Effect of discretization order on preconditioning and convergence of a high-order unstructured Newton-GMRES solver for the Euler equations

Nejat, Amir; Ollivier‐Gooch, Carl

doi:10.1016/j.jcp.2007.10.024

Cited by 38 publications

(24 citation statements)

References 43 publications

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“…Of course, shocks and vortices still require an adequate local mesh to catch all the structures but high-order schemes prove to be efficient in reducing the numerical diffusion. For instance the k-exact reconstruction [3,4,34] increases the method accuracy using quadratic or cubic polynomial approximations [32,35,36]. Nevertheless, traditional TVD (Total Variation Diminishing) limiting procedures drastically reduce the order of accuracy despite the construction of alternative limiters [45,33] to enhance the quality of the solution.…”

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confidence: 99%

a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

2017

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To cite this version:Stéphane Clain, Raphaël Loubère, Gaspar Machado. a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations. 2016. a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equationsWe propose a new family of finite volume high-accurate numerical schemes devoted to solve one-dimensional steady-state hyperbolic systems. High-accuracy (up to the sixth-order presently) is achieved thanks to polynomial reconstructions while stability is provided with an a posteriori MOOD method which control the cell polynomial degree for eliminating non-physical oscillations in the vicinity of discontinuities. Such a procedure demands the determination of a chain detector to discriminate between troubled and valid cells, a cascade of polynomial degrees to be successively tested when oscillations are detected, and a parachute scheme corresponding to the last, viscous, and robust scheme of the cascade. Experimented on linear, Burgers', and Euler equations, we demonstrate that the schemes manage to retrieve smooth solutions with optimal order of accuracy but also irregular solutions without spurious oscillations.

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confidence: 99%

a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

2017

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“…Therein, the choice of the preconditioner is crucial for the convergence process. Our main focus is on the incomplete LU-factorization (ILU) and Gauss-Seidel (GS) preconditioners that are widely-used in solvers in the numerical simulation of compressible flows [1,6,18,31,35,40].…”

Section: Point-block Preconditionersmentioning

confidence: 99%

“…It is known from experience, as reported in, e.g., [29,31], that a benefit from second order methods can only be expected after a certain number of time steps have been elapsed. Usually in the early iterations a first order implementation of a matrixvector product is faster and more robust.…”

Section: Numerical Experimentsmentioning

confidence: 99%

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Iterative Solvers for Discretized Stationary Euler Equations

Pollul

Reusken

2010

Notes on Numerical Fluid Mechanics and Multidisciplinary Design

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In this paper we treat subjects which are relevant in the context of iterative methods in implicit time integration for compressible flow simulations. We present a novel renumbering technique, some strategies for choosing the time step in the implicit time integration, and a novel implementation of a matrix-free evaluation for matrix-vector products. For the linearized compressible Euler equations, we present various comparative studies within the QUADFLOW package concerning preconditioning techniques, ordering methods, time stepping strategies, and different implementations of the matrix-vector product. The main goal is to improve efficiency and robustness of the iterative method used in the flow solver.

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“…For the equations, we offer a choice of twodimensional domains on which the problem can be posed, along with boundary conditions and other aspects of the problem, and a choice of finite element discretizations on a quadrilateral element mesh, whereas the discrete NavierStokes equations require a method such as the generalized minimum residual method (GMRES), which is designed for non symmetric systems [9,19]. The key for fast solution lies in the choice of effective preconditioning strategies.…”

Section: Introductionmentioning

confidence: 99%

Resolution of Unsteady Navier-stokes Equations with the C a,b Boundary condition

El-Mekkaoui¹,

Elkhalfi²,

Elakkad³

2013

IJACSA

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Abstract-in this work, we introduce the unsteady incompressible Navier-Stokes equations with a new boundary condition, generalizes the Dirichlet and the Neumann conditions. Then we derive an adequate variational formulation of timedependent Navier-Stokes equations. We then prove the existence theorem and a uniqueness result. A Mixed finite-element discretization is used to generate the nonlinear system corresponding to the Navier-Stokes equations. The solution of the linearized system is carried out using the GMRES method. In order to evaluate the performance of the method, the numerical results are compared with others coming from commercial code like Adina system.

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Effect of discretization order on preconditioning and convergence of a high-order unstructured Newton-GMRES solver for the Euler equations

Cited by 38 publications

References 43 publications

a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

Iterative Solvers for Discretized Stationary Euler Equations

Resolution of Unsteady Navier-stokes Equations with the C a,b Boundary condition

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