The Elastoplast Discontinuous Galerkin (EDG) method for the Navier–Stokes equations

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Abstract

The present work details the Elastoplast (this name is a translation from the French “sparadrap”, a concept first applied by Yves Morchoisne for Spectral methods [1]) Discontinuous Galerkin (EDG) method to solve the compressible Navier–Stokes equations. This method was first presented in 2009 at the ICOSAHOM congress with some Cartesian grid applications. We focus here on unstructured grid applications for which the EDG method seems very attractive. As in the Recovery method presented by van Leer and Nomura in 2005 for diffusion, jumps across element boundaries are locally eliminated by recovering the solution on an overlapping cell. In the case of Recovery, this cell is the union of the two neighboring cells and the Galerkin basis is twice as large as the basis used for one element. In our proposed method the solution is rebuilt through an L2 projection of the discontinuous interface solution on a small rectangular overlapping interface element, named Elastoplast, with an orthogonal basis of the same order as the one in the neighboring cells. Comparisons on 1D and 2D scalar diffusion problems in terms of accuracy and stability with other viscous DG schemes are first given. Then, 2D results on acoustic problems, vortex problems and boundary layer problems both on Cartesian or unstructured triangular grids illustrate stability, precision and versatility of this method.

Introduction

Discontinuous Galerkin methods have become the subject of considerable research over the last decade due to their ability to give high order solutions in complex applications. Albeit well suited to the discretization of first order hyperbolic problems such as wave propagation phenomena, their extension to elliptic problems such as diffusion, is far less natural and still an up-to-date subject.

We can classify these extensions into two categories. In the first one, the scheme is devised through a mixed formulation by introducing an equation for the gradient that takes into account the jump of the solution at interfaces. The scheme needs to be stabilized by either interior penalty terms or numerical viscosity terms with parameters to be adjusted. Depending on the formulation, the resulting scheme is either compact or non compact.

Among the main contributors to this first category, we can cite Bassi and Rebay with their BR1 and BR2 methods for the compressible Navier–Stokes equations [2], [3], Cockburn and Shu with the LDG method [4], Peraire and Person with the CDG method [5], Douglas and Dupont [6], Brezzi et al. [7] with the symmetric interior penalty (IP) method. In [8], Gassner et al. show the link between their diffusive generalized Riemann solver and the IP approach.

In the same spirit, Luo et al. [9] use a BGK-based DG method to compute together convective and dissipative fluxes at the interface for the Navier–Stokes equations, using a gas-kinetic function. In this category, one should also refer to the relaxation system model introduced by Nishikawa [10].

A second category is based on local reconstruction or recovery of the solution to smooth the discontinuities. van Leer et al. [11], [12], [11], [12] was the first to propose the so-called Recovery method where the viscous fluxes at element boundaries are computed by merging the adjacent elements and defining on this new element a locally smooth P2k+1 recovered solution that is in the weak sense indistinguishable from the piecewise discontinuous Pk solution. This method eliminates the artificial introduction of penalty terms and the tuning of parameters. An impediment is the construction of the local merging basis and the need to solve a linear problem at each interface which can be awkward if we use an adaptive strategy on unstructured grids. Along the same lines is the PNPM schemes of Dumbser [13], where the DG fluxes are computed through a reconstruction of higher order than the local DG polynomials together with a time-accurate one-step time discretization.

In this paper, we develop a new DG method for the compressible Navier–Stokes equations where jumps across element boundaries are eliminated in the computation of the viscous fluxes using an L2 projection of the piecewise Pk discontinuous solution on the Pk basis of overlapping rectangular elements: so, we propose to label this method the Elastoplast DG method (EDG) method. This method, previously presented in [14], is a sequel to the shift cell technique that uses the Green formula [15] that reconstructs the gradient by projection on the shift cell basis. The main motivation for developing the Elastoplast method, which is closely related to the Modified Recovery method introduced and studied in [16], is to devise a simpler numerical procedure easily implemented on unstructured grids. Another similar method is the RDG method proposed by Luo et al. [17] that was developed independently. RDG and EDG were devised for an easier and more flexible implementation of van Leer’s Recovery method on unstructured grids. Both methods generate a locally smooth solution of the same order. In RDG, this is done through reconstruction over the union of the two adjacent cells, in EDG, this is done through a L2 projection onto a Cartesian DG basis attached to an overlapping rectangular cell. RDG leads to the inversion of a symmetric system while EDG explicitly computes the coefficients with Gauss integration. For smooth solutions, RDG and EDG should provide similar solutions but EDG’s greater locality can be useful in the case of very stiff problems.

This paper is devoted to a complete presentation of the method and a detailed numerical evaluation of its performances for DG polynomials of order 2, and especially for unstructured triangular grids. A comparison between order 1 and 2 is made in Section 4.0.4

Section snippets

Governing equations

The governing equations to be solved are the 2D time-dependent dimensionless Navier–Stokes equations for a viscous compressible flow which express conservation of mass, momentum and energy,tW+·F(W)-·D(W,W)=0where W=(ρ,ρU,ρE) is the conservation variable vector with classical notations, F represents the inviscid fluxes:F=ρU,ρUU+pI,U(ρE+p)and D represents the diffusive and heat fluxes:D=(0,τ¯¯,τ¯¯·U+λT).Above, ρ is the density, U=(u,v) the velocity, τ¯¯ the shear stress tensor, p the

Numerical results for a scalar diffusion equation

For both Cartesian and unstructured computations, we used the same functional space Vh. The 1D cases are computed on 2D Cartesian grids with two cells in the degenerated direction and periodic boundary conditions.

The first two test cases concern the scalar heat equation and are chosen to assess the performance in terms of precision and stability of the present approach both compared with more classical DG methods (1D case) and in 2D, on triangular regular versus irregular grids. As a model

Full Navier–Stokes with source term convergence study

The experimental order of convergence of the Navier–Stokes with EDG is investigated in the case of an inhomogeneous problem (Eq. (27)) as in Gassner et al. [8].tW+·F(W)-·D(W,W)=SS is such thatW=sin(k(x+y)-ωt)+csin(k(x+y)-ωt)+csin(k(x+y)-ωt)+c(sin(k(x+y)-ωt)+c)2is an exact solution to (Eq. (27)). Exact boundary conditions are applied by imposing the exact solution at the ghost cell Gauss quadrature points. For the computation, the coefficients wereμ=0.01γ=1.4Pr=0.72k=2.0ω=0.5c=4.0The

Conclusion

The EDG scheme for an accurate discretization of the viscous fluxes in the Navier–Stokes equations has been presented for a general DG formulation and numerically evaluated in a particular case both on Cartesian and unstructured triangular grids. Discontinuities are removed at each interface between elements by an L2 projection on a staggered rectangular element whatever the shape elements is, called Elastoplast.

The main advantage of the proposed scheme is its simplicity to implement either on

Acknowledgements

The authors would like to thank Dr. Germain Billet for many fruitful discussions and Marie-Claire LePape who authorized us to reproduce her results.

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