A numerical investigation of matrix-free implicit time-stepping methods for large CFD simulations

Sarshar, Arash; Pickering, Brent P.; McCall, Andrew J.; Sandu, Adrian; Roy, Christopher J.

doi:10.1016/j.compfluid.2017.09.014

Cited by 12 publications

(10 citation statements)

References 40 publications

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“…However, the use of full-matrix operators was still required for preconditioning purposes, and thus only a limited reduction of the overall memory footprint has been achieved. In [8] a matrix-free approach is employed in the context of several time integration strategies with applications to unsteady, laminar two-dimensional problems. Recently the use of a matrix-free implementation was explored and compared to a matrix-based approach in the context of incompressible unsteady turbulent flows, see [9,10].…”

Section: Introductionmentioning

confidence: 99%

p-Multigrid matrix-free discontinuous Galerkin solution strategies for the under-resolved simulation of incompressible turbulent flows

Franciolini¹,

Botti²,

Colombo³

et al. 2018

Preprint

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In recent years several research efforts focused on the development of high-order discontinuous Galerkin (dG) methods for scale resolving simulations of turbulent flows. Nevertheless, in the context of incompressible flow computations, the computational expense of solving large scale equation systems characterized by indefinite Jacobian matrices has often prevented from dealing with industrially-relevant computations. In this work we seek to improve the efficiency of Rosenbrock-type linearly-implicit Runge-Kutta methods by devising robust, scalable and memory-lean solution strategies. In particular, we introduce memory saving p-multigrid preconditioners coupling matrix-free and matrix-based Krylov iterative smoothers. The p-multigrid preconditioner relies on cheap block-diagonal smoother's preconditioners on the fine space to reduce assembly costs and memory allocation, and ensures an adequate resolution of the coarsest space of the multigrid iteration using Additive Schwarz precondioned smoothers to obtain satisfactory convergence rates and optimal parallel efficiency of the method. In addition, the use of specifically crafted rescaled-inherited coarse operators to overcome the excess of stabilization provided by the standard inheritance of the fine space operators is explored. Extensive numerical validation is performed. The Rosenbrock formulation is applied to test cases of growing complexity: the laminar unsteady flow around a two-dimensional cylinder at Re = 200 and around a sphere at Re = 300, the transitional flow problem of the ERCOFTAC T3L test case suite with different levels of free-stream turbulence. As proof of concept, the numerical solution of the Boeing Rudimentary Landing Gear test case at Re = 10 6 is reported. A good agreement of the solutions with experimental data is documented, as well as strong memory savings and execution time gains with respect to state-of-the art solution strategies.

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Section: Introductionmentioning

confidence: 99%

p-Multigrid matrix-free discontinuous Galerkin solution strategies for the under-resolved simulation of incompressible turbulent flows

Franciolini¹,

Botti²,

Colombo³

et al. 2018

Preprint

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“…Recall Eqs. (10), (12) and (13), in order to calculate the common inviscid and viscous fluxes, information from neighbor elements will be needed. Therefore, in the procedure of calculating R i,m and with the complex-step derivative approximation, adding a disturbance to the corresponding entry in q j will be done n dof times.…”

Section: Details On the Parallel Implementationmentioning

confidence: 99%

“…Linearly implicit Rosenbrock methods avoid solving nonlinear systems every stage in some IRK methods, such as singly diagonally implicit Runge-Kutta methods (SDIRK) [12], and solve only one linear system at each stage [13]. Different variations of Rosenbrocktype IRK methods can be found in Refs.…”

Section: Introductionmentioning

confidence: 99%

On the parallel implementation and performance study of high-order Rosenbrock-type implicit Runge-Kutta methods for the FR/CPR solutions of the Navier-Stokes equations

Wang

2018

2018 AIAA Aerospace Sciences Meeting

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The Rosenbrock-type implicit Runge-Kutta (ROIRK) methods only require one Jacobian matrix evaluation per time step rather than per stage as other types of implicit Runge-Kutta (IRK) methods need. This feature makes ROIRK attractive for numerical simulations using implicit methods. We present the parallel implementation of several matrix-based ROIRK methods with flux reconstruction/correction procedure reconstruction (FR/CPR) formulations for solving the 3D Navier-Stokes equations. In this study, METIS has been utilized to partition the mesh in the preprocessing. The complex-step derivative approximation is employed to evaluate the Jacobi matrix, accurate to machine zero. The GMRES solver in the PETSc library is used to iteratively solve the linear system. The ROIRK methods have demonstrated high order of accuracy in numerical simulations. The scalability study reveals that the matrix-based ROIRK methods have good parallel efficiency. With the block Jacobi preconditioner, it is observed that the linear systems resulting from ROIRK3-3 are stiffer than those from ROIRK2-2 and ROIRK4-6. This makes the scalability of ROIRK3-3 worse than ROIRK2-2 and ROIRK4-6 taking the number of stages into account.

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“…They observed that the third-order ROW methods tested are more efficient than ESDIRK3. Sarshar et al [46] conducted a comparative study of various matrix-free implicit time-stepping methods on solving two-dimensional (2D) Navier-Stokes equations including SDIRK, Rosenbrock, ROW, and ROK using a secondorder spatial discretization. It is found that ROK can be a competitive method compared to other implicit time integrations.…”

Section: Introductionmentioning

confidence: 99%

Comparison of ROW, ESDIRK, and BDF2 for Unsteady Flows with the High-Order Flux Reconstruction Formulation

Wang

2020

J Sci Comput

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We conduct a comparative study of the Jacobian-free linearly implicit Rosenbrock-Wanner (ROW) methods, the explicit first stage, singly diagonally implicit Runge-Kutta (ESDIRK) methods, and the second-order backward differentiation formula (BDF2) for the high-order flux reconstruction/correction procedure via reconstruction (FR/CPR) solution of the unsteady Navier-Stokes equations. Pseudo-transient continuation is employed to solve the nonlinear equation at each stage of ESDIRK (excluding the first stage) and each step of BDF2. A Jacobian-free implementation of the restarted generalized minimal residual method (GMRES) solver is employed with a low storage element-Jacobi preconditioner to solve the linear system at each stage of ROW and each pseudo time iteration of ESDIRK and BDF2. Several numerical experiments, including both laminar and turbulent flow simulations, are conducted to carry out the comparison. We observe that the multistage ROW2 and ESDIRK2 are more efficient than the multistep BDF2, and higher-order implicit time integrators are more efficient than lower-order ones. In general, the ESDIRK method allows a larger physical time step size for unsteady flow simulation than the ROW method when the element-Jacobi preconditioner is employed, especially for wall-bounded flows; and the ROW method can be more efficient than the ESDIRK method when the time step size is refined.

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A numerical investigation of matrix-free implicit time-stepping methods for large CFD simulations

Cited by 12 publications

References 40 publications

p-Multigrid matrix-free discontinuous Galerkin solution strategies for the under-resolved simulation of incompressible turbulent flows

p-Multigrid matrix-free discontinuous Galerkin solution strategies for the under-resolved simulation of incompressible turbulent flows

On the parallel implementation and performance study of high-order Rosenbrock-type implicit Runge-Kutta methods for the FR/CPR solutions of the Navier-Stokes equations

Comparison of ROW, ESDIRK, and BDF2 for Unsteady Flows with the High-Order Flux Reconstruction Formulation

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