On the efficiency of a matrix-free linearly implicit time integration strategy for high-order Discontinuous Galerkin solutions of incompressible turbulent flows

Franciolini, Matteo; Crivellini, Andrea; Nigro, Alessandra

doi:10.1016/j.compfluid.2017.10.008

Cited by 28 publications

(29 citation statements)

References 40 publications

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“…The right plot shows that MF-TSP outperforms ME for large time steps corresponding to accuracy levels of order of 10 −2 − 10 −3 , whereas their computational efforts are roughly the same for smaller time steps and higher precision levels. In this respect, however, the CPU ratio entries of Table 1 highlight a trend in terms of performance comparison, as reported in the work of Franciolini et al, 26 which depends on the spatial resolution: MF-TSP is slightly less expensive than ME using P6 and P4 polynomial approximations, whereas the opposite trend is found for P1 and P2 elements.…”

Section: Effect Of the Polynomial Degree On The Performances Of Mf Ansupporting

confidence: 64%

“…These results demonstrate that the MF implementation of the MEBDF scheme is beneficial for large time steps and for high-order accurate DG discretizations, when the size of the blocks in the Jacobian is large and the cost to compute the system matrix and to perform the matrix-vector product is much more expensive than a residual evaluation. In this article, only 2D computations are considered, but the findings here and the analysis reported in the work of Franciolini et al 26 allow to conclude that better performances are expected for 3D computations.…”

Section: Effect Of the Polynomial Degree On The Performances Of Mf Anmentioning

confidence: 64%

“…The MF approach for the numerical solution of partial differential equations has been developed in other works and has been used to solve compressible and incompressible flows . It has been also applied in the context of the high‐order DG discretization …”

Section: Introductionmentioning

confidence: 99%

“…[20][21][22] It has been also applied in the context of the high-order DG discretization. [23][24][25][26] In this work, we adopt the MF approach for unsteady computations solving the DG discretized compressible Navier-Stokes equations with the high-order accurate Modified Extended BDF (MEBDF) method. This implicit time integration method belongs to the family of predictor-corrector methods and involves three nonlinear stages: the first two are predictor stages and the last one is a corrector stage that uses an implicit k-step formula of order k + 1.…”

Section: Introductionmentioning

confidence: 99%

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Matrix‐free modified extended BDF applied to the discontinuous Galerkin solution of unsteady compressible viscous flows

Nigro

Bartolo

Crivellini

et al. 2018

Numerical Methods in Fluids

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Summary In this work, a time‐accurate integration of the discontinuous Galerkin space‐discretized Navier‐Stokes equations is performed exploiting the matrix‐free (MF) approach to speed up the solution process of the modified extended backward differentiation formulae (MEBDF) schemes. MEBDF are high‐order accurate implicit multistep schemes composed by three nonlinear stages. The proposed algorithm consists in solving the resulting nonlinear system of each stage with a preconditioned MF Newton/Krylov method using a frozen preconditioner strategy to improve its efficiency. Numerical results for compressible inviscid and viscous test cases, both with a known analytical solution, aim at assessing the performance of the proposed MF‐MEBDF algorithm, by comparing it with the one obtained by using its matrix‐explicit counterpart or the explicit strong stability preserving Runge‐Kutta scheme. In particular, the influence of some relevant physical (low‐speed flows) and discretization (aspect ratio, polynomial degree) aspects on the performance of the different time integration schemes are investigated, highlighting the pros and cons of the proposed algorithm and its effectiveness in solving nonstiff and stiff systems. Furthermore, the scalability in parallel computations of the proposed algorithm is investigated and, finally, its potential for efficient long‐time simulations is demonstrated computing a laminar vortex shedding behind a circular cylinder at different Reynolds numbers and by comparing its effectiveness with that of the strong stability preserving Runge‐Kutta scheme.

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Section: Effect Of the Polynomial Degree On The Performances Of Mf Ansupporting

confidence: 64%

Section: Effect Of the Polynomial Degree On The Performances Of Mf Anmentioning

confidence: 64%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Matrix‐free modified extended BDF applied to the discontinuous Galerkin solution of unsteady compressible viscous flows

Nigro

Bartolo

Crivellini

et al. 2018

Numerical Methods in Fluids

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“…The main challenge toward fast implicit solvers allowing the use of arbitrarily large time step sizes is the development of preconditioners and multigrid smoothers that provide robust convergence rates independent of h and k and, at the same time, can be implemented in a matrix‐free way with optimal (theoretical) computational complexity of

scriptO ((), k^{d + 1})

, such as the algorithms used in the present work. For example, implicit solvers often rely on matrix‐based methods with a complexity of

scriptO ((), k^{2 d})

for matrix‐vector products . Progress toward implicit solvers with optimal complexity–type preconditioners and multigrid smoothers has been made recently in the works of Bastian et al and Pazner and Persson …”

Section: Motivationmentioning

confidence: 99%

A matrix‐free high‐order discontinuous Galerkin compressible Navier‐Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows

Fehn

Wall

Kronbichler

2018

Numerical Methods in Fluids

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Summary Both compressible and incompressible Navier‐Stokes solvers can be used and are used to solve incompressible turbulent flow problems. In the compressible case, the Mach number is then considered as a solver parameter that is set to a small value, M ≈0.1, in order to mimic incompressible flows. This strategy is widely used for high‐order discontinuous Galerkin (DG) discretizations of the compressible Navier‐Stokes equations. The present work raises the question regarding the computational efficiency of compressible DG solvers as compared to an incompressible formulation. Our contributions to the state of the art are twofold: Firstly, we present a high‐performance DG solver for the compressible Navier‐Stokes equations based on a highly efficient matrix‐free implementation that targets modern cache‐based multicore architectures with Flop/Byte ratios significantly larger than 1. The performance results presented in this work focus on the node‐level performance, and our results suggest that there is great potential for further performance improvements for current state‐of‐the‐art DG implementations of the compressible Navier‐Stokes equations. Secondly, this compressible Navier‐Stokes solver is put into perspective by comparing it to an incompressible DG solver that uses the same matrix‐free implementation. We discuss algorithmic differences between both solution strategies and present an in‐depth numerical investigation of the performance. The considered benchmark test cases are the three‐dimensional Taylor‐Green vortex problem as a representative of transitional flows and the turbulent channel flow problem as a representative of wall‐bounded turbulent flows. The results indicate a clear performance advantage of the incompressible formulation over the compressible one.

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An Implicit Discontinuous Galerkin Method with Reduced Memory Footprint for the Simulation of Turbulent Flows

Crivellini

Franciolini

Nigro

2019

Direct and Large-Eddy Simulation XI

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On the efficiency of a matrix-free linearly implicit time integration strategy for high-order Discontinuous Galerkin solutions of incompressible turbulent flows

Cited by 28 publications

References 40 publications

Matrix‐free modified extended BDF applied to the discontinuous Galerkin solution of unsteady compressible viscous flows

Matrix‐free modified extended BDF applied to the discontinuous Galerkin solution of unsteady compressible viscous flows

A matrix‐free high‐order discontinuous Galerkin compressible Navier‐Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows

An Implicit Discontinuous Galerkin Method with Reduced Memory Footprint for the Simulation of Turbulent Flows

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