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

Franciolini, Matteo; Botti, Lorenzo Alessio; Colombo, Alessandro; Crivellini, Andrea

doi:10.1016/j.compfluid.2020.104558

Cited by 20 publications

(16 citation statements)

References 43 publications

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“…The performance of the Newton-Krylov method substantially depends on the preconditioner. In the context of the Jacobian-free implementation, the element-Jacobi preconditioner is arguably the simplest one with acceptable performance for many applications among the low-storage preconditioners, such as the matrix-free LU-SGS (Lower-Upper Symmetric-Gauss-Seidel) preconditioner [53], and p-multigrid preconditioner [54]. In this study, only the element-Jacobi preconditioner is considered and it is updated at the starting stage of each physical time step.…”

Section: Iterative Methodsmentioning

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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Section: Iterative Methodsmentioning

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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“…A commonly used ASM preconditioner strategy for DG discretizations consists in employing an ILU decomposition in each subdomain matrix suitably extended to include 3 and 4, respectively the matrix rows of ghost elements, that is, neighbors of local mesh elements that belong to a different subdomain. This implies that the local matrix is extended to encompass the stencil of the DG discretizations, see [43] for additional details. We consider a similar strategy for HHO discretizations: each subdomain matrix is extended to include the matrix rows of ghost faces, that is, faces of the local mesh elements that belong to a different subdomain.…”

Section: Scalabilitymentioning

confidence: 99%

“…The purpose of applying iterative solvers to coarse problems is twofold: on the one hand, a coarser operator translates into a global sparse matrix of smaller size with fewer non-zero entries, resulting in cheaper matrix-vector products; on the other hand, coarse level iterations are best suited to smooth out the low-frequency components of the error, that are hardly damped by fine level iterations. In the context of DG discretizations, p-multilevel solvers have been fruitfully utilized in practical applications, see, e.g., [12,42,43,48,54]. h-, p-and hp-multigrid solvers for DG discretizations of elliptic problems have been considered in [4], where uniform convergence with respect to the number of levels for the W-cycle iteration has been proved, and in [19].…”

Section: Introductionmentioning

confidence: 99%

p-Multilevel Preconditioners for HHO Discretizations of the Stokes Equations with Static Condensation

Botti

Pietro

2021

Commun. Appl. Math. Comput.

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We propose a p-multilevel preconditioner for hybrid high-order (HHO) discretizations of the Stokes equation, numerically assess its performance on two variants of the method, and compare with a classical discontinuous Galerkin scheme. An efficient implementation is proposed where coarse level operators are inherited using $$L^2$$ L 2 -orthogonal projections defined over mesh faces and the restriction of the fine grid operators is performed recursively and matrix-free. Both h- and k-dependency are investigated tackling two- and three-dimensional problems on standard meshes and graded meshes. For the two HHO formulations, featuring discontinuous or hybrid pressure, we study how the combination of p-coarsening and static condensation influences the V-cycle iteration. In particular, two different static condensation procedures are considered for the discontinuous pressure HHO variant, resulting in global linear systems with a different number of unknowns and matrix non-zero entries. Interestingly, we show that the efficiency of the solution strategy might be impacted by static condensation options in the case of graded meshes.

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“…We henceforth refer to them as h-multigrid. In HO-methods, alternatively to mesh-coarsening, we can represent the high-order errors on lower orders, effectively coarsening the polynomial-order p. The resulting p-multigrid has been extensively used for the last decade for elliptic, Euler and compressible Navier-Stokes equations [10,11,12,13], including RANS. Pure h-multigrid has also been used in HOmethods for Navier-Stokes equations [15,16,14].…”

Section: Introductionmentioning

confidence: 99%

HP-Multigrid for RANS-Modeled Turbulent Flows in a High-Order Flux Reconstruction Framework

Joshi¹,

Hurtado-de-Mendoza²,

Kou³

et al. 2021

14th WCCM-ECCOMAS Congress

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High-order (HO) methods are of concerted academic and industrial interest in recent years due to their improved accuracy and their capability to deal with complex geometries [1]. Of particular note is the flux reconstruction method [2], which unifies several existing HO schemes into a simpler and computationally efficient approach that has been shown to work on all element types (including simplices) in two and three dimensions. There is considerable interest to apply HO methods to industrially relevant problems. At the same time, accurate and robust turbulence modeling techniques are essential for reliable results. As outlined in the National Aeronautics and Space Administration's CFD vision 2030 study, Large Eddy Simulation (LES) still remains impractical for industrial cases-therefore, Reynolds-Averaged Navier Stokes (RANS) and hybrid RANS-LES methods hold high significance in the near future [4]. Achieving fastest convergence to steady-state is important in the context of RANS simulations, for which several convergence acceleration techniques are being investigated. Multigrid methods are an industry standard in Finite Volume (FV) type schemes and are increasingly being applied to HO methods in the form of p-multigrid [22]. They exploit the polynomial hierarchy of the solution space to represent errors on a coarser resolution. A natural extension of this idea is hp-multigrid, where we can augument the classical h-multigrid to the polynomial hierarchy [23]. In this paper we illustrate the application of high-order flux reconstruction methods to simulate compressible, turbulent flows on body-fitted meshes. The case in point is the turbulent flow over a flat plate [24]. Turbulence is modeled through the RANS approach using the one-equation Spalart-Allmaras model. Grid-coarsening for the h-levels is performed by removing every other line in each direction from the original mesh. The system is driven to a steady-state solution using hp-multigrid convergence acceleration with local time-stepping using an explicit Runge-Kutta time-marcher. We show that the augumented h-multigrid is highly effective with a 10X to 24X drop in convergence time.

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

Cited by 20 publications

References 43 publications

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

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

p-Multilevel Preconditioners for HHO Discretizations of the Stokes Equations with Static Condensation

HP-Multigrid for RANS-Modeled Turbulent Flows in a High-Order Flux Reconstruction Framework

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