855 research outputs found
An optimal order interior penalty discontinuous Galerkin discretization of the compressible Navier-Stokes equations
In this article we propose a new symmetric version of the
interior penalty discontinuous Galerkin finite element
method for the numerical approximation
of the compressible Navier-Stokes equations. Here, particular
emphasis is devoted to the construction of an optimal numerical
method for the evaluation of certain target functionals of practical
interest, such as the lift and drag coefficients of a body immersed in a
viscous fluid. With this in mind, the key ingredients in the
construction of the method include: (i) An adjoint consistent imposition of
the boundary conditions; (ii) An adjoint consistent reformulation of the
underlying target functional of practical interest; (iii) Design of appropriate
interior--penalty stabilization terms. Numerical experiments presented
within this article clearly indicate the optimality of the proposed
method when the error is measured in terms of both the L2-norm, as well as
for certain target functionals. Computational comparisons with
other discontinuous Galerkin schemes proposed in the literature,
including the second scheme
of Bassi and Rebay, the standard SIPG method
outlined in [Hartmann,Houston-2006], and an NIPG variant of the new scheme
will be undertaken
A matrix-free high-order discontinuous Galerkin compressible Navier-Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows
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, , in order to mimic incompressible flows.
This strategy is widely used for high-order discontinuous Galerkin
discretizations of the compressible Navier-Stokes equations. The present work
raises the question regarding the computational efficiency of compressible DG
solvers as compared to a genuinely incompressible formulation. Our
contributions to the state-of-the-art are twofold: Firstly, we present a
high-performance discontinuous Galerkin solver for the compressible
Navier-Stokes equations based on a highly efficient matrix-free implementation
that targets modern cache-based multicore architectures. 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 discontinuous Galerkin
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
Error estimation and adjoint-based adaptation in aerodynamics
In this article we give an overview of recent developments
in error estimation and in residual-based and goal-oriented
(adjoint-based) adaptation for Discontinuous Galerkin discretizations
of sub- and supersonic viscous compressible flows. We also give an
outlook on the planned continuation of this research in the EU project
ADIGMA
An Optimal Order Interior Penalty Discontinuous Galerkin Discretization of the Compressible Navier-Stokes Equations
In this article we propose a new symmetric version of the interior penalty discontinuous Galerkin finite element method for the numerical approximation of the compressible Navier-Stokes equations. Here, particular emphasis is devoted to the construction of an optimal numerical method for the evaluation of certain target functionals of practical interest, such as the lift and drag coefficients of a body immersed in a viscous fluid. With this in mind, the key ingredients in the construction of the method include: (i) An adjoint consistent imposition of the boundary conditions; (ii) An adjoint consistent reformulation of the underlying target functional of practical interest; (iii) Design of appropriate interior-penalty stabilization terms. Numerical experiments presented within this article clearly indicate the optimality of the proposed method when the error is measured in terms of both the L_2-norm, as well as for certain target functionals. Computational comparisons with other discontinuous Galerkin schemes proposed in the literature, including the second scheme of Bassi & Rebay, cf. [11], the standard SIPG method outlined in [25], and an NIPG variant of the new scheme will be undertaken
A high-order semi-explicit discontinuous Galerkin solver for 3D incompressible flow with application to DNS and LES of turbulent channel flow
We present an efficient discontinuous Galerkin scheme for simulation of the
incompressible Navier-Stokes equations including laminar and turbulent flow. We
consider a semi-explicit high-order velocity-correction method for time
integration as well as nodal equal-order discretizations for velocity and
pressure. The non-linear convective term is treated explicitly while a linear
system is solved for the pressure Poisson equation and the viscous term. The
key feature of our solver is a consistent penalty term reducing the local
divergence error in order to overcome recently reported instabilities in
spatially under-resolved high-Reynolds-number flows as well as small time
steps. This penalty method is similar to the grad-div stabilization widely used
in continuous finite elements. We further review and compare our method to
several other techniques recently proposed in literature to stabilize the
method for such flow configurations. The solver is specifically designed for
large-scale computations through matrix-free linear solvers including efficient
preconditioning strategies and tensor-product elements, which have allowed us
to scale this code up to 34.4 billion degrees of freedom and 147,456 CPU cores.
We validate our code and demonstrate optimal convergence rates with laminar
flows present in a vortex problem and flow past a cylinder and show
applicability of our solver to direct numerical simulation as well as implicit
large-eddy simulation of turbulent channel flow at as well as
.Comment: 28 pages, in preparation for submission to Journal of Computational
Physic
Robust and efficient discontinuous Galerkin methods for under-resolved turbulent incompressible flows
We present a robust and accurate discretization approach for incompressible
turbulent flows based on high-order discontinuous Galerkin methods. The DG
discretization of the incompressible Navier-Stokes equations uses the local
Lax-Friedrichs flux for the convective term, the symmetric interior penalty
method for the viscous term, and central fluxes for the velocity-pressure
coupling terms. Stability of the discretization approach for under-resolved,
turbulent flow problems is realized by a purely numerical stabilization
approach. Consistent penalty terms that enforce the incompressibility
constraint as well as inter-element continuity of the velocity field in a weak
sense render the numerical method a robust discretization scheme in the
under-resolved regime. The penalty parameters are derived by means of
dimensional analysis using penalty factors of order 1. Applying these penalty
terms in a postprocessing step leads to an efficient solution algorithm for
turbulent flows. The proposed approach is applicable independently of the
solution strategy used to solve the incompressible Navier-Stokes equations,
i.e., it can be used for both projection-type solution methods as well as
monolithic solution approaches. Since our approach is based on consistent
penalty terms, it is by definition generic and provides optimal rates of
convergence when applied to laminar flow problems. Robustness and accuracy are
verified for the Orr-Sommerfeld stability problem, the Taylor-Green vortex
problem, and turbulent channel flow. Moreover, the accuracy of high-order
discretizations as compared to low-order discretizations is investigated for
these flow problems. A comparison to state-of-the-art computational approaches
for large-eddy simulation indicates that the proposed methods are highly
attractive components for turbulent flow solvers
Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations I: Method Formulation
In this article we consider the development of discontinuous Galerkin finite element methods for the numerical approximation of the compressible Navier-Stokes equations. For the discretization of the leading order terms, we propose employing the generalization of the symmetric version of the interior penalty method, originally developed for the numerical approximation of linear self-adjoint second-order elliptic partial differential equations. In order to solve the resulting system of nonlinear equations, we exploit a (damped) Newton-GMRES algorithm. Numerical experiments demonstrating the practical performance of the proposed discontinuous Galerkin method with higher-order polynomials are presented
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