Published by De Gruyter December 6, 2017

Some Remarks About Conservation for Residual Distribution Schemes

Rémi Abgrall

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2017-0056

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Abstract

We are interested in the discretisation of the steady version of hyperbolic problems. We first show that all the known schemes (up to our knowledge) can be rephrased in a common framework. Using this framework, we then show they flux formulation, with an explicit construction of the flux, and thus are locally conservative. This is well known for the finite volume schemes or the discontinuous Galerkin ones, much less known for the continuous finite element methods. We also show that Tadmor’s entropy stability formulation can naturally be rephrased in this framework as an additional conservation relation discretisation, and using this, we show some connections with the recent papers [13, 20, 18, 19]. This contribution is an enhanced version of [4].

Keywords: Hyperbolic Problems; Finite Volume Methods; Finite Element Methods; Flux Formulation

MSC 2010: 65M08; 76M10

Funding statement: The author has been funded in part by the SNSF project 200021_153604 “High fidelity simulation for compressible materials”.

A A DG RDS Scheme

Let us consider problem (1.1) defined on Ω⊂ℝ2. In this case, the approximation can be discontinuous across edges: uh∈𝒱h.

In a first step, we consider a conformal triangulation of Ω using triangles. This is not essential but simplifies a bit the text. The 3D case can be dealt with in a similar way.

In K, we say that the degrees of freedom are located at the vertices, and we represent the approximated solution in K by the degree one interpolant polynomial at the vertices of K. Let us denote by uh this piecewise linear approximation, that is in principle discontinuous at across edges. In the following, we use the notations described in Figure 5.

Figure 5

Geometrical elements for defining the scheme.

In [10], the degrees of freedom are located at the midpoint of the edges that connect the centroid of K and its vertices. This choice was motivated by the fact that the ℙ1 basis functions associated to these nodes are orthogonal in L2⁢(K). This property enables us to reinterpret the DG schemes as RD schemes, and hence to adapt the stabilization techniques of RD to DG. In particular, we are able to enforce an L∞ stability property. However, this method was a bit complex, and it is not straightforward to generalize it to more general elements than triangles.

The geometrical idea behind the version that we describe now is to forget the RD interpretation of the DG scheme and to let the geometrical localization of the degrees of freedom move to the vertices of the element.

With this in mind, we define two types of total residuals:

A total residual per element K, i.e.
ΦK⁢(uh)=∮∂⁡K𝐟⁢(uh)⋅𝐧⁢𝑑γ.
A total residual per edge Γ, i.e.
ΦΓ⁢(uh)=∮Γ[𝐟⁢(u)⋅𝐧]⁢𝑑γ,
where [𝐟⁢(u)⋅𝐧] represents the jump of the function 𝐟⁢(u)⋅𝐧 across Γ. Here, if 𝐧 is the outward unit normal to K (see Figure 5), which enables us to define a right side and a left side. Hence we set
[𝐟⁢(u)⋅𝐧]=(𝐟⁢(uR)-𝐟⁢(uL))⋅𝐧.
We notice that ΦΓ only depends on the values of u on each side of Γ.

The idea is to split the total residuals into sub-residuals so that a monotonicity preserving scheme can be defined. Here, we choose the Rusanov scheme, but other choices could be possible. Thus we consider:

For the element K and any vertex σ∈K,
ΦσK=ΦK3+αK⁢(uσ-u¯)
with
u¯=13⁢∑σ′∈Kuσ′,
and αK≥max𝐱∈K⁡∥∇⁡𝐟⁢(uh⁢(𝐱))∥ where ∥⋅∥ is any norm in ℝ2, for example the Euclidean norm.
For the edge Γ, any σ∈Γ,
ΦσΓ⁢(uh)=ΦΓ⁢(uh)4+αΓ⁢(uσ-u¯)
with
u¯=14⁢∑σ′∈K+∪K-uσ′,
where and αΓ≥maxK=K+,K-⁡max𝐱∈∂⁡K∩Γ⁡∥∇⁡𝐟⁢(uh⁢(𝐱))∥, see Figure 5 for a definition of K±.

We have the following conservation relations:

∑σ∈KΦσK⁢(uh)=ΦK⁢(uh),∑σ∈ΓΦσΓ⁢(uh)=ΦΓ⁢(uh).

The choice αK≥max𝐱∈K⁡∥∇⁡𝐟⁢(uh⁢(𝐱))∥ and αΓ≥maxK=K+,K-⁡max𝐱∈∂⁡K∪Γ⁡∥∇u⁡𝐟⁢(uh⁢(𝐱))∥ are justified by the following standard argument. If we set Q=K or Γ, we can rewrite the two residuals as

ΦσQ⁢(uh)=∑σ′∈Qcσ⁢σ′Q⁢(uσ-uσ′)

with cσ⁢σ′Q≥0 under the above mentioned conditions. Indeed, using

uh-uσ=∑σ′∈K(uσ-uσ)⁢φσ′,

we get (for Q=K for example)

ΦσK⁢(uh)=ΦK⁢(uh)3+αK⁢(uσ-u¯)=13⁢∮∂⁡K(𝐟⁢(uh)-𝐟⁢(uσ))⋅𝐧⁢𝑑γ+αK⁢(uσ-u¯)=∑σ′∈K13⁢[∮∂⁡K(∫01∇⁡𝐟⁢(s⁢uh+(1-s)⁢uσ)⁢φσ′⁢(𝐱)⁢𝑑s)⋅𝐧⁢𝑑γ-αK]⁢(uσ-uσ′),

which proves the result.

Using standard arguments, as defining uh as the limit of the solution of

(A.1)uσn+1=uσn-ωσ⁢(∑K,σ∈KΦσK⁢(uh,n)+∑Γ,σ∈ΓΦσΓ⁢(uh,n))

with

ωσ⁢(∑K,σ∈Kcσ⁢σ′K+∑Γ,σ′∈Γcσ⁢σ′Γ)≤1,

we see that we have a maximum principle.

It is possible to construct a scheme that is formally second-order accurate by setting

ΦσK,⋆⁢(uh)=βσK⁢ΦK⁢(uh) and ΦσΓ,⋆⁢(uh)=βσΓ⁢ΦK⁢(uh)

with

xσK=ΦσK⁢(uh)ΦK⁢(uh),xσΓ=ΦσΓ⁢(uh)ΦΓ⁢(uh),

and

βσK=max⁡(xσK,0)∑σ′∈Kmax⁡(xσ′K,0),βσΓ=max⁡(xσΓ,0)∑σ′∈Kmax⁡(xσ′Γ,0).

As in the “classical” RD framework, the coefficients β are well defined thanks to the conservation relations (3.2). The scheme is written as (A.2) where the residuals ΦσK⁢(uh) (respectively, ΦσΓ⁢(uh)) are replaced by ΦσK,⋆⁢(uh) (respectively, ΦσΓ,⋆⁢(uh).

The solution uh is defined as follows: find uh linear in each triangle K such that for any degree of freedom σ (i.e. vertex of the triangulation),

(A.2)∑K,σ∈KΦσK,⋆⁢(uh)+∑Γ,σ∈ΓΦσΓ,⋆⁢(uh)=0.

We have a first order approximation just by replacing the “starred” residuals by the first order ones. System (A.2) is solved by an iterative method such as (A.1).

Acknowledgements

I would also like to thank Anne Burbeau (CEA-DEN) for her critical reading of the first draft of this paper. Her input has helped to improve the readability of this paper. The two referees and the editor are also warmly thanked for their patience, their comments and ability to trace typos. The remaining mistakes are mine.

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Received: 2017-08-10

Revised: 2017-11-11

Accepted: 2017-11-21

Published Online: 2017-12-06

Published in Print: 2018-07-01