Explicit Hyperbolic Reconstructed Discontinuous Galerkin Methods for Time-Dependent Problems

Transcription

1 AIAA AVIATION Forum June Atlanta Georgia 218 Fluid Dynamics Conference / Explicit Hyperbolic Reconstructed Discontinuous Galerkin Metods for Time-Dependent Problems Jialin Lou Lingquan Li and Hong Luo Nort Carolina State University Raleig NC United States and Hiroaki Nisikawa National Institute of Aerospace Hampton VA United States Downloaded by Hiroaki Nisikawa on July ttp://arc.aiaa.org DOI: / Tis paper reports a new finding tat time-accurate explicit time-stepping scemes can be constructed by te reconstructed discontinuous Galerkin metod applied to a yperbolic diffusion formulation tat is consistent wit a diffusion equation only in te steady state. A naive discretization of te yperbolic diffusion system is known to be time inconsistent but a special Galerkin discretization wit a matrix basis used as a test function is found to yield a time-accurate semi-discrete system tat can be integrated in time by explicit time-stepping scemes. Te mecanism beind te unexpected property is discussed. Te presented scemes are more efficient tan conventional discontinuous Galerkin scemes wit iger-order accuracy acieved wit a fewer degrees of freedom. Two reconstruction tecniques are considered for te efficient construction: least-squares reconstruction and variational reconstruction. Unsteady computations for pure diffusion problems and advection-diffusion problems are presented to assess accuracy and performance of te newly developed ig-order explicit yperbolic reconstructed discontinuous Galerkin scemes. Numerical experiments demonstrate tat te explicit yperbolic reconstructed discontinuous Galerkin scemes acieve te designed optimal order of accuracy for bot solutions and teir derivatives on regular and irregular grids for unsteady problems. I. Introduction Tis paper presents explicit time-accurate ig-order yperbolic reconstructed discontinuous Galerkin (rdg scemes. Te yperbolic rdg metod is a special discretization metod tat combines te yperbolic diffusion formulation [1] and te rdg discretization metods [2 7]. Te yperbolic diffusion formulation is a first-order system form of a diffusion equation wit extra variables called te gradient variables added to form a system and wit pseudo time terms added to render te system yperbolic. Te rdg metod is a general framework for constructing efficient ig-order scemes wit reconstruction tecniques aving te finite-volume (FV and DG scemes as special cases. As we ave sown in our previous developments [8 12] te two approaces can be combined in a systematic manner to simplify te discretization of diffusion terms improve gradient accuracy accelerate iterative convergence and acieve iger-order accuracy tan DG metods wit fewer numbers of degrees of freedom. Specifically we ave developed yperbolic rdg scemes for diffusion wit scalar and tensor diffusion coefficients [8 9] nonlinear diffusion [1] advection-diffusion equations [11] and te Navier-Stokes equations [12]. For unsteady problems we followed Refs.[13 14] and employed implicit time-stepping scemes. Te major reason for implicit-time stepping lies in te fact tat te yperbolic diffusion formulations rely on te equivalent to te original diffusion/viscous equations in te pseudo steady state and terefore are not necessarily time-accurate wen te pseudo time is treated as te pysical time. However as a matter of fact te yperbolic metod is a metod for spatial discretizations: P.D Student Department of Mecanical and Aerospace Engineering Student Member AIAA. P.D Student Department of Mecanical and Aerospace Engineering Student Member AIAA. Professor Department of Mecanical and Aerospace Engineering Associate Fellow AIAA. Associate Researc Fellow National Institute of Aerospace Associate Fellow AIAA. 1 of 13 Copyrigt 218 by autors. Publised by te American Institute of Aeronautics American and Institute Astronautics of Inc. Aeronautics wit permission. and Astronautics

2 Downloaded by Hiroaki Nisikawa on July ttp://arc.aiaa.org DOI: / form a first-order diffusion system wit a pseudo time term suc tat te system is yperbolic in te pseudo time discretize it by upwind scemes and drop te pseudo time term to obtain a spatial discretization. Seemingly ten it can be combined wit any time-stepping sceme. However as we will discuss later te resulting scemes would lose te coupling among te variables in te discrete equations if explicit timestepping scemes are used. In effect suc scemes reduce to conventional diffusion scemes and lose all te benefits of te yperbolic metod. To keep te coupling in te discrete level implicit-time stepping scemes needed to be employed were a steady solver is used for solving fully coupled unsteady residual equations. In tis paper we report a quite remarkable finding tat time-accurate explicit time-stepping scemes arise automatically from te rdg or DG metod applied to te yperbolic diffusion formulation. Te yperbolic diffusion formulation [1] is similar to te classical yperbolic eat equations of Cattaneo [15] and Vernotte [16] wit a relaxation time parameter T r. Tese classical equations establis te equivalence to te original second-order diffusion equation in te limit T r tus resulting in a system wit stiff source terms. On te oter and te yperbolic metod considered ere is different from tese classical models in tat te yperbolic formulation is constructed wit a pseudo time term and designed to be equivalent to te diffusion term in te pseudo steady state for arbitrary relaxation time. Te relaxation time can be taken as O(1 designed to optimize iterative convergence [1] and tus tere are no stiff source terms. Because of te large relaxation time te yperbolic formulation will not be time-accurate wen te pseudo-time term treated as te pysical-time term. It is possible to recover time accuracy by taking a very small relaxation time and it is useful to be able to employ explicit time-stepping scemes wic are simpler and computationally inexpensive compared wit implicit counterparts especially wen problems are not stiff e.g. no boundary layers are involved. In tis regard Toro and Montecinos [17] analyzed a similar yperbolic diffusion system and derived an upper bound on T r below wic time accuracy can be obtained wit explicit scemes. Specifically tey derived te upper bound as T r = O( 1+r/2 were is a mes spacing and r is te accuracy order (e.g. r = 2 for second-order accurate scemes and demonstrated its validity for scemes up to sevent-order accuracy. In tis paper we sow tat explicit time-accurate yperbolic scemes can be constructed witout any careful adjustment to te relaxation time. Rater surprisingly we ave found tat te rdg or DG discretization applied to te yperbolic diffusion formulation is automatically time-accurate wit explicit timestepping wit arbitrary relaxation time of O(1. As will be sown te rdg/dg formulation automatically generates appropriate time derivatives for te gradient variables and creates a semi-discrete system tat can be accurately integrated in time wit explicit scemes. It is also demonstrated tat te resulting explicit yperbolic rdg scemes can yield te same order of accuracy for te primal solution and its gradients on irregular grids. Tis is one of te unique features of te yperbolic metod as demonstrated in many previous papers and would be igly desired for unstructured-grid applications wit bot explicit and implicit timestepping scemes. For reconstruction bot ybrid least-squares (LS [18] and variational reconstruction (VR [19] are considered. As demonstrated in Ref.[11] te developed scemes are more efficient tan te underlying DG scemes. Unsteady computations are presented for pure diffusion problems and advection-diffusion equations. Numerical results demonstrate tat te yperbolic rdg metod is a cost-effective ig-order metod and encourage applications to te incompressible and compressible Navier-Stokes equations. Te paper is organized as follows. Te difficulty in te construction of explicit yperbolic diffusion scemes is illustrated in Section II. Te fact tat it is not at all difficult in te case of te yperbolic rdg scemes is discussed in Section III. Numerical experiments are reported in Section IV. Concluding remarks and a plan of future work are given in Section V. II. Difficulity in Explicit Hyperbolic Diffusion Scemes Consider te time-dependent diffusion equation t = ν 2 ϕ x 2 (1 2 of 13

3 were ϕ denotes a scalar function tat can be referred to as velocity potential and ν is a constant. A classical yperbolic diffusion formulation [15 16] is given by t = ν v x x v x = 1 t ɛ ( (2 Downloaded by Hiroaki Nisikawa on July ttp://arc.aiaa.org DOI: / were ɛ is a small constant and v x is an additional variable denoted as te gradient of te primary variable ϕ. Tis system is equivalent to te original diffusion equation for a sufficiently small ɛ. Montecinos and Toro [17 2] derived an upper bound of ɛ wic preserves a design order of accuracy and developed ig-order scemes for te yperbolic formulation. In tis formulation explicit time integration scemes can be easily employed because te yperbolic formulation is equivalent to te diffusion equation in te partial-differentialequation level. However improved accuracy in te solution gradients on irregular grids as demonstrated in Ref.[21] is not confirmed in tis approac. In te yperbolic metod [1] te same yperbolic system is employed: t = ν v x x but te relaxation time T r is defined as v x t = 1 T r ( T r = L2 r ν L r = 1 2π (4 wic as been derived by requiring Fourier modes to propagate for fast steady convergence [1]. As discussed in Ref.[1] tis relaxation time is too large to ensure te consistency wit te original diffusion equation but te equivalence is guaranteed in te steady state. Te steady equivalence is te key idea in te yperbolic metod by wic spatial discretizations wit special features (e.g. iger-order accuracy in te gradient and convergence acceleration can be constructed. In te yperbolic metod time-accurate scemes are constructed based on te following form: τ + t = ν v x x v x τ = 1 T r ( were τ is a pseudo time variable. In te pseudo steady state (or as soon as te pseudo time derivatives are dropped we recover te equivalence wit te original usteady diffusion equation. Te system being yperbolic in τ we can discretize te spatial part by upwind metods and drop te pseudo time derivatives to obtain: dϕ dt = ν ( vx x = 1 T r ( were te subscript indicates te discrete approximation. Ten time-accurate scemes are constructed by implicit time-integration scemes e.g. by te backward Euler sceme: ϕ n+1 ϕ n = ν t ( n+1 vx x = 1 T r ( were t is a time step and n indicates te time level. Tese equations are fully coupled and need to be solved simultaneously at every time step to obtain ϕ n+1 and (v x n+1. Te variable coupling is essential n+1 (3 (5 (6 (7 3 of 13

4 Downloaded by Hiroaki Nisikawa on July ttp://arc.aiaa.org DOI: / to ensuring te properties of yperbolic scemes: same order of accuracy for ϕ and v x and fast iterative convergence in solving te unsteady residual equations. On te oter and if te forward Euler sceme is employed we ave ϕ n+1 ϕ n = ν t ( vx x n = 1 T r ( In tis case te equations are decoupled: ϕ n+1 is immediately updated by te first equation alone and tere is no update sceme for te oter variable (v x n+1. In effect te second equation as become redundant. One may solve ( x ϕ v x n+1 = for (v x n+1 but tis is simply a gradient reconstruction and terefore te equal order of accuracy for ϕ and v x wic is one of te advantages of te yperbolic metod is likely to be lost on irregular grids. In sort tis is just a conventional diffusion sceme wit gradient reconstruction. Te above argument explains wy implicit time integration scemes ave been employed exclusively in te yperbolic metod in all previous papers. Below it is discussed tat explicit time-stepping scemes can be constructed in a rater straigtforward manner in te yperbolic DG metod. III. Explicit Hyperbolic Reconstruction Discontinuous Galerkin Metods A. DG Discretization Consider te yperbolic diffusion system in te vector form: were n. U τ + T U t + F x = S (9 x ( ( ( ( ϕ 1 νv U = T = F x = x S =. (1 v x ϕ/t r v x /T r We begin by formulating te DG metod for te yperbolic diffusion system. We assume tat te domain Ω is subdivided into a collection of non-overlapping arbitrary elements Ω e and ten introduce te following broken Sobolev space V n: { V n = v [ L 2 (Ω ] k : v Ωe [ Vn k ] Ωe Ω} (11 wic consists of discontinuous vector polynomial functions of degree n and were k is te dimension of te unknown vector and V n is te space of all polynomials of degree n. To formulate te DG metod we introduce te following weak formulation wic is obtained by multiplying Eq. (9 by a test function W integrating over an element Ω e and ten performing an integration by parts: find U V p suc as W U dω + W TU dω + W F k n k dγ τ Ω i t Ω i Γ i (12 W F k dω = W SdΩ W V n x k Ω i Ω i were U and W are represented by piecewise polynomial functions of degrees p wic are discontinuous between te cell interfaces and n k te unit outward normal vector to te Γ i : te boundary of Ω i. Te standard DG solution U witin te element Ω i can be expressed as (8 U (x t τ = C(xV(t τ (13 were C is a basis matrix and V is a vector of unknown polynomial coefficients. In te implementation of te DG metods in tis paper modal based DG metods are adopted. Te numerical polynomial solutions are represented using a Taylor series expansion at te cell center and normalized to improve te conditioning of te system matrix. For instance according to te Taylor expansion in 1D one would ave ϕ = ϕ + ϕ c x xb 2 + ϕ c xx x 2 B 3 + ϕ c xxx x 3 B 4 + (14 4 of 13

5 Downloaded by Hiroaki Nisikawa on July ttp://arc.aiaa.org DOI: / were te ϕ represents te averaged quantity of ϕ and te superscript c stands for te central values. Te basis functions are given as follows B 1 = 1 B 2 = x x c x B 3 = 1 (B 22 1Ωe B 2 2dΩ 2 B 4 = 1 (B 32 1Ωe B Ω e 6 2dΩ 3 (15 Ω e were x =.5(x max x min =.5. (16 A more efficient construction can be devised [11] in wic te basis functions form a matrix C. A furter discussion about C and V will be given later in te next section. If we set te test function W as te transpose of te basis matrix C ten we obtain te following: τ C T CVdΩ + C T TCVdΩ + C T C F k n k dγ Ω i t Ω i Γ i Ωi T F k dω = C T SdΩ. (17 x k Ω i Te integrals are evaluated by Gaussian quadrature rules of appropriate orders and te flux at te interface will be computed art eac quadrature point for a given set of solution values obtained from te polynomials defined on two elements saring te interface. In tis paper te following upwind flux is employed for te diffusion term [22]: F i+1/2 = 1 2 (F L + F R 1 ν (U R U L (18 2 T r were i + 1/2 indicates an interface and te subscripts L and R indicate te solutions at te interface evaluated by te polynomials on te left and rigt elements respectively. Te resulting sceme is a DG metod of degree n or in sort notation DG(P n. By simply increasing te degree n of te polynomials te DG metods of corresponding iger order are obtained. B. Efficient Hyperbolic rdg Discretization Compared wit reconstructed FV metods te DG metods would require more degrees of freedom additional domain integration and more Gauss quadrature points for te boundary integration wic leads to more computational costs and storage requirements. Inspired by te reconstructed DG metods from Dumbser et al. in te frame of P n P m sceme[23 25] termed rdg(p n P m in tis paper least-squares based and variational reconstruction based rdg metods are designed to acieve ig order of accuracy wile reducing te computational cost. Te rdg metod is a general framework tat contains te FV and DG metods as special cases tus allowing for a direct efficiency comparison. For rdg(p n P m metod wit m > n a iger-order reconstructed numerical solution is constructed over an element Ω i : U R (x t τ = C R (xv R (t τ (19 were te superscript R indicates reconstructed polynomials and iger-order derivatives (iger tan n- t and up to m-t are reconstructed from te underlying P n polynomial. Tis iger-order numerical solution U R is used for flux and source term computations in order to raise te order of accuracy. Tere are tree approaces to te reconstruction. One is a least-squares reconstruction metod and anoter is a variational reconstruction metod. Te variational reconstruction generates a globally coupled system of equations for gradients by minimizing jumps in te solution and derivatives at element interfaces [19]. Te resulting linear system is iteratively solved along wit te solution iteration and terefore te cost is comparable to a least-squares reconstruction. See Ref.[19] for details. Te metod based on te leastsquares/variational reconstruction is expressed by rdg(p n P m. Te tird approac wic is unique in te yperbolic diffusion formulation considered ere is to directly use te gradient variables and teir moments to evaluate te iger-order derivatives in te primary solution polynomial. Or equivalently tis approac can be tougt of as defining te primary solution as P m and use te iger-order moments to represent te gradient variables in te yperbolic diffusion formulation. Tis is te key idea to effectively reducing te number of discrete unknowns despite te increase in te variables in te yperbolic diffusion formulation. Te metod based on tis approac is expressed by DG(P P m. Naturally rdg(p n P m and DG(P P m can be combined to generate efficient scemes as we will discuss later. Below we describe tis efficient construction of yperbolic rdgv scemes for te combination of DG(P P 1 and DG(P in one dimension as an example. Furter details can be found in Ref.[11]. 5 of 13

6 By moving te tird and fourt terms to te rigt-and-side (r..s. in Eq. (17 we arrive at V M τ τ + M V t t = R(UR (2 were M τ and M t are te mass matrices defined as M τ = C T CdΩ Ω i M t = C T TCdΩ Ω i (21 and R is te residual vector defined as C R = Ωi T F k (U R + C T S(U R dω C T F k (U R n k dγ. (22 x k Ω i Γ i Downloaded by Hiroaki Nisikawa on July ttp://arc.aiaa.org DOI: / Te mass matrix M t olds te key to te construction of explicit time-accurate scemes as we will discuss later. Boundary conditions are enforced weakly troug te numerical flux in a similar manner as in te previous work [26]. Based on different rdg metods effective yperbolic rdg scemes can be constructed. Te format A + B is used to indicate te discretization metod for te system were A refers to te discretization metod for ϕ and B refers to te discretization metod for its derivatives. Various coices and combinations for A and B are compared in Ref.[11]. As an example consider DG(P P 1 +DG(P wit ( ϕ V = (23 ϕ x x C = ( B 1 B 2. (24 B 1 x 1 As we can see ere te basis matrix C as a off-diagonal term to include te connection between ϕ and its derivatives leading to a coupled system. Note tat te derivatives of ϕ are determined as solutions to te yperbolic diffusion system wereas conventional P 1 DG metods determine tem as solutions to discrete equations derived by te weak formulation. See Ref.[11] for various oter ig-order scemes and te corresponding non-diagonal basis matrices. C. Explicit Hyperbolic rdg Sceme We now sow tat te yperbolic rdg metod leads to a semi-discrete system for wic explicit timestepping scemes can be applied. Dropping te pseudo time terms in Eq.(2 we obtain te following system of ordinary differential equations in te pysical time: were te mass matrix M t is given for DG(P P 1 +DG(P as ( M t = C T B1 TCdΩ = 2 B 1 B 2 dω = Ω i Ω i B 1 B 2 B2 2 M t dv dt = R(UR (25 (. (26 3 /12 Remarkably a nonzero entry as been created in M t (2 2 from T wit T(2 2 = due to te non-diagonal entry in C. Tat is a pysical time derivative as been created in te second equation for te gradient variable. Te system can be integrated in time by explicit time-stepping scemes; te solution and gradient variables are bot updated wit teir own rigt and sides. Tis is a time-accurate semi-discrete system because te pseudo time term as been dropped. Apparently te time accurate discrete equation for te gradient variable as been derived automatically by te DG differentiation: multiply te first equation (i.e. te diffusion equation by B 2 and ten integrate it. Te process is equivalent to differentiation and it generates a discrete approximation to te spatial derivative of te diffusion equation. But tat is not te 6 of 13

7 wole story. To see wat is going on we fully discretize te semi-discrete system (25 on a uniform grid wit a spacing using Simpson s rule for te volume integral: Downloaded by Hiroaki Nisikawa on July ttp://arc.aiaa.org DOI: / dϕ i dt d(v x i dt = ν [ ] (vx i+1/2 (v x i 1/2 (27 = ν (v x i+1/2 2(v x i + (v x i 1/2 (/ [ ] ϕi+1/2 ϕ i 1/2 2 (v x j (28 T r were V i = (ϕ i (v x i te subscripts i 1/2 and i + 1/2 indicate te left and rigt interfaces of a cell i and values from te numerical flux (18. In te second equation te time derivative and te first term on te rigt and side are te approximation to te spatial derivative of te diffusion equation created by te DG differentiation. However te second term on te rigt and side is te one tat dominates because of te factor 1/ 2. Terefore te semi-discrete system is in fact a consistent approximation to te following system: t = ν v x x v x = 1 ( (29 t ɛ were ɛ = 2 T r 12 = O(2 (3 wic is small enoug to preserve up to second-order accuracy [172] and tus te resulting discrete sceme is time accurate. A few remarks are in order. First te DG differentiation part is equivalent to a conventional P 1 DG sceme applied to te scalar diffusion equation and terefore te same order of accuracy cannot be expected for bot ϕ i and (v x i on irregular grids; (v x i would be one order lower. Te same order of accuracy observed ( in numerical experiments supports te above argument: te leading term is not ν xx v x but 1 ɛ In oter words te strong coupling in te differential-equation-level is retained in te discrete level and te sceme is solving te yperbolic diffusion system wit a sufficiently small relaxation parameter ɛ. Altoug similar te above sceme is different from tose of Montecinos and Toro [17 2] in te numerical flux (18 were our T r is O(1 wile teir T r is O( 1+r/2. Te latter makes te dissipation coefficient O(1/ for second-order scemes and it looks very muc like conventional diffusion scemes. Secondly te time accurate semi-discrete system is a feature unique to te rdg/dg metods; it cannot appen in FV metods nor in te scemes of Montecinos and Toro [172] wic is not based on te weak formulation (tus te relaxation time needed to be adjusted directly. Furtermore it is a unique feature coming from using te non-diagonal basis matrix C tat arises from te efficient construction as a test function were te gradient variable is used to form a iger-order polynomial for ϕ [11]. Te feature may be lost if a different test function is cosen e.g. te diagonal part of C. Finally te above argument is specific to DG(P P 1 +DG(P and it remains to be extended to iger-order scemes. In tis paper te time consistency of te semi-discrete system (25 in iger-order scemes is verified by numerical experiments. To demonstrate te semi-discrete system (25 for unsteady problems in tis paper we employ te following explicit tree-stage tird-order TVD Runge-Kutta sceme [2728](TVDRK3 for time integration:. V (1 = V n + tm t 1 R(V n V (2 = 3 4 Vn [U(1 + tm t 1 R(V (1 ] V n+1 = 1 3 Vn [V(2 + tm t 1 R(V (2 ]. (31 In summary explicit yperbolic DG scemes can be constructed in a straigtforward manner because te Galerkin formulation creates a pysical time derivative for te gradient variable v x and results in a semi-discrete system ready for explicit time stepping. Furter details suc as stability properties remain to be investigated; ere we demonstrate numerically tat ig-order explicit scemes actually work and deliver design orders of accuracy for unsteady problems. 7 of 13

8 IV. Numerical Examples A. 1D unsteady eat equation Consider te following 1D eat conduction problem: t = ν 2 ϕ x 2 x [ 1]. (32 u( t = u(1 t = u(x = C sin(πx Downloaded by Hiroaki Nisikawa on July ttp://arc.aiaa.org DOI: / wit Te exact solution is given by ν =.6 C = 5. (33 ϕ(x t = C exp( νπ 2 t sin(πx. (34 In tis case tree different yperbolic rdg scemes DG(P P 1 +DG(P DG(P P 2 +rdg-ls(p P 1 DG(P P 2 +rdg-vr(p P 1 are used to perform computations up to t = 1 wit a fixed time step = 1 4. DG(P P 1 +DG(P is a first-order sceme wile DG(P P 2 +rdg-ls(p P 1 and DG(P P 2 +rdg- VR(P P 1 are second-order scemes [1126]. Error convergence results are sown in Figure 1 wit uniform grids wit and 64 elements. Clearly te design order of accuracy as been confirmed for all scemes. Furtermore te same order of accuracy as been acieved in te gradient variable for all scemes. error Log( DG(PP1+DG(P DG(PP2+rDG_LS(PP1 DG(PP2+rDG_VR(PP1 error v x DG(PP1+DG(P DG(PP2+rDG_LS(PP1 DG(PP2+rDG_VR(PP Log( Figure 1: Grid refinement study on regular grids for 1D unsteady eat equation. B. 1D unsteady advection diffusion problem Consider te following exact solution to te 1D unsteady linear advection diffusion problem wit te exact solution given as were t + a x = ν 2 ϕ x [ 2] (35 x2 ϕ(x t = 1 4t + 1 exp ( (x at x 2 ν(4t + 1 (36 a = 1 4 ν =.1 x =.5. (37 Te parameters are cosen to yield an advection dominant problem. Te initial Gaussian bump will travel wit a constant velocity wit a small diffusion effect. Te numerical flux is constructed as a sum of te 8 of 13

9 Downloaded by Hiroaki Nisikawa on July ttp://arc.aiaa.org DOI: / upwind diffusion flux (18 and an upwind advective flux as described in Ref.[29]. For tis test case we test te fourt-order sceme DG(P P 3 +rdg-vr(p P 2 in addition to te tree scemes considered before. Tese scemes are known to acieved one-order iger-order of accuracy in te primal solution for advectiondominated cases [11]. A grid refinement test as been carried out to verify te spatial order of accuracy in tis unsteady case. A small pysical time step t = 1 9 as been set for all te grids wit a fixed final time t end = 1 4. Te grids are uniform wit nelem = and 256. Diriclet boundary conditions ave been applied on bot ends. All te presented metods use te same degrees of freedom 2 wic is equivalent to a conventional P 1 DG metod. Note also tat a FV sceme would ave 2 unknowns per element for te yperbolic diffusion formulation; owever te developed yperbolic rdg scemes are not FV scemes because of te coupling generated by te non-diagonal basis matrix C T. Te numerical results are sown in Figure 2. As can be observed DG(P P 1 +DG(P acieves first-order accuracy in te gradient and one-order iger-order of accuracy in te primal solution as expected. Unexpectedly DG(P P 2 +rdg- LS(P P 1 and DG(P P 2 +rdg-vr(p P 1 acieve asymptotically second-order accuracy in te gradient and fourt-order accuracy in te primal solution. Furtermore DG(P P 3 +rdg-vr(p P 2 gives fourt-order accuracy in bot te solution and te gradient. Secondly a comparison between te developed scemes and a conventional DG (Direct DG [3] in tis study is sown in Figure 3. Here periodic boundary conditions are enforced wit t = 1 9 t end = 1 3 on a uniform mes (nelem = 32. Clearly te presented yperbolic rdg scemes can outperform te conventional counterpart better resolving te peak of te Gaussian profile. Efficiency of te yperbolic rdg scemes are evident from te fact tat all scemes use 2 degrees of freedom per element including te P 1 DDG. error DG(PP1+DG(P DG(PP2+rDG_LS(PP1 DG(PP2+rDG_VR(PP1 DG(PP3+rDG_VR(PP2 Slope Log( error v x DG(PP1+DG(P DG(PP2+rDG_LS(PP1 DG(PP2+rDG_VR(PP1 DG(PP3+rDG_VR(PP2 Slope Log( Figure 2: Grid refinement study on regular grids for 1D unsteady case. C. 2D unsteady advection diffusion problem In tis case we extend te previous case to two dimensions. Te analytical solution is given by ϕ(x y t = 1 ( 4t + 1 exp (x at x 2 + (y bt y 2 (x y [ 2] [ 2] (38 ν(4t + 1 were Te following scemes are tested and compared: x = y = 1. a = b = 1 5 ν =.1. (39 DG(P P 1 +DG(P and DG(P P 2 +DG(P 1 DG(P P 2 +rdg-ls(p P 1 and DG(P P 3 +rdg-ls(p 1 P 2 DG(P P 2 +rdg-vr(p P 1 and DG(P P 3 +rdg-vr(p P 2 DG(P P 3 +rdg-vr(p 1 P 2 and DG(P P 4 +rdg-vr(p 1 P 3. 9 of 13

10 Exact DG(PP1+DG(P DG(PP2+rDG_LS(PP1 DG(PP2+rDG_VR(PP1 DG(PP3+rDG_VR(PP2 DDG(P1 v x 5 Exact DG(PP1+DG(P DG(PP2+rDG_LS(PP1 DG(PP2+rDG_VR(PP1 DG(PP3+rDG_VR(PP2 DDG(P x x Figure 3: Grid refinement study on regular grids for 1D unsteady case. Downloaded by Hiroaki Nisikawa on July ttp://arc.aiaa.org DOI: / Note tat te yperbolic rdg scemes wit rdg(p 1 P 2 and rdg(p 1 P 3 use six degrees of freedom per element wic is equivalent to te P 2 DG sceme DG(P P 2 +DG(P 1. Two sets of triangular meses are used in te test namely regular and irregular grids. Te sample of eac type of grids are sown in Figure 4. Y X Y X Figure 4: Te sample regular and irregular grids generated from a Cartesian grid by subdivision and nodal perturbation. Te pysical time step is set as t = 1 3 wit t end = 1. Diriclet boundary conditions are applied on all te boundary faces. Te numerical results are sown in Table 1 and Figures 5 to 6. All te presented scemes are sown to provide te design or iger order of accuracy in tis unsteady case. In particular DG(P P 3 +rdg-vr(p P 2 acieves sligtly iger tan fourt-order accuracy for bot te solution and te gradient on irregular grids wit only tree degrees of freedom per element wic is equivalent to a conventional P 1 DG sceme tat is second-order in te solution and first-order in te gradient. V. Conclusions and Outlook Contrary to expectations we ave demonstrated tat rdg metods applied to a yperbolic diffusion system are time-accurate wit explicit time-stepping scemes. Te yperbolic diffusion formulation used ere is formally equivalent to te diffusion equation only in te steady state. However it as been discovered tat te reconstructed discontinuous Galerkin discretization automatically creates a valid time-accurate 1 of 13

11 -2 error -4 DG(PP1+DG(P DG(PP2+DG(P1 DG(PP2+rDG_LS(PP1 DG(PP3+rDG_LS(P1P2 DG(PP2+rDG_VR(PP1 DG(PP3+rDG_VR(PP2 DG(PP3+rDG_VR(P1P2 DG(PP4+rDG_VR(P1P3 Slope 4 Slope 5 error v x -2-4 DG(PP1+DG(P DG(PP2+DG(P1 DG(PP2+rDG_LS(PP1 DG(PP3+rDG_LS(P1P2 DG(PP2+rDG_VR(PP1 DG(PP3+rDG_VR(PP2 DG(PP3+rDG_VR(P1P2 DG(PP4+rDG_VR(P1P3 Slope 4 Slope Log(DoF -1/ Log(DoF -1/2 Figure 5: Grid refinement study on regular grids for 2D unsteady case. Downloaded by Hiroaki Nisikawa on July ttp://arc.aiaa.org DOI: / error DG(PP1+DG(P DG(PP2+DG(P1 DG(PP2+rDG_LS(PP1 DG(PP3+rDG_LS(P1P2 DG(PP2+rDG_VR(PP1 DG(PP3+rDG_VR(PP2 DG(PP3+rDG_VR(P1P2 DG(PP4+rDG_VR(P1P3 Slope 4 Slope Log(DoF -1/2 error v x -2-4 DG(PP1+DG(P DG(PP2+DG(P1 DG(PP2+rDG_LS(PP1 DG(PP3+rDG_LS(P1P2 DG(PP2+rDG_VR(PP1 DG(PP3+rDG_VR(PP2 DG(PP3+rDG_VR(P1P2 DG(PP4+rDG_VR(P1P3 Slope 4 Slope Log(DoF -1/2 Figure 6: Grid refinement study on irregular grids for 2D unsteady case. semi-discrete system tat can be integrated in time by explicit time-stepping scemes. Tis special property as been found due to te non-diagonal nature of te basis matrix used in te efficient construction of te yperbolic rdg metod. Te off-diagonal entries create te pysical time derivatives troug te weak formulation and also scale te relaxation properly to recover time accuracy. In effect no special tecnique nor modification is necessary; explicit time-stepping scemes can be constructed in a rater straigtforward manner. Several ig-order yperbolic rdg scemes ave been constructed and demonstrated for an unsteady advection-diffusion equation. Tese scemes are very efficient acieving up to fourt-order accuracy on irregular grids wit only tree or six degrees of freedom per element in two dimensions wile conventional DG scemes can acieve only up to tird-order accuracy wit six degrees of freedom per element. Also te developed yperbolic rdg scemes acieve te same order of accuracy in te solution and te gradient on irregular grids in te diffusion limit wile conventional DG scemes yield one-order lower order of accuracy in te gradient. Tese yperbolic rdg scemes provide attractive alternatives to solve unsteady problems wit explicit time-stepping scemes. An undergoing effort is being put on extending te metod to nonlinear equations to sow its potential for applications to te Navier-Stokes equation on fully 3D unstructured grids. Acknowledgments Tis work as been funded by te U.S. Army Researc Office under te contract/grant number W911NF wit Dr. Mattew Munson as te program manager. 11 of 13

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