Active Flux Schemes for Advection Diffusion

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1 AIAA Aviation - Jne, Dallas, TX nd AIAA Comptational Flid Dynamics Conference AIAA - Active Fl Schemes for Advection Diffsion Hiroaki Nishikawa National Institte of Aerospace, Hampton, VA 3, USA Downloaded by Hiroaki Nishikawa on Jne 3, DOI:./.- In this paper, we etend the third-order active-fl diffsion scheme introdced in Ref.[AIAA Paper -9] to advection diffsion problems. It is shown that a third-order active-fl advection-diffsion scheme can be constrcted by adding the advective term as a sorce term to the diffsion scheme. The soltion gradient, which is compted simltaneosly to thirdorder accracy by the diffsion scheme, is sed to epress the advective term as a scalar sorce term. To solve the residal eations efficiently, Newton s method is employed rather than eplicit psedo-time iterations, which reires a large nmber of residal evalations. For nsteady comptations, it leads to a third-order implicit time-stepping scheme with Newton s method. Nmerical eperiments show that the reslting advection-diffsion scheme achieves third-order accrate and the Newton solver converges very rapidly for both steady and nsteady problems.. Introdction Active fl schemes have been developed for hyperbolic systems of conservation laws in Refs.[, ], bilt pon Scheme V of Van Leer [3], as a viable alternative to high-order methods. Active fl schemes are finitevolme-based compact high-order schemes. These schemes are sbstantially different from other high-order schemes and have attractive featres for a practical implementation. First, active fl schemes do not rely on a typical one-dimensional fl across a control-volme face, bt incorporate mlti-dimensional physics into the residal. The nmerical fl at a face is determined not by solving a one-dimensional Riemann problem, bt calclated by the method of spherical mean, which is an eact soltion to a mlti-dimensional initialvale problem. It is eivalent to a soltion to the characteristic eations in one dimension. Second, the memory reirement is mch redced compared with discontinos Galerkin methods de to sharing of degrees of freedom among elements. In addition to cell-averages, active fl schemes carry point-vales at faces; the latter are shared by adjacent cells, ths reslting in degrees of freedom per cell for third-order accracy in one dimension, and approimately 3. in two and three dimensions. In order to make an impact on practical trblent-flow simlations, however, the method needs to be etended to diffsion and viscos terms. It is, in fact, straightforward to constrct an eplicit active-fl diffsion scheme based on the Taylor epansion as shown in Ref.[], which can actally be shown to be eivalent to the recovery approach in Ref.[]. However, a severe stability restriction imposed on the eplicit time step discorages the se for practical problems. To overcome the limitation, an implicit time-stepping scheme was constrcted in Ref.[]. The constrction is made straightforward by the hyperbolic formlation of diffsion [7]. It has also been shown in Ref.[] that the activefl schemes lose third-order accracy nless sorce terms are discretized in a compatible manner and that the scheme can become inconsistent nless the bondary condition is implemented correctly. In this paper, we etend the implicit active-fl diffsion scheme to advection diffsion problems. It shold be pointed ot that the etension is not as straightforward as adding the diffsion scheme to the advection scheme. Sch a naive etension will destroy the third-order accracy easily. This is a well-known isse for schemes that reire compatible discretizations, inclding the residal-distribtion method [8], the third-order edge-based finite-volme method [9], and the active-fl method. One way to ensre the compatibility is to formlate the advection-diffsion eation as a single hyperbolic system []. Then, the constrction of the active-fl scheme will be trivially simple for the advection-diffsion eation. However, this strategy is crrently not applicable to the compressible Navier-Stokes eations becase a complete characteristic decomposition has not been discovered yet for hyperbolic formlations of the compressible Navier-Stokes eations []. As a practical alternative, we propose a strategy of adding the advective term to the diffsion scheme as a sorce term. The idea is applied to the comptation of the face vales, and the cell-averages are pdated by the sal finite-volme Associate Research Fellow (hiro@nianet.org), National Institte of Aerospace, Eploration Way, Hampton, VA 3 USA of 9 Copyright by Hiroaki Nishikawa. Pblished by the, Inc., with permission.

2 method with the sm of advective and diffsive fles to garantee discrete conservation. We demonstrate that the reslting advection-diffsion scheme yields third-order accrate soltion and gradients for both steady and nsteady advection diffsion problems. This work also aims at overcoming very slow convergence by the psedotime iterations employed in the previos work [] for solving a system of globally copled residal eations. It is shown that the convergence is significantly improved by Newton s method: five and two Newton iterations (i.e., only five and two residal evalations) are sfficient for steady and nsteady problems, respectively... Advection-Diffsion Eation Consider the advection-diffsion eation:. Hyperbolic Advection-Diffsion System τ + a = ν + s, (.) Downloaded by Hiroaki Nishikawa on Jne 3, DOI:./.- where τ denotes a psedo time, a is a constant advection speed, ν is a constant diffsion coefficient, and s = s (, ) is a sorce term. The sorce term incldes forcing fnctions as well as a physical time derivative, e.g., the backward difference formla (BDF), for time accrate comptations. In the discssion that follows, we focs on the psedo steady state, bt it is eivalent to advancing one time step in the physical time. Hence, the algorithm presented is eally applicable to steady and nsteady comptations. The active fl scheme is already available individally for the advective term [] and the diffsive term []. In some methods, a third-order advection-diffsion scheme can be easily constrcted by adding the advection and diffsion schemes. Eamples inclde the finite-volme methods and the discontinos Galerkin methods. However, in other methods, sch a simple constrction is known to destroy the formal accracy of the individal schemes, for eample, in the residal-distribtion method [8] and in the third-order edge-based finite-volme method [9]. The active fl scheme is a mitre of both as it consists of two steps: the characteristic fl comptation step and the finite-volme discretization step. The simple constrction is applicable to the finitevolme discretization step, bt not to the fl comptation step. The latter reires a carefl constrction to preserve the accracy as discssed in details for a hyperbolic system with sorce terms in Ref.[]. In this paper, we propose a strategy for constrcting active fl schemes for the advection-diffsion eation based on two alternative forms of the advection-diffsion eation... Conservative Form We write the advection-diffsion eation as a first-order hyperbolic system: where = [ p ], f = τ + f = s, (.) [ a νp /T r ], s = [ s s ], (.3) where s = p/t r. Note that the system is eivalent to the advection-diffsion eation in the psedo steady state for any nonzero vale of the relaation time, T r. The eivalence in the steady state is the key idea first introdced in Ref.[7] for constrcting diffsion schemes. Therefore, the relaation time has no physical importance, and can be determined solely by nmerical consideration, e.g., fast convergence to the steady state. A typical choice is the following: T r = L r ν, L r = π, (.) which can be derived for pre diffsion by reiring Forier modes to propagate to enhance the iterative convergence as discssed in Ref.[]. The same choice has been derived based on a similar argment applied to a first-order finite-volme scheme as described in Ref.[9]..3. Sorce Form The conservative form (.) is sitable for the finite-volme discretization step of the active fl scheme. However, the fl comptation step relies on the soltion of the characteristic eations, and therefore reires the of 9

3 complete eigen-strctre for the hyperbolic advection-diffsion system (.). The complete eigen-strctre is available for the advection-diffsion as described in Ref.[], bt crrently not available for the compressible Navier-Stokes eations. To avoid the difficlty, we propose a method based on the following form of the advection-diffsion eation: τ = ν p + s ap, (.) τ p = T r ( p), (.) where the advective term has been epressed by p, which will be eivalent to in the psedo steady state, and taken to the right hand side. A major advantage of this particlar form is that it is eactly in the form of a hyperbolic system with sorce terms, for which the active fl scheme is already available []. The eigenvales of the system are eactly the same as those of the hyperbolic diffsion system: ν ν λ =, λ =, (.7) T r T r Downloaded by Hiroaki Nishikawa on Jne 3, DOI:./.- and the eigenvectors are also the same []. Therefore, the fl comptation step is eactly the same as described in Ref.[] for pre diffsion problems, ecept that the sorce term incldes the advective term as ap. Note that since it reires only the eigen-strctre of the diffsive term, it can be etended to the compressible Navier-Stokes eations for which a complete eigen-strctre is known for a hyperbolized viscos term. 3. Active Fl Scheme for Advection-Diffsion Eation To discretize the hyperbolic advection-diffsion system (.) by the active fl scheme, we begin by storing the cell-averages within each cell and the point-vales at each face in a one-dimensional grid. In each cell, we constrct adratic polynomials, () and p(), by interpolating the face vales and reiring that the cell-average of the polynomial redces to the cell-averaged soltion. Integrating the system (.) over a space- (psedo-)time control volme of the cell j and the psedo time interval [k, k + ], we obtain ( k+ j k j )h = τ [ ] j +h/ f j+/ f j/ + τ s d, (3.) j h/ where h = j+/ j/ is the mesh spacing and τ = τ k+ τ k is the psedo time step. Note that f j+/ and f j/ are time-averaged fles, which can be evalated by Trapezoidal rle withot degrading accracy [], and that the sorce term is independent of psedo time. It leads to the following psedo-time marching scheme: where k+ j = k j τ h Res j, (3.) Res j = f j+/ f j/ j +h/ j h/ s d. (3.3) The sorce term discretization is performed eactly with the adratic representation of the soltion. The time-averaged fl is compted by Trapezoidal rle: f j+/ = a k j+/ + k+ j+/ ν pk j+/ + pk+ j+/ k j+/ + k+ j+/ T r, (3.) and similarly for f j/. The face vales at k + are compted by solving the characteristic eations. Here, we employ the sorce form in Section.3. It is in the form of the hyperbolic diffsion system with sorce terms. Therefore, the algorithm developed in Ref.[] is directly applicable. The soltion to the characteristic eations 3 of 9

4 of the hyperbolic diffsion system with sorce terms is given by k+ j+/ = [ ( R ) + ( L ) + L r (p( R ) p( L )) + j+/ s w d + λ R λ where j+/ L s w d ], (3.) p k+ j+/ = [ ( p( R ) + p( L ) + Lr ( R ) ( L ) + j+/ s w d j+/ )] s w d, (3.) λ R λ L L = j+/ + j/, R = j+3/ + j+/, (3.7) s w = L r s + (s ap), s w = L r s (s ap). (3.8) Note that the psedo-time step τ is taken locally with the CFL nmber. in the above formlas. The derivation of the above formlas is described in details in Ref.[]. For nsteady comptations, the physical time derivative discretized by the third-order backward difference formla (BDF3) is added to the sorce term s. See Ref.[] for details. Downloaded by Hiroaki Nishikawa on Jne 3, DOI:./.-. Implicit Solver For both steady or nsteady problems, it is necessary to solve the residal eation: Res(U h ) =, (.) where U h is a global nmerical soltion vector containing all cell-averages and face-vales. The global residal vector Res(U h ) is defined by a set of the cell-residals (3.3) and the face-residals. The face-residal is defined by Res j+/ = k+ j+/ k j+/, (.) p k+ j+/ pk j+/ for the face j + /. As in the previos stdy, the residal eation can be solved by the psedo-time iteration (3.) with the eplicit face-vale evalation as in Eations (3.) and (3.). Thanks to the hyperbolic formlation for diffsion, the nmber of psedo-time iterations ehibits linear increase with respect to the grid size, instead of adratic that is typical for diffsion, as demonstrated in Ref.[]. However, it still reires hndreds of residal evalations for convergence and it is highly desirable to improve the solver convergence. To this end, we constrct Newton s method: where the correction U h is obtained by solving the linearized system, U k+ h = U k h + U h, (.3) Res U h U h = Res(U k h). (.) The residal Jacobian on the left hand side is constrcted eactly, and the system is relaed by the Gass-Seidel relaation with the nder-relaation parameter of. to redce the linear residal by one order of magnitde in the L norm. Typically, we perform the Newton iteration to redce the residal by si and two orders of magnitde for steady and nsteady problems, respectively. The advantage of the linear (instead of adratic) increase in the nmber of iterations by the hyperbolic formlation is epected now in the nmber of linear relaations. The implicit active-fl advection-diffsion scheme may be compared with the third-order residaldistribtion scheme developed in Ref.[]. Both schemes are based on the hyperbolic advection-diffsion system and third-order accrate for both the soltion and the gradient on irreglar grids. The implicit solver is not eactly Newton s method for the third-order residal-distribtion scheme, bt the residal Jacobian is block tri-diagonal and the convergence is as rapid as Newton s method in practice. On the other hand, or solver is eactly Newton s method, bt the Jacobian is not compact having more than two off-diagonal blocks. Note that we have completely ignored the psedo-time derivative, and ths no psedo-time stepping is performed. Therefore, the reslting nsteady scheme is not a dal time-stepping scheme. It is an implicit time-stepping scheme with the nsteady residal eation solved by Newton s method. of 9

5 . Reslts.. Steady Problem We consider a steady advection-diffsion problem taken from Refs.[,, 3]: a = ν + s, (.) with the bondary conditions () = and () =, and the forcing term, s = νπ a [a cos(π) + νπ sin(π)]. (.) Downloaded by Hiroaki Nishikawa on Jne 3, DOI:./.- The eact soltion is given by () = ep( a/ν) ep(a/ν a/ν) ep( a/ν) + ν sin(π). (.3) a The parameters are set as a/ν =.,.,.,,. The Dirichlet bondary condition is imposed weakly at both ends as described in Ref.[]. Therefore, the soltion vales are determined by the nmerical scheme for all cells and faces, inclding the bondary faces. Steady convergence is taken to be achieved when the L norm of the residal is redced by si orders of magnitde. The initial soltion is set by a randomly pertrbed eact soltion. Comptations have been performed for a series of grids:, 3,, and 8 cells. The grids are niformly spaced for a/ν =.,.,., and slightly stretched for a/ν =,. Reslts are shown in Figres -. As shown in Figres, 3,, 7, and 9, the nmerical soltions are very accrate throghot he bondaries even on the coarsest grid of cells. Figres,,, 8, show that thirdorder accracy has been verified for both the soltion and the gradient for all cases. In the case of a/ν =, Figre shows that forth-order accracy is achieved for the soltion and the gradient at faces and the gradient in the cells. Althogh one order higher order of accracy has been known to occr for a finite-volme hyperbolic advection-diffsion scheme when advection dominates [9], it is not immediately clear how it happens to the active-fl scheme. These reslts show that the implicit solver converged within five Newton iterations and the nmber of linear relaations increases linearly with the nmber of nodes, not adratically which is typical for diffsion problems. Note that these reslts demonstrate significant improvements over the eplicit psedo-time stepping scheme as it only reires abot five residal evalations to obtain the soltion, rather than hndreds of residal evalations [9]. 8 c f Eact c f Eact....8 Figre. a/ν =.: Steady soltion on the -cell grid. (c, pc) are the cell-averaged vales, and (f, pf) are the point vales at the face. Log(L error) L(c ceact) L(pc pceact) L(f feact) L(pf pfeact) Slope Slope. Log(h) (Si order redction in residal) Linear Relaation Figre. a/ν =.: L Error and iterative convergence reslts for a steady problem. Linear Relaation.. Unsteady Problem We consider the following nsteady advection-diffsion problem, t + a = ν, (.) of 9

6 8 c f Eact c f Eact....8 Figre 3. a/ν =.: Steady soltion on the -cell grid. (c, pc) are the cell-averaged vales, and (f, pf) are the point vales at the face. Log(L error) 9 L(c ceact) L(pc pceact) L(f feact) L(pf pfeact) Slope Slope. Log(h) (Si order redction in residal) Linear Relaation Figre. a/ν =.: L Error and iterative convergence reslts for a steady problem. Linear Relaation Downloaded by Hiroaki Nishikawa on Jne 3, DOI:./ c f Eact c f Eact....8 Figre. a/ν =.: Steady soltion on the -cell grid. (c, pc) are the cell-averaged vales, and (f, pf) are the point vales at the face. with the following bondary conditions: Log(L error) 9 L(c ceact) L(pc pceact) L(f feact) L(pf pfeact) Slope Slope. Log(h) (Si order redction in residal) Linear Relaation Figre. a/ν =.: L Error and iterative convergence reslts for a steady problem. () =, (.) () = U cos(ωt), (.) Linear Relaation where U and ω are arbitrary constants. The eact soltion eists to this problem; it can be fond in Ref.[]. For nmerical reslts, the parameters have been set as follows: a =., ν =.3, U =., ω = π. (.7) Again, the bondary condition is imposed weakly and ths the soltions are compted by the nmerical scheme at all cells and faces. At every physical time step, the initial soltion is set as the soltion at the previos physical time step, and the Newton sb-iteration is taken to be converged when the L norm of the nsteady residal is redced by two orders of magnitde. To start p the comptation, we se BDF over the first step, BDF in the net step, and BDF3 thereafter. Ideally, it wold be best to perform the first two steps with a small enogh time step not to introdce large errors, bt nmerical reslts show that the low-order errors in the first two steps do not greatly impact the accracy of the soltion at a later time. We tested the scheme for a given grid of cells with randomly distribted nodes by refining the time step:./ m, where m =,,, 3. The final time is., i.e., si periods in the nsteady bondary condition. Reslts of 9

7 c f Eact c f Eact....8 Figre 7. a/ν = : Steady soltion on the -cell grid. (c, pc) are the cell-averaged vales, and (f, pf) are the point vales at the face. Log(L error) 9 L(c ceact) L(pc pceact) L(f feact) L(pf pfeact) Slope Slope. Log(h) (Si order redction in residal) Linear Relaation Figre 8. a/ν = : L Error and iterative convergence reslts for a steady problem. Linear Relaation Downloaded by Hiroaki Nishikawa on Jne 3, DOI:./ c f Eact c f Eact....8 Figre 9. a/ν = : Steady soltion on the -cell grid. (c, pc) are the cell-averaged vales, and (f, pf) are the point vales at the face. Log(L error) 9 L(c ceact) L(pc pceact) L(f feact) L(pf pfeact) Slope Slope. Log(h) (Si order redction in residal) Linear Relaation Figre. a/ν = : L Error and iterative convergence reslts for a steady problem. are shown in Figre, which confirms the formal third-order time accracy. The accracy deteriorates in the finest grid apparently becase the spatial error begins to dominate. Also shown is the iterative convergence for the Newton sb-iteration. Only two iterations, ths two residal evalations, are reired at every physical time step. This is a tremendos improvement over the eplicit psedo-time stepping scheme in Ref.[], which reires hndreds of residal evalations per physical time step. The cell-averaged soltion and gradient at t =.,.,.,.7 are plotted in Figres, 3,,, respectively. These reslts show that the activefl advection-diffsion scheme enables highly accrate nsteady simlations on a rather coarse grid even for irreglar grids. Linear Relaation. Conclding Remarks In this paper, we have etend the third-order active-fl diffsion scheme introdced in Ref.[] to advectiondiffsion problems. We constrcted a third-order active-fl advection-diffsion scheme by adding the advective term as a sorce term to the diffsion scheme. The soltion gradient, which is compted simltaneosly to third-order accracy by the diffsion scheme, is sed to epress the advective term as a scalar sorce term. For nsteady comptations, we employ a third-order implicit time-stepping scheme with the nsteady residal eations solved by Newton s method. Third-order accracy in the soltion as well as in the gradient, and rapid 7 of 9

8 Downloaded by Hiroaki Nishikawa on Jne 3, DOI:./.- Log(L error)... L(c ceact) L(pc pceact). L(f feact) L(pf pfeact) Slope Slope... Log( t) Average sb iteration ( order redction in residal) t Figre. Nmerical reslts for BDF3. Time accracy on the left and the nmber of sb-iterations on the right. convergence by Newton s method have been demonstrated for steady and nsteady problems. Ftre work shold focs on etensions to two and three dimensions as well as to the Navier-Stokes eations. The idea of treating the advective term as a sorce term can be etended to the compressible Navier-Stokes eations by a new hyperbolic formlation for the Navier-Stokes eations proposed in Ref.[] Nmerical soltion Eact soltion..... Nmerical soltion Eact soltion Figre. Unsteady soltion at t =.. t =. and cells Figre 3. Unsteady soltion at t =.. t =. and cells. Acknowledgments This work has been fnded by the NASA NRA/ROA- Contract NNXAJ7A. References Timothy A. Eymann and Philip L. Roe. Active fl schemes for systems. In Proc. of th AIAA Comptational Flid Dynamics Conference, AIAA Paper -38, Hawaii,. Timothy A. Eymann and Philip L. Roe. Mltidimensional active fl schemes. In Proc. of st AIAA Comptational Flid 8 of 9

9 . Nmerical soltion Eact soltion. Nmerical soltion Eact soltion Figre. Unsteady soltion at t =.. t =. and cells Figre. Unsteady soltion at t =.7. t =. and cells. Downloaded by Hiroaki Nishikawa on Jne 3, DOI:./.- Dynamics Conference, AIAA Paper 3-9, San Diego, California, Jne 3. 3 B. van Leer. Towards the ltimate conservative difference scheme. IV. a new approach to nmerical convection. J. Compt. Phys., 3:7 99, 977. M. Sn and P. L. Roe. Viscos active fl method. Unpblished, Jne 3. B. van Leer and S. Nomra. Discontinos Galerkin for diffsion. In Proc. of 7th AIAA Comptational Flid Dynamics Conference, AIAA Paper -8, Toronto,. H. Nishikawa and P. L. Roe. Active fl for diffsion. In Proc. of 7th AIAA Theoretical Flid Mechanics Conference, AIAA Aviation and Aeronatics Form and Eposition, AIAA Paper -9, Atlanta, GA,. 7 H. Nishikawa. A first-order system approach for diffsion eation. I: Second order residal distribtion schemes. J. Compt. Phys., 7:3 3, 7. 8 H. Nishikawa and P. L. Roe. On high-order flctation-splitting schemes for Navier-Stokes eations. In C. Groth and D. W. Zingg, editors, Comptational Flid Dynamics, pages Springer-Verlag,. 9 H. Nishikawa. First, second, and third order finite-volme schemes for advection-diffsion. J. Compt. Phys., 73:87 39,. H. Nishikawa. A first-order system approach for diffsion eation. II: Unification of advection and diffsion. J. Compt. Phys., 9:3989,. H. Nishikawa. New-generation hyperbolic Navier-Stokes schemes: O(/h) speed-p and accrate viscos/heat fles. In Proc. of th AIAA Comptational Flid Dynamics Conference, AIAA Paper -33, Honoll, Hawaii,. A. Mazaheri and H. Nishikawa. Very efficient high-order hyperbolic schemes for time-dependent advection-diffsion problems: Third-, forth-, and sith-order. Compters and Flids, :3 7,. 3 A. Mazaheri and H. Nishikawa. First-order hyperbolic system method for time-dependent advection-diffsion problems. NASA-TM 87, March. Katate Masatska. I do like CFD, VOL., Second Edition H. Nishikawa. Alternative formlations for first-, second-, and third-order hyperbolic Navier-Stokes schemes. In Proc. of nd AIAA Comptational Flid Dynamics Conference,, Reston, VA,. sbmitted for pblication. 9 of 9