reconstructed discontinuous galerkin methods for hyperbolic...

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Ninth International Conference on Computational Fluid Dynamics (ICCFD9), Istanbul, Turkey, July 11-15, 2016

ICCFD9-xxxx

Reconstructed Discontinuous Galerkin Methods for Hyperbolic

Diffusion Equations on Unstructured Grids Xiaodong Liu1, Jialin Lou1, Hong Luo1, & Hiroaki Nishikawa2

1Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695, USA

2 National Institute of Aerospace, Hampton, VA 23666, USA

Corresponding author: [email protected]

Abstract: Reconstructed Discontinuous Galerkin (rDG) methods are presented for solving diffusion equations based on a first-order hyperbolic system (FOHS) formulation. The idea is to combine the advantages of the FOHS formulation and the rDG methods in an effort to develop a more reliable, accurate, efficient, and robust method for solving the diffusion equations. The developed hyperbolic DG methods can be made to have the same number of degrees-of-freedom as the conventional DG methods. A number of test cases for different diffusion equations are presented to assess accuracy and performance of the newly developed hyperbolic rDG methods in comparison with the standard BR2 DG method. Numerical experiments demonstrate that the hyperbolic rDG methods are able to achieve the designed optimal order of accuracy for both solutions and their derivatives on regular, irregular, and heterogeneous girds, and outperform the BR2 method in terms of the magnitude of the error, the order of accuracy, the size of time steps, and the CPU times required to achieve steady state solutions, indicating that the developed hyperbolic rDG methods provide an attractive and probably an even superior alternative for solving the diffusion equations. Keywords: Reconstructed Discontinuous Galerkin Methods, First-Order Hyperbolic System Formulation, and Unstructured Grids.

1 Introduction The discontinuous Galerkin (DG) methods1-17 have recently become popular for the solution of systems of conservation laws. Nowadays, they are widely used in computational fluid dynamics, computational acoustics, and computational magneto-hydrodynamics. The DG methods combine two advantageous features commonly associated to the finite element (FE) and finite volume (FV) methods. As in classical finite element methods, the order of accuracy is obtained by means of high-order polynomial approximation within an element rather than by wide stencils as in the finite volume methods. The physics of wave propagation is, however, accounted for by solving the Riemann problems that arise from the discontinuous representation of the solution at element interfaces. In this respect, the DG methods are therefore similar to the finite volume methods. The DG methods have many attractive features: 1) They have several useful mathematical properties with respect to conservation, stability, and convergence; 2) They can be easily extended to higher-order (>2nd) approximation; 3) They are well suited for complex geometries since they can be applied on unstructured grids. In addition, the methods can also handle non-conforming elements, where the grids are allowed to have hanging nodes; 4) The methods are highly parallelizable, as they are compact and each element is independent. Since the elements are discontinuous, and the inter-element communications are minimal, domain decomposition can be efficiently employed; 5) They can easily handle adaptive strategies, since refining or coarsening a grid can be achieved without considering the continuity restriction commonly associated with the conforming elements. The methods allow easy implementation of hp-refinement, for example, the order of accuracy, or shape, can vary from element to element. However, the DG methods have a number of their own weaknesses. In particular, how to

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effectively control spurious oscillations in the presence of strong discontinuities, how to reduce the computing costs for the DG methods, and how to efficiently solve elliptic problems or discretize diffusion terms in the parabolic equations remain the three unresolved and challenging issues in the DG methods. The DG methods have been recognized as expensive in terms of both computational costs and storage requirements. Indeed, compared to the FE and FV methods, the DG methods (DGMs) require solutions of systems of equations with more unknowns for the same grids. In order to reduce high costs associated with the DG methods, Dumbser et al.18-20 have introduced a new family of reconstructed DGM, termed PnPm schemes and referred to as rDG(PnPm) in this paper, where Pn indicates that a piecewise polynomial of degree of n is used to represent a DG solution, and Pm represents a reconstructed polynomial solution of degree of m (m≥n) that is used to compute the fluxes. The rDG(PnPm) schemes21-25 are designed to enhance the accuracy of the DGM by increasing the order of the underlying polynomial solution. The beauty of rDG(PnPm) schemes is that they provide a unified formulation for both FVM and DGM, and contain both classical FVM and standard DGM as two special cases of rDG(PnPm) schemes. When n=0, i.e. a piecewise constant polynomial is used to represent a numerical solution, rDG(P0Pm) is nothing but classical high order FV schemes, where a polynomial solution of degree m (m≥1) is reconstructed from a piecewise constant solution. When m=n, the reconstruction reduces to the identity operator, and rDG(PnPn) scheme yields a standard DG(Pn) method. For n>0, and n>m, a new family of numerical methods from third-order of accuracy upwards is obtained. A Hierarchical WENO-based rDG method26,27 is designed not only to reduce the high computing costs of the DGM, but also to avoid spurious oscillations in the vicinity of strong discontinuities, thus effectively overcoming the first two shortcomings of the DG methods. The DG methods are indeed a natural choice for the solution of the hyperbolic equations, such as the compressible Euler equations. However, the DG formulation is far less certain and advantageous for elliptic problems or parabolic equations such as the compressible Navier-Stokes equations, where diffusive fluxes exist and which require the evaluation of the solution derivatives at the interfaces. Taking a simple arithmetic mean of the solution derivatives from the left and right is inconsistent, because it does not take into account a possible jump of the solutions. A number of numerical methods have been proposed in the literature to address this issue, such as those by Bassi and Rebay7,28,29, Cockburn and Shu30, Baumann and Oden31, Peraire and Persson32, and many others. Arnold et al.33 have analyzed a large class of DGM for second-order elliptic problems in a unified formulation. All these methods have introduced in some way the influence of the discontinuities in order to define correct and consistent diffusive fluxes. Gassner et al34 introduced a numerical scheme based on the exact solution of the diffusive generalized Riemann problem for the DGM. van Leer et al35-37 proposed a recovery-based DG method for the diffusion equation using the recovery principle, that recovers a smooth continuous solution that in the weak sense is indistinguishable from the discontinuous discrete solution. Luo et al. have developed a reconstructed discontinuous Galerkin method using an inter-cell reconstruction38 for the solution of the compressible Navier-Stokes equations. Unfortunately, all these methods seem to require substantially more computational effort than the classical continuous finite element methods, which are naturally more suited for the discretization of elliptic problems. An alternative approach for viscous discretization is to reformulate the viscous terms as a hyperbolic system using the first-order hyperbolic system (FOHS) formulation developed by Nishikawa39-44 over the last several years. In the FOHS formulation, the diffusion equations are first formulated as a first-order system (FOS) by including derivative quantities as additional variables. The system is then rendered to be hyperbolic, which is the distinguished feature of the FOHS method from other FOS methods, by adding pseudo time derivatives to the first-order system. It thus generates a system of pseudo-time evolution equations for the solution and the derivatives in the partial differential equation (PDE) level, not in the discretization level as in DGM. The hyperbolic formulation in the PDE level allows a dramatic simplification in the discretization because well-established methods for hyperbolic systems can be directly applied to the viscous terms. In the pseudo steady state or simply by dropping the pseudo-time derivative after the discretization, a consistent and superior discretization of the diffusion equations is obtained. The FOS method is especially attractive in the context of the DGM, since it allows the use of inviscid algorithms for the viscous terms and thus greatly simplifies the discretization of the compressible Navier Stokes equations.

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The objective of the effort discussed in this paper is to develop a higher-order rDG method for solving diffusion equations based on the first-order hyperbolic system formulation, termed hyperbolic rDG methods in this paper. The idea behind the hyperbolic rDG methods is to combine the advantages of the FOHS formulation and the rDG methods in an effort to develop a more reliable, accurate, efficient, and robust method for solving the diffusion equations and ultimately the incompressible and compressible Navier-Stokes equations on fully irregular, adaptive, anisotropic, unstructured grids. By using the solution derivatives handily available in the FOHS formulation and Taylor basis in the DG formulation, the hyperbolic DG methods can be designed to have the same number of degrees-of-freedom as the conventional DG methods. A number of test cases for different diffusion equations are presented to assess accuracy and performance of the newly developed hyperbolic rDG methods in comparison with the standard BR2 DG method. Numerical experiments demonstrate that the hyperbolic rDG methods are able to achieve the designed optimal order of accuracy for both solutions and their derivatives on regular, irregular, and heterogeneous girds, and outperform the BR2 method in terms of the magnitude of the error, the order of accuracy, the size of time steps, and the CPU times required to achieve steady state solutions, indicating that the developed hyperbolic rDG methods provide an attractive and probably an even superior alternative for solving the diffusion equations. The remainder of this paper is organized as follows. A FOHS formulation for diffusion equations is described in Section 2. The rDG methods for solving the hyperbolic diffusion equations are presented in Section 3. Extensive numerical experiments are reported in Section 4. Concluding remarks are given in Section 5. 2 First-Order Hyperbolic System formulation for Diffusion Equations The FOHS formulation39 was introduced by Nishikawa as a radical approach for solving diffusion equation. Consider the following diffusion equation in two dimensions −∇i D∇ϕ( ) = f , (2.1) where ϕ denotes a scalar function, which can be referred to as velocity potential, and D denotes a symmetric positive definite diffusion tensor, which can be expressed as

D =Dxx x, y( ) Dxy x, y( )Dxy x, y( ) Dyy x, y( )

!

"

##

$

%

&&

(2.2)

Eq. (2.1) can be rewritten as a first-order system of equations by introducing an auxiliary vector variable V, which is the gradient of the velocity potential ϕ , i.e., velocity vector,

−∂∂x

Dxxvx +Dxyvy( )− ∂∂y

Dxyvx +Dyyvy( ) = f

vx =∂ϕ∂x

vy =∂ϕ∂y

#

$

%%%

&

%%%

(2.3)

This formulation is known for decades as the mixed formulation in the FE community and widely used in many discretization methods including the DG methods. For example, Bassi and Rebay derived the well-known and widely used BR1 and BR2 methods from this formulation28-29. Note that the system (2.1) is elliptic in space, which is equivalent to the original equation (2.1). In the FOHS formulation, the system is made hyperbolic by adding pseudo-time derivatives with respect to all variables as follows:

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∂ϕ∂t

=∂∂x

Dxxvx +Dxyvy( )+ ∂∂y

Dxyvx +Dyyvy( )+ f

∂vx∂t

=1Tr

∂ϕ∂x

− vx#

$%

&

'(

∂vy∂t

=1Tr

∂ϕ∂y

− vy#

$%

&

'(

)

*

++++

,

++++

(2.4)

where t is the pseudo time andTr is a free parameter called the relaxation time. In the FOHS formulation, our interest is to obtain the steady-state solution of the pseudo-time system (2.4), which is the solution to the diffusion equation (2.1). The FOHS can be written in vector form as

∂U∂t

= ∂Fx∂x

+∂Fy∂y

+ S (2.5)

where

U =ϕuv

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥,Fx =

−dxu − dcv−ϕ /Tr0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥,Fy =

−dcu − dyv

0−ϕ /Tr

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

,S =f

−u /Tr−v /Tr

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

(2.6)

Since D is a symmetric positive definite tensor, consider an unit vector at an arbitrary direction, n = [ nx ny ]

T , one would have

ν = nTDn = dxnx2 + 2dcnxny + dyny

2 > 0 (2.7) Consider the Jacobian of the flux projected along n ,

An =∂ F in( )∂U

=∂ Fxnx + Fyny( )

∂U=

0 −dxnx − dcny −dcnx − dyny−nx /Tr 0 0−ny /Tr 0 0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

(2.8)

It has the following eigenvalues

λ1 = − νTr,λ2 =

νTr,λ3 = 0 (2.9)

The first two nonzero eigenvalues indicate that the system describes a wave propagating isotropically. The third eigenvalue corresponds to the inconsistency damping mode. The relaxation time Tr does not affect the steady solution, and thus can be defined solely for the purpose of accelerating the convergence to the steady state. For simplicity, Tr is defined as

Tr = Lr2 , Lr =

12π

(2.10)

where the length scale Lr , according to Nishikawa’s work39-44, has been defined to maximize the effect of propagation. The absolute Jacobian An is constructed by the right-eigenvector matrix, Rn , and the diagonal eigenvalue-matrix Λn ,

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Rn =12

1 −1 0nx / νTr nx / νTr −2 dcnx + dyny( )ny / νTr ny / νTr 2 dxnx + dcny( )

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

,Λn =−λ 0 00 λ 00 0 0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

(2.11)

where λ = λ1 = λ2 = ν /Tr . Hence,

An = Rn Λn Rn−1 = λ

1 0 00 nx dxnx + dcny( ) ny dxnx + dcny( )0 nx dcnx + dyny( ) ny dcnx + dyny( )

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

(2.12)

3 Reconstructed Discontinuous Galerkin Methods Discontinuous Galerkin methods (DGMs) are a family of numerical method that combines the advantages of classic finite volume (FV) methods and finite element methods. With the method of lines, it would expressed the solution as

uh = uj t( )Bj x, y( )j=1

N

∑ (3.1)

where Bj is basis function of polynomials of degree P . In the traditional DMG, termed nodal

discontinues Galerkin methods, numerical polynomial solutions Uh in each element are expressed using either standard Lagrange finite element or hierarchical node-based basis function. Multiply the basis function Bi to Eq. (2.5), and integrate by parts would yield

ddt

UhBi dΩ + F in∂Ω!∫Ωe

∫ BidΓ − F i∇Bi dΩ =Ωe∫ SBi dΩ, i = 1,",N

Ωe∫ (3.2)

In short, we could express it as the semi-discretized form

M dUdt

= R (3.3)

where M is the mass matrix, whose entries are

mij = BiBj dΩ i, j = 1,!,N

Ωe∫ (3.4)

and R is the residual vector, defined as

ri = Fx

∂Bi∂x

+ Fy∂Bi∂y

+ SBi⎛⎝⎜

⎞⎠⎟dΩ− FnBi dΓ i = 1,!,N

∂Ω"∫Ωe∫ (3.5)

In the implementation of the DGMs in this paper, however, modal DG is adopted. The numerical polynomial solutions are represented using a Taylor series expansion at the cell center and normalized to improve the conditioning of the system matrix. For example, the quadratic polynomial solutions could be expressed as follows,

Uh = Uc +∂U∂x c

x − xc( ) + ∂U∂y c

y − yc( )

+ ∂2U∂x2 c

x − xc( )22

+ ∂2U∂y2 c

y − yc( )22

+ ∂2U∂x∂y c

x − xc( ) y − yc( ) (3.6)

which could be further expressed as cell-averaged values, i.e., !U , and the derivatives at the center.

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Uh = !UB1 +∂U∂x c

ΔxB2 +∂U∂y c

ΔyB3

+ ∂2U∂x2 c

Δx2B4 +∂2U∂y2 c

Δy2B5 +∂2U∂x∂y c

ΔxΔyB6

(3.7)

where Δx = 0.5 xmax − xmin( )， Δy = 0.5 ymax − ymin( ) , and Δz = 0.5 zmax − zmin( ) , and xmax , xmin , ymax , ymin , zmax , zmin are the maximum and minimum coordinates in the cell Ωe in x-, y-, z-directions respectively. The basis functions are given as follows,

B1 = 1,B2 =

x − xcΔx

,B3 =y − ycΔy

,B4 =B22

2− 1Ωe

B22

2dΩe,Ωe

∫

B5 =B32

2− 1Ωe

B32

2dΩe,B6 = B2B3 −

1Ωe

B2B3 dΩeΩe∫Ωe

∫ (3.8)

Compared with reconstructed FV methods, the DGM would require more degrees of freedom, additional domain integration, and more Gauss quadrature points for the boundary integration, which leads to more computational costs and storage requirements. Inspired by the DGM from Dumbser et al. in the frame of PnPm scheme18-20, termed rDG(PnPm) in this paper, a hierarchical WENO-based rDG method26-27 is designed to achieve high order of accuracy while reducing the computational cost. As a matter of fact, rDG method could provide a unified formulation for both FV and DG methods. The standard FV and DG method would be nothing but special cases in rDG frame work, and thus allow for a direct efficiency comparison. Based on different DGMs, some effective discretization hyperbolic DG methods will be presented to deal with the derived first-order hyperbolic system (FOHS). The format A +B is used to indicate the discretization method for the system, where A refers to the discretization method for ϕ and B refers to the discretization method for its derivatives. Note that what make the hyperbolic DG methods based on FOHS different from standard DGMs is that the system is in the partial differential equation (PDE) level other than in the discretization level.

• DG(Pk)+DG(Pk) method This method is designed to be (k+1)th order accurate for variable ϕ and (k+1)th order accurate for the derivatives of variable. It is obvious that DG(P0)+DG(P0), DG(P1)+DG(P1), and DG(P2)+DG(P2) all belong to this kind of method. For example, DG(P0)+DG(P0) means that DG(P0) is adopted to discretize both ϕ and its derivatives.

• DG(P0Pk)+DG(Pj)method This method is designed to be (k+1)th order accurate for variable ϕ and (j+1)th order accurate for variables u and v . Let's take DG(P0P1)+DG(P0) as an example. It is well known that the first derivatives of ϕ are nothing but u and v . Thus, the variables u and v could be used directly to achieve the linear interpolation of variable ϕ . Because herein, the gradient of ϕ does not come from the reconstruction, it is called DG(P0P1). In our work, DG(P0P1)+DG(P0), DG(P0P1)+DG(P1), DG(P0P2)+DG(P1) have been calculated and validated.

• DG(P0Pk)+rDG(Pj-1Pj)method This method is designed to be (k+1)th order accurate for variable and (j+1)th order accurate for derivatives of variable. It is necessary to point out that this type of method is superior to the previous methods in terms of computational cost. Let's take DG(P0P1) + rDG(P0P1) as an example. Herein, DG(P0P1) method has been given detailed explanation when we introduce the second type of method. rDG(P0P1), also known as second order finite volume method, means that the first derivative of u or v is obtained by least-square reconstruction. In our work, DG(P0P1) + rDG(P0P1), DG(P0P2) + rDG(P0P1), DG(P0P2) + rDG(P1P2) and DG(P0P3) + rDG(P1P2) have been calculated and validated.

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• rDG(Pk-1Pk)+rDG (Pk-1Pk)method

This method is designed to be (k+1)th order accurate for variable and (k+1)th order accurate for derivative variables. Let's take rDG(P0P1) + rDG(P0P1) as an example. Least-square reconstruction is adopted to obtain the first derivative of every variable. Obviously, as mentioned above, there is no need to construct the first derivative of ϕ . Therefore this method is not cost-effective. In our work, rDG(P0P1)+ rDG(P0P1) and rDG(P1P2)+rDG(P1P2) have been calculated and validated.

In this work, Gaussian quadrature formulas are used to compute the integration. Therefore, the number of Gauss integration points for face and domain integration is listed in Table 1, from which we could see that hyperbolic rDG methods could achieve high order of accuracy while using less degrees of freedom and also less Gauss quadrature points compared with standard DGMs.

Table 1 The list of number of Gauss quadrature points for face and domain integral for different schemes.

Scheme Number of Gauss points for face integral

Number of Gauss points for domain integral

DG(P0)+DG(P0) 1 1 DG(P1)+DG(P1) 2 3 DG(P2)+DG(P2) 3 6

DG(P0P1)+DG(P0) 2 1 DG(P0P1)+DG(P1) 2 3 DG(P0P2)+DG(P1) 2 3

DG(P0P2)+rDG(P0P1) 2 1 DG(P0P2)+rDG(P0P1) 2 1 DG(P0P2)+rDG(P1P2) 2 4 DG(P0P3)+rDG(P1P2) 3 4 rDG(P0P1)+rDG(P0P1) 2 1 rDG(P1P2)+rDG(P1P2) 3 4

The hyperbolic formulation in FOHS would allow hyperbolic DGM to use well-established methods for hyperbolic system. Instead of using BR2, DDG or other usual diffusion scheme for DGMs, the numerical flux is computed by the upwind flux as

Fij =

12FL + FR( ) inij − 12 An UR −UL( ) (3.9)

where nij is the unit directed area vector, and An is defined as the normal flux Jacobian along nij . To advance the solution in time, explicit three-stage TVD Runge-Kutta (TVDRK3) scheme and implicit BDF1 scheme have been employed in this work. The time step is determined as

Δt = CFL 2Ωe

ν / LrAij +Ω j /Tr( )j∑ (3.10)

For steady problem, local time step would be used to accelerate the convergence process. 4 Numerical Examples Several steady modal diffusion problems in a unit square are considered in this section. All the cases would be computed on three types of grids. The first and second types are regular and irregular triangular grids with 9×9, 17×17, 33×33, 65×65, and 129×129 nodes. The irregular gird is generated from the regular grid by random diagonal swapping and nodal perturbation. The third type is a set of heterogeneous grids with 12×11, 23×21, 45×41, and 99×81 nodes. The second grid of every type is shown in Figure 1.

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Figure 1 The second mesh of every type, that is, 17×17 regular grid, 17×17 irregular grid, and 23×21

heterogeneous grid. In our wok, the quality of these 3 types of grids is investigated as follows. For one triangle, there exist one ratio as,

r = RoutRin

(4.1)

where, Rout denotes the radius of the circumscribed circle, and Rin refers to the radius of the inscribed circle. It is well known that the equilateral triangle is the best choice for numerical computation, and its ratio is equal to 2.0. Therefore, in this work, the quality of meshes would be measured through

R = r − 2.02.0

(4.2)

Obviously, if R = 0 , the triangle is equilateral; however larger R means that the triangle meshes deviates greatly from the equilateral triangle. For the first type of mesh, R keeps same all through the calculation domain. We just pay attention to the last 2 types of meshes. The contours distributions of R of the first mesh from the latter 2 types of meshes are shown in Figure 2. It could be observed that, for the first heterogeneous mesh, maximum of R reached 80, which are almost 2 orders of magnitude higher than the one of the irregular mesh. This demonstrates the bad quality of the heterogeneous mesh, and also will demonstrate the robustness and accuracy of developed methods.

Figure 2 Distribution of R of the most coarse irregular mesh and the most coarse heterogeneous mesh.

4.1 Case I. Scalar coefficient without source term

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To begin with, a steady diffusion problem in a unit square is considered with the exact solution given by

ϕ x, y( ) = sinh π x( )sin π y( )sinh π y( )sin π x( )sinh π( ) (4.3)

and D = I, f = 0 (4.4) In this work, the case is computed with the aforementioned methods respectively on each mesh. The initial solution is set to be 1.0 everywhere and the steady state is considered to be reached when the residuals drop below 10-15 in the L2 norm. The grid refinement study is carried out for regular, irregular and heterogeneous mesh to show that this kind method is viable, accurate and robust. L2 norm of the difference between the numerical results and the exact ones is used as the error measurement. All norms are computed by Gaussian quadrature. The grid refinement study results for regular, irregular and heterogeneous meshes are shown in Figure 3 through Figure 5.

Log(h)

Log(

L 2 err

or

)

-2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8

-8

-7

-6

-5

-4

-3

-2

DG(P0)+DG(P0)DG(P1)+DG(P1)DG(P2)+DG(P2)DG(P0P1)+DG(P0)DG(P0P1)+DG(P1)DG(P0P2)+DG(P1)rDG(P0P1)+rDG(P0P1)rDG(P1P2)+rDG(P1P2)DG(P0P1)+rDG(P0P1)DG(P0P2)+rDG(P0P1)DG(P0P2)+rDG(P1P2)DG(P0P3)+rDG(P1P2)Slope 1Slope 2Slope 3Slope 4

Log(h)

Log(

L 2 err

or u

)

-2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8

-6

-5

-4

-3

-2

-1

DG(P0)+DG(P0)DG(P1)+DG(P1)DG(P2)+DG(P2)DG(P0P1)+DG(P0)DG(P0P1)+DG(P1)DG(P0P2)+DG(P1)rDG(P0P1)+rDG(P0P1)rDG(P1P2)+rDG(P1P2)DG(P0P1)+rDG(P0P1)DG(P0P2)+rDG(P0P1)DG(P0P2)+rDG(P1P2)DG(P0P3)+rDG(P1P2)Slope 1Slope 2Slope 3

Figure 3 Gird refinement study of case 1 on regular grids.

Log(h)

Log(

L 2 err

or

)

-2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8

-8

-7

-6

-5

-4

-3

-2


Log(h)

Log(

L 2 err

or u

)

-2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8

-6

-5

-4

-3

-2

-1


Figure 4 Gird refinement study of case 1 on irregular grids.

10

Log(h)

Log(

L 2 err

or

)

-2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1

-7

-6

-5

-4

-3

-2

DG(P0)+DG(P0)DG(P1)+DG(P1)DG(P2)+DG(P2)DG(P0P1)+DG(P0)DG(P0P1)+DG(P1)DG(P0P2)+DG(P1)rDG(P0P1)+rDG(P0P1)rDG(P1P2)+rDG(P1P2)DG(P0P1)+rDG(P0P1)DG(P0P2)+rDG(P0P1)DG(P0P2)+rDG(P1P2)Slope 1Slope 2Slope 3

Log(h)

Log(

L 2 err

or u

)

-2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1-6

-5

-4

-3

-2

-1


Figure 5 Gird refinement study of case 1 on heterogeneous grids.

With DG(P0P2)+rDG(P0P1), only 2nd order of accuracy could be achieved for variable ϕ . This is consistent with the Nishikawa’s paper41. In addition, DG(P0P3)+rDG(P1P2) is not stable on the heterogeneous mesh. Therefore, WENO reconstruction has to be used to guarantee the stability of such scheme with the cost of achieving only 3rd order of accuracy for variable ϕ . Other than these 2 schemes, the numerical results demonstrate that for other aforementioned methods, the designed order of accuracy could be achieved for both unknowns and their derivatives.

Time steps

Log1

0(re

sidu

al)

100 101 102 103-14

-12

-10

-8

-6

-4

-2

0

DG(P0)+DG(P0)+BDF1DG(P0)+DG(P0)+TVDRK3DG(P0P1)+DG(P0)+BDF1DG(P0P1)+DG(P0)+TVDRK3rDG(P0P1)+rDG(P0P1)+BDF1rDG(P0P1)+rDG(P0P1)+TVDRK3DG(P0P1)+rDG(P0P1)+BDF1DG(P0P1)+rDG(P0P1)+TVDRK3DG(P0P2)+rDG(P0P1)+BDF1DG(P0P2)+rDG(P0P1)+TVDRK3

CPU time (s)

Log1

0(re

sidu

al)

10-1 100 101 102 103-14

-12

-10

-8

-6

-4

-2

0

DG(P1)+DG(P1)+BDF1DG(P1)+DG(P1)+TVDRK3DG(P2)+DG(P2)+BDF1DG(P2)+DG(P2)+TVDRK3DG(P0P1)+DG(P1)+BDF1DG(P0P1)+DG(P1)+TVDRK3DG(P0P2)+DG(P1)+BDF1DG(P0P2)+DG(P1)+TVDRK3rDG(P1P2)+rDG(P1P2)+BDF1rDG(P1P2)+rDG(P1P2)+TVDRK3DG(P0P2)+rDG(P1P2)+BDF1DG(P0P2)+rDG(P1P2)+TVDRK3DG(P0P3)+rDG(P1P2)+BDF1DG(P0P3)+rDG(P1P2)+TVDRK3

Figure 6 Convergence history of case 1 on 127×127 regular mesh using hyperbolic FV methods.

11

Time steps

Log1

0(re

sidu

al)

100 101 102 103-14

-12

-10

-8

-6

-4

-2

0


CPU time (s)

Log1

0(re

sidu

al)

10-1 100 101 102-14

-12

-10

-8

-6

-4

-2

0


Figure 7 Convergence history of case 1 on 127×127 irregular mesh using hyperbolic FV methods.

Time steps

Log1

0(re

sidu

al)

100 101 102 103-14

-12

-10

-8

-6

-4

-2

0

DG(P0)+DG(P0)+BDF1DG(P0)+DG(P0)+TVDRK3DG(P0P1)+DG(P0)+BDF1DG(P0P1)+DG(P0)+TVDRK3DG(P0P1)+rDG(P0P1)+BDF1DG(P0P1)+rDG(P0P1)+TVDRK3DG(P0P2)+rDG(P0P1)+BDF1DG(P0P2)+rDG(P0P1)+TVDRK3

CPU time (s)

Log1

0(re

sidu

al)

10-1 100 101 102-14

-12

-10

-8

-6

-4

-2

0

DG(P0)+DG(P0)+BDF1DG(P0)+DG(P0)+TVDRK3DG(P0P1)+DG(P0)+BDF1DG(P0P1)+DG(P0)+TVDRK3DG(P0P1)+rDG(P0P1)+BDF1DG(P0P1)+rDG(P0P1)+TVDRK3DG(P0P2)+rDG(P0P1)+BDF1DG(P0P2)+rDG(P0P1)+TVDRK3

Figure 8 Convergence history of case 1 on 99×81 heterogeneous mesh using hyperbolic FV methods.

Also, to compare different time marching scheme, the case on the finest mesh of each type is tested using both BDF1 and TVDRK3 with convergence history shown in Figure 6 through Figure 11.

Time steps

Log1

0(re

sidu

al)

100 101 102 103 104-14

-12

-10

-8

-6

-4

-2

0


CPU time (s)

Log1

0(re

sidu

al)

10-1 100 101 102 103-14

-12

-10

-8

-6

-4

-2

0


Figure 9 Convergence history of case 1 on 127×127 regular mesh using hyperbolic DG methods.

12

Time steps

Log1

0(re

sidu

al)

100 101 102 103 104-14

-12

-10

-8

-6

-4

-2

0


CPU time (s)

Log1

0(re

sidu

al)

10-1 100 101 102 103-14

-12

-10

-8

-6

-4

-2

0


Figure 10 Convergence history of case 1 on 127×127 irregular mesh using hyperbolic DG methods.

Time steps

Log1

0(re

sidu

al)

100 101 102 103 104-14

-12

-10

-8

-6

-4

-2

0

DG(P1)+DG(P1)+BDF1DG(P1)+DG(P1)+TVDRK3DG(P2)+DG(P2)+BDF1DG(P2)+DG(P2)+TVDRK3DG(P0P1)+DG(P1)+BDF1DG(P0P1)+DG(P1)+TVDRK3DG(P0P2)+DG(P1)+BDF1DG(P0P2)+DG(P1)+TVDRK3DG(P0P2)+rDG(P1P2)+BDF1DG(P0P2)+rDG(P1P2)+TVDRK3

CPU time (s)

Log1

0(re

sidu

al)

10-1 100 101 102 103-14

-12

-10

-8

-6

-4

-2

0

DG(P1)+DG(P1)+BDF1DG(P1)+DG(P1)+TVDRK3DG(P2)+DG(P2)+BDF1DG(P2)+DG(P2)+TVDRK3DG(P0P1)+DG(P1)+BDF1DG(P0P1)+DG(P1)+TVDRK3DG(P0P2)+DG(P1)+BDF1DG(P0P2)+DG(P1)+TVDRK3DG(P0P2)+rDG(P1P2)+BDF1DG(P0P2)+rDG(P1P2)+TVDRK3

Figure 11 Convergence history of case 1 on 99×81 heterogeneous mesh using hyperbolic DG methods.

Next, the comparison between hyperbolic DGM and BR2 is carried out on regular, irregular and heterogeneous meshes respectively, and the numerical results are shown from Figure 12 to Figure 14. It is obvious that, DG(P1)(BR2) could achieve 2nd order of accuracy for variable ϕ , and 1st order of accuracy for gradient of variable. Likewise, DG(P2)(BR2) could achieve 3rd order of accuracy for variable, and 2nd order of accuracy for gradient of variable. Several hyperbolic DGMs are presented to compare with BR2 scheme. Note that for heterogeneous grids, DG(P0P3) + rDG(P1P2) is carried out with WENO reconstruction to remain stability. Results show that, our method is comparable to the BR2 method in terms of computational cost and accuracy.

13

A

A

A

AB

B

B

B

Log(h)

Log(

L2 e

rror

)

-2 -1.5 -1

-7

-6

-5

-4

-3

-2

DG(P0P1) + DG(P0)DG(P0P1) + DG(P1)DG(P0P2) + DG(P1)DG(P0P1) + rDG(P0P1)DG(P0P3) + rDG(P1P2)DG(P1)(BR2)DG(P2)(BR2)Slope 1Slope 2Slope 3Slope 4

AB

AA

AA B

B

B

B

Log(h)

Log(

L2 e

rror

u)

-2 -1.5 -1

-5

-4

-3

-2

-1

DG(P0P1) + DG(P0)DG(P0P1) + DG(P1)DG(P0P2) + DG(P1)DG(P0P1) + rDG(P0P1)DG(P0P3) + rDG(P1P2)DG(P1)(BR2)DG(P2)(BR2)Slope 1Slope 2Slope 3

AB

Figure 12 Order of accuracy for using hyperbolic DGM and standard BR2 of case 1 on regular mesh.

A

A

A

A

B

B

B

B

Log(h)

Log(

L2 e

rror

)

-2 -1.5 -1-8

-7

-6

-5

-4

-3

-2

-1

DG(P0P1) + DG(P0)DG(P0P1) + DG(P1)DG(P0P2) + DG(P1)DG(P0P1) + rDG(P0P1)DG(P0P3) + rDG(P1P2)DG(P1)(BR2)DG(P2)(BR2)Slope 1Slope 2Slope 3Slope 4

AB

AA

AA B

B

B

B

Log(h)

Log(

L2 e

rror

u)

-2 -1.5 -1

-5

-4

-3

-2

-1


AB

Figure 13 Order of accuracy for using hyperbolic DGM and standard BR2 of case 1 on irregular mesh.

A

A

A

AB

B

B

Log(h)

Log(

L2 e

rror

)

-2 -1.5 -1

-6

-5

-4

-3

-2


AB

AA

AA

B

B

B

Log(h)

Log(

L2 e

rror

u)

-2 -1.5 -1

-5

-4

-3

-2

-1


AB

Figure 14 Order of accuracy for using hyperbolic DGM and standard BR2 of case 1 on heterogeneous mesh.

Moreover, hyperbolic DG would also outperform BR2 in terms of the time steps and CPU time required to reach convergence. The simulation is carried out by hyperbolic DG method and BR2 method using TVDRK3 on the second mesh of each type, with the results shown in Figure 15 through Figure 17. Clearly, hyperbolic DGMs would be way less time consuming than standard BR2 scheme.

14

A

A

A

A

A

A

B

BB

BB

B

Time steps

Log1

0(re

sidu

al)

0 5000 10000

-15

-10

-5

0DG(P1) + DG(P1)DG(P2) + DG(P2)DG(P0P1) + DG(P0)DG(P0P2) + DG(P1)DG(P1), BR2DG(P2), BR2

AB

A

A

A

A

A

A

B

BB

BB

BB

BB

CPU time

Log1

0(re

sidu

al)

0 20 40-15

-10

-5


AB

Figure 15 Convergence history of case 1 on 17×17 regular mesh using hyperbolic DG methods and BR2.

A

AA

AA

AA

AA

AA

A

B

B B B B B B B B B B B B

Time steps

Log1

0(re

sidu

al)

0 5000 10000 15000 20000 25000-15

-10

-5

0 DG(P1) + DG(P1)DG(P2) + DG(P2)DG(P0P1) + DG(P0)DG(P0P2) + DG(P1)DG(P1), BR2DG(P2), BR2

AB

A

A

A

A

A

B

BB

BB

BB

BB

BB

BB

BB

CPU time

Log1

0(re

sidu

al)

0 50 100 150 200-15

-10

-5


AB

Figure 16 Convergence history of case 1 on 17×17 irregular mesh using hyperbolic DG methods and BR2.

A

A A A A A A A

B

B B B B B B B

Time steps

Log1

0(re

sidu

al)

0 10000 20000 30000 40000

-14

-12

-10

-8

-6

-4

-2

0

DG(P1) + DG(P1)DG(P2) + DG(P2)DG(P0P1) + DG(P0)DG(P0P2) + DG(P1)DG(P1), BR2DG(P2), BR2

AB

A

A A A A A A A A A A A A A A A A A A

B

B B B B B B B

CPU time

Log1

0(re

sidu

al)

0 50 100 150 200 250 300 350 400

-14

-12

-10

-8

-6

-4

-2

0

DG(P1) + DG(P1)DG(P2) + DG(P2)DG(P0P1) + DG(P0)DG(P0P2) + DG(P1)DG(P1), BR2DG(P2), BR2

AB

Figure 17 Convergence history of case 1 on 23×21 irregular mesh using hyperbolic DG methods and BR2.

4.2 Case II. Scalar coefficient with source term A steady diffusion problem with source term is considered in this case. The exact solution is given by ϕ x, y( ) = 2cos π x( )sin 2π y( ) + 2 (4.5)

15

and D = I, f = 10π 2 cos π x( )sin 2π y( ) (4.6)

Log(h)

Log(

L 2 err

or

)

-2 -1.5 -1

-8

-7

-6

-5

-4

-3

-2

-1


Log(h)

Log(

L 2 err

or u

)

-2 -1.5 -1-7

-6

-5

-4

-3

-2

-1



Log(h)

Log(

L 2 err

or

)

-2 -1.5 -1

-8

-7

-6

-5

-4

-3

-2

-1


Log(h)

Log(

L 2 err

or u

)

-2 -1.5 -1-7

-6

-5

-4

-3

-2

-1



Log(h)

Log(

L 2 err

or

)

-2 -1.5 -1-7

-6

-5

-4

-3

-2

-1


Log(h)

Log(

L 2 err

or u

)

-1.5 -1-6

-5

-4

-3

-2

-1


Figure 20 Gird refinement study of case 2 on heterogeneous grids.

Similarly, the simulations are performed in those three types of grids using hyperbolic DGMs, with the numerical results shown in Figure 18 through Figure 20. Note that in this case, DG(P0P1)+DG(P0) and DG(P0P2)+DG(P1) can only achieve 1st order and 2nd

16

order of accuracy for ϕ respectively. However, as for source term free case, i.e., like the first case, both methods could have super-convergence by having one order higher for ϕ . One may need to upwind the source term as well to have super-convergence in the cases with non-zero source term. All other hyperbolic DGMs could deliver the designed order including DG(P0P3)+rDG(P1P2), which would deliver 4th order of accuracy for ϕ and 3rd order for the derivatives with relatively less computational cost than standard high order DGMs. 4.3 Case III. Tensor coefficient without source term To test the developed methods further, the following model diffusion problem is tested, which have the exact solution given by

ϕ x, y( ) = 1− tanh x − 0.5( )2 + y − 0.5( )20.01

⎛

⎝⎜⎞

⎠⎟ (4.7)

and the diffusion tensor D given by

D =x +1( )2 + y2 −xy

−xy y +1( )2⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

(4.8)

Clearly, this case would have non-zero source term. Again, grid refinement study is carried out for all three types of meshes, with the result shown on Figure 21 through Figure 23. Similar to the second case, DG(P0P1)+DG(P0) and DG(P0P2)+DG(P1) would not have super-convergence in ϕ while all other hyperbolic DGMs delivering designed order of accuracy for both variables and their derivatives, indicating the presented scheme is robust and can provide an attractive alternatives for using DGM for diffusion problems. 5 Concluding Remarks Reconstructed discontinuous Galerkin methods have been developed for solving diffusion equations based on a first-order hyperbolic system formulation. By judiciously using a Taylor basis in the DG formulation and solution derivatives handily available in the FOHS formulation, the resultant hyperbolic DG methods have the same number of degrees of freedom as the standard DG methods. A number of test cases for different diffusion equations have been presented to assess accuracy and performance of the newly developed hyperbolic rDG methods in comparison with the standard BR2 DG method. Numerical experiments demonstrated that the hyperbolic rDG methods are able to achieve the designed optimal order of accuracy for both solutions and their derivatives on regular, irregular, and heterogeneous girds, and outperform the BR2 method in terms of the magnitude of the error, the order of accuracy, the size of time steps, and the CPU times required to achieve steady state solutions, indicating that the developed hyperbolic rDG methods provide an attractive and probably an even superior alternative for solving the diffusion equations. Our future work is to extend the developed hyperbolic rDG methods for solving the advection-diffusion equations, and the incompressible and compressible Navier-Stokes equations on fully irregular, adaptive, anisotropic, unstructured grids.

17

Log(h)

Log(

L 2 err

or

)

-2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0


Log(h)

Log(

L 2 err

or u

)

-2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8

-4

-3

-2

-1

0



Log(h)

Log(

L 2 err

or

)

-2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0


Log(h)

Log(

L 2 err

or u

)

-2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8

-4

-3

-2

-1

0



Log(h)

Log(

L 2 err

or

)

-2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2-5

-4

-3

-2

-1


Log(h)

Log(

L 2 err

or u

)

-2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2-4

-3

-2

-1

0


Figure 23 Gird refinement study of case 3 on heteroneous grids.

Acknowledgments This work has been funded by the U.S. Army Research Office under the contract/grant number W911NF-16-1-0108 with Dr. Matthew Munson as the program manager.

18

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reconstructed discontinuous galerkin methods for hyperbolic...

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