Finite volume discretization of heat equation and...

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Finite volume discretization of heat equation andcompressible Navier-Stokes equations with weakDirichlet boundary condition on triangular grids

Praveen Chandrashekar

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Abstract A vertex-based finite volume method for Laplace operator on tri-angular grids is proposed in which Dirichlet boundary conditions are imple-mented weakly. The scheme satisfies a summation-by-parts (SBP) propertyincluding boundary conditions which can be used to prove energy stability ofthe scheme for the heat equation. A Nitsche-type penalty term is proposedwhich gives improved accuracy. The scheme exhibits second order convergencein numerical experiments. For the compressible Navier-Stokes equations weconstruct a finite volume scheme in which Dirichlet boundary conditions onthe velocity and temperature are applied in a weak manner. Using the centeredkinetic energy preserving flux, the scheme is shown to be consistent with theglobal kinetic energy equation. The SBP discretization of viscous and heat con-duction terms together with penalty terms are combined with upwind fluxesin a Godunov-MUSCL scheme. Numerical results on some standard test casesfor compressible flows are given to demonstrate the performance of the scheme.

Keywords Finite volume, triangular grids, SBP, energy stability, compress-ible Navier-Stokes, kinetic energy preservation

1 Introduction

Finite volume methods might be cell-centered or vertex-centered dependingon the spatial location of the solution. In the latter case, a dual finite volumehas to be constructed around each vertex, including vertices on the bound-ary. On triangular/tetrahedral grids, the vertex-based scheme has a flavour offinite element method using P1 basis functions, especially for the discretiza-tion of Laplacian term [1]. A difficulty that presents itself with vertex-centered

TIFR Center for Applicable Mathematics, Bangalore-560065, IndiaE-mail: [email protected]

2 Praveen Chandrashekar

scheme is the implementation of boundary conditions since there are degreesof freedom located on the boundaries. For the Poisson or heat equation, onecan directly set the Dirichlet boundary condition for vertices on the boundaryand update only the interior vertices using the finite volume method. This isthe strong implementation of the boundary condition. For compressible Navier-Stokes equations, the no-slip boundary condition can be implemented strongly;but there are additional variables due to density and energy. The usual practiceis to update all the quantities for a boundary vertex using the finite volumemethod and then reset the velocity to satisfy the no-slip condition. The de-grees of freedom in a compressible model are the mass density, momentumdensity and energy density while boundary conditions are usually provided onvelocity and temperature, which are not degrees of freedom but are derivedquantities from the actual degrees of freedom.

In a weak implementation of Dirichlet boundary conditions, one updatesboundary points also using the finite volume method which should implicitlyaccount for the boundary conditions. The boundary vertex value is not resetto the boundary condition value as in the strong implementation, so thatthe solution on the boundary vertices do not exactly agree with the Dirichletconditions. It is not necessary to exactly satisfy the boundary conditions sinceanyway the interior solution is only approximate. The error in the boundarysolution should be acceptable as long as it is of the same order as the errorof the interior solution and the global error converges at expected rates. Itmust also be remembered that the solution of a finite volume method denotescell average values and not point values. The cell average value is a secondorder approximation to the solution at the centroid of the finite volume. Fora dual cell around a boundary vertex, the cell lies only on one side of theboundary and the cell centroid does not lie on the boundary. The solutionat a boundary vertex in a finite volume method thus represents the averagevalue in the boundary cell and is not the value of the solution at the boundaryvertex.

The guiding principle in the construction of a scheme with weakly imple-mented boundary conditions is the satisfaction of an energy estimate that isconsistent with the energy estimate of the exact solution of the partial differ-ential equation. Thus the boundary conditions are chosen so that the result-ing scheme is stable in L2-norm. At the mathematical level, energy stabilityis obtained by using integration by parts and its discrete counterpart is thesummation-by-parts (SBP) property [2,3]. This can also be characterized interms of the skew-symmetry and symmetry properties of the difference approx-imations to convective (first order) and diffusive (second order) partial differ-ential operators. In order to obtain energy stability in the presence of boundaryconditions, the simultaneous approximation term (SAT) approach [4] has beenused which imposes the boundary conditions in a weak manner. On Cartesianand structured grids, SBP-SAT schemes have been developed for hyperbolicand parabolic problems, including the Euler and Navier-Stokes equations [5,6]. For a linear convection-diffusion equation, weak boundary conditions in asecond order SBP-SAT scheme are found to give more accurate solutions on

Title Suppressed Due to Excessive Length 3

coarse meshes while on finer meshes, the strong implementation was more ac-curate [7] and similar behavior was observed for the solution of Navier-Stokesequations. Weak imposition of boundary conditions is found to lead to fasterconvergence of solutions to steady state problems while in the case of strongimposition, the residuals may not converge to machine precision [8].

Mimetic schemes [9] are constructed to satisfy the equations of integralcalculus like divergence theorem and related identities, which endows themwith SBP properties. Dirichlet boundary conditions have been implementedstrongly in mimetic methods on logically rectangular grids for Poisson equa-tion [10]. A kinetic energy preserving scheme for incompressible Navier-Stokesequations on logically rectangular grids is proposed in [11]. Stable SBP schemeson unstructured grids for hyperbolic problems using vertex-based finite vol-ume method have been studied in [12] where boundary conditions are imposedweakly through the fluxes using a local characteristic decomposition that sepa-rates incoming and outgoing waves. Higher order mimetic methods which haveSBP property have been developed in [13] for diffusion equation but boundaryconditions and their influence on stability property was not analyzed. For thediscretization of second order terms on unstructured grids, there are few workswhich have studied the energy stability property. Svard and Nordstrom [14,15] study the vertex-based discretization of Laplacian on unstructured gridsusing an edge-based scheme which is shown to be stable since resulting matrixhas negative eigenvalues. For boundary vertices, they construct two dual cellsto obtain an estimate of the normal derivative of the solution which is used tocompute the flux across the boundary faces. In numerical experiments they findthat the scheme does not converge even on a uniform triangulated Cartesiangrid, while convergence is obtained only with equilateral grids. An alternateapproach to the edge-based scheme is to use a P1 Galerkin approximationon triangles that leads to a compact scheme for the Laplacian operator [1,16]; it is exact for affine functions and leads to a convergent scheme for theLaplace operator on triangular grids. Consistency of the schemes with respectto energy evolution can be considered as a secondary conservation principlesatisfied by the scheme. A recent review of discrete conservation principleson unstructured grids leading to second conservation properties can be foundin [17].

In this work we restrict ourselves to triangular grids in two spatial di-mensions and consider the vertex-based finite volume scheme. We construct ascheme for the Laplacian which is similar to the P1 Galerkin approach but weincorporate boundary conditions within the gradient approximations used forthe Laplacian which implicitly accounts for an SAT approach. This allows usto show a SBP property for the Laplacian approximation including boundaryconditions which also leads to a discrete energy equation for the solution ofheat equation. We also propose the addition of a Nitsche-type penalty term [18]for Dirichlet boundary conditions which enhances the accuracy of the scheme;the penalty term is not necessary for the stability of the scheme. Through nu-merical experiments on the heat equation, we show that the solutions convergeat a rate of O(h2) where h is a typical size of the triangles. These ideas are


i j

k

nTk

nTi

nTj

T

Fig. 1 Definition of edge normals

extended to the compressible Navier-Stokes equations for the discretizationof viscous terms and heat conduction terms appearing in the momentum andenergy equation. No-slip and isothermal boundary conditions are implementedin a weak manner and Nitsche-type penalty terms are also used in the momen-tum and energy equations. Using a kinetic energy preserving central flux [19],we show that the scheme is consistent for the global kinetic energy evolution.The SBP discretizations of viscous and heat conduction terms are then com-bined with a numerical flux function in a Godunov finite volume scheme forwhich higher order accuracy can be obtained via the MUSCL approach. Thediscretization of diffusive terms can be implemented in existing finite volumecodes with little extra modifications. The presented schemes can be naturallyextended to three dimensions on tetrahedral grids. However it is not easy todevelop similar methods for hybrid grids. In two dimensions we have onlysucceeded in constructing an SBP scheme on grids containing triangles andparallelograms.

The rest of the paper is organized as follows. Section (2) introduces thenew SBP discretization for the Laplacian and shows the SBP property. Sec-tion (3) presents the finite volume scheme for Poisson equation and its solv-ability is shown. The scheme is then applied to heat equation in section (4)and an energy equation is demonstrated for the semi-discrete scheme. Nu-merical experiments on steady and unsteady heat conduction problems aregiven to demonstrate the convergence properties. Section (5) discusses theapplication of the new scheme to the compressible Navier-Stokes equationsfor which a global kinetic energy balance equation is demonstrated which isconsistent with the true balance equation. Then the SBP scheme is combinedwith a Godunov-MUSCL scheme. Finally, many numerical results are shownfor compressible flows in two dimensions and for axisymmetric rotating flow.

2 Finite volume approximation of Laplacian

Consider a discretization of the domain Ω by triangles. The nodes of thetriangles are denoted by the letters i, j, k etc. For each triangle we define the


i j

k

ne

Te

∂Ω

Ω

Fig. 2 A boundary triangle

outward normal vectors as shown in figure (1) with the magnitude of the vectorequal to the length of the corresponding edge; e.g., nTi denotes the outwardnormal to the face in triangle T which is opposite to the vertex i. We willdistinguish between two types of triangles refered to as interior and boundarytriangles. An interior triangle does not have any of its faces on the Dirichletboundaries while a boundary triangle has atleast one face on some Dirichletboundary. The derivative is approximated on a triangle by the Green-Gausstheorem ∫

T

∇udx =

∫∂T

unds =∑e∈∂T

∫e

unds (1)

The surface integral is approximated by the trapezoidal rule. Then for aninterior triangle, the gradient approximation is given by

∇huT =1

|T |

[ui + uj

2nTk +

uj + uk2

nTi +uk + ui

2nTj

](2)

= − 1

2|T |[uin

Ti + ujn

Tj + ukn

Tk

](3)

The second expression follows from the fact that nTi +nTj +nTk = 0. The firstform given by equation (2) is more fundamental since it is a direct consequenceof the Green-Gauss theorem and will be useful for enforcing Dirichlet boundaryconditions. If we construct the P1 interpolant on the triangle and take itsgradient, then the result is identical to the above expressions. The gradient ona boundary triangle Te adjacent to a boundary edge e, see figure (2), is givenby equation (1) but we make use of the Dirichlet boundary condition for theintegral on the boundary edge e; This leads to the following approximation forthe gradient on Te

∇huTe =1

|Te|

[fi + fj

2nTe

k +uj + uk

2nTei +

uk + ui2

nTej

](4)

Note that we have used the Dirichlet boundary condition only for the integralon the boundary edge nTk in the Green-Gauss formula; for the other edges,the solution values ui, uj are used. Since the boundary conditions are to beimplemented weakly during the solution of the PDE, the solution values ui, ujneed not exactly coincide with the boundary values fi, fj . This is an important


i

T

nTi

∂Ωi

T

nTi

(a) (b)

Fig. 3 Definition of dual finite volume for (a) interior vertex and (b) boundary vertex

point about the scheme we present and must be remembered when readingthe rest of the paper. We make the following definitions

i ∈ T = all vertices i belonging to triangle TT ∈ i = all triangles T having vertex i

Γ = all boundary edgesΓi = all boundary edges having vertex i

For an interior vertex i the set Γi is empty. Around each vertex i, we constructthe dual cell by joining the cell centroid to the mid-point of the edges, seefigure (3a). For a boundary vertex, the cell is closed by the boundary edges asshown in figure (3b). Integrate the Laplacian over a dual cell Ai to obtain∫

Ai

∆udx =

∫∂Ai\Γ

∂u

∂nds+

∫∂Ai∩Γ

∂u

∂nds

The flux integral on the right involves contributions from the interior bound-aries of the dual cell Ai and the boundaries of Ω if the dual cell is adjacentto the boundary. For the part of the boundary of Ai which is inside a triangleT , we estimate ∇u ≈ ∇huT while for a boundary edge integral on an edge e,the gradient from the adjacent triangle Te is used. This leads to the followingapproximation to the Laplacian

(∆hu)i =1

Ai

[1

2

∑T∈i∇huT · nTi +

1

2

∑e∈Γi

∇huTe · ne

](5)

For two functions u, v defined on Ω with u = f , v = g on ∂Ω, we have therelation ∫

Ω

v∆udx = −∫Ω

∇u · ∇vdx+

∫∂Ω

g∂u

∂nds (6)

We next show that the discrete approximation to the Laplacian given by equa-tion (5) satisfies an analogous property which is usually called summation-by-parts property. This is also crucial for showing the existence of solution toPoisson problem and energy stability for the time dependent problem likeheat equation.


Theorem 1 Let u, v be two functions defined on Ω with u = f , v = g on ∂Ω.Then the discrete approximation given by equation (5) satisfies the summation-by-parts property∑

i

vi(∆hu)iAi = −∑T

(∇huT · ∇hvT )|T |+∑e∈Γ

(gi + gj

2

)∇huTe · ne (7)

where the vertices i, j form the boundary edge e.

Proof: Multiply equation (5) by vi and sum over all vertices.∑i

vi(∆hu)iAi =1

2

∑i

∑T∈i

vi∇uT · nTi +1

2

∑i

∑e∈Γi

vi∇uTe · ne

Interchanging the order of the summation, we get∑i

vi(∆hu)iAi =1

2

∑T

∑i∈T

vi∇uT · nTi +∑e∈Γ

(vi + vj

2

)∇uTe · ne (8)

The second term involves boundary contributions and we note that vi neednot be equal to the boundary value gi. We now consider the first term on theright hand side of equation (8). For an interior triangle T , we use the secondform of the gradient formula, equation (3), to get

1

2

∑i∈T

vi∇huT · nTi = ∇huT ·1

2

∑i∈T

vinTi = −∇huT · ∇hvT |T | (9)

For a boundary triangle Te adjacent to a boundary edge e, we need to accountfor the Dirichlet conditions. Assume that the edge e is bounded by the verticesi, j while k is the third vertex of the triangle Te so that ne = nTe

k , see figure (2);using the equations (2), (3), (4) we obtain

1

2

∑i∈Te

vi∇huTe ·nTei = −∇huTe ·

[vi + vj

2nTe

k +vj + vk

2nTei +

vk + vi2

nTej

]|Te|

Adding and subtracting Dirichlet boundary values, we can write the aboveformula as

1

2

∑i∈Te

vi∇huTe · nTei

= −∇huTe ·[gi + gj

2nTe

k +vj + vk

2nTei +

vk + vi2

nTej

]|Te|

−∇huTe ·[vi + vj

2nTe

k −gi + gj

2nTe

k

]= −∇huTe · ∇hvTe |Te| − ∇huTe ·

[vi + vj

2ne −

gi + gj2

ne

](10)


Using (9) and (10) we have shown that

1

2

∑i∈T

vi∇huT ·nTi = −∑T

∇huT ·∇hvT |T |−∑e∈Γ∇huTe ·

[vi + vj

2ne −

gi + gj2

ne

](11)

From (8) and (11) we obtain the desired result and the proof is complete. Dueto the incorporation of the boundary condition within the gradient approxi-mation, we are able to replace the term

vi+vj2 in equation (8) with the value

gi+gj2 which is then consistent with the exact equation (6).

Remark Interchanging the role of u and v, we obtain the following SBP for-mula∑

i

ui(∆hv)iAi = −∑T

(∇huT · ∇hvT )|T |+∑e∈Γ

(fi + fj

2

)∇hvTe · ne (12)

If both u, v vanish on the boundary, i.e., f = g = 0, then∑i

vi(∆hu)iAi =∑i

ui(∆hv)iAi

which shows that the discrete Laplacian ∆h is self adjoint. In the analysis ofnumerical schemes, this property is important to show optimal convergencerates of the error in L2 norm but we do not have a theoretical proof for theschemes presented here.

Remark The use of the gradient approximation given by equation (4) wasimportant in the above proof. If we use the second form of the gradient inequation (3) and strongly implement the boundary conditions, then the gra-dient for a boundary triangle would be

∇uTe = − 1

2|Te|[fin

Ti + fjn

Tj + ukn

Tk

]But using this approximation does not lead to the summation by parts prop-erty as given in equation (7).

3 Finite volume method for Poisson equation

In this section we consider the Dirichlet problem for the Poisson equation

−∆u = s in Ω

u = f on ∂Ω

so that Γ = ∂Ω = ΓD. We will use the finite volume approximation to theLaplacian given by equation (5) and in addition add a penalty term inspired


by the Nitsche method and discontinuous Galerkin methods. The finite volumemethod is given by

−Ai∆hui +∑e∈Γi

Cp2he

(ui − fi)|ne| = Aisi (13)

where Cp ≥ 0 controls the penalty term and he is a mesh length associatedwith the edge e. In the computations, we take he to be the height of thetriangle adjacent to the edge e, i.e.,

he =2|Te||ne|

The penalty term does not destroy the consistency of the method since if wesubstitue the exact solution in (13) then the penalty term vanishes. Note thatthe scheme (13) includes the boundary conditions and is applied at all thevertices. We next state the existence and uniqueness of solution to the abovefinite volume scheme.

Theorem 2 Consider the finite volume scheme given by equation (13).

1. If Cp > 0, then it has a unique solution.2. If Cp = 0 and if all boundary vertices belong to at atleast one interior

triangle, then it has a unique solution.3. If Cp = 0 and if at least one boundary vertex belongs to at atleast one

interior triangle, then it has a unique solution.

Proof: Equation (13) is a linear equation for the ui and it will have a uniquesolution if we can show that in the case of the homogeneous problem f = s = 0,the only solution is the zero solution. Hence assuming homogeneous data,multiply equation (13) by ui and sum over all vertices

−∑i

Aiui∆hui +∑i

∑e∈Γi

Cp2he

u2i |ne| = 0

Using the summation by parts property (7), this equation becomes∑T

‖∇huT ‖2|T |+∑e∈Γ

Cp2he

(u2i + u2j )|ne| = 0

We now consider different cases in the theorem.

1. Assume that Cp > 0. Then

∇huT = 0 ∀ T and ui = 0 ∀ i ∈ Γ

On an interior triangle T we have

∇huT = − 1

2|T |[(uj − ui)nTj + (uk − ui)nTk ] = 0


j i

k l

re

Te

T1

T2

∂Ω

Ω

Fig. 4 Example of boundary vertex i belonging to an interior triangle T1

j i

k l

re

Te T1

T2

∂Ω

Ω

Fig. 5 Example of a boundary vertex i belonging to only boundary triangles

and since nTj , nTk are linearly independent, we conclude that ui = uj = uk.On a boundary triangle Te adjacent to a boundary edge e as in figure (2)we have from equation (4)

∇huTe =1

|T |

[0 + 0

2nTk +

0 + uk2

nTi +uk + 0

2nTj

]= − uk

2|T |nTk = 0

which implies that uk = 0. These two facts are enough to conclude thatui = 0 ∀ i.

2. Assume that Cp = 0. Then we can only conclude that

∇uT = 0 ∀ T

For an interior triangle we again obtain that all the three vertex values areequal. Consider a boundary vertex i which belongs to an interior triangleT1 and a boundary triangle Te as shown in figure (4). Since T1 is an interiortriangle, from the condition ∇huT1 = 0 we conclude that ui = uk whilefrom ∇huTe = 0, i.e.,

∇huTe =1

|T |

[0 + 0

2nTk +

uj + uk2

nTi +uk + ui

2nTj

]= 0

we conclude that ui + uk = 0 which together imply that ui = 0. Hence wecan conclude that ui = 0 ∀ i.

3. Assume that Cp = 0. Then we can only conclude that

∇uT = 0 ∀ T

For an interior triangle we again obtain that all the three vertex values areequal. Now consider a boundary vertex i that belongs to only boundarytriangles and does not belong to any interior triangle, as in figure (5).From ∇huTe = 0, we can conclude that ui +uk = 0 and uj +uk = 0 whichimplies that ui = uj = −uk. Similarly from ∇huT1 = 0 we conclude that


ui = ur = −uk. But since T2 is an interior triangle, ∇huT2 = 0 impliesur = uk and hence we obtain ui = uj = ur = 0. From this we can concludethat ui = 0, ∀ i.

Remark We see that the addition of a penalty term leads to a simple proof ofexistence. However if Cp = 0, the difficulty in the proof is in case (3) wherea vertex i belongs to only boundary triangles. However, geometrically it isnot possible that all boundary vertices belong to only boundary triangles. Forexample, in figure (5) the vertex r will belong to an interior triangle and theproof can be completed by connecting vertex i to vertex r by moving alongboundary edges.

4 Discretization of heat equation

We consider the heat equation

ut = ∇ · (µ∇u) in Ω

u = f on ΓD

µ∂u

∂n= g on ΓN

µ∂u

∂n+ αu = h on ΓR

(14)

with µ > 0 and α ≥ 0. Here the boundary is composed of three parts corre-sponding to Dirichlet (ΓD), Neumann (ΓN ) and mixed (ΓR) boundary con-ditions. Multiplying the above equation by u and integration over Ω gives theenergy equation

d

dt

∫Ω

1

2u2dx = −

∫Ω

µ|∇u|2dx+

∫ΓD

µf∂u

∂nds+

∫ΓN

ugds+

∫ΓR

(uh−αu2)ds

(15)Note that we have to use the integration by parts rule to arrive at the aboveresult. In the case of homogeneous boundary conditions, f = g = h = 0, weobtain

d

dt

∫Ω

1

2u2dx = −

∫Ω

µ|∇u|2dx−∫ΓR

αu2ds ≤ 0

which shows the well-posedness of the problem.

4.1 Finite volume method

We now consider the discretization of the operator appearing in equation (14).Define the operator

Lu = ∇ · (µ∇u)


By performing an integration over the dual volume Ai and using the boundaryconditions on u we obtain∫Ai

(Lu)dx =

∫∂Ai

µ∇u·nds+

∫∂Ai∩ΓD

µ∇u·nds+

∫∂Ai∩ΓN

gds+

∫∂Ai∩ΓR

(h−αu)ds

The finite volume approximation of this operator is given by

Ai(Lhu)i =1

2

∑T∈i

µT∇huT · nTi +1

2

∑e∈ΓD

i

µTe∇huTe · ne

+1

2

∑e∈ΓN

i

gi|ne|+1

2

∑e∈ΓR

i

(hi − αui) |ne| (16)

where the term ∇huTe for a triangle Te adjacent to a Dirichlet boundaryedge e is computed using the approximation given by equation (4) while fora boundary triangle adjacent to any other type of boundary, the gradient iscomputed from equation (3). Using the above approximation, we formulatethe finite volume scheme including a Nitsche type penalty term for Dirichletboundary condition as

Aiduidt

= Ai(Lhu)i −∑e∈ΓD

i

CpµTe

2he(ui − fi)|ne| (17)

To derive the energy equation, we multiply the above equation by ui and sumover all vertices to obtain

d

dt

∑i

1

2u2iAi = −

∑T

µT ‖∇huT ‖2|T |+∑e∈ΓD

(fi + fj

2

)µTe∇huTe · ne

+∑e∈ΓN

(uigi + ujgj

2

)|ne|+

∑e∈ΓR

[(uihi + ujhj

2

)− α

(u2i + u2j

2

)]|ne|

−∑e∈ΓD

CpµTe

2he[ui(ui − fi) + uj(uj − fj)] (18)

The last term in the above equation is due to the Dirichlet penalty terms whichdoes not destroy the consistency of the method while the remaining terms areconsistent with the continuous energy equation (15).

Theorem 3 Assume that atleast one of ΓD, ΓR is non-empty. Then underhomogeneous data f = g = h = 0, the semi-discrete finite volume schemegiven by (17) is stable in the energy norm.

Proof: In the case of homogeneous data, the energy equation (18) becomes

d

dt

∑i

1

2u2iAi = −

∑T

µT ‖∇huT ‖2|T |−∑e∈ΓD

CpµTe

2he(u2i+u

2j )|ne|−

∑e∈ΓR

α

(u2i + u2j

2

)|ne| ≤ 0


nTj

nTk

nTi

T

i j

k

Ω

Fig. 6 Corner cell

Moreover, using the condition that at least one of ΓD or ΓR is non-empty, itis easy to check that the right hand side in the above equation is zero if andonly if ui = 0 ∀ i and hence the scheme is stable.

4.2 Need for penalty term

The Laplace equation was shown to have a unique solution even in the absenceof penalty term. For the heat equation however, one can converge to wrongsolution in some cases in the absence of penalty term. Consider the followingproblem

ut = ∆u, in Ω

u(x, 0) = 0, in Ω

u(x, t) = 1, on ∂Ω

The time asymptotic value of the solution to the above problem reaches theconstant value of unity. Consider a grid where the corner boundary vertexi belongs to a single boundary triangle as shown in figure (6). We start thecomputations by setting the initial value to be zero at all the vertices. Thenthe equation for the corner point is

Aiduidt

=1

2∇huT · (nTi + nTj + nTk ) = 0

and the solution at vertex i remains zero for all future times while the exactsolution tends to unity. The addition of a penalty term will avoid this situationsince the penalty term drives the solution at all boundary points towards thecorrect boundary value. It is of course possible to avoid such situations by re-triangulating the grid but the penalty term gives a simpler solution withoutchanging the grid.

4.3 Other type of dual cells

The most common type of dual cell which we have described here is obtainedby joining the cell centroid to the edge midpoints. However there are other ways


ne1

ne2

T1

T2

f = 1

f = 0

Ω

Fig. 7 Example of discontinuous boundary condition

to construct the dual cell. In the containment dual or voronoi approach [20],the circumcenter of the triangles is joined to the edge midpoints. If the circum-center lies outside the triangle, as can happen for an obtuse angled triangle,the midpoint of the largest side is used. For a right-angled triangle, the cir-cumcenter lies at the midpoint of the hypotenuse, which is the largest side ofthe right-angled triangle. In the case of stretched triangle as would result inboundary layer meshes, the voronoi dual cell becomes a quadrilateral or closeto a rectangle. Such cells have edges aligned with the flow and are advanta-geous in boundary layers where the streamwise gradients are smaller comparedto the wall normal gradients [21]. No matter which point inside the triangle isused to form the dual cells, we always have the following identity satisfied

nTij + nTik =1

2nTi

due to which all of our results on summation by parts and energy stabilitystill hold. The area of the dual cells Ai has to be appropriately computedbased on how the dual cell was constructed.

4.4 Numerical implementation

The numerical implementation of the above schemes can be achieved by loop-ing over all the edges in the triangulation and accumulating the contribution ofthe edge to the adjacent triangles. For boundary edges, the boundary conditioncan be implemented based on the type of boundary and the contribution addedto the adjacent triangle. The gradient equation (2) is used for every triangleexcept for a triangle adjacent to a Dirichlet boundary for which equation (4) isused. Note that this allows the boundary condition to be discontinuous acrossthe boundary edges. For example, in the situation depicted in figure (7) theDirichlet boundary condition is discontinuous at the corner; for the two edgese1 and e2, the contributions to the corresponding triangles are computed as

∇uT1 = ∇uT1 +1 + 1

2ne1 and ∇uT2 = ∇uT2 +

0 + 0

2ne2


which are the integrals over the edges, and the integrals make sense even ifthe boundary condition is discontinuous. In the case of strongly implementedboundary conditions, there is ambiguity on what value to use for the cor-ner point, which does not arise in our scheme. Other boundary integrals inequation (16) are similarly computed which allows the Neumann and mixedboundary conditions g, h to be also discontinuous across the boundary edges.

4.5 Test I: smooth solution

We consider the heat equation on the unit square with homogeneous Dirichletboundary conditions and initial condition given by

u(x, y, 0) = sin(2πx) sin(2πy) + sin(4πx) sin(4πy) (19)

whose exact solution is

u(x, y, t) = e−8π2t sin(2πx) sin(2πy) + e−32π

2t sin(4πx) sin(4πy) (20)

The numerical solution is computed up to the time t = 0.05 and the errorsin maximum norm and L2 norm at the final time are computed. At this finaltime, the maximum value of the solution has decreased by about a factor ofabout 100. For the L2 norm we use the definition

‖u‖2 =

√∑i

[ui − u(xi, yi)]2Ai (21)

where ui is the numerical solution in dual cell Ai and u(xi, yi, t) is the exactsolution evaluated at the vertex (xi, yi). Three types of grids are considered asshown in figure (8); the grids in the figures may be refered to as uniform, quasi-uniform and non-uniform respectively. Note that the quasi-uniform and non-uniform grids are randomly generated. Grids are generated with 10, 15, . . . , 50points on each side of the unit square for studying the convergence of errors.Figure (9) shows the convergence of the solution error with mesh size on theuniform and non-uniform meshes. In all the cases, the error decreases at arate close to O(h2) indicating second order accuracy. The errors are smaller inthe case of uniform grids and the convergence is monotonic while in the caseof the quasi-uniform grids, the error converges in a non-monotonic manner,though asymptotically, the convegence seems to be atleastO(h2). On the quasi-uniform grids, as the grid becomes finer, the errors are smaller than in thecase of non-uniform grids, which indicated some cancellation mechanism dueto the more regular structure of the quasi-uniform grids as compared to non-uniform grids. Since the boundary conditions are exactly enforced, the strongboundary condition shows better accuracy in L∞ on the uniform grids, whilethe L2 error norms are nearly same indicating that the average error is similarin both cases. On the quasi-uniform and non-uniform grids, both strong andweak implementations show similar error levels in both norms.


(a) (b) (c)

Fig. 8 Grids used for Test I: (a) uniform grid, (b) quasi-uniform grid, (c) non-uniform grid

10−2

10−1

100

10−4

10−3

10−2

10−1

100

Mesh size h

Err

or n

orm

Strong, L∞ error

Weak, L∞ error

Strong, L2 error

Weak, L2 error

Second order

10−2

10−1

100

10−4

10−3

10−2

10−1

100

Mesh size h

Err

or n

orm

Strong, L∞ error

Weak, L∞ error

Strong, L2 error

Weak, L2 error

Second order

(a) (b)

10−2

10−1

100

10−4

10−3

10−2

10−1

100

Mesh size h

Err

or n

orm

Strong, L∞ error

Weak, L∞ error

Strong, L2 error

Weak, L2 error

Second order

(c)

Fig. 9 Error convergence for Test I: (a) Uniform grid, (b) quasi-uniform grids, (c) non-uniform grid

4.6 Test II: discontinuous diffusivity

We consider the Laplace equation with spatially varying diffusion coefficient [13]as in equation (14) on a unit square with

µ =

µ1 x < 1

2

µ2 x > 12


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) (b)

Fig. 10 Problem with discontinuous coefficient: (a) Test II (b) Test III

Dirichlet boundary conditions are applied on the left and right boundarieswhile zero Neumann conditions are prescribed on the bottom and top bound-aries. The exact solution is given by

u =

µ2x+2µ1µ2

12 (µ1+µ2)+4µ1µ2

0 ≤ x ≤ 12

µ1x+2µ1µ2+12 (µ2−µ1)

12 (µ1+µ2)+4µ1µ2

12 ≤ x ≤ 1

which is continuous across the material discontinuity surface x = 12 but the

derivatives are discontinuous. In the numerical computations, we use µ1 = 1and µ2 = 4. The Dirichlet boundary condition is obtained from the above exactsolution. The grid is constructed so that there are grid points located alongx = 1

2 and the grid edges of the primary grid do not cross this discontinuityline. The results are shown in figure (10a) and the scheme recovers the exactsolution (upto machine precision) on any such grid because the approximationof the Laplacian is exact for any affine function. Even though the coefficientis discontinuous, the solution is continuous and the vertex-centered methodscan satisfy this continuity condition due to the presence of degrees of freedomlocated on the discontinuity surface which is not the case with cell centeredmethods.

4.7 Test III: discontinuous diffusivity

We again consider the Laplace equation with the same distribution of dif-fusivity coefficient as the previous problem. The exact solution is chosen tobe

u =

a+ bx+ cy 0 ≤ x ≤ 1

2

a− bµ1−µ2

2µ2+ bµ1

µ2x+ cy 1

2 ≤ x ≤ 1

In this case, the tangential flux is discontinuous across the material interfaceand many methods which assume continuity of the tangential flux give wrong


results [13]. In the vertex-centered method developed here, the material dis-continuity is not a finite volume face and we do not have to compute the fluxacross the material interface. The Dirichlet conditions are taken from the ex-act solution and applied on all the four sides of the unit square domain. Forthe computations, we take a = b = c = 1, µ1 = 1 and µ2 = 4. The scheme isable to recover the exact solution as shown in figure (10b).

5 Compressible Navier-Stokes equations

We next extend the ideas presented till now to the case of compressible flows.The compressible Navier-Stokes equations represent the conservation laws formass, momentum and energy, and can be written as

∂U

∂t+∇ · F = ∇ ·G (22)

where U is the vector of conserved variables, F are the inviscid fluxes and Gare viscous fluxes

U =

ρmE

=

ρρuE

, F =

ρupI + ρu⊗ u

(E + p)u

, G =

0σ

σ · u− q

(23)

In the above equations ρ is the density, u is velocity, p is the pressure, E istotal energy per unit volume, while σ and q are the shear stress tensor andheat flux vector respectively. The pressure is related to the other quantitiesthrough perfect gas relation and takes the form

E =p

γ − 1+

1

2ρ|u|2

while the shear stress and heat flux are given by the Newtonian and Fourierconstitutive laws, respectively.

σ =1

2µ(∇u + (∇u)>)− 2

3µ(∇ · u)I, q = −κ∇T

where I is the unit tensor.

5.1 Kinetic energy equation

We will assume Dirichlet boundary conditions on the velocity over the wholeboundary. Thus the boundary conditions on the velocity are

u = f on Γ (24)

In practice, the normal velocity f · n would be zero representing a stationarywall, while the tangential velocity could have a non-zero component whichwould model a sliding wall or a rotating wall for axisymmetric rotating flows.


The kinetic energy per unit volume is k = 12ρ|u|

2; the evolution of the totalkinetic energy is given by

d

dt

∫Ω

kdx = −∫Γ

[(p+ k)(f ·n)− f · σ ·n]ds+

∫Ω

[p(∇ · u)− σ : ∇u] dx (25)

It can be shown that for Newtonian fluid [22], the last term in the aboveequation destroys kinetic energy, i.e.,

−σ : ∇u ≤ 0

In the case of homogeneous boundary conditions f = 0, we have the followingstability estimate for the kinetic energy

d

dt

∫Ω

kdx ≤∫Ω

p(∇ · u)dx

The kinetic energy can change due to change of internal energy by the com-pression/expansion of the fluid and vice versa but the viscous terms alwayslead to destruction of kinetic energy. If the compressibility effect is negligiblethen the total kinetic energy decreases with time due to viscous dissipation.

5.2 Finite volume method

Consider a division of the domain by triangles and around each vertex, con-struct the dual finite volume as described in section (2). For each dual cell, theinviscid and viscous fluxes must be computed on the faces of the dual cell. Forthe viscous fluxes which require derivatives of velocity and temperature, weuse a scheme similar to the heat equation, i.e., the viscous fluxes are computedtriangle-wise. The inviscid fluxes are computed edge-wise using the solutionat the vertices forming the edge. Apart from the notations introduced in sec-tion (2), we introduce the notation j ∈ i to denote the set of all vertices jwhich are connected to vertex i by an edge. Integrating the conservation lawover a dual cell, we get1

AidUidt

= −∑j∈i

∫∂Ai∩∂Aj

F · nds+∑T∈i

∫∂Ai∩T

G · nds−∫

∂Ai∩Γ

F · nds+

∫∂Ai∩Γ

G · nds

≈ −∑j∈i

Fij ·∫

∂Ai∩∂Aj

nds+∑T∈i

GT ·∫

∂Ai∩T

nds

−∑e∈Γi

∫e∩∂Ai

F · nds+∑e∈Γi

∫e∩∂Ai

G · nds

= −∑j∈i

Fij · nij +1

2

∑T∈i

GT · nTi −∑e∈Γi

Fi,e ·ne2

+∑e∈Γi

GTe · ne2

1 The notation ∂Ai ∩ T denotes the portion of the boundary ∂Ai which is inside T .


The normal vectors nij , nTi , etc. have dimensions of length. The quantityFij ·nij is the flux across the face common to the cells Ai and Aj . Writing theabove scheme for the mass, momentum and energy equations, we obtain thefinite volume discretization as follows

Aidρidt

= −∑j∈i

F ρij −∑e∈Γi

F ρi,e (26)

Aidmi

dt= −

∑j∈i

Fmij −∑e∈Γi

Fmi,e +1

2

∑T∈i

σT · nTi +1

2

∑e∈Γi

σTe · ne (27)

AidEidt

= −∑j∈i

FEij −∑e∈Γi

FEi,e +1

2

∑T∈i

uT · σT · nTi

+1

2

∑e∈Γi

fi · σTe · ne −1

2

∑T∈i

qT · nTi −1

2

∑e∈Γi

qTe · ne (28)

Here F ρij ,Fmij , F

Eij are the inviscid fluxes of mass, momentum and energy across

interior faces of the dual finite volume while F ρi,e,Fmi,e, F

Ei,e are the inviscid

fluxes across the boundary edges, all of which are yet to be specified. Thelength of the cell faces has been absorbed inside the definition of the fluxes.The viscous fluxes are computed for each triangle; for this, the derivatives ofvelocity and temperature are computed on each triangle using the scheme givenfor heat equation, while applying Dirichlet boundary condition for boundarytriangles. For an interior triangle, the velocity gradient is given by

∇huT =1

|T |

[ui + uj

2⊗ nTk +

uj + uk2

⊗ nTi +uk + ui

2⊗ nTj

](29)

while for a triangle adjacent to a Dirichlet boundary edge e whose vertices arei, j, the gradient is given by

∇huTe =1

|Te|

[fi + fj

2⊗ nTe

k +uj + uk

2⊗ nTe

i +uk + ui

2⊗ nTe

j

](30)

The notation u ⊗ v denotes the dyadic product whose components are (u ⊗v)ij = uivj . The coefficient of thermal viscosity and conductivity are averagedon each triangle; then the shear stress σT and heat flux qT can be computedon each triangle from the constitutive laws of Newton and Fourier

σT = µT[∇huT + (∇huT )>

2− 2

3(∇h · uT )I

], qT = −κT∇hTT (31)

5.3 Discrete kinetic energy balance

The time derivative of kinetic energy can be written in terms of density andmomentum derivatives as

dkidt

= −|ui|2

2

dρidt

+ ui ·dmi

dt


The energy equation is not necessary to derive the kinetic energy equation.Using these equations, we can derive the discrete global kinetic energy balanceequation by summing up over all the vertices. We first notice that F ρij = −F ρjiand Fmij = −Fmji . We also introduce the symbol Γa to denote all the edges inthe grid. The global kinetic energy equation is derived in several steps below.We begin by adding up the kinetic energy equation from all the cells.

d

dt

∑i

kiAi = −∑i

∑j∈i

(ui · Fmij −

|ui|2

2F ρij

)−∑i

∑e∈Γi

(ui · Fmi,e −

|ui|2

2F ρi,e

)+

1

2

∑i

∑T∈i

ui · σT · nTi +1

2

∑i

∑e∈Γi

ui · σTe · ne

= −∑e∈Γa

[(ui − uj) · Fmij −

|ui|2 − |uj |2

2F ρij

]

−∑e∈Γ

[ui · Fmi,e + uj · Fmj,e −

|ui|2

2F ρi,e −

|uj |2

2F ρj,e

]+

1

2

∑T

∑i∈T

ui · σT · nTi +∑e∈Γ

ui + uj2

· σTe · ne

= I + II + III (32)

where the last two terms have been denoted as III. Let F ρij be any consistentmass flux; then the momentum flux is taken to be of the form [19]

Fmij = pijnij + uijFρij , uij =

1

2(ui + uj) (33)

The first term in the right hand side of equation (32) can be written as

(ui−uj)·Fmij−|ui|2 − |uj |2

2F ρij = (ui−uj)·

[Fmij −

ui + uj2

F ρij

]= pij(ui−uj)·nij

For the pressure we assume pij = 12 (pi+pj); the pressure terms can be rewrit-

ten as

−∑e∈Γa

pij(ui − uj) · nij = −∑e∈Γa

[pi2

(ui − uj) · nij +pj2

(ui − uj) · nij]

= −∑i

1

2pi∑j∈i

(ui − uj) · nij

where i, j are the two vertices of the edge e.


∂Ωi rs

ne1ne2

Fig. 11 Dual cell around a boundary vertex

Case 1 (Interior vertex) For an interior vertex i we collect all the terms con-taining pi and add piui ·

∑j∈i nij which is zero since

∑j∈i nij = 0.

−1

2pi∑j∈i

(ui − uj) · nij = −1

2pi∑j∈i

(ui − uj) · nij + piui ·∑j∈i

nij

= pi∑j∈i

ui + uj2

· nij = pi(∇h · u)iAi

The last step provides an approximation to the divergence at vertex i byGreen’s theorem which is denoted as (∇h · u)i.

Case 2 (Boundary vertex) Now we consider a boundary vertex i as shown infigure (11).

−1

2pi∑j∈i

(ui − uj) · nij

= −1

2pi∑j∈i

(ui − uj) · nij + piui ·

∑j∈i

nij +1

2ne1 +

1

2ne2

︸︷︷︸

=0

= pi∑j∈i

ui + uj2

· nij +1

2piui · (ne1 + ne2)

= pi

∑j∈i

ui + uj2

· nij +1

2fi · (ne1 + ne2)

+1

2piui · (ne1 + ne2)

−1

2pifi · (ne1 + ne2)

= pi(∇h · u)iAi +1

2piui · (ne1 + ne2)− 1

2pifi · (ne1 + ne2) (34)

The term inside the square brackets in the penultimate step gives an ap-proximation to the divergence (∇h · u)i at the boundary vertex i via Green’stheorem. Hence we obtain

I =∑i

pi(∇h · u)iAi +∑e∈Γ

1

2(piui + pjuj) · ne −

∑e∈Γ

1

2(pifi + pjfj) · ne (35)


The above equation suggests that for a boundary edge e, the contributions ofthe momentum flux to the two vertices i and j should be of the form

Fmi,e =1

2pine +

1

2ρiui(fi · ne), F ρi,e =

1

2ρi(fi · ne)

Fmj,e =1

2pjne +

1

2ρjuj(fj · ne), F ρj,e =

1

2ρj(fj · ne)

(36)

In the momentum flux, notice that the Dirichlet condition is used only forthe normal velocity fi · ne and fj · ne while the momentum is still evaluatedwith the solution values ui, uj which is necessary to obtain the correct kineticenergy equation. Then

II = −∑e∈Γ

1

2(piui + pjuj) · ne (37)

The last two terms in equation (32) can be manipulated similar to the heatequation case to obtain

III =1

2

∑T

∑i∈T

ui · σT · nTi +∑e∈Γ

(ui + uj

2

)· σTe · ne

= −∑T

(σT : ∇huT )|T |+∑e∈Γ

(fi + fj

2

)· σTe · ne (38)

Using (35), (37), (38), the discrete kinetic energy equation (32) leads to

d

dt

∑i

kiAi = −∑e∈Γ

[(ki + pi)fi + (kj + pj)fj

2

]· ne +

∑e∈Γ

(fi + fj

2

)· σTe · ne

+∑i

pi(∇h · u)iAi −∑T

(σT : ∇huT )|T |

This equation is consistent with the continuous kinetic energy equation (25).In the case of homogeneous velocity conditions fi ≡ 0, we obtain

d

dt

∑i

kiAi =∑i

pi(∇h · u)iAi −∑T

(σT : ∇uT )|T | ≤∑i

pi(∇h · u)iAi

which is again consistent with the exact solution and shows a non-linear sta-bility property of the scheme in terms of a bound on the growth of kineticenergy.

5.4 Godunov-MUSCL finite volume scheme

The central scheme discussed in the previous section is useful when performingDNS of compressible flows where the mesh is sufficiently fine to have a stablescheme without additional dissipation [19]. This can also be characterised interms of the local mesh Peclet number being of order unity. For unresolved


simulations where the Peclet number would be large, and also for simulatingEuler equations which do not contain any physical viscosity, some numericaldissipation is necessary to stabilize the scheme. The inviscid flux Fij · nij canbe computed using a numerical flux function

Fij · nij =

F ρijFmijFEij

= F (Ui, Uj ,nij) (39)

which includes explicit or implicit numerical dissipation. There is a wide choiceof flux functions that are available based on central or upwind principles [23].We will combine such a numerical flux together with the SBP discretizationof viscous and heat conduction terms in the momentum and energy equationsand penalty terms for the Dirichlet boundary conditions for the velocity andtemperature. The semi-discrete scheme then has the following form where theusual flux contributions are not indicated but can be seen in (26), (27), (28)

Aidρidt

= . . .

Aidmi

dt= . . .+

∑e∈Γns

i

CpµTe

2he(fi − ui)|ne|

AidEidt

= . . .+∑

e∈Γ isoti

CpκTe

2he(Tw,i − Ti)|ne|

We have added penalty terms into the momentum and energy equations fornoslip Γns and isothermal Γ isot boundaries. To achieve higher order accuracythe MUSCL approach [24] is used in which the numerical flux is calculated as

Fij · nij = F (Vij , Vji,nij) (40)

where we have written the flux in terms of the primitive variables

V = [p, u, T ]> (41)

The quantities Vij , Vji are the reconstructed values of V from i and j to themid-point of the edge ij. We perform the reconstruction in terms of the primi-tive variables since it is easier to enforce positivity of pressure and temperaturein the reconstructed values Vij , Vji. It is also useful for implementing bound-ary conditions involving pressure or temperature, which are the more commontype of boundary conditions that one encounters in practice. The interfacevalues Vij are obtained using a MUSCL-type edge-based reconstruction whichcan be written as [25]

Vij = Vi +sij4

[(1− κsij)∆−Vij + (1 + κsij)∆+Vij ]

where

∆−Vij = 2(∇hV )i · rij − (Vj − Vi), ∆+Vij = Vj − Vi, rij = rj − ri


The gradients of primitive variables at the vertices (∇hV )i are computed using

(∇hV )i =1

Ai

∑T∈i

(∇hV T )|T |

The triangle gradients ∇hV T are evaluated using formulae like (2), (4) forscalar quantities like pressure and temperature, and (29), (30) for the velocityvector; note that these approximations incorporate the boundary conditions ifspecified on any portion of the boundary. The triangle gradients for velocityand temperature are anyway required for the viscous terms and we use them tocompute the vertex gradients with only a little additional work. The quantitysij is a limiter which enforces monotonicity of the reconstruction by reducing itto first order in the presence of shocks or steep gradients. In the computations,we use the van Albada limiter which is given by

sij = L(∆−Vij , ∆+Vij), L(a, b) = max

(0,

2ab+ ε

a2 + b2 + ε

), 0 < ε 1

An alternate is to use a multi-dimensional reconstruction approach [26] whichcan be written as

Vij = Vi +φi2

(∇hV )i · rij

where the limiter function φi is evaluated using the min-max or Barth-Jespersenlimiter, see [27] or the more differentiable limiter of Venkatakrishnan [28].

5.5 Numerical results

In the previous sections, the finite volume scheme was presented in a semi-discrete form where time was still continuous, leading to a system of ODEsin time. In the computations below, we use the three stage strong stabilitypreserving Runge-Kutta scheme of Shu-Osher [29] to advance the solutionin time. For steady state problems, a matrix-free LUSGS scheme is used toachieve faster convergence.

5.5.1 Plane Couette flow

The domain consists of [0, 5]× [0, 1] and the top surface y = 1 is moved witha velocity of (1, 0). Isothermal conditions are applied on the top and bottomsurfaces. At the left domain, the velocity is specified as u = y. At the outlet aconstant pressure boundary condition is applied through the numerical fluxes.The Roe flux is used for convective flux and the penalty parameter in caseof weak boundary conditions is Cp = 10. The exact incompressible solutionis u = y but we solve this problem using our compressible solver at an inletMach number of 0.1. A uniform mesh of 25× 25 which is triangulated shownin figure (12). The initial condition is taken to be zero velocity so that solutionevolves due to the boundary conditions. The maximum error in the velocity at


Fig. 12 Grid used for Couette flow and plane Poiseille flow

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Res

idua

l

Number of iterations

Densityx momentumy momentum

Energy

1e-10

1e-08

1e-06

0.0001

0.01

1

100

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Res

idua

l



Energy

(a) (b)

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

100

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Res

idua

l



Energy

(c)

Fig. 13 Convergence of residual for Couette flow: (a) strong, (b) weak without penalty (c)weak with penalty

the walls is of the order of 10−6− 10−5. While the velocity on the boundary isnot exactly equal to the boundary velocity in the case of weak implementation,the L2 norm of the error in the velocity is 4.44 × 10−5 while in the case ofstrong boundary condition it is 6.38 × 10−5. The convergence of the residualis shown in figure (13); the strong case does not converge below 10−6 whilethe weak case without penalty also saturates but at a lower residual while theaddition of penalty term leads to full convergence.

5.5.2 Plane Poiseille flow

This problem involves flow between two parallel plates driven by a constantpressure gradient for which an exact incompressible solution is available whichis a parabolic velocity profile. Again we cannot exactly approach this solutionsince we are using a compressible solver. The domain, mesh and numerical


parameters are identical to the plane couette flow problem. A pressure differ-ence is specified between the inlet and outlet boundaries while the top andbottom boundaries have noslip and isothermal conditions. The L2 norm of theerror in the velocity is 5.91 × 10−4 and 7.10 × 10−4 in the case of weak andstrong implementations respectively. If the weak conditions are used withoutthe Nitsche penalty the error is 4.31×10−3 which shows the necessity of usinga penalty term. The residuals converge to machine precision for the weak casewith penalty term, while full convergence is not obtained in the strong caseand even for the weak case when the penalty terms are not used.

5.5.3 Lid-driven cavity

The lid driven cavity is a classic test case for incompressible Navier-Stokesequations. The cavity is a unit square with the top surface being moved ata constant velocity of (1, 0) while the other sides are stationary. The centralkinetic energy preserving flux [19] is used without any dissipation in the flux.We use conditions corresponding to a mach number of less than 0.1 in ourcompressible code and the Reynolds number is 1000. Isothermal boundaryconditions are imposed on all the four sides of the cavity. A uniform triangulargrid with 128 nodes on each side of the cavity and a total of 21180 nodes isused. The velocity magnitude and velocity vectors are shown in figure (14) nearthe top right corner of the cavity. The horizontal and vertical velocity profilesalong the vertical and horizontal centerlines are shown in figure (15) and arecompared with solution from [30]. The temperature should remain constantwhile in the numerical solution the maximum variation in temperature is 0.3% in the whole domain; the variation in temperature is larger in this casedue to compressibility effects at the top corner points where the velocity isdiscontinuous. The strong implementation of the boundary conditions leadsto similar results (not shown) but the residuals do not converge while theyconverge for the weak implementation.

5.5.4 Laminar boundary layer

This problem corresponds to viscous flow over a flat plate which leads tothe development of a boundary layer near the plate surface. The Reynoldsnumber corresponding to the plate length is 105 while the Mach number ofthe incoming flow is taken to be 0.1. The computation domain is rectangularas shown in figure (16) which also shows the grid used for the computations.There is an initial inlet portion of the domain on which slip boundary conditionis imposed followed by the no-slip boundary corresponding to the flat plate.Adiabatic conditions are used on the flat plate boundary. At the top and outlet,the free-stream pressure is specified while at the inlet the free-stream valuesare used together with the numerical flux function to compute the flux. Theinitial condition corresponds to uniform flow of (u∞, 0) which does not satisfythe no-slip condition on the flat plate. Once steady state solution is obtained,the maximum variation in temperature is 0.17% in the whole computational


Velocity Magnitude

0.950.90.850.80.750.70.650.60.550.50.450.40.350.30.250.20.150.10.05

Fig. 14 Lid drivern cavity at Re=1000 and weak boundary conditions: Velocity magnitudeand velocity vectors

−0.2 0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

u

y

GhiaTaxis

0 0.2 0.4 0.6 0.8 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

x

v

GhiaTaxis

(a) (b)

Fig. 15 Lid drivern cavity at Re=1000 using weak boundary conditions and central KEPflux: (a) x velocity, (b) y velocity

Inlet portion Flat plate

Fig. 16 Domain and grid for laminar boundary layer problem


0

1

2

3

4

5

6

7

0 0.2 0.4 0.6 0.8 1 1.2

u/ui

nf

eta

BlasiusFVM

0

1

2

3

4

5

6

7

0 0.2 0.4 0.6 0.8 1 1.2

v*sq

rt(2*

Rex

)/uin

f

eta

BlasiusFVM

(a) (b)

Fig. 17 Laminar boundary layer with weak boundary conditions: (a) x velocity (b) y ve-locity

domain. We compare the numerical solution of the velocities with the blasiussemi-analytical solution in figure (17) in the standard non-dimensional units.These results are taken from the center point of the plate in the downstreamdirection. It is clear that the solutions agree very well with the analyticalsolutions and the weak implementation of the boundary conditions is able todrive the flow towards the no-slip condition.

5.5.5 Flow over NACA0012 airfoil

We consider laminar flow over the standard NACA0012 airfoil at a freestreamReynolds number of 500, Mach number of 0.8 and 10 degree angle of attack.The grid consists of 10458 vertices with 198 vertices on the airfoil surface. Fig-ure (18) shows the comparison of velocity vectors near the leading edge of theairfoil for strong and weak boundary conditions. The two solutions are nearlyidentical and the weak boundary conditions are effective in enforcing the noslipcondition. Figure (19a) shows the contours of Mach number for the weak casewhich shows the wake arising due to separation of the flow. Figure (19b) showsthe comparison of the skin friction coefficient for the two different boundaryconditions which also compares well with published literature2. Laminar flowa Reynolds number of 5000 and zero degrees angle of attack is also computedusing the two boundary conditions. Figure (20a) shows the velocity vectorsnear the leading edge of the airfoil indicating a more prominent boundarylayer than the previous case, and the satisfaction of the noslip condition togood accuracy. The skin friction coefficient is compared for the two boundaryconditions which also compare favourably with published results.

2 B. Fortunato and M. Vinicio, In Proceedings of 14th ICNMFD, Lecture Notes in Physics,Springer-Verlag Berlin 1995, pp. 259.


Fig. 18 Laminar flow past NACA-0012 airfoil, Re=500, α = 10 deg: velocity vectors for(a) strong and (b) weak boundary conditions

Mach

1.11.0510.950.90.850.80.750.70.650.60.550.50.450.40.350.30.250.20.150.10.05 0 0.2 0.4 0.6 0.8 1

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x/c

Cf

StrongWeakFortunato

(a) (b)

Fig. 19 Laminar flow past NACA-0012 airfoil, Re=500, α = 10 deg: (a) Mach number forweak BC case (b) skin friction

5.5.6 Axisymmetric rotating flow inside an annulus

This test case involves an annulus filled with Argon gas (γ = 0.67, Pr = 0.668,R = 208 J/(kg·K), µ = 2.273×10−5 kg/(m·s)) and rotating at a high speed of3140 radians/sec which is approximately 1000 revolutions/sec. The inner andouter radii are 0.1 m and 0.2 m respectively while the height of the cylinder is0.1 m. A triangular grid with 100 points on each side and a total of 10000 pointsis used. The temperature boundary conditions are shown in figure (21a). Thereare five equations to be solved since the swirl velocity is also a function of radialand axial coordinates. The equations can be written in conservation form andlook similar to the two dimensional equations except that the radial coordinateis present in all the terms of the equation. Dirichlet conditions on all the threevelocity components and the temperature are imposed weakly together withpenalty terms. The KFVS scheme [31] is used for the numerical flux function


0 0.2 0.4 0.6 0.8 1

−0.1

−0.05

0

0.05

0.1

0.15

x/cC

f

StrongWeakVenkatakrishnan

(a) (b)

Fig. 20 Laminar flow past NACA-0012 airfoil at Re=5000, α = 0 deg: (a) Velocity vectorsfor weak boundary conditions (b) skin friction

Radial

Axi

al

0.1 0.12 0.14 0.16 0.18 0.20

0.02

0.04

0.06

0.08

0.1T=290 K

T=310 K

T=300 K T=300 K

T

309308307306305304303302301300299298297296295294293292291

(a) (b)

Fig. 21 High speed rotating annulus: (a) domain and temperature boundary conditions,(b) temperature

which is known to be robust in terms of positivity preservation. For the secondorder reconstruction, the min-max limiter is used wich ensures positivity ofreconstructed pressure and temperature. The strong implementation could notsucceed due to loss of positivity property of density/pressure near the cornersof the annulus. For the weak implementation, the temperature solution isshown in figure (21b) which indicates that the isothermal conditions havebeen well approximated by the weak scheme, while the density is shown infigure (22b). There is a large nearly exponential radial gradient in density andpressure due to the high speed rotation and the density is very small nearthe inner cylinder. The non-uniform temperature of the walls sets up a weaksecondary flow which is shown in figure (22a).


Density

0.000240.000220.00020.000180.000160.000140.000120.00018E-056E-054E-05

(a) (b)

Fig. 22 High speed rotating annulus: (a) secondary flow streamlines (b) density

6 Summary and conclusions

We have proposed a novel method for finite volume approximation of Laplaceoperator on triangular grids which has the summation-by-parts property on tri-angular grids in two dimensions with proper consideration of Dirichlet bound-ary conditions. The new scheme implements Dirichlet boundary conditions ina weak manner and leads to an energy stability property for the time depen-dent heat equation. We also propose the use of a Nitsche-type penalty termwhich is found to improve the accuracy of the scheme. The weak implementa-tion gives solutions which are as accurate as the strong boundary conditions.The general idea for the approximation of Laplacian type operator as pre-sented here can be used in other problems also. These ideas are extended tothe compressible Navier-Stokes equations and we prove kinetic energy stabilityproperty of the finite volume scheme. Such schemes can be useful for DNS ofcompressible flows where kinetic energy consistency plays an important role incorrectly capturing the energy cascade mechanism. For unresolved simulations,the scheme needs to contain some explicit or implicit numerical dissipation.We use the SBP discretization of viscous and heat transfer terms in combi-nation with a numerical flux function for the inviscid fluxes and show thatthis leads to accurate solutions on many standard test problems. The weakimplementation of the boundary conditions on velocity and temperature alsopromotes better iterative convergence for steady state problems which is notthe case with strong boundary conditions. These schemes can be extended totetrahedral grids in three dimensions.

Acknowledgements

The author was supported by the Airbus Chair on Mathematics of ComplexSystems at TIFR-CAM, Bangalore, in carrying out this work.


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