Modified extended BDF scheme for the discontinuous Galerkin solution of unsteady compressible flows

Nigro, Alessandra; Ghidoni, Antonio; Rebay, Stefano; Bassi, Francesco

doi:10.1002/fld.3944

Cited by 28 publications

(14 citation statements)

References 35 publications

(56 reference statements)

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“…The simulations have been computed using the exact Riemann solver of Gottlieb and Groth to approximate the inviscid numerical fluxes. Following the analysis shown in the work of Nigro et al, the linear solver tolerance, which is a relative residual L 2 ‐norm, and the Newton tolerance have been carefully set for each test‐case to minimize the computational effort without affecting the accuracy of the solution. For the isentropic vortex and the laminar MS, both with a known analytic solution, the linear‐solver tolerance was fixed to 10 −2 , and the Newton algorithms were stopped when the L 2 ‐norm of the difference between the solutions computed by two successive Newton iterations was one order less than the solution error.…”

Section: Resultsmentioning

confidence: 99%

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: Resultsmentioning

confidence: 99%

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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“…Several high-order implicit time integrators, relying on multistage and multistep schemes, are already available. For example, Explicit Singly Diagonally Implicit Runge-Kutta (ESDIRK) schemes [14,15], are A-stable up to order five, Modified Extended BDF (MEBDF) [16][17][18], are A-stable up to order four, and Two Implicit Advanced…”

Section: Introductionmentioning

confidence: 99%

Linearly implicit Rosenbrock-type Runge–Kutta schemes applied to the Discontinuous Galerkin solution of compressible and incompressible unsteady flows

et al. 2015

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a b s t r a c tIn this work we investigate the use of linearly implicit Rosenbrock-type Runge-Kutta schemes to integrate in time high-order Discontinuous Galerkin space discretizations of the Navier-Stokes equations. The final goal of this activity is the application of such schemes to the high-order accurate, both in space and time, simulation of turbulent flows. Besides being able to overcome the severe time step restriction of explicit schemes, Rosenbrock schemes have the attractive feature of requiring just one Jacobian matrix evaluation per time step, thus reducing the overall computational effort. Several high-order (up to sixth order) Rosenbrock schemes available in the literature have been implemented and evaluated on benchmark test cases of both compressible and incompressible flows. For the sake of completeness, the sets of coefficients of the schemes here considered have been reported in an Appendix to the paper. An implementation of Rosenbrock schemes for systems of equations with a solution dependent block diagonal matrix multiplying the time derivative is here proposed and described in detail. This can occur, for example, if sets of working variables different from the conservative ones are used in the compressible Navier-Stokes equations. In particular, we have found useful to employ primitive variables based on the logarithms of pressure and temperature in order to ensure the positivity of all thermodynamic variables at the discrete level. The best performing Rosenbrock scheme resulting from our analysis has then been applied to the Implicit Large Eddy Simulation of the transitional flow around the Selig-Donovan SD7003 airfoil.

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“…Nigro et. al have seen success applying multi-stage, multi-step modified extended BDF (MEBDF) and two implicit advanced step-point (TIAS) schemes to the compressible Euler and Navier-Stokes equations [21,22]. Additionally, in [3], Bassi et al have used linearly implicit Rosenbrock-type to integrate DG discretizations for various fluid flow problems.…”

Section: Introductionmentioning

confidence: 99%

Stage-parallel fully implicit Runge–Kutta solvers for discontinuous Galerkin fluid simulations

Pazner¹,

Persson²

2017

Journal of Computational Physics

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In this paper, we develop new techniques for solving the large, coupled linear systems that arise from fully implicit Runge-Kutta methods. This method makes use of the iterative preconditioned GMRES algorithm for solving the linear systems, which has seen success for fluid flow problems and discontinuous Galerkin discretizations. By transforming the resulting linear system of equations, one can obtain a method which is much less computationally expensive than the untransformed formulation, and which compares competitively with other time-integration schemes, such as diagonally implicit Runge-Kutta (DIRK) methods. We develop and test several ILU-based preconditioners effective for these large systems. We additionally employ a parallel-in-time strategy to compute the Runge-Kutta stages simultaneously. Numerical experiments are performed on the Navier-Stokes equations using Euler vortex and 2D and 3D NACA airfoil test cases in serial and in parallel settings. The fully implicit Radau IIA Runge-Kutta methods compare favorably with equal-order DIRK methods in terms of accuracy, number of GMRES iterations, number of matrix-vector multiplications, and wall-clock time, for a wide range of time steps.

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Modified extended BDF scheme for the discontinuous Galerkin solution of unsteady compressible flows

Cited by 28 publications

References 35 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

Linearly implicit Rosenbrock-type Runge–Kutta schemes applied to the Discontinuous Galerkin solution of compressible and incompressible unsteady flows

Stage-parallel fully implicit Runge–Kutta solvers for discontinuous Galerkin fluid simulations

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