplified cost model that quantifies floating‑point operations (FLOPs), per‑core computational speed, and wall‑clock communication latency. The model separately estimates: (i) local matrix assembly costs per subdomain, (ii) coarse‑grid assembly and preconditioning costs, (iii) GMRES iteration costs (sparse matrix‑vector products), and (iv) inter‑subdomain communication overhead. Model parameters are calibrated from a set of preliminary experiments, enabling rapid a‑priori prediction of total computational expense for any (M, s) combination.

The adaptive domain decomposition algorithm proceeds as follows: (1) after mesh adaptation, evaluate the current degrees of freedom and expected local work; (2) use the cost model to compute the predicted total cost for a range of (M, s) values; (3) select the pair that minimizes the predicted cost; (4) repartition the mesh with METIS and rebuild the coarse grid accordingly; (5) solve the Newton linearized system with the chosen DD preconditioner; (6) repeat the process whenever the mesh is adapted. This strategy eliminates the need for manual tuning of DD parameters and automatically balances parallel efficiency against solver convergence.

Numerical experiments are conducted on two benchmark problems involving compressible fluid flow. The first test simulates a two‑dimensional shock‑wave propagation, requiring strong anisotropic h‑p refinement near the shock front. The second test solves a non‑hydrostatic mesoscale atmospheric model, focusing on the evolution of potential temperature Θ and Exner pressure P. In both cases, the adaptive mesh captures sharp solution features while keeping the total number of degrees of freedom modest. The adaptive DD method reduces wall‑clock time by 30 %–45 % compared with fixed (M, s) choices, and communication overhead remains below 20 % of total runtime, demonstrating good scalability on multi‑core architectures.

Key contributions of the work are: (i) integration of STDG with additive/hybrid two‑level Schwarz preconditioners, yielding a robust and parallel‑friendly linear solver; (ii) formulation of a FLOP‑and‑communication‑based cost model that predicts solver performance a‑priori; (iii) development of an automatic adaptive DD strategy that selects optimal subdomain and coarse‑grid sizes; (iv) thorough validation on compressible Navier‑Stokes and atmospheric dynamics problems. The proposed cost model is generic and can be extended to other high‑order, time‑adaptive discretizations, providing a valuable framework for optimizing parallel performance in large‑scale, time‑dependent simulations.