Abstract The Flux Reconstruction (FR) approach to high-order methods is a flexible androbust framework that has proven to be a promising alternative to the traditional Discontinu-ous Galerkin (DG) schemes on parallel architectures like Graphical Processing Units (GPUs)since it pairs exceptionally well with explicit time-stepping methods. The FR formulationwas originally proposed by Huynh (AIAA Pap 2007-4079:1–42, 2007). Vincent et al. (J SciComput 47(1):50–72, 2011) later developed a single parameter family of correction func-tions which provide energy stable schemes under this formulation in 1D. These schemes,known as Vincent–Castonguay–Jameson–Huynh (VCJH) schemes, offer control over prop-erties like stability, dispersion and dissipation through the variation of the VCJH parameter.Classical schemes like nodal-DG and Spectral Difference (SD) can also be recovered underthis formulation. Following the development of the FR approach in 1D, Castonguay et al. (JSci Comput 51(1):224–256, 2012) andWilliams et al. (J Comput Phys 250:53–76, 2013) andWilliams and Jameson (J Sci Comput 59(3):721–759, 2014) developed correction functionsthat give rise to energy stable FR formulations for triangles and tetrahedra. For the caseof tensor product elements like quadrilaterals and hexahedra however, a simple extensionof the 1D FR approach utilizing the 1D VCJH correction functions was possible and hasbeen adopted by several authors (Castonguay in High-order energy stable flux reconstructionschemes for fluid flow simulations on unstructured grids, 2012; Witherden et al. in ComputFluids 120:173–186, 2015; Comput Phys Commun 185(11):3028–3040, 2014). But whethersuch an extension of the 1D approach to tensor product elements is stable remained an openquestion. A direct extension of the 1D stability analysis fails due to certain key difficultieswhich necessitate the formulation of a norm different from the one utilized for stability anal-ysis in 1D and on simplex elements. We have recently overcome these issues and shownthat the VCJH schemes are stable for the linear advection equation on Cartesian meshes for
any non-negative value of the VCJH parameter (Sheshadri and Jameson in J Sci Comput67(2):769–790, 2016; J Sci Comput 67(2):791–794, 2016). In this paper, we have extendedthe stability analysis to the advection–diffusion equation, demonstrating that the tensor prod-uct FR formulation is stable on Cartesian meshes for the advection–diffusion case as well.The analysis in this paper also provides additional insights into the dependence on the VCJHparameter of the diffusion and stability characteristics of these schemes. Several numericalexperiments that support the theoretical results are included.
High-orderDiscontinuousGalerkin-typemethods in theory promise increased computationalefficiency and flexibility over their low-order counterparts. The DG method was first intro-duced in the context of the neutron transport equation byReed andHill [1]. Several variants ofthe DGmethod have been proposed thereafter in the context of Computational Fluid Dynam-ics (CFD). The Local Discontinuous Galerkin (LDG) method was proposed by Cockburnand Shu [2] for second-order PDEs where the second-order system is broken down into twofirst-order systems and the additional variable introduced is eliminated through judiciouschoices for the interface fluxes. Compact versions of this method for 2D and 3D knownas Compact Discontinuous Galerkin (CDG) methods were then developed by Peraire andPersson [3]. Other notable schemes in this context include the Interior Penalty method [4]and the Bassi-Rebay-2 method [5]. In order to alleviate the cost of large linear solves nec-essary for implicit implementations, the Hybrid Discontinuous Galerkin (HDG) methodswere proposed by Cockburn et al. [6]. These methods reduce the number of global degreesof freedom to single counts on each element interface. All DG methods share the commonfeature with Continuous Galerkin methods of solving the variational or the weak form ofthe PDE. However, the DG formulation is more suitable for convection dominated problemsthat are commonly encountered in CFD. These DG methods are well studied and detailedaccounts of them can be found in the textbooks by Cockburn et al. [7] and Hesthaven andWarburton [8].
Another notable class of high-order unstructured methods is the Spectral Differencemethod originally proposed by Kopriva and Kolias [9] and later generalized to triangularelements by Liu et al. [10]. These methods are similar to nodal DG methods but solve thedifferential or the strong form of the PDE.
The FR formulation provides a unifying framework for high-order discontinuous FiniteElement Methods (FEMs) for utilization with explicit time-stepping schemes and are verywell suited for highly parallel architectures like GPUs [11]. Originally formulated by Huynh,Vincent et al. [12] later proposed a family of correction functions that give rise to stablenumerical schemes in 1D by following the stability analysis of SDmethods by Jameson [13].Classic schemes like DG and SD can be recovered as special cases under this family ofschemes, at least for linear problems. The connections between FR and DG methods havebeen examined in detail by Allaneau and Jameson [14], De Grazia et al. [15] and Zwanenburget al. [16].More recently, the connections between the twomethods for the cases of curvilinearand irregular meshes were established by Mengaldo et al. [17,18].
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A closely related scheme called the Lifting Collocation Penalty (LCP) method wasdeveloped by Wang and Gao [19]. These methods are also sometimes referred to as FluxReconstruction, while some authors refer to them collectively as Correction Procedure viaReconstruction. In this article, we only refer to the FR approach that uses theVCJH correctionfunctions, also known as VCJH schemes in 1D.
For the VCJH schemes, Jameson et al. [20] extended the 1D stability analysis to nonlinearproblems and discussed several strategies to minimize aliasing driven instabilities. Morerecently, Mengaldo et al. [21] further studied the aliasing properties and proposed de-aliasingtechniques suitable for this approach. The FR approach has also been extended to triangularelements by Castonguay et al. [22] who proposed a stable family of correction functionsfor triangles in the context of the linear advection equation. Williams et al. made furtherextensions to tetrahedral elements along with proofs of stability on triangles and tetrahedrafor linear advection as well as advection–diffusion equations [23,24].
Parallel to these developments, simple tensor-product like extensions of the 1D FRapproach utilizing the 1D VCJH correction functions were developed for quadrilateral andhexahedral elements [25]. But the question of whether such an extension is stable remainedopen until recently, due to a few major difficulties which required the formulation of a normdifferent from the one utilized in 1D or for the simplex cases.We recently overcame these bar-riers for the case of the linear advection problem [26,27] and demonstrated that the schemesare stable on Cartesian meshes for the linear advection equation whenever the VCJH param-eter is non-negative. In this paper, we extend the results to the advection–diffusion equationand show that the VCJH schemes are stable on Cartesian meshes for any non-negative valueof the VCJH parameter while using Lax Friedrichs (LF) for the common numerical advectiveflux at the interfaces, and Local Discontinuous Galerkin (LDG) formulation for the diffusiveparts. These choices for the interface flux calculation have a broad applicability and recoverseveral other popular choices as special cases. We use the results for the advection part of theadvection–diffusion equation from [27] and show that the contributions from the diffusiveflux are stable.
We also include several numerical experiments which support the intuition provided bythe results of the stability analysis. We start in Sect. 2 by providing a brief description of theFR approach for the linear advection–diffusion equation on quadrilateral elements as wellas the 1D VCJH correction functions. In Sect. 3, we setup and prove the stability of theseschemes while restricting ourselves to Cartesian meshes. In Sect. 4, we show the numericalresults supporting our observations.
2 Flux Reconstruction Methodology
2.1 Preliminaries
Consider the 2D conservation law
∂u
∂t+ ∇ · f (u,∇u) = 0 in �, (2.1)
where � is a bounded connected subset of R2 with boundary Γ composed of a finite unionof parts of hyperplanes. u is a conserved scalar quantity referred to as the solution and fis a vector representing the flux of the scalar quantity u. For the linear advection–diffusionequation, f is assumed to be a linear flux of the form
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f = au − b∇u with f =(
FG
), a =
(ax
ay
)and b > 0. (2.2)
Consider a partition TN of � into N non-empty, non-overlapping, conforming quadrilateral
elementsΩk with boundaries Γk such that Γk =4⋃
i=1F i
k whereFik are straight lines represent-
ing the faces (or edges) of the element Ωk . Furthermore, we restrict ourselves to non-mortarelements, i.e., if F i
k ∩ Γk′ �= ∅ for k′ �= k, then F ik ∩ Γ j = ∅, ∀ j �= k, k′ and F i
k ∩ Γ = ∅.Similar to the classical LDG method, the second-order PDE in (2.1) is solved by decom-
posing it into a system of two first-order PDEs as
∂u
∂t+ ∇ · f (u, q) = 0 (2.3)
q − ∇u = 0 (2.4)
where u and q (solution gradient) are the unknown variables with f = au − bq. The firstcomponent of the flux is the advective part of the flux, while the second represents thediffusive flux. i.e., fadv = au and fdi f = −bq.
To facilitate a uniform implementation of the method, each element Ωk can be mapped toa standard square reference domain defined by ΩS = {
(ξ, η)| − 1 ≤ ξ, η ≤ 1}as follows:
xk = �k(ξ, η) =4∑
i=1
Ni (ξ, η)vik (2.5)
Here xk represents the physical coordinates (x, y) of an arbitrary point in the element Ωk ,vik denote the physical coordinates of the 4 vertices of Ωk , and Ni (ξ, η) are bilinear shapefunctions defined on ΩS . Figure 1 shows an example of such a mapping. Further, let theJacobian matrix associated with the geometric transformation �k be denoted by Jk and itsdeterminant by Jk . In addition, the physical quantities u, q and f are also transformed to thereference domain, and the transformed quantities are denoted using a hat over their respectivesymbols in the physical domain. The transformation equations for these quantities as well asa few others which will assist in our stability proof are as follows:
(−1,− 1()1 ,−1)
(1,1)(−1,1)
v1k
v2k
v3k
v4k
x
y
ξ
η
Fig. 1 Mapping between the physical domain (on the left) and the reference element (on the right). This isshown for a general quadrilateral element for purpose of illustration of the machinery of the FR formulation.However, note that we restrict ourselves to Cartesian grids in this paper while analyzing stability
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uk = Jkuk (2.6)
qk = ∇u = JkJTk qk (2.7)
f k = JkJ−1k f k (2.8)
∇ · f k = Jk∇ · f k (2.9)
f k · qk = J 2k f k · qk (2.10)∫
ΩS
f k · qkdΩS = Jk
∫Ωk
f k · qkdΩk (2.11)
∫ΓS
uk( f k · n)dΓS =∫Γk
Jkuk( f k · n)dΓk (2.12)
Using these equations we can see that the conservation law can be written in the referencedomain as follows:
∂ u D
∂t+ ∇ · f = 0 (2.13)
qD − ∇u = 0 (2.14)
Unlike the case of linear simplex elements, Jk varies from point to point within an elementfor a general linear quadrilateral element. This varying geometric Jacobian makes the sta-bility analysis extremely tedious. Therefore, we restrict ourselves to Cartesian meshes whileanalyzing the stability of the tensor product formulation in this paper. The Jacobian matrixis a constant for each element in such a mesh and we have
∂xk
∂η= ∂yk
∂ξ= 0 (2.15)
where xk and yk are the horizontal and vertical coordinates of a point in the physical domaininside the kth element, i.e., Ωk . Additional notation can be introduced to exploit this andsimplify the algebra during the stability analysis. Let Jxk = ∂xk
∂ξand Jyk = ∂yk
∂η. We then have
Fk = Jyk Fk; Gk = Jxk Gk; uk = Jxk Jyk uk = Jkuk (2.16)
2.2 FR Procedure
This section provides the details of the FR approach used to solve the 2D linear advection–diffusion equation. A detailed description of the procedure in 1D can be found in the articleby Castonguay et al. [28]. As mentioned in the previous section, the advection–diffusionequation is first decomposed into a system of first-order PDEs which are then transformed toa standard reference domain within each element to get (2.13) and (2.14). For constructing ascheme of (p +1)th order accuracy, a set of (p +1)2 points referred to as solution points arechosen in the reference domain. There are several choices for the location of these solutionpoints in the reference domain. A possible choice is the tensor product of p + 1 1D Gauss–Legendre points.
The transformed solution within each element, i.e., uk , is then represented using a tensorproduct of degree-p 1D Lagrange polynomial bases defined on the set of 1D solution points,i.e.,
u D =p∑
i=0
p∑j=0
u Di j li (ξ)l j (η), (2.17)
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Fig. 2 Figure showing thesolution and flux points in thereference element for a p = 3scheme
where li (ξ) and l j (η) are the 1D Lagrange polynomials associated with the solution points ξi
and η j respectively, and u Di j is the value of the transformed solution at (ξi , η j ) in the reference
domain ΩS . Since u D is a transformed quantity, it is understood to be associated with acertain generic element Ωk . Therefore the subscript k is dropped in order to maintain brevityof notation. Also, similar to the DG method, the solution u is allowed to be discontinuousacross elements. Such discontinuous quantities are represented with a superscript D for thepurpose of clarity.
Next, a set of p + 1 points referred to as flux points are chosen along each boundary edgeof the quadrilateral element. These flux points are chosen to align with the solution points inthe reference domain, i.e., they are chosen to be the 1D Gauss–Legendre points along eachedge if such an option is used for the solution points. Figure 2 shows the solution and fluxpoint locations in the reference domain for a scheme of fourth order accuracy (p = 3).
At each timestep, startingwith the discontinuous solution in the reference domain (for eachelement), the gradient of the solution (∇u D) is computed by taking the gradient of (2.17).However, this gradient is of degree p−1. In order to obtain a gradient of degree p, this gradientis then corrected using the information from the neighboring elements. The first step in thisregard is to extrapolate the solution u D to the flux points on the edges. This extrapolation isperformed by evaluating the appropriate Lagrange polynomials at the element boundaries.For example, the interface solution at the left interface is obtained as
u DL (η) =
p∑i=0
p∑j=0
u Di j li (ξ = −1)l j (η), (2.18)
where u DL is a 1D polynomial (in η) representing the solution on the left boundary. A similar
extrapolation is performed in the neighboring elements. The solution obtained at the interfacesin the reference domain is then transformed to the physical domain in each element in orderto facilitate the computation of a common numerical solution at these interfaces. Finally,the common numerical solution at an interface is computed using an appropriate method.Common choices for this include Central Flux (CF), Local Discontinuous Galerkin (LDG),Compact Discontinuous Galerkin (CDG), Internal Penalty, Bassi Rebay 1 (BR1), and BassiRebay 2 (BR2) approaches. The LDG scheme in particular is identical to CDG in 1D and
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can recover CF and BR1 schemes in 1D, 2D and 3D, and we use the LDG scheme in thispaper for our analysis. In the LDG method the common solution u∗ at the j th flux point onan interface is obtained as
u∗f, j = {{u D
f, j }} − β · [[u Df, j ]], (2.19)
where
{{u}} = uin + uout
2, [[u]] = uinnin + uinnout ,
and β is a directional parameter that allows upwinding or downwinding. Here the subscript‘in’ refers to the element we are currently computing in and ‘out’ refers to the neighboringelement. n refers to the normal at the interface pointing outward from the (appropriate)element. Finally, u∗ is transformed back to the reference domain using (2.6) to obtain u∗.
The next step involves computing the correction to the gradient which is added to ∇u D
to obtain qD
qD = ∇u D + ∇uC (2.20)
This correction to the gradient denoted by ∇uC is given by
(∇uC)ξ
= dhL(ξ)
dξ
p∑j=0
(u∗ − u D)L j l j (η) + dh R(ξ)
dξ
p∑j=0
(u∗ − u D)R j l j (η) (2.21)
(∇uC)η
= dhL(η)
dη
p∑j=0
(u∗ − u D)B j l j (ξ) + dh R(η)
dη
p∑j=0
(u∗ − u D)Tj l j (ξ) (2.22)
where if one wants to recover the VCJH schemes, hL and h R denote the left and right 1DVCJH correction functions of degree p + 1 respectively. These correction functions areintroduced in detail in Sect. 2.3. L , R, B, T represent the left (ξ = −1), right (ξ = 1),bottom (η = −1) and top (η = 1) edges respectively. (.)L j denotes the value at the j th fluxpoint on the left boundary. Finally, l j denotes the j th member of the 1D Lagrange basis ofdegree p defined on the edge. Note that we have used hL and h R as the correction functionsfor η-component (2.22) as well because the correction along the η direction is performed inthe same 1D sense as that in the ξ direction.
It can be seen that the correction in (2.21) and (2.22) is equivalent to performing thecorrection along 1D lines in both the horizontal and vertical directions. Note that, unlike in1D, this correction is not directly applied to the solution u, i.e., u is not made continuousacross element interfaces. Instead, only the normal component of the corrected gradient qD
is rendered continuous at element interfaces. Therefore, correction is applied to the gradientdirectly without ever changing u D . Before proceeding further, it is worth mentioning that qD
is a pth degree polynomial and is represented in a similar fashion to u D , i.e.,
qD =p∑
i=0
p∑j=0
qDi j li (ξ)l j (η), (2.23)
This correction procedure is then repeated for the flux. First, the discontinuous flux fDis
computed at all the solution points using u D and qD at those points. This fDis a pth degree
polynomial and is represented using a Lagrangian basis similar to (2.23), i.e.,
fD =
p∑i=0
p∑j=0
fDi j li (ξ)l j (η), (2.24)
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This flux is discontinuous across element interfaces. In order to obtain a conservative scheme,it needs to be corrected such that it is continuous in the normal direction across elementinterfaces. To compute this common numerical flux which enforces normal continuity at theinterfaces, the corrected gradient is first evaluated at the element boundaries. This gradientis then transformed to the physical domain using (2.7). Using this and the solution at theinterfaces in the physical domain that was computed previously, the flux in the physicaldomain f (u D, qD) is calculated at all the flux points. Using these interface fluxes fromneighboring elements, we compute a common numerical flux by employing an appropriatemethod. In general, the advective and diffusive parts of the flux are handled separately in thisregard. For linear problems like the linear advection–diffusion equation, a common choicefor the advective flux is the local Lax Friedrichs (LF) formulation. The common numericalflux using the LF method is given by
f ∗adv = {{ f D
adv}} + λ
2
(max
u∈[u Din ,u D
out ]
∣∣∣∣∂ f adv
∂u· n
∣∣∣∣)
[[u D]] (2.25)
where {{.}} and [[.]] operators are similar to those of (2.19) and 0 ≤ λ ≤ 1. The LF formulationis capable of recovering both the upwind and central flux formulations as special cases andtherefore offers broad applicability. For the diffusive flux, any of the aforementionedmethodsused for computing a common solution can be utilized. If the LDG method is elected, thenthe common numerical flux is given by
f ∗di f = {{ f di f }} + τ [[u]] + β[[ f di f ]] (2.26)
where the jump operator for the flux is given by
[[ f ]] = f in · nin + f out · nout
and τ is a parameter that penalizes the jump in the solution. Note that the subscript ‘dif’is dropped above for brevity of notation. Finally f ∗ is transformed back to the referencedomain to obtain f
∗. As an example, this can be done on the left boundary using(
f∗ · n
)L j
= JL j
(f ∗ · n)
L j(2.27)
where JL j is the edge-Jacobian at the j th flux point on the left boundary. The edge-Jacobianis an edge-based scaling factor which is just equal to the edge length in the Cartesian case.Therefore (2.27) can be rewritten for the case of Cartesian meshes as(
f∗ · n
)L j
= Jy(f ∗ · n)
L j(2.28)
where Jy is the edge length of the left (and right) edge.
Now the discontinuous flux polynomial fD
has to be corrected such that it takes the
common numerical flux value f∗at the element interfaces. To achieve this, f
Dis evaluated
at the element interfaces and a correction to the flux denoted by fCis computed using
FC = −hL(ξ)
p∑j=0
(( f
∗ − fD) · n)
L jl j (η) + h R(ξ)
p∑j=0
(( f
∗ − fD) · n)
R jl j (η)
(2.29)
GC = −hL(η)
p∑j=0
(( f
∗ − fD) · n)
B jl j (ξ) + h R(η)
p∑j=0
(( f
∗ − fD) · n)
Tjl j (ξ)
(2.30)
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This correction is then added to the discontinuous component to obtain the total flux f ,i.e.,
f = fD + f
C(2.31)
which is continuous in the normal direction across elements. Finally, the transformed solutionat the next time step in the kth element is calculated using
∂ u Dk
∂t= −∇ · fk (2.32)
For brevity, we introduce some additional notation:
Δα j = (( f
∗ − fD) · n)
α jΔu
α j= (u∗ − u D)α j (2.33)
where α refers to one of the four interfaces L , R, B, T .
2.3 VCJH Correction Functions
The 1D VCJH correction functions are described in detail in the article by Vincent et al. [12]Here we provide a brief description of these functions along with a few important propertieswhich will be used in the stability analysis later. The 1D VCJH correction functions hL andh R can be written as follows:
hL = (−1)p
2
[L p − ηp L p−1 + L p+1
1 + ηp
]h R = 1
2
[L p + ηp L p−1 + L p+1
1 + ηp
](2.34)
whereηp = c
2(2p + 1)(ap p!)2 (2.35)
and ap is the leading coefficient of the pth Legendre polynomial L p defined on [−1, 1]. cis a free parameter referred to as the VCJH parameter. Several different schemes like theDG, Spectral Difference (SD) and Huynh’s g2 scheme [29] can be recovered by varying thisparameter. For example, setting c = 0 allows us to recover the classical nodal DG method(for linear fluxes). Table 1 shows the values of c which recover other notable schemes. c− isthe value above which the FR approach is guaranteed to be stable in 1D. cDG , cSD and cHU
are the values of c that recover the DG, SD and Huynh’s g2 method [29,30] respectively, andclarge is just a relatively large value to compare with and is chosen to be 10.
Figure 3 shows the left and right 1D correction functions for a p = 4 (5th order) schemecorresponding to c = 0, i.e., for the FR scheme that recovers the nodal DG method. Figure 4shows the correction functions for a p = 4 scheme for different values of c.
The VCJH functions have the following properties:
hL(ξ) = h R(−ξ) (2.36)
Table 1 Table provides thevalues of the VCJH parameter cfor recovering different existingschemes
Parameter type Value
c− −299225
cDG 0
cSD8
496125
cHU1
39690
c− is the smallest value of cwhich can provide a stablescheme
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Fig. 3 Figure shows onepossible choice for the left andright correction functionsemployed for a p = 4 FRscheme. These correctionfunctions are polynomials ofdegree p + 1, i.e., degree 5. Thecorrection functions shown herecorrespond to a VCJH parameterc = 0 which recovers the nodalDG method for linear problems
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
c−/2 cDG cSD cHU clarge
Fig. 4 Figure shows the left correction function hL (ξ) for various values of c
hL(−1) = 1 hL(1) = 0 (2.37)1∫
−1
dlidξ
hL dξ = cd plidξ p
d p+1hL
dξ p+1 (2.38)
The first property states that the correction process is symmetric. The second property ensuresthe left correction functions alter only the left interface value. The third property ensures thatthe FR approach using VCJH correction functions gives rise to a stable family of schemes in1D.
3 Stability Analysis of the FR Approach on Quadrilateral Meshes
In this section, we prove that the FR approach for the linear advection–diffusion equationon Cartesian meshes is stable when the VCJH parameter c ≥ 0. We prove this for popularchoices for the common numerical solution and fluxes at the interfaces. It appears that the
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proof can be extended to other similar choices, but which are out of the scope of this work.We begin by providing an outline of the proof.
3.1 Outline of the Analysis
– The overall approach is to study the time evolution of a suitable Sobolev norm of thesolution and determine the various factors that may cause the scheme to become unstable.
– As a first step, expressions for the different pieces that form the modified Sobolev normof the gradient of the solution qD are obtained by manipulating the second PDE (2.14)or its expanded version (2.20). Lemmas 3.1–3.4 contain the expressions for the L2 normof either qD or one of its various derivatives which appear in the Sobolev norm.
– Lemmas 3.1–3.4 are then combined in an appropriate manner to obtain an expression forthe Sobolev norm of the gradient of the solution qD in 3.5.
– Similar manipulations need to be performed for the first PDE (2.13) to obtain the expres-sion for the Sobolev norm of the solution u D . However, this was already completed forthe linear advection case in our recent work [26,27]. Although this only dealt with theadvective flux, themanipulations used to obtain the expression of the Sobolev norm of u D
do not use any specific property of the advective flux and are therefore applicable to theadvective–diffusive flux as well. The results of that paper and its Erratum are borrowed,and the diffusive terms are separated out to obtain Lemma 3.6.
– Finally, the above Lemmas are combined to obtain an expression for the Sobolev normof the solution. The various terms that affect the evolution of this norm of the solutionare analyzed, and it is shown that they all give rise to stable contributions (under certaincommon conditions) whenever the VCJH parameter c is non-negative.
3.2 Proof of Stability
Lemma 3.1 For the tensor product FR formulation utilizing the 1D VCJH correction func-tions described in Sect. 2, the following holds.
Jk
∫Ωk
f Ddi f,k · qD
k dΩk =∫
ΩS
fDdi f · ∇u DdΩS +
∫ΓS
Δu( fDdi f · n)dΓS
−∫
ΩS
∂ F Ddi f
∂ξhL(ξ)
p∑j=0
ΔuL j
l j (η)dΩS
︸ ︷︷ ︸A1
−∫
ΩS
∂ F Ddi f
∂ξh R(ξ)
p∑j=0
ΔuR j
l j (η)dΩS
︸ ︷︷ ︸A2
−∫
ΩS
∂G Ddi f
∂ηhL(η)
p∑j=0
ΔuB j
l j (ξ)dΩS
︸ ︷︷ ︸A3
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−∫
ΩS
∂G Ddi f
∂ηh R(η)
p∑j=0
ΔuTj
l j (ξ)dΩS
︸ ︷︷ ︸A4
(3.1)
Proof Substituting (2.20) into (2.11), we obtain
Jk
∫Ωk
f Ddi f,k · qD
k dΩk =∫
ΩS
fDdi f · ∇u DdΩS +
∫ΩS
fDdi f · ∇uC dΩS (3.2)
Consider the second term on the RHS of (3.2). Using (2.21) and (2.22), we get∫
ΩS
fDdi f · ∇uC dΩS =
∫ΩS
F Ddi f
dhL(ξ)
dξ
p∑j=0
ΔuL j
l j (η)dΩS
+∫
ΩS
F Ddi f
dh R(ξ)
dξ
p∑j=0
ΔuR j
l j (η)dΩS
+∫
ΩS
G Ddi f
dhL(η)
dη
p∑j=0
ΔuB j
l j (ξ)dΩS
+∫
ΩS
G Ddi f
dh R(η)
dη
p∑j=0
ΔuTj
l j (ξ)dΩS (3.3)
Now using integration by parts for the first term on the RHS of (3.3) and utilizing (2.37)gives
1∫−1
1∫−1
F Ddi f
dhL(ξ)
dξ
p∑j=0
ΔuL j
l j (η)dξdη =1∫
−1
[F D
di f hL(ξ)]ξ=1ξ=−1
p∑j=0
ΔuL j
l j (η)dη
−∫
ΩS
∂ F Ddi f
∂ξhL(ξ)
p∑j=0
ΔuL j
l j (η)dΩS
= −1∫
−1
F Ddi fL
p∑j=0
ΔuL j
l j (η)dη
−∫
ΩS
∂ F Ddi f
∂ξhL(ξ)
p∑j=0
ΔuL j
l j (η)dΩS
=∫
ΓS,L
Δu( fDdi f · n)dΓS
−∫
ΩS
∂ F Ddi f
∂ξhL(ξ)
p∑j=0
ΔuL j
l j (η)dΩS (3.4)
Similar manipulations can be applied to the other three terms on the RHS of (3.3). Com-bining these results gives us Lemma 3.1. ��
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J Sci Comput (2018) 74:1757–1785 1769
Lemma 3.2 For the tensor product FR formulation utilizing the 1D VCJH correction func-tions described in Sect. 2, the following holds.
Jk
∫Ωk
∂ p f Ddi f,k
∂ξ p· ∂ pqD
k
∂ξ pdΩS =
∫ΩS
∂ p fDdi f
∂ξ p· ∂ p(∇u D)
∂ξ pdΩS
+ 2
c
∫ΩS
hL(ξ)∂ F D
di f
∂ξ
p∑j=0
ΔuL j
l j (η)dΩS
︸ ︷︷ ︸A1
+ 2
c
∫ΩS
h R(ξ)∂ F D
di f
∂ξ
p∑j=0
ΔuR j
l j (η)dΩS
︸ ︷︷ ︸A2
− 2∂ pG D
di fB
∂ξ p
p∑j=0
ΔuB j
∂ pl j (ξ)
∂ξ p
− 2
1∫−1
∂
∂η
(∂ pG D
di f
∂ξ p
)hL(η)
p∑j=0
ΔuB j
∂ pl j (ξ)
∂ξ pdη
︸ ︷︷ ︸B1
+ 2∂ pG D
di fT
∂ξ p
p∑j=0
ΔuTj
∂ pl j (ξ)
∂ξ p
−2
1∫−1
∂
∂η
(∂ pG D
di f
∂ξ p
)h R(η)
p∑j=0
ΔuTj
∂ pl j (ξ)
∂ξ pdη
︸ ︷︷ ︸B2
(3.5)
Proof We begin by taking the pth derivative w.r.t. ξ of f Ddi f,k and qD
k , and then take theirdot product. Since the Jacobian matrix Jk is a constant in the element Ωk , similar to (2.10),we get
J 2k
∂ p f Ddi f,k
∂ξ p· ∂ pqD
k
∂ξ p= ∂ p f
Dk
∂ξ p· ∂ p qD
k
∂ξ p(3.6)
Integrating this over ΩS gives
Jk
∫Ωk
∂ p f Ddi f,k
∂ξ p· ∂ pqD
k
∂ξ pdΩk =
∫ΩS
∂ pvDk
∂ξ p· ∂ p qD
k
∂ξ pdΩS (3.7)
Using (2.20), (2.21) and (2.22), we get
Jk
∫Ωk
∂ p f Ddi f,k
∂ξ p· ∂ pqD
k
∂ξ pdΩk =
∫ΩS
∂ p fDdi f
∂ξ p· ∂ p(∇u D)
∂ξ pdΩS
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1770 J Sci Comput (2018) 74:1757–1785
+∫
ΩS
∂ p F Ddi f
∂ξ p
d p+1hL(ξ)
dξ p+1
p∑j=0
ΔuL j
l j (η)dΩS
+∫
ΩS
∂ p F Ddi f
∂ξ p
d p+1h R(ξ)
dξ p+1
p∑j=0
ΔuR j
l j (η)dΩS
+∫
ΩS
∂ pG Ddi f
∂ξ p
∂hL(η)
∂η
p∑j=0
ΔuB j
∂ pl j (ξ)
∂ξ pdΩS
+∫
ΩS
∂ pG Ddi f
∂ξ p
∂h R(η)
∂η
p∑j=0
ΔuTj
∂ pl j (ξ)
∂ξ pdΩS (3.8)
Consider the second term on the RHS of (3.8). Note that the integrand is a constant for theξ -integral, i.e.,
∫ΩS
∂ p F Ddi f
∂ξ p
d p+1hL(ξ)
dξ p+1
p∑j=0
ΔuL j
l j (η)dΩS = 2
1∫−1
∂ p F Ddi f
∂ξ p
d p+1hL(ξ)
dξ p+1
p∑j=0
ΔuL j
l j (η)dη
(3.9)Since F D
di f is a tensor product of pth degree Lagrange polynomials in ξ and η, we can utilizethe third property of the VCJH correction functions, i.e., (2.38) in the following form
∂ p F Ddi f
∂ξ p
d p+1hL(ξ)
dξ p+1 = 1
c
1∫−1
∂ F Ddi f
∂ξhL(ξ)dξ (3.10)
Substituting this into (3.9), we get
∫ΩS
∂ p F Ddi f
∂ξ p
d p+1hL(ξ)
dξ p+1
p∑j=0
ΔuL j
l j (η)dΩS = 2
c
∫ΩS
∂ F Ddi f
∂ξhL(ξ)
p∑j=0
ΔuL j
l j (η)dΩS (3.11)
The third term on the RHS of (3.8) can be manipulated similarly. Now consider the fourthterm on the RHS of (3.8). Performing integration by parts for the η-integral, we obtain
∫ΩS
∂ pG Ddi f
∂ξ p
∂hL(η)
∂η
p∑j=0
ΔuB j
∂ pl j (ξ)
∂ξ pdΩS
=1∫
−1
[∂ pG D
di f
∂ξ phL(η)
]η=1
η=−1
p∑j=0
ΔuB j
∂ pl j (ξ)
∂ξ pdξ
−1∫
−1
1∫−1
∂
∂η
(∂ pG D
di f
∂ξ p
)hL(η)
p∑j=0
ΔuB j
∂ pl j (ξ)
∂ξ pdξdη
= −1∫
−1
∂ pG Ddi f,B
∂ξ p
p∑j=0
ΔuB j
∂ pl j (ξ)
∂ξ pdξ − 2
1∫−1
∂
∂η
(∂ pG D
di f
∂ξ p
)hL(η)
p∑j=0
ΔuB j
∂ pl j (ξ)
∂ξ pdη
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J Sci Comput (2018) 74:1757–1785 1771
= −2∂ pG D
di f,B
∂ξ p
p∑j=0
ΔuB j
∂ pl j (ξ)
∂ξ p− 2
1∫−1
∂
∂η
(∂ pG D
di f
∂ξ p
)hL(η)
p∑j=0
ΔuB j
∂ pl j (ξ)
∂ξ pdη
(3.12)
The fifth term on the RHS of (3.8) can be manipulated similarly. Substituting these resultsinto (3.8), we obtain Lemma 3.2. ��Remark One of the key difficulties that is encountered in the stability analysis on tensor
product elements is the non-vanishing of terms similar to ∂ p(∇u D)∂ξ p . In 1D and on triangles
and tetrahedra, the gradient of the solution is a degree p−1 polynomial and its pth derivativevanishes. However, in the case of quadrilaterals, this is not the case, and therefore, a differentnorm is necessary for tractability. See [31] for details.
Lemma 3.3 For the tensor product FR formulation utilizing the 1D VCJH correction func-tions described in Sect. 2, the following holds.
Jk
∫Ωk
∂ p f Ddi f,k
∂ηp· ∂ pqD
k
∂ηpdΩk =
∫ΩS
∂ p fDdi f
∂ηp· ∂ p(∇u D)
∂ηpdΩS
+ 2
c
∫ΩS
hL(η)∂G D
di f
∂η
p∑j=0
ΔuB j
l j (ξ)dΩS
︸ ︷︷ ︸A3
+ 2
c
∫ΩS
h R(η)∂G D
di f
∂η
p∑j=0
ΔuTj
l j (ξ)dΩS
︸ ︷︷ ︸A4
− 2∂ p F D
di fL
∂ηp
p∑j=0
ΔuL j
∂ pl j (η)
∂ηp
− 2
1∫−1
∂
∂ξ
(∂ p F D
di f
∂ηp
)hL(ξ)
p∑j=0
ΔuL j
∂ pl j (η)
∂ηpdξ
︸ ︷︷ ︸B3
+ 2∂ p F D
di fR
∂ηp
p∑j=0
ΔuR j
∂ pl j (η)
∂ηp
− 2
1∫−1
∂
∂ξ
(∂ p F D
di f
∂ηp
)h R(ξ)
p∑j=0
ΔuR j
∂ pl j (η)
∂ηpdξ
︸ ︷︷ ︸B4
(3.13)
Proof The derivation of this Lemma is similar to that of Lemma 3.2. Instead of taking the ξ
derivative, we begin by taking the pth derivative w.r.t η and proceed in a similar fashion tothe proof of Lemma 3.2. ��
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1772 J Sci Comput (2018) 74:1757–1785
Lemma 3.4 For the tensor product FR formulation utilizing the 1D VCJH correction func-tions described in Sect. 2, the following holds.
Jk
∫Ωk
( ∂2p f Ddi f,k
∂ξ p∂ηp
)·(
∂2pqDk
∂ξ p∂ηp
)dΩk = 4
c
1∫−1
∂
∂ξ
( ∂ p F Ddi f
∂ηp
)hL (ξ)
p∑j=0
ΔuL j
∂ pl j (η)
∂ηp dξ
︸ ︷︷ ︸B3
+ 4
c
1∫−1
∂
∂ξ
( ∂ p F Ddi f
∂ηp
)h R(ξ)
p∑j=0
ΔuR j
∂ pl j (η)
∂ηp dξ
︸ ︷︷ ︸B4
+ 4
c
1∫−1
∂
∂η
( ∂ pG Ddi f
∂ξ p
)hL (η)
p∑j=0
ΔuB j
∂ pl j (ξ)
∂ξ p dη
︸ ︷︷ ︸B1
+ 4
c
1∫−1
∂
∂η
( ∂ pG Ddi f
∂ξ p
)h R(η)
p∑j=0
ΔuTj
∂ pl j (ξ)
∂ξ p dη
︸ ︷︷ ︸B2
(3.14)
Proof We begin by taking the (2p)th derivative ∂2p(.)∂ξ p∂ηp of f D
di f,k and qDk , and then take their
dot product. Since the Jacobian matrix Jk is a constant in the element Ωk , similar to (2.10),we obtain
J 2k
∂2p f Ddi f,k
∂ξ p∂ηp· ∂2pqD
k
∂ξ p∂ηp= ∂2p f
Dk
∂ξ p∂ηp· ∂2p qD
k
∂ξ p∂ηp(3.15)
Integrating this over the domain ΩS , we get
Jk
∫Ωk
∂2p f Ddi f,k
∂ξ p∂ηp· ∂2pqD
k
∂ξ p∂ηpdΩk =
∫ΩS
∂2p fDk
∂ξ p∂ηp· ∂2p qD
k
∂ξ p∂ηpdΩS (3.16)
Using (2.20), (2.21) and (2.22), and noting that ∂2p(∇u D)∂ξ p∂ηp = 0 since ∇u D is a vector of degree
p − 1 polynomials, we obtain
Jk
∫Ωk
∂2p f Ddi f,k
∂ξ p∂ηp · ∂2pqDk
∂ξ p∂ηp dΩk =∫
ΩS
∂2p F Ddi f
∂ξ p∂ηpd p+1hL (ξ)
dξ p+1
p∑j=0
ΔuL j
∂ pl j (η)
∂ηp dΩS
+∫
ΩS
∂2p F Ddi f
∂ξ p∂ηpd p+1h R(ξ)
dξ p+1
p∑j=0
ΔuR j
∂ pl j (η)
∂ηp dΩS
+∫
ΩS
∂2pG Ddi f
∂ξ p∂ηpd p+1hL (η)
dηp+1
p∑j=0
ΔuB j
∂ pl j (ξ)
∂ξ p dΩS
+∫
ΩS
∂2pG Ddi f
∂ξ p∂ηpd p+1h R(η)
dηp+1
p∑j=0
ΔuTj
∂ pl j (ξ)
∂ξ p dΩS (3.17)
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J Sci Comput (2018) 74:1757–1785 1773
Note that all the integrands in the integrals on the RHS of (3.17) are constants. Therefore wecan rewrite it as
Jk
∫Ωk
∂2p f Ddi f,k
∂ξ p∂ηp· ∂2pqD
k
∂ξ p∂ηpdΩk
= 4∂2p F D
di f
∂ξ p∂ηp
d p+1hL(ξ)
dξ p+1
p∑j=0
ΔuL j
∂ pl j (η)
∂ηp
+ 4∂2p F D
di f
∂ξ p∂ηp
d p+1h R(ξ)
dξ p+1
p∑j=0
ΔuR j
∂ pl j (η)
∂ηp
+ 4∂2pG D
di f
∂ξ p∂ηp
d p+1hL(η)
dηp+1
p∑j=0
ΔuB j
∂ pl j (ξ)
∂ξ p
+ 4∂2pG D
di f
∂ξ p∂ηp
d p+1h R(η)
dηp+1
p∑j=0
ΔuTj
∂ pl j (ξ)
∂ξ p(3.18)
Using (2.38), the first term on the RHS of (3.18) can be rewritten as follows:
4∂2p F D
di f
∂ξ p∂ηp
d p+1hL(ξ)
dξ p+1
p∑j=0
ΔuL j
∂ pl j (η)
∂ηp= 4
c
1∫−1
∂
∂ξ
(∂ p F D
di f
∂ηp
)hL(ξ)
p∑j=0
ΔuL j
∂ pl j (η)
∂ηpdξ
(3.19)Performing similar manipulations on the other three terms on the RHS of (3.18) gives usLemma 3.4. ��Lemma 3.5 For the tensor product FR formulation utilizing the 1D VCJH correction func-tions described in Sect. 2, the following holds.
− bJk‖qDk ‖2 =
∫ΩS
fDdi f · ∇u DdΩS +
∫ΓS
Δu( fDdi f · n)dΓS
+ c∑
e
[∂ p
(f
Ddi f · n)
∂φ p
∂ pu D
∂φ p
]e︸ ︷︷ ︸
C1
+ c∑
e
[∂ p
(f
Ddi f · n)
∂φ p
∂ pΔu
∂φ p
]e
− c
1∫−1
∂ p
∂ξ p
(∂G Ddi f
∂η
)∂ pu D
∂ξ pdη
︸ ︷︷ ︸C2
− c
1∫−1
∂ p
∂ηp
(∂ F Ddi f
∂ξ
)∂ pu D
∂ηpdξ
︸ ︷︷ ︸C3
(3.20)
where
‖qDk ‖2 =
∫Ωk
qDk · qD
k dΩk
+ c
2
∫Ωk
∂ pqDk
∂ξ p· ∂ pqD
k
∂ξ pdΩk + c
2
∫Ωk
∂ pqDk
∂ηp· ∂ pqD
k
∂ηpdΩk
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1774 J Sci Comput (2018) 74:1757–1785
+c2
4
∫Ωk
(∂2pqD
k
∂ξ p∂ηp
)·(
∂2pqDk
∂ξ p∂ηp
)dΩk
is a Sobolev norm of qD in the elementΩk , and the summation over e represents the summationover the 4 edges of the square reference domain ΩS. φ = ξ on the top and bottom edges andφ = η on the left and right edges.
Proof We begin by multiplying Lemmas 3.2 and 3.3 by c2 and Lemma 3.4 by c2
4 and addingthem to Lemma 3.1. The terms marked A1, A2, A3, A4, B1, B2, B3, B4 in Lemmas 3.1–3.4cancel. Noting that f D
di f,k = −bqDk , we finally obtain
− bJk‖qDk ‖2 =
∫ΩS
fDdi f · ∇u DdΩS +
∫ΓS
Δu( fDdi f · n)dΓS
+ c
2
∫ΩS
∂ p fDdi f
∂ξ p· ∂ p(∇u D)
∂ξ pdΩS
+ c
2
∫ΩS
∂ p fDdi f
∂ηp· ∂ p(∇u D)
∂ηpdΩS
−c∂ pG D
di fB
∂ξ p
p∑j=0
ΔuB j
∂ pl j (ξ)
∂ξ p+ c
∂ pG Ddi fT
∂ξ p
p∑j=0
ΔuTj
∂ pl j (ξ)
∂ξ p
−c∂ p F D
di fL
∂ηp
p∑j=0
ΔuL j
∂ pl j (η)
∂ηp+ c
∂ p F Ddi fR
∂ηp
p∑j=0
ΔuR j
∂ pl j (η)
∂ηp(3.21)
The third term on the RHS of (3.21) can be rewritten as follows:
c
2
∫ΩS
∂ p fDdi f
∂ξ p· ∂ p(∇u D)
∂ξ pdΩS
= c
2
∫ΩS
∂ pG Ddi f
∂ξ p
∂ p
∂ξ p
(∂ u D
∂η
)dΩS
= c
1∫−1
∂ pG Ddi f
∂ξ p
∂ p
∂ξ p
(∂ u D
∂η
)dη (3.22)
The first step in (3.22) is due to the fact that ∂ p
∂ξ p
(∂ u D
∂ξ
) = 0, since u D is a polynomial ofdegree p. The second step is because the integrand is a constant w.r.t ξ . Applying integrationby parts to the RHS of (3.22) lets us further rewrite this:
c
2
∫ΩS
∂ p fDdi f
∂ξ p· ∂ p(∇u D)
∂ξ pdΩS
= c
[∂ pG D
di f
∂ξ p
∂ pu D
∂ξ p
]η=1
η=−1− c
1∫−1
∂ p
∂ξ p
(∂G Ddi f
∂η
)∂ pu D
∂ξ pdη
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J Sci Comput (2018) 74:1757–1785 1775
= c
[∂ p
(f
Ddi f · n)
∂ξ p
∂ pu D
∂ξ p
]T
+ c
[∂ p
(f
Ddi f · n)
∂ξ p
∂ pu D
∂ξ p
]B
− c
1∫−1
∂ p
∂ξ p
(∂G Ddi f
∂η
)∂ pu D
∂ξ pdη (3.23)
Similar manipulations can be performed on the fourth term on the RHS of (3.21). Next, thefifth term on the RHS of (3.21) can be simplified as follows:
− c∂ pG D
di fB
∂ξ p
p∑j=0
ΔuB j
∂ pl j (ξ)
∂ξ p= −c
∂ pG Ddi fB
∂ξ p
∂ pΔuB
∂ξ p= c
[∂ p( f
Ddi f · n)
∂ξ p
∂ pΔu
∂ξ p
]B
(3.24)
Substitution of these results into (3.21) gives us Lemma 3.5. ��Lemmas 3.1–3.5 dealt with obtaining an expression for the Sobolev normof qD bymanipulat-ing the second PDE in the two PDE system (2.3). Similar manipulations need to be performedon the first PDE to obtain a suitable expression for the time evolution of the Sobolev normof u D . We performed these already for the linear advection case [26,27]. Although Lemmas3.1–3.4 in [26] were obtained for the linear advection equation, the conservation law (2.3) isthe same, with the only difference being that f is now given by f = fadv + fdi f , whereasin [26], f = fadv . This discrepancy does not make any difference for the arguments usedto obtain Lemmas 3.1–3.4 in [26] since no particular property of the advective flux was uti-lized in obtaining them. Therefore, we quote Equation (3.28) of [26] which is obtained bycombining Lemmas 3.1–3.4 of that article:
1
2Jk
d
dt
(‖u D‖2k) = −
∫ΩS
u D(∇ · fD)dΩS −
∫ΓS
u D( fC · n)dΓS
−c
1∫−1
∂ pu D
∂ηp
∂ p
∂ηp
(∂ F D
∂ξ
)dξ − c
1∫−1
∂ pu D
∂ξ p
∂ p
∂ξ p
(∂G D
∂η
)dη
−c∂ pu D
∂ξ p
∣∣∣∣η=−1
( p∑j=0
ΔB j
∂ pl j (ξ)
∂ξ p
)− c
∂ pu D
∂ξ p
∣∣∣∣η=1
( p∑j=0
ΔTj
∂ pl j (ξ)
∂ξ p
)
−c∂ pu D
∂ηp
∣∣∣∣ξ=−1
( p∑j=0
ΔL j
∂ pl j (η)
∂ηp
)− c
∂ pu D
∂ηp
∣∣∣∣ξ=1
( p∑j=0
ΔR j
∂ pl j (η)
∂ηp
)
(3.25)
where
‖u D‖2k =∫Ωk
(u Dk )2dΩk + c
2
∫Ωk
((∂ pu D
k
∂ξ p
)2
+(
∂ pu Dk
∂ηp
)2)dΩk + c2
4
∫Ωk
(∂2pu D
k
∂ξ p∂ηp
)2
dΩk
is a Sobolev norm of the solution over element k, i.e., in Ωk . Noting that f = fadv + fdi f ,we can separate out the advective and diffusive parts in (3.25). In [26] and its Erratum [27],we have already shown that the advection terms (i.e., those arising out of fadv) lead to non-positive contributions ensuring stability. Therefore, let us group them together as ‘Advectiveterms’ and focus our attention on the diffusion terms alone. This leads us to our next Lemma.
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1776 J Sci Comput (2018) 74:1757–1785
Lemma 3.6 For the tensor product FR formulation utilizing the 1D VCJH correction func-tions described in Sect. 2, the following holds.
1
2Jk
d
dt
(‖u D‖2k) = Advection terms
−∫
ΩS
u D(∇ · fDdi f )dΩS −
∫ΓS
u D( fCdi f · n)dΓS
− c
1∫−1
∂ pu D
∂ηp
∂ p
∂ηp
(∂ F D
di f
∂ξ
)dξ
︸ ︷︷ ︸C3
− c
1∫−1
∂ pu D
∂ξ p
∂ p
∂ξ p
(∂G D
di f
∂η
)dη
︸ ︷︷ ︸C2
− c∑
e
[∂ pu D
∂φ p
∂ p
∂φ p
(f
∗di f · n)]
e
+ c∑
e
[∂ pu D
∂φ p
∂ p
∂φ p
(f
Ddi f · n)]
e︸ ︷︷ ︸C1
(3.26)
Proof We begin by rewriting (3.25) by grouping the terms due to the advective part of theflux fadv as ‘Advective terms’ and focusing only on the diffusion terms:
1
2Jk
d
dt
(‖u D‖2k) = Advection terms −
∫ΩS
u D(∇ · fDdi f )dΩS −
∫ΓS
u D( fCdi f · n)dΓS
−c
1∫−1
∂ pu D
∂ηp
∂ p
∂ηp
(∂ F D
di f
∂ξ
)dξ − c
1∫−1
∂ pu D
∂ξ p
∂ p
∂ξ p
(∂G D
di f
∂η
)dη
−c∂ pu D
∂ξ p
∣∣∣∣η=−1
( p∑j=0
(( f
∗di f − f
Ddi f ) · n)
B j
∂ pl j (ξ)
∂ξ p
)
−c∂ pu D
∂ξ p
∣∣∣∣η=1
( p∑j=0
(( f
∗di f − f
Ddi f ) · n)
Tj
∂ pl j (ξ)
∂ξ p
)
−c∂ pu D
∂ηp
∣∣∣∣ξ=−1
( p∑j=0
(( f
∗di f − f
Ddi f ) · n)
L j
∂ pl j (η)
∂ηp
)
−c∂ pu D
∂ηp
∣∣∣∣ξ=1
( p∑j=0
(( f
∗di f − f
Ddi f ) · n)
R j
∂ pl j (η)
∂ηp
)(3.27)
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J Sci Comput (2018) 74:1757–1785 1777
The fifth diffusion term on the RHS of (3.27) can be manipulated as follows:
−c∂ pu D
∂ξ p
∣∣∣∣η=−1
( p∑j=0
(( f
∗di f − f
Ddi f ) · n)
B j
∂ pl j (ξ)
∂ξ p
)
= −c
[∂ pu D
∂ξ p
∂ p
∂ξ p
(( f
∗di f − f
Ddi f ) · n
)]B
= −c
[∂ pu D
∂ξ p
∂ p
∂ξ p
(f
∗di f · n)]
B+ c
[∂ pu D
∂ξ p
∂ p
∂ξ p
(f
Ddi f · n)]
B(3.28)
Similar manipulations can be performed on the last 3 diffusion terms on the RHS of (3.27).Substituting these results into (3.27) gives us Lemma 3.6. ��
At this point, we are ready to state the main result of this paper and discuss the conditionsunder which the additional diffusion terms lead to stable contributions providing us with astable numerical scheme.
Theorem 3.7 If the tensor product FR approach employing the 1D VCJH correctionfunctions described in Sect. 2 is utilized for solving the 2D linear advection–diffusion equa-tion (2.1)with periodic boundary conditions on a Cartesian mesh, and if the following choicesare made for computing common values at the element interfaces:
1. Local Lax Friedrichs approach (2.25) for the common numerical advective flux f ∗adv with
0 ≤ λ ≤ 12. LDG formulation for the common solution u∗ (2.19) and the common diffusive flux
f ∗di f (2.26) with τ ≥ 0,
then the following holdsd
dt‖u D‖2W 2p,2 ≤ 0 when c ≥ 0 (3.29)
for a partial Sobolev norm of the solution defined as
‖u D‖2W 2p,2 =N∑
k=1
∫Ωk
[(u D
k )2 + c
2
((∂ pu D
k
∂ξ p
)2
+(
∂ pu Dk
∂ηp
)2)+ c2
4
(∂2pu D
k
∂ξ p∂ηp
)2]dΩk
(3.30)
Proof Subtracting Lemma 3.5 from 3.6 and noting that the terms marked C1, C2, C3 cancel,we obtain
1
2Jk
d
dt
(‖u D‖2k) + bJk‖qD
k ‖2 = Advection terms −∫
ΩS
u D(∇ · fDdi f )dΩS
−∫ΓS
u D( fCdi f · n)dΓS
−∫
ΩS
fDdi f · ∇u DdΩS −
∫ΓS
Δu( fDdi f · n)dΓS
− c∑
e
[∂ pu D
∂φ p
∂ p
∂φ p
(f
∗di f · n)]
e
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1778 J Sci Comput (2018) 74:1757–1785
− c∑
e
[∂ p
(f
Ddi f · n)
∂φ p
∂ pΔu
∂φ p
]e
(3.31)
Consider the first and third diffusion terms on the RHS of the (3.31) together. They canbe simplified as follows:
−∫
ΩS
u D(∇ · fDdi f )dΩS −
∫ΩS
fDdi f · ∇u DdΩS
= −∫
ΩS
∇ · (u D fDdi f )dΩS = −
∫ΓS
u D( fDdi f · n)dΓS (3.32)
The second diffusion term on the RHS of (3.31) can be manipulated to obtain
−∫ΓS
u D( fCdi f · n)dΓS = −
∫ΓS
u D(( f
∗di f − f
Ddi f ) · n)
dΓS
= −∫ΓS
u D( f∗di f · n)dΓS +
∫ΓS
u D( fDdi f · n)dΓS (3.33)
Substituting (3.32) and (3.33) into (3.31), we get
1
2Jk
d
dt
(‖u D‖2k) + bJk‖qD
k ‖2 = Advection terms
−∫ΓS
u D( f∗di f · n)dΓS −
∫ΓS
Δu( fDdi f · n)dΓS
− c∑
e
[∂ pu D
∂φ p
∂ p
∂φ p
(f
∗di f · n)]
e
− c∑
e
[∂ p
(f
Ddi f · n)
∂φ p
∂ pΔu
∂φ p
]e
(3.34)
Transforming the RHS of (3.34) to the physical domain, we obtain
1
2Jk
d
dt
(‖u D‖2k) + bJk‖qD
k ‖2 = Jk(Advection termsphy)
− Jk
∫Γk
u Dk ( f ∗
di f,k · n)dΓk − Jk
∫Γk
(u∗k − u D
k )( f Ddi f,k · n)dΓk
− cJk
∑e
[J 2p+1ψk
∂ pu Dk
∂ψ p
∂ p
∂ψ p
(f ∗
di f,k · n)]e
− cJk
∑e
[J 2p+1ψk
∂ p(f D
di f,k · n)∂ψ p
∂ p(u∗k − u D
k )
∂ψ p
]e
(3.35)
where ψ = x on the top and bottom edges, ψ = y on the left and right edges of the element,and Advection termsphy refers to the version of advection terms after transformation to thephysical domain. Note that Jk cancels throughout the equation. After removing Jk , we can
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J Sci Comput (2018) 74:1757–1785 1779
aggregate (3.35) over all the elements in the domain Ω to obtain
1
2
d
dt
(‖u D‖2) = −b‖qD‖2 +N∑
k=1
Advection termsphy
−N∑
k=1
∫Γk
u Dk ( f ∗
di f,k · n)dΓk −N∑
k=1
∫Γk
(u∗k − u D
k )( f Ddi f,k · n)dΓk
− cN∑
k=1
∑e
[J 2p+1ψk
∂ pu Dk
∂ψ p
∂ p
∂ψ p
(f ∗
di f,k · n)]e
− cN∑
k=1
∑e
[J 2p+1ψk
∂ p(f D
di f,k · n)∂ψ p
∂ p(u∗k − u D
k )
∂ψ p
]e
(3.36)
In [26] and its erratum [27], it was shown that the summation of the terms coming fromthe advective part, i.e., Advection termsphy , can be rewritten as a summation over theelement edges. Furthermore, the contribution of Advection termsphy was shown to be non-positive for any cartesian mesh whenever c ≥ 0. Unsurprisingly, the terms appearing afterAdvection termsphy on the RHS of (3.36) can be similarly expressed as a summation overall the edges of the mesh. To evaluate this, let us consider one generic vertical edge in themesh and accumulate all contributions from these additional terms.
Here we use the expression for common flux from (2.26) and for the common solutionfrom (2.19). Let us denote the terms coming from the element on the left side of the edge withthe subscript − and the ones from the right with + subscript. First, consider the contributionof the first diffusion term on the RHS of (3.36) to a vertical edge:
−∫e
u D−[( F D
di f,− + F Ddi f,+
2
)+ τ(u D− − u D+) + βx (F D
di f,− − F Ddi f,+)
]dy
−∫e
u D+[
−( F D
di f,− + F Ddi f,+
2
)− τ(u D− − u D+) − βx (F D
di f,− − F Ddi f,+)
]dy
= −∫e
(u D− − u D+)
[( F Ddi f,− + F D
di f,+2
)
+ τ(u D− − u D+) + βx (F Ddi f,− − F D
di f,+)
]dy (3.37)
Similarly, contributions of the second diffusion term on the RHS of (3.36) to a vertical edgeare as follows:
−∫e
F Ddi f,−
[(u D− + u D+
2
)− βx (u
D− − u D+) − u D−]
dy
−∫e
−F Ddi f,+
[(u D− + u D+
2
)− βx (u
D− − u D+) − u D+]
dy
= −∫e
[(u D+ − u D−
2
)(F D
di f,− + F Ddi f,+) − βx (u
D− − u D+)(F Ddi f,− − F D
di f,+)
]dy (3.38)
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1780 J Sci Comput (2018) 74:1757–1785
Combining these contributions from the first two diffusion terms in the RHS of (3.36) byadding (3.37) and (3.38) gives
−∫e
τ(u D− − u D+)2dy ≤ 0 for τ ≥ 0 (3.39)
Proceeding in a similar manner for the third and fourth diffusion terms, the contribution to avertical edge from these terms can be obtained as
− cτ J 2p+1y
(∂ pu D−∂y p
− ∂ pu D+∂y p
)2
≤ 0 for τ ≥ 0 and c ≥ 0 (3.40)
Equations (3.39) and (3.40) give the contributions of the diffusion terms to a single verticalelement interface or edge. Similarly, we can obtain the contributions of these terms to ahorizontal edge or interface. Noting the fact that for a periodic domain, all the edges canbe treated like internal edges or interfaces, we can then write these diffusion terms as asummation over all element interfaces or edges:
Nedges∑e=1
[−
∫e
τ(u− − u+)2dψ − cτ J 2p+1ψ
(∂ pu D−∂ψ p
− ∂ pu D+∂ψ p
)2](3.41)
where the summation here is over all the element edges in the domainΩ ,ψ = x for horizontaledges and ψ = y for vertical edges. As noted previously, our earlier work ([26,27]) showsthat the advection terms can be written as a summation over element interfaces or edges:
Advection termsphy =Nedges∑e=1
[−
∫e
λ
2|a · n|(u− − u+)2dψ
︸ ︷︷ ︸�adv
−cλ
2|a · n|J 2p+1
ψ
(∂ pu D−∂ψ p
− ∂ pu D+∂ψ p
)2
︸ ︷︷ ︸�extra
]
(3.42)where the part from�adv was obtained from terms similar to those that appear in the stabilityanalysis for triangles, while �extra was the extra component that appears for tensor productelements alone and was handled in [26]. Both these terms are shown to be non-positive orstable whenever c ≥ 0 (λ is assumed to be in the range 0 ≤ λ ≤ 1).
By substituting the results of (3.41) and (3.42) into (3.36), we get
1
2
d
dt
(‖u D‖2) = −b‖qD‖2
+Nedges∑e=1
[−
∫e
λ
2|a · n|(u− − u+)2dψ
−λ
2c|a · n|J 2p+1
ψ
(∂ pu D−∂ψ p
− ∂ pu D+∂ψ p
)2
−∫e
τ(u− − u+)2dψ − cτ J 2p+1ψ
(∂ pu D−∂ψ p
− ∂ pu D+∂ψ p
)2]. (3.43)
We see that both the advection and diffusion terms on the RHS of (3.36) give non-positivecontributions. Combining this with the fact that b > 0 and therefore −b‖qD‖2 is also non-positive, we can conclude that
1
2
d
dt
(‖u D‖2) ≤ 0, (3.44)
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J Sci Comput (2018) 74:1757–1785 1781
thus proving the stability of the FR scheme for the linear advection–diffusion equation withb > 0 on all cartesian meshes whenever c ≥ 0, 0 ≤ λ ≤ 1 and τ ≥ 0 ��3.3 Insights Gained from the Stability Analysis
Apart from the stability result stated in Theorem 3.7, (3.43) allows us to gain some additionalinsights about the behaviour of theVCJH schemes. Someof themajor observations or insightsare as follows:
1. In contrast to the stability analysis on triangles and tetrahedra (and 1D), we obtain extraterms that have an explicit dependence on the VCJH parameter c. This helps us analyzethe trends of stability and dissipation properties as this parameter varies. Many of thesetrendswere only observed through numerical experiments so far or studied using separateVon-Neumann analysis.
2. Since the terms multiplying the VCJH parameter c are non-positive, as c is increased,these terms increase the rate at which the Sobolev norm of the solution decreases, i.e.,the dissipation of the numerical scheme increases with c.
3. When c is negative, the terms containing c turn into unstable contributions and thereis a competing effect with the other stabilizing terms which are not dependent on c.Therefore, the scheme can be rendered unstable by decreasing c arbitrarily (below 0),which is observed in numerical experiments.
4 Numerical Experiments
In this section, we solve both the 2D linear advection and advection–diffusion equationsnumerically using the FR approach described in Sect. 2 in order to support our analysis inthe previous section. We show results for both upwind and central fluxes. In the followingnumerical experiments, the computational domain is chosen as Ω = {
(x, y)| − 5 ≤ x, y ≤5}. A 20 × 20 uniform Cartesian quadrilateral mesh is used for all the computations. The
advection velocity is chosen as a = [1 1]T and the diffusion coefficient b is chosen tobe either 0 (for pure advection) or 0.01 for advection–diffusion. The initial condition is acentered Gaussian bump, i.e.,
u(x, 0) = e−(x2+y2) (4.1)
A periodic boundary condition is used and the time period for this setup is 10. A fourth-order Runge–Kutta method (RK44) with a constant time-step (Δt = 0.01) is used for time-stepping. The LDG formulation parameters used for this simulation are β = 0 and τ = 1.For a purely upwind flux, we set λ = 1 and for central flux, we use λ = 0.
4.1 2D Linear Advection Equation
Figure 5 shows the evolution of the L2 energy of the solution with time for schemes withdifferent values of the VCJH parameter c. For linear advection alone, the stability analysisshowed that the evolution of the Sobolev norm of the solution is given by
1
2
d
dt
(‖u D‖2) =Nedges∑e=1
[−
∫e
λ
2|a ·n|(u− −u+)2dψ −c
λ
2|a ·n|J 2p+1
ψ
(∂ pu D−∂ψ p
− ∂ pu D+∂ψ p
)2]
(4.2)
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1782 J Sci Comput (2018) 74:1757–1785
(a) (b)
(d)(c)
Fig. 5 Plots of the evolution of the L2 norm of the solution obtained using the FR approach with differentvalues of the VCJH parameter c. a Results with upwind flux while b results obtained using central interfaceflux. The bottom two plots are zoomed in versions of (b). aUpwind flux, b central flux, c central fluxmagnified1, d central flux magnified 2
For upwind flux (λ = 1), the numerical results show that the dissipation increases as cincreases. This is evident from (4.2) where we can see that c multiplies a non-positive term.As c increases, this term becomes increasingly negative, and thus the energy of the solutiondecays at a faster rate. For the case of central flux (λ = 0), the numerical experiments showthat the L2 energy stays almost constant but varies slightly in a bounded fashion. This isobserved in the zoomed-in plots where we see oscillatory behavior for the L2 energy of thesolution. Our stability analysis (see (4.2)) showed that the Sobolev norm of the solution doesnot change when λ = 0. But the L2 energy of the solution can change in a bounded fashiondue to energy exchange between the solution and its derivatives while the Sobolev normremains constant. Therefore this numerical experiment provides additional insights whilesupporting our analysis.
4.2 2D Linear Advection–Diffusion Equation
Figures 6 and 7 show the plots of the evolution of the L2 energy of the solution for upwind andcentral interface fluxes respectively. The physical diffusion present in the problem dominatesthe energy dissipation. However, the plots show that the trend with respect to the VCJHparameter c is similar to the case of the linear advection equation, both for upwind as well
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J Sci Comput (2018) 74:1757–1785 1783
Fig. 6 Plots of the evolution of the L2 norm of the solution obtained from the FR approach using upwindinterface fluxes for different values of the VCJH parameter c. The plot on the right shows a zoomed versionof the one on the left
Fig. 7 Plots of the evolution of the L2 norm of the solution obtained from the FR approach using centralinterface fluxes for different values of the VCJH parameter c. The plot on the right shows a zoomed versionof the one on the left
as central fluxes. For the upwind flux case, this is supported by the form in which c affectsthe energy dissipation (see (3.36)). The effect of varying τ appears to be insignificant whencompared to the physical diffusion and upwinding parameters and is therefore not shownseparately.
5 Summary and Conclusions
In this article, we investigated the stability of the FR approach for solving the 2D linearadvection–diffusion equation on Cartesian meshes. The results obtained extend our earlierwork [26,27] on the linear advection equation and answer one of the last open questionsregarding the linear stability of the FR approach using VCJH correction functions. Thecentral result of the paper is that the tensor product FR formulation is stable for solving the2D linear advection–diffusion equation on Cartesian meshes whenever the VCJH parameterc is non-negative. A summary of the assumptions utilized in this analysis are as follows:
1. The boundary conditions are periodic.
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1784 J Sci Comput (2018) 74:1757–1785
2. The domain Ω is discretized using a Cartesian mesh.3. A tensor product formulation of the 1D VCJH scheme is utilized to obtain the numerical
solution u D .4. The local Lax Friedrichs (LF) formulation is used to compute the common advective flux
at the element interfaces. This has broad applicability since the LF formula is capable ofrecovering both upwind and central fluxes as special cases.
5. The common solution and the diffusive part of the common numerical flux are computedusing the LDG approach which can recover the Bassi-Rebay 1 (BR1) and Central Flux(CF) approaches as special cases.
Under these assumptions, we show that a suitably formulated Sobolev norm of the solutionis non-increasing in time for c ≥ 0. This is the general operating regime for the FR approach,and all the classical schemes recovered by the FR approach lie in this regime.When c becomesnegative, a competing effect is introduced that eventually leads to instability as c is decreased.
Several authors have previously investigated the linear and nonlinear stability of the FRformulation on 1D and simplex elements [20–24,29]. The two complexities that set apartquadrilateral elements from their 1D and triangular counterparts are the variation of thetransformation Jacobian matrix within each element, and the fact that the (p+1)th derivativeof the solution of a pth degree tensor product formulation does not vanish.We overcame theseissues previously for the case of the linear advection equation [26,27] by formulating a newnorm appropriate for tensor product elements. Here we extend this to the advection–diffusionequation which is solved as a system of two equations.
In addition to the stability result, the explicit dependence established between the energyof the solution and the VCJH parameter in this paper confirms the connections between DGand FRDG (the DG scheme obtained through FR) that have been demonstrated earlier byseveral authors [14–18].We also conclude that the dissipation of the schemes increases with cin general when an upwind flux is used, and the numerical experiments support our findings.For the central flux case, the numerical experiments show that the L2 energy can vary slightlyin a bounded and oscillatory fashion while the Sobolev norm remains unchanged. Further, wealso observe that the trend with c is reversed when compared with the upwind case. Finally, itappears that an extension of this analysis to 3D Cartesian meshes with hexahedral elementsshould be straightforward, but this is outside the scope of this paper.
Acknowledgements The authors would like to thank the Stanford Graduate Fellowship (SGF) and the AirForce Office of Scientific Research (Grant FA-9550-14-1-0186) for supporting their research.
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