Multigrid reduction-in-time convergence for advection problems: A Fourier analysis perspective

A long-standing issue in the parallel-in-time community is the poor convergence of standard iterative parallel-in-time methods for hyperbolic partial differential equations (PDEs), and for advection-dominated PDEs more broadly. Here, a local Fourier analysis (LFA) convergence theory is derived for the two-level variant of the iterative parallel-in-time method of multigrid reduction-in-time (MGRIT). This closed-form theory allows for new insights into the poor convergence of MGRIT for advection-dominated PDEs when using the standard approach of rediscretizing the fine-grid problem on the coarse grid. Specifically, we show that this poor convergence arises, at least in part, from inadequate coarse-grid correction of certain smooth Fourier modes known as characteristic components, which was previously identified as causing poor convergence of classical spatial multigrid on steady-state advection-dominated PDEs. We apply this convergence theory to show that, for certain semi-Lagrangian discretizations of advection problems, MGRIT convergence using rediscretized coarse-grid operators cannot be robust with respect to CFL number or coarsening factor. A consequence of this analysis is that techniques developed for improving convergence in the spatial multigrid context can be re-purposed in the MGRIT context to develop more robust parallel-in-time solvers. This strategy has been used in recent work to great effect; here, we provide further theoretical evidence supporting the effectiveness of this approach

Parallel-in-time integration of the shallow water equations on the rotating sphere using Parareal and MGRIT

Despite the growing interest in parallel-in-time methods as an approach to accelerate numerical simulations in atmospheric modelling, improving their stability and convergence remains a substantial challenge for their application to operational models. In this work, we study the temporal parallelization of the shallow water equations on the rotating sphere combined with time-stepping schemes commonly used in atmospheric modelling due to their stability properties, namely an Eulerian implicit-explicit (IMEX) method and a semi-Lagrangian semi-implicit method (SL-SI-SETTLS). The main goal is to investigate the performance of parallel-in-time methods, namely Parareal and Multigrid Reduction in Time (MGRIT), when these well-established schemes are used on the coarse discretization levels and provide insights on how they can be improved for better performance. We begin by performing an analytical stability study of Parareal and MGRIT applied to a linearized ordinary differential equation depending on the choice of a coarse scheme. Next, we perform numerical simulations of two standard tests to evaluate the stability, convergence and speedup provided by the parallel-in-time methods compared to a fine reference solution computed serially. We also conduct a detailed investigation on the influence of artificial viscosity and hyperviscosity approaches, applied on the coarse discretization levels, on the performance of the temporal parallelization. Both the analytical stability study and the numerical simulations indicate a poorer stability behaviour when SL-SI-SETTLS is used on the coarse levels, compared to the IMEX scheme. With the IMEX scheme, a better trade-off between convergence, stability and speedup compared to serial simulations can be obtained under proper parameters and artificial viscosity choices, opening the perspective of the potential competitiveness for realistic models.Comment: 35 pages, 23 figure

Parallel-in-Time Simulation of an Electrical Machine using MGRIT

We apply the multigrid-reduction-in-time (MGRIT) algorithm to an eddy current simulation of a two-dimensional induction machine supplied by a pulse-width-modulation signal. To resolve the fast-switching excitations, small time steps are needed, such that parallelization in time becomes highly relevant for reducing the simulation time. The MGRIT algorithm is well suited for introducing time parallelism in the simulation of electrical machines using existing application codes, as MGRIT is a non-intrusive approach that essentially uses the same time integrator as a traditional time-stepping algorithm. We investigate effects of spatial coarsening on MGRIT convergence when applied to two numerical models of an induction machine, one with linear material laws and a full nonlinear model. Parallel results demonstrate significant speedup in the simulation time compared to sequential time stepping, even for moderate numbers of processors.Comment: 14 page

Parallel-in-time integration of kinematic dynamos

The precise mechanisms responsible for the natural dynamos in the Earth and Sun are still not fully understood. Numerical simulations of natural dynamos are extremely computationally intensive, and are carried out in parameter regimes many orders of magnitude away from real conditions. Parallelization in space is a common strategy to speed up simulations on high performance computers, but eventually hits a scaling limit. Additional directions of parallelization are desirable to utilise the high number of processor cores now available. Parallel-in-time methods can deliver speed up in addition to that offered by spatial partitioning but have not yet been applied to dynamo simulations. This paper investigates the feasibility of using the parallel-in-time algorithm Parareal to speed up initial value problem simulations of the kinematic dynamo, using the open source Dedalus spectral solver. Both the time independent Roberts and time dependent Galloway-Proctor 2.5D dynamos are investigated over a range of magnetic Reynolds numbers. Speedups beyond those possible from spatial parallelisation are found in both cases. Results for the Galloway-Proctor flow are promising, with Parareal efficiency found to be close to 0.3. Roberts flow results are less efficient, but Parareal still shows some speed up over spatial parallelisation alone. Parallel in space and time speed ups of ∼300 were found for 1600 cores for the Galloway-Proctor flow, with total parallel efficiency of ∼0.16

Time integration for complex fluid dynamics

2021 Fall.Includes bibliographical references.Efficient and accurate simulation of turbulent combusting flows in complex geometry remains a challenging and computationally expensive proposition. A significant source of computational expense is in the integration of the temporal domain, where small time steps are required for the accurate resolution of chemical reactions and long solution times are needed for many practical applications. To address the small step sizes, a fourth-order implicit-explicit additive Runge-Kutta (ARK4) method is developed to integrate the stiff chemical reactions implicitly while advancing the convective and diffusive physics explicitly in time. Applications involving complex geometry, stiff reaction mechanisms, and high-order spatial discretizations are challenged by stability issues in the numerical solution of the nonlinear problem that arises from the implicit treatment of the stiff term. Techniques for maintaining a physical thermodynamic state during the numerical solution of the nonlinear problem, such as placing constraints on the nonlinear solver and the use of a nonlinear optimizer to find valid thermodynamic states, are proposed and tested. Verification and validation are performed for the new adaptive ARK4 method using lean premixed flames burning hydrogen, showing preservation of 4th-order error convergence and recovery of literature results. ARK4 is then applied to solve lean, premixed C3H8-air combustion in a bluff-body combustor geometry. In the two-dimensional case, ARK4 provides a 70× speedup over the standard explicit four-stage Runge-Kutta method and, for the three-dimensional case, three-orders-of-magnitude-larger time step sizes are achieved. To further increase the computational scaling of the algorithms, parallel-in-time (PinT) techniques are explored. PinT has the dual benefit of providing parallelization to long temporal domains as well as taking advantage of hardware trends towards more concurrency in modern high-performance computing platforms. Specifically, the multigrid reduction-in-time (MGRIT) method is adapted and enhanced by adding adaptive mesh refinement (AMR) in time. This creates a space-time algorithm with efficient solution-adaptive grids. The new MGRIT+AMR algorithm is first verified and validated using problems dominated by diffusion or characterized by time periodicity, such as Couette flow and Stokes second problem. The adaptive space-time parallel algorithm demonstrates up to a 13.7× speedup over a time-sequential algorithm for the same solution accuracy. However, MGRIT has difficulties when applied to solve practical fluid flows, such as turbulence, governed by strong hyperbolic partial differential equations. To overcome this challenge, the multigrid operations are modified and applied in a novel way by exploiting the space-time localization of fine turbulence scales. With these new operators, the coarse-scale errors are advected out of the temporal domain while the fine-scale dynamics iterate to equilibrium. This leads to rapid convergence of the bulk flow, which is important for computing macroscopic properties useful for engineering purposes. The novel multigrid operations are applied to the compressible inviscid Taylor-Green vortex flow and the convergence of the low-frequency modes is achieved within a few iterations. Future work will be focused on a performance study for practical highly turbulent flows

Preconditioning of Hybridizable Discontinuous Galerkin Discretizations of the Navier-Stokes Equations

The incompressible Navier-Stokes equations are of major interest due to their importance in modelling fluid flow problems. However, solving the Navier-Stokes equations is a difficult task. To address this problem, in this thesis, we consider fast and efficient solvers. We are particularly interested in solving a new class of hybridizable discontinuous Galerkin (HDG) discretizations of the incompressible Navier-Stokes equations, as these discretizations result in exact mass conservation, are locally conservative, and have fewer degrees of freedom than discontinuous Galerkin methods (which is typically used for advection dominated flows). To achieve this goal, we have made various contributions to related problems, as I discuss next. Firstly, we consider the solution of matrices with 2x2 block structure. We are interested in this problem as many discretizations of the Navier-Stokes equations result in block linear systems of equations, especially discretizations based on mixed-finite element methods like HDG. These systems also arise in other areas of computational mathematics, such as constrained optimization problems, or the implicit or steady state treatment of any system of PDEs with multiple dependent variables. Often, these systems are solved iteratively using Krylov methods and some form of block preconditioner. Under the assumption that one diagonal block is inverted exactly, we prove a direct equivalence between convergence of 2x2 block preconditioned Krylov or fixed-point iterations to a given tolerance, with convergence of the underlying preconditioned Schur-complement problem. In particular, results indicate that an effective Schur-complement preconditioner is a necessary and sufficient condition for rapid convergence of 2x2 block-preconditioned GMRES, for arbitrary relative-residual stopping tolerances. A number of corollaries and related results give new insight into block preconditioning, such as the fact that approximate block-LDU or symmetric block-triangular preconditioners offer minimal reduction in iteration over block-triangular preconditioners, despite the additional computational cost. We verify the theoretical results numerically on an HDG discretization of the steady linearized Navier--Stokes equations. The findings also demonstrate that theory based on the assumption of an exact inverse of one diagonal block extends well to the more practical setting of inexact inverses. Secondly, as an initial step towards solving the time-dependent Navier-Stokes equations, we investigate the efficiency, robustness, and scalability of approximate ideal restriction (AIR) algebraic multigrid as a preconditioner in the all-at-once solution of a space-time HDG discretization of the scalar advection-diffusion equation. The motivation for this study is two-fold. First, the HDG discretization of the velocity part of the momentum block of the linearized Navier-Stokes equations is the HDG discretization of the vector advection-diffusion equation. Hence, efficient and fast solution of the advection-diffusion problem is a prerequisite for developing fast solvers for the Navier-Stokes equations. The second reason to study this all-at-once space-time problem is that the time-dependent advection-diffusion equation can be seen as a ``steady'' advection-diffusion problem in (d+1)-dimensions and AIR has been shown to be a robust solver for steady advection-dominated problems. We present numerical examples which demonstrate the effectiveness of AIR as a preconditioner for time-dependent advection-diffusion problems on fixed and time-dependent domains, using both slab-by-slab and all-at-once space-time discretizations, and in the context of uniform and space-time adaptive mesh refinement. A closer look at the geometric coarsening structure that arises in AIR also explains why AIR can provide robust, scalable space-time convergence on advective and hyperbolic problems, while most multilevel parallel-in-time schemes struggle with such problems. As the final topic of this thesis, we extend two state-of-the-art preconditioners for the Navier-Stokes equations, namely, the pressure convection-diffusion and the grad-div/augmented Lagrangian preconditioners to HDG discretizations. Our preconditioners are simple to implement, and our numerical results show that these preconditioners are robust in h and only mildly dependent on the Reynolds numbers