# Finite volume discretization of heat equation and compressible Navier–Stokes equations with weak Dirichlet boundary condition on triangular grids

## Abstract

A vertex-based finite volume method for Laplace operator on triangular grids is proposed in which Dirichlet boundary conditions are implemented weakly. The scheme satisfies a summation-by-parts (SBP) property including boundary conditions which can be used to prove energy stability of the scheme for the heat equation. A Nitsche-type penalty term is proposed which gives improved accuracy. The scheme exhibits second order convergence in numerical experiments. For the compressible Navier–Stokes equations we construct a finite volume scheme in which Dirichlet boundary conditions on the velocity and temperature are applied in a weak manner. Using the centered kinetic energy preserving flux, the scheme is shown to be consistent with the global kinetic energy equation. The SBP discretization of viscous and heat conduction terms together with penalty terms are combined with upwind fluxes in a Godunov-MUSCL scheme. Numerical results on some standard test cases for compressible flows are given to demonstrate the performance of the scheme.

### Keywords

Finite volume Triangular grids SBP Energy stability Compressible Navier–Stokes Kinetic energy preservation## 1 Introduction

Finite volume methods might be cell-centered or vertex-centered depending on the spatial location of the solution. In the latter case, a dual finite volume has to be constructed around each vertex, including vertices on the boundary. On triangular/tetrahedral grids, the vertex-based scheme has a flavour of finite element method using \(P_1\) basis functions, especially for the discretization of Laplacian term [1]. A difficulty that presents itself with vertex-centered scheme is the implementation of boundary conditions since there are degrees of freedom located on the boundaries. For the Poisson or heat equation, one can directly set the Dirichlet boundary condition for vertices on the boundary and update only the interior vertices using the finite volume method. This is the *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 practice is to update all the quantities for a boundary vertex using the finite volume method and then reset the velocity to satisfy the no-slip condition. The degrees of freedom in a compressible model are the mass density, momentum density and energy density while boundary conditions are usually provided on velocity and temperature, which are not degrees of freedom but are derived quantities from the actual degrees of freedom.

In a *weak* implementation of Dirichlet boundary conditions, one updates boundary points also using the finite volume method which should implicitly account for the boundary conditions. The boundary vertex value is not reset to the boundary condition value as in the strong implementation, so that the solution on the boundary vertices do not exactly agree with the Dirichlet conditions. It is not necessary to exactly satisfy the boundary conditions since anyway the interior solution is only approximate. The error in the boundary solution should be acceptable as long as it is of the same order as the error of the interior solution and the global error converges at expected rates. It must also be remembered that the solution of a finite volume method denotes cell average values and not point values. The cell average value is a second order approximation to the solution at the centroid of the finite volume. For a dual cell around a boundary vertex, the cell lies only on one side of the boundary and the cell centroid does not lie on the boundary. The solution at a boundary vertex in a finite volume method thus represents the average value in the boundary cell and is not the value of the solution at the boundary vertex.

The guiding principle in the construction of a scheme with weakly implemented boundary conditions is the satisfaction of an energy estimate that is consistent with the energy estimate of the exact solution of the partial differential equation. Thus the boundary conditions are chosen so that the resulting scheme is stable in \(L_2\)-norm. At the mathematical level, energy stability is obtained by using integration by parts and its discrete counterpart is the summation-by-parts (SBP) property [2, 3]. This can also be characterized in terms of the skew-symmetry and symmetry properties of the difference approximations to convective (first order) and diffusive (second order) partial differential operators. In order to obtain energy stability in the presence of boundary conditions, the simultaneous approximation term (SAT) approach [4] has been used which imposes the boundary conditions in a weak manner. On Cartesian and structured grids, SBP-SAT schemes have been developed for hyperbolic and parabolic problems, including the Euler and Navier–Stokes equations [5, 6]. For a linear convection–diffusion equation, weak boundary conditions in a second order SBP-SAT scheme are found to give more accurate solutions on coarse meshes while on finer meshes, the strong implementation was more accurate [7] and similar behavior was observed for the solution of Navier–Stokes equations. Weak imposition of boundary conditions is found to lead to faster convergence of solutions to steady state problems while in the case of strong imposition, the residuals may not converge to machine precision [8].

Mimetic schemes [9] are constructed to satisfy the equations of integral calculus like divergence theorem and related identities, which endows them with SBP properties. Dirichlet boundary conditions have been implemented strongly in mimetic methods on logically rectangular grids for Poisson equation [10]. A kinetic energy preserving scheme for incompressible Navier–Stokes equations on logically rectangular grids is proposed in [11]. Stable SBP schemes on unstructured grids for hyperbolic problems using vertex-based finite volume method have been studied in [12] where boundary conditions are imposed weakly through the fluxes using a local characteristic decomposition that separates incoming and outgoing waves. Higher order mimetic methods which have SBP property have been developed in [13] for diffusion equation but boundary conditions and their influence on stability property was not analyzed. For the discretization of second order terms on unstructured grids, there are few works which have studied the energy stability property. Svard and Nordstrom [14, 15] study the vertex-based discretization of Laplacian on unstructured grids using an edge-based scheme which is shown to be stable since resulting matrix has negative eigenvalues. For boundary vertices, they construct two dual cells to obtain an estimate of the normal derivative of the solution which is used to compute the flux across the boundary faces. In numerical experiments they find that the scheme does not converge even on a uniform triangulated Cartesian grid, while convergence is obtained only with equilateral grids. An alternate approach to the edge-based scheme is to use a \(P_1\) Galerkin approximation on 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 the Laplace operator on triangular grids. Consistency of the schemes with respect to energy evolution can be considered as a secondary conservation principle satisfied by the scheme. A recent review of discrete conservation principles on unstructured grids leading to second conservation properties can be found in [17].

In this work we restrict ourselves to triangular grids in two spatial dimensions and consider the vertex-based finite volume scheme. We construct a scheme for the Laplacian which is similar to the \(P_1\) Galerkin approach but we incorporate boundary conditions within the gradient approximations used for the Laplacian which implicitly accounts for an SAT approach. This allows us to show a SBP property for the Laplacian approximation *including boundary conditions* which also leads to a discrete energy equation for the solution of heat 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 numerical experiments on the heat equation, we show that the solutions converge at a rate of \(O(h^2)\) where *h* is a typical size of the triangles. These ideas are extended to the compressible Navier–Stokes equations for the discretization of viscous terms and heat conduction terms appearing in the momentum and energy equation. No-slip and isothermal boundary conditions are implemented in a weak manner and Nitsche-type penalty terms are also used in the momentum 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 combined with a numerical flux function in a Godunov finite volume scheme for which higher order accuracy can be obtained via the MUSCL approach. The discretization of diffusive terms can be implemented in existing finite volume codes with little extra modifications. The presented schemes can be naturally extended to three dimensions on tetrahedral grids. However it is not easy to develop similar methods for hybrid grids. In two dimensions we have only succeeded in constructing an SBP scheme on grids containing triangles and parallelograms.

The rest of the paper is organized as follows. Section 2 introduces the new SBP discretization for the Laplacian and shows the SBP property. Section 3 presents the finite volume scheme for Poisson equation and its solvability is shown. The scheme is then applied to heat equation in Sect. 4 and an energy equation is demonstrated for the semi-discrete scheme. Numerical experiments on steady and unsteady heat conduction problems are given to demonstrate the convergence properties. Section 5 discusses the application of the new scheme to the compressible Navier–Stokes equations for which a global kinetic energy balance equation is demonstrated which is consistent with the true balance equation. Then the SBP scheme is combined with a Godunov-MUSCL scheme. Finally, many numerical results are shown for compressible flows in two dimensions and for axisymmetric rotating flow.

## 2 Finite volume approximation of Laplacian

*i*,

*j*,

*k*etc. For each triangle we define the outward normal vectors as shown in Fig. 1 with the magnitude of the vector equal to the length of the corresponding edge; e.g., \({\mathbf {n}}_i^T\) denotes the outward normal to the face in triangle

*T*which is opposite to the vertex

*i*. We will distinguish between two types of triangles referred to as

*interior*and

*boundary*triangles. An interior triangle does not have any of its faces on the Dirichlet boundaries while a boundary triangle has atleast one face on some Dirichlet boundary. The derivative is approximated on a triangle by the Green–Gauss theorem

*e*, see Fig. 2, is given by Eq. (1) but we make use of the Dirichlet boundary condition for the integral on the boundary edge

*e*; This leads to the following approximation for the gradient on \(T_e\)

*Since the boundary conditions are to be implemented weakly during the solution of the PDE, the solution values*\(u_i,u_j\)

*need not exactly coincide with the boundary values*\(f_i\), \(f_j\). This is an important point about the scheme we present and must be remembered when reading the rest of the paper. We make the following definitions

*i*the set \(\varGamma _i\) is empty. Around each vertex

*i*, we construct the dual cell by joining the cell centroid to the mid-point of the edges, see Fig. 3a. For a boundary vertex, the cell is closed by the boundary edges as shown in Fig. 3b. Integrate the Laplacian over a dual cell \(A_i\) to obtain

*T*, we estimate \(\nabla u \approx \nabla _h u^T\) while for a boundary edge integral on an edge

*e*, the gradient from the adjacent triangle \(T_e\) is used. This leads to the following approximation to the Laplacian

*u*,

*v*defined on \(\varOmega \) with \(u=f\), \(v=g\) on \(\partial \varOmega \), we have the relation

**Theorem 1**

*Let*

*u*,

*v*

*be two functions defined on*\(\varOmega \)

*with*\(u=f\), \(v=g\)

*on*\(\partial \varOmega \).

*Then the discrete approximation given by Eq.*(5)

*satisfies the summation-by-parts property*

*where the vertices*

*i*,

*j*

*form the boundary edge*

*e*.

*Proof*

*T*, we use the second form of the gradient formula, Eq. (3), to get

*e*, we need to account for the Dirichlet conditions. Assume that the edge

*e*is bounded by the vertices

*i*,

*j*while

*k*is the third vertex of the triangle \(T_e\) so that \({{\mathbf {n}}_e}={\mathbf {n}}_k^{T_e}\), see Fig. 2; using the Eqs. (2), (3), (4) we obtain

*Remark*

*u*and

*v*, we obtain the following SBP formula

*u*,

*v*vanish on the boundary, i.e., \(f=g=0\), then

*Remark*

## 3 Finite volume method for Poisson equation

*e*. In the computations, we take \(h_e\) to be the height of the triangle adjacent to the edge

*e*, i.e.,

**Theorem 2**

*Consider the finite volume scheme given by Eq.*(13).

- 1.
*If*\(C_p> 0\),*then it has a unique solution*. - 2.
*If*\(C_p=0\)*and if all boundary vertices belong to at atleast one interior triangle, then it has a unique solution*. - 3.
*If*\(C_p=0\)*and if at least one boundary vertex belongs to at atleast one interior triangle, then it has a unique solution*.

*Proof*

- 1.Assume that \(C_p> 0\). ThenOn an interior triangle$$\nabla _h u^T = 0 \quad \forall \quad T \qquad {\text {and}} \qquad u_i = 0 \quad \forall \quad i \in \varGamma $$
*T*we haveand since \({\mathbf {n}}_j^T\), \({\mathbf {n}}_k^T\) are linearly independent, we conclude that \(u_i = u_j = u_k\). On a boundary triangle \(T_e\) adjacent to a boundary edge$$ \nabla _h u^T = -\frac{1}{2|T|}\left[ (u_j-u_i){\mathbf {n}}_j^T+(u_k-u_i){\mathbf {n}}_k^T\right] =0 $$*e*as in Fig. 2 we have from Eq. (4)which implies that \(u_k = 0\). These two facts are enough to conclude that \(u_i = 0 \ \forall \ i.\)$$ \nabla _h u^{T_e} = \frac{1}{|T|}\left[ \frac{0 + 0}{2} {\mathbf {n}}_k^T + \frac{0 + u_k}{2} {\mathbf {n}}_i^T + \frac{u_k + 0}{2} {\mathbf {n}}_j^T \right] = -\frac{u_k}{2|T|} {\mathbf {n}}_k^T = 0$$ - 2.Assume that \(C_p=0\). Then we can only conclude thatFor an interior triangle we again obtain that all the three vertex values are equal. Consider a boundary vertex$$ \nabla u^T = 0 \quad \forall \quad T $$
*i*which belongs to an interior triangle \(T_1\) and a boundary triangle \(T_e\) as shown in Fig. 4. Since \(T_1\) is an interior triangle, from the condition \(\nabla _h u^{T_1}=0\) we conclude that \(u_i = u_k\) while from \(\nabla _h u^{T_e}=0\), i.e.,we conclude that \(u_i + u_k = 0\) which together imply that \(u_i = 0\). Hence we can conclude that \(u_i = 0 \ \forall \ i\).$$ \nabla _h u^{T_e} = \frac{1}{|T|}\left[ \frac{0 + 0}{2} {\mathbf {n}}_k^T + \frac{u_j + u_k}{2} {\mathbf {n}}_i^T + \frac{u_k + u_i}{2} {\mathbf {n}}_j^T \right] = 0 $$ - 3.Assume that \(C_p=0\). Then we can only conclude thatFor an interior triangle we again obtain that all the three vertex values are equal. Now consider a boundary vertex$$\nabla u^T = 0 \quad \forall \quad T $$
*i*that belongs to only boundary triangles and does not belong to any interior triangle, as in Fig. 5. From \(\nabla _h u^{T_e}=0\), we can conclude that \(u_i + u_k = 0\) and \(u_j + u_k=0\) which implies that \(u_i = u_j = -u_k\). Similarly from \(\nabla _h u^{T_1}=0\) we conclude that \(u_i = u_r = -u_k\). But since \(T_2\) is an interior triangle, \(\nabla _h u^{T_2}=0\) implies \(u_r = u_k\) and hence we obtain \(u_i = u_j = u_r = 0\). From this we can conclude that \(u_i = 0\), \(\forall \,i\).\(\square \)

*Remark*

We see that the addition of a penalty term leads to a simple proof of existence. However if \(C_p=0\), the difficulty in the proof is in case (3) where a vertex *i* belongs to only boundary triangles. However, geometrically it is not possible that all boundary vertices belong to only boundary triangles. For example, in Fig. 5 the vertex *r* will belong to an interior triangle and the proof can be completed by connecting vertex *i* to vertex *r* by moving along boundary edges.

## 4 Discretization of heat equation

*u*and integration over \(\varOmega \) gives the energy equation

### 4.1 Finite volume method

*u*we obtain

*e*is computed using the approximation given by Eq. (4) while for a boundary triangle adjacent to any other type of boundary, the gradient is computed from Eq. (3). Using the above approximation, we formulate the finite volume scheme including a Nitsche type penalty term for Dirichlet boundary condition as

**Theorem 3**

*Assume that atleast one of*\(\varGamma ^D\), \(\varGamma ^R\)*is non-empty*. *Then under homogeneous data*\(f=g=h=0\), *the semi-discrete finite volume scheme given by* (17) *is stable in the energy norm*.

*Proof*

### 4.2 Need for penalty term

*i*belongs to a single boundary triangle as shown in Fig. 6. We start the computations by setting the initial value to be zero at all the vertices. Then the equation for the corner point is

*i*remains zero for all future times while the exact solution tends to unity. The addition of a penalty term will avoid this situation since the penalty term drives the solution at all boundary points towards the correct boundary value. It is of course possible to avoid such situations by re-triangulating the grid but the penalty term gives a simpler solution without changing the grid.

### 4.3 Other type of dual cells

### 4.4 Numerical implementation

*g*,

*h*to be also discontinuous across the boundary edges.

### 4.5 Test I: smooth solution

### 4.6 Test II: discontinuous diffusivity

### 4.7 Test III: discontinuous diffusivity

## 5 Compressible Navier–Stokes equations

*U*is the vector of conserved variables,

*F*are the inviscid fluxes and

*G*are viscous fluxes

*p*is the pressure,

*E*is total energy per unit volume, while \(\sigma \) and \({\mathbf {q}}\) are the shear stress tensor and heat flux vector respectively. The pressure is related to the other quantities through perfect gas relation and takes the form

### 5.1 Kinetic energy equation

### 5.2 Finite volume method

*i*by an edge. Integrating the conservation law over a dual cell, we get

^{1}

*e*whose vertices are

*i*,

*j*, the gradient is given by

### 5.3 Discrete kinetic energy balance

*III*. Let \(F^{\rho }_{ij}\) be any consistent mass flux; then the momentum flux is taken to be of the form [19]

*i*,

*j*are the two vertices of the edge

*e*.

**Case 1**(

*Interior vertex*) For an interior vertex

*i*we collect all the terms containing \(p_i\) and add \(p_i {\mathbf {u}}_i \cdot \sum _{j\in i} {\mathbf {n}}_{ij}\) which is zero since \(\sum _{j\in i} {\mathbf {n}}_{ij} = 0\).

*i*by Green’s theorem which is denoted as \((\tilde{\nabla }_h \cdot {\mathbf {u}})_i\).

**Case 2**(

*Boundary vertex*) Now we consider a boundary vertex

*i*as shown in Fig. 11.

*i*via Green’s theorem. Hence we obtain

*e*, the contributions of the momentum flux to the two vertices

*i*and

*j*should be of the form

### 5.4 Godunov-MUSCL finite volume scheme

*combine such a numerical flux together with the SBP discretization of viscous and heat conduction terms in the momentum and energy equations and penalty terms for the Dirichlet boundary conditions for the velocity and temperature*. The semi-discrete scheme then has the following form where the usual flux contributions are not indicated but can be seen in (26), (27), (28)

*V*from

*i*and

*j*to the mid-point of the edge

*ij*. We perform the reconstruction in terms of the primitive variables since it is easier to enforce positivity of pressure and temperature in the reconstructed values \(V_{ij}\), \(V_{ji}\). It is also useful for implementing boundary conditions involving pressure or temperature, which are the more common type of boundary conditions that one encounters in practice. The interface values \(V_{ij}\) are obtained using a MUSCL-type edge-based reconstruction which can be written as [25]

### 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 ODEs in time. In the computations below, we use the three stage strong stability preserving Runge–Kutta scheme of Shu–Osher [29] to advance the solution in time. For steady state problems, a matrix-free LUSGS scheme is used to achieve faster convergence.

#### 5.5.1 Plane Couette flow

#### 5.5.2 Plane Poiseille flow

This problem involves flow between two parallel plates driven by a constant pressure gradient for which an exact incompressible solution is available which is a parabolic velocity profile. Again we cannot exactly approach this solution since we are using a compressible solver. The domain, mesh and numerical parameters are identical to the plane couette flow problem. A pressure difference is specified between the inlet and outlet boundaries while the top and bottom boundaries have no-slip and isothermal conditions. The \(L_2\) norm of the error in the velocity is \(5.91\times 10^{-4}\) and \(7.10\times 10^{-4}\) in the case of weak and strong implementations respectively. If the weak conditions are used without the Nitsche penalty the error is \(4.31\times 10^{-3}\) which shows the necessity of using a penalty term. The residuals converge to machine precision for the weak case with penalty term, while full convergence is not obtained in the strong case and even for the weak case when the penalty terms are not used.

#### 5.5.3 Lid-driven cavity

#### 5.5.4 Laminar boundary layer

#### 5.5.5 Flow over NACA0012 airfoil

^{2}Laminar flow a Reynolds number of 5000° and 0° angle of attack is also computed using the two boundary conditions. Figure 20a shows the velocity vectors near the leading edge of the airfoil indicating a more prominent boundary layer than the previous case, and the satisfaction of the no-slip condition to good accuracy. The skin friction coefficient is compared for the two boundary conditions which also compare favourably with published results.

#### 5.5.6 Axisymmetric rotating flow inside an annulus

## 6 Summary and conclusions

We have proposed a novel method for finite volume approximation of Laplace operator on triangular grids which has the summation-by-parts property on triangular grids in two dimensions with proper consideration of Dirichlet boundary conditions. The new scheme implements Dirichlet boundary conditions in a weak manner and leads to an energy stability property for the time dependent heat equation. We also propose the use of a Nitsche-type penalty term which is found to improve the accuracy of the scheme. The weak implementation gives solutions which are as accurate as the strong boundary conditions. The general idea for the approximation of Laplacian type operator as presented here can be used in other problems also. These ideas are extended to the compressible Navier–Stokes equations and we prove kinetic energy stability property of the finite volume scheme. Such schemes can be useful for DNS of compressible flows where kinetic energy consistency plays an important role in correctly 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 combination with a numerical flux function for the inviscid fluxes and show that this leads to accurate solutions on many standard test problems. The weak implementation of the boundary conditions on velocity and temperature also promotes better iterative convergence for steady state problems which is not the case with strong boundary conditions. These schemes can be extended to tetrahedral grids in three dimensions.

## Footnotes

## Notes

### Acknowledgments

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

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