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

TL;DR

This work develops and validates an adaptive framework for solving time-dependent PDEs with space-time discontinuous Galerkin discretization. It combines anisotropic hp-mesh adaptation with two-level Schwarz domain-decomposition preconditioners for GMRES, and introduces a computational-cost model that jointly accounts for floating-point operations, parallel speed, and inter-core communications to optimally choose the number of subdomains $M$ and coarse elements $s$ per time level. An adaptive domain-decomposition algorithm uses history-based fits of flop and speed data and communication costs to predict wall-clock times and select $(M^{\mathrm{opt}}_m, s^{\mathrm{opt}}_m)$ that minimize total runtime, demonstrated on rising thermal bubble and Kelvin-Helmholtz benchmarks. The results show that the adaptive approach outperforms fixed partitionings, with the cost model providing reliable predictions and a practical path toward scalable, efficient simulations of compressible flow using STDGM. The methodology is generalizable to other discretizations and preconditioners, enabling broader applicability of adaptive DD strategies in time-dependent problems.

Abstract

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.

Figures (15)

• Figure 1: Example of the numbering of mesh elements, bold lines correspond to the boundaries among subdomains ${\Omega}_{m}^{i}$, $i=1,\dots,M_{m}$.
• Figure 2: Example of a triangulation ${{\mathscr T}_{h,m}}$ and the block structures of the corresponding matrix $\pmb{A}_m$. Left: triangles with polynomial approximation degrees and the domain partition into four subdomains (blue lines). Center: element blocks of $\pmb{A}_m$ (small red boxes) and the subdomain-blocks $\pmb{A}_m^i$, $i=1,\dots,4$ (large blue boxes). Right: domain decomposition having four subdomains (colored) and the coarse grid ${{\mathscr T}_{H,m}}$ with 16 coarse elements (thick lines).
• Figure 3: Computational cost model using 5 cores $I=1,\dots,5$, parallel computations (green boxes) followed by communication among cores (magenta arrows), and subsequent steps of the algorithm (blue arrows).
• Figure 4: Number of the floating point operations for the matrix factorization: dependence of $\mathsf{flops}_{\mathrm{fac}}^{m,i}$ on the system size ${N_{m}^i}$ corresponding to triangular ${\mathscr T}_{h,m}^i$ (left) and polygonal ${{\mathscr T}_{H,m}}$(right) grids.
• Figure 5: Number of the floating point operations for the solution substitution: dependence of $\mathsf{flops}_{\mathrm{sub}}^{m,i}$ on the system size ${N_{m}^i}$ corresponding to triangular ${\mathscr T}_{h,m}^i$ (left) and ${{\mathscr T}_{H,m}}$ polygonal (right) grids.

Theorems & Definitions (5)

• Definition 3.1
• Remark 3.2
• Remark 4.1
• Remark 5.1
• Remark 6.1