Analysis and Implementation of Recovery-Based Discontinuous Galerkin for Diffusion 1 Marcus Lo 2 and Bram van Leer 3

Transcription

1 9th AIAA Computational Fluid Dnamics - 5 June 009, San Antonio, Texas AIAA Analsis and Implementation of Recover-Based Discontinuous Galerkin for Diffusion Marcus Lo and Bram van Leer 3 Department of Aerospace Engineering Universit of Michigan khlo@umich.edu, bram@umich.edu Abstract AIAA Progress in the analsis and implementation of the Recover-based DG method (RDG-x) for diffusion operators is reported. Among the new theoretical insights in the method we present a general stabilit proof, and an arbitrar-order penalt-term formulation in one dimension. Regarding implementation we discuss the savings afforded b pre-computing at each interface the smooth recover basis, which is invariant so long as the grid remains fixed. Furthermore, we discuss various versions of the RDG scheme in standard form (RDG-x), and the extension of RDG to nonlinear diffusion terms such as appear in the Navier-Stokes equations. In addition we discuss recover based on more than two neighboring cells, and not just for diffusion. A section with numerical results compares different implementations of RDG-x with RDG-x. Introduction Since its introduction at the 7th AIAA Computational Fluid Dnamics Conference, Toronto, 005[], and further reporting at the 8th conference in Miami, 007[], the development of the Recover-based Discontinuous Galerkin (RDG) method has led to further theoretical and practical insights, in part owing to the opportunit we had to port RDG into the CFD software package HOME of HPerComp (Westlake, CA) under an AFOSR-funded STTR contract, but also to the analsis, testing and applications contributed b other researchers [3, 4, 5]. In this paper we report the latest developments in RDG. We first review the definition and chief properties of RDG (Sec. ). Then follow separate sections on progress in the analsis (Sec. 3) and the implementation (Sec. 4) of the method. Of particular importance are the availabilit of at least an Ansatz to a general stabilit proof (Sec. 3.), and various was to implement recover in the standard DG form resulting after partiall integrating onl once (RDG-x, Sec. 4.). Next, a separate section (Sec. 5) is devoted to recover based on more c 009 b Marcus Lo and Bram van Leer. Published b the American Institute of Aeronautics and Astronautics, Inc., with permission. Doctoral candidate, member AIAA 3 Professor, Fellow AIAA Copright 009 b Bram van Leer and Marcus Lo. Published b the American Institute of Aeronautics and Astronautics, Inc., with permission.

2 than two neighboring cells, and not just for diffusion. The last section contains a numerical comparison of different implementations of RDG-x with RDG-x; the computations are done on a Cartesian grid, which brings out the accurac differences most clearl. Recover and Discontinuous Galerkin Recover is a name we gave 4 to a weak interpolation technique for locall reconstructing, as well as possible with the available degrees of freedom, a function of which an L -projection is known on the meshes of a computational net; the recovered function must have the same L projection. The latter propert provides the qualit control of the interpolation, just as in strong interpolation function- and/or derivative-values must be matched b the interpolant in points. Recover is especiall useful when including diffusion in a DG discretization, as the solution and its gradient are discontinuous at the interfaces where diffusive fluxes must be computed. In the RDG approach the diffusive fluxes are computed from a smooth, locall recovered solution that in the weak sense is indistinguishable from the discontinuous discrete solution[, ]. This eliminates the introduction of ad hoc penalt or coupling terms found in traditional DG methods; in the simple case of -D diffusion the recover principle can be shown to generate a string of penalt-like terms not encountered in standard DG methods (see Section 3.). The most attractive feature of RDG is the simplicit and generalit of the recover principle, which immediatel allows one to create and further develop DG diffusion schemes. In addition, these schemes are of higher accurac than an other DG diffusion schemes currentl in use, while boasting the smallest eigenvalues, hence the largest stabilit range for explicit time-marching; see Hunh, 009 [3]. This is particularl clear when comparing the methods on a uniform Cartesian mesh - Hunh analzed and tested 6 of them. For example, if the numerical solution u lies in a piecewise polnomial space that is the tensor product of one-dimensional polnomial spaces of degree p, then the order of accurac of RDG is 3p + or 3p + for p even or odd, respectivel 5. The largest negative-real eigenvalue for p = 3, when RDG is 0th-order accurate, equals -34, to be compared with -340 for Bassi-Reba- (6th-order accurate)[7], and -878 for CDG (8th-order accurate)[8]. While all these methods revert to the order of accurac p + on an irregular grid, the pedigree of RDG remains visible in the lower error coefficient and smaller eigenvalues. There are two basic versions of RDG: RDG-x and RDG-x. RDG-x is the original version and starts from a weak form of the diffusion equation 6, U t = D U, () 4 The use of the term recover in CFD is due to Bill Morton [6], who indicated with it the reconstruction of a shock discontinuit from the structure of a captured numerical shock. This is trul the opposite of what we are doing in RDG for diffusion: recovering a smooth function from a discontinuous solution. 5 An earlier supposition[] that the order might be exponential (,4,8,... for p=0,,,...), turned out to be false. 6 The diffusion equation is taken linear in this and other sections; Sec. 4. deals with the nonlinear-sstem case.

3 obtainable after multiplication with a test function and integration b parts twice over the interior of an element with boundar : (v k ) j U t dω = D {(v k ) j U U (v k ) j } ˆn d Ω + D U (v k ) j dω, k =,..., K. () Here U is the true solution, D is a constant scalar diffusion coefficient and the v k are the test functions, also used as the basis functions; these are polnomial on and zero elsewhere. In a DG method U is replaced in the above equation b a numerical approximation u that in each element lies in the span of the basis {v k }. This numerical solution is discontinuous at ; if the boundar integral in () is evaluated on the inside of the boundar, the method will be inconsistent since there will be no coupling with the neighboring cells. In RDG the coupling is provided b replacing u on a facet of the boundar b f, a smooth solution recovered from the information in the two abutting elements. The recovered solution f j,j+, centered at the interface between elements and +, is uniquel determined b making it indistinguishable from u in the weak sense in the contributing elements, that is, (v k ) j f j,j+ dω = (v k ) j u dω, k =,..., K, (3) (v k ) j+ f j,j+ dω = (v k ) j+ u dω, k =,..., K. (4) + + In order to satisf these equations, f must lie in the space of the recover basis w k, with k =,.., K; for a detailed description of this basis see []. Each face of generates a different recovered solution f j,.. ; thus, there are as man surrogates for u or U as there are faces to the element. Inserting these into the boundar integral ields the RDG scheme: (v k ) j u t dω = D {(v k ) j f j,.. f j,.. (v k ) j } ˆn d Ω Ω + D u (v k ) j dω, k =,..., K. (5) It is worth noticing that replacing u b an of the f j,.., or an average of these, in the volume integral on the right-hand side does not change the value of the integral (see []). For future reference we also give the weak form of the diffusion equation resulting after integrating b parts once: (v k ) j U t dω = D (v k ) j U ˆn d Ω D U (v k ) j dω, k =,..., K; (6) all traditional DG diffusion schemes are based on this form, which is wh we have called it standard. In [9] we indicated how to implement the DG form (6) using recover, and called the resulting discretization RDG-x. We will discuss several implementation options in Sec

4 3 Theoretical results 3. A stabilit proof To assess the stabilit of RDG schemes, RDG-x in particular, we replace the test function v k in Eqn. (5) b the solution u, which combines all test functions, and integrate over the entire computational domain V with boundar S; this ields an equation for the evolution of the energ of the solution on the domain: V t ( ) u dv = D S (u f f u) ˆn ds + D [u f f u] ˆn de E + D u u dv. (7) Here E represents the union of all interior cell faces; the jump [ ] and normal ˆn at an interface are matched such that if ˆn is chosen to point from cell j to cell j, [ ] is defined as ( ) j ( ) j. Note that we have replaced U and U on the domain boundar b their recovered values f and f; it is f, not u, to which the boundar condition must be applied. We now undo the second partial integration so as to arrive at the following equation: V t ( ) u dv = D S V {u f + (u f) u} ˆn ds + D [u f + (u f) u] ˆn de E D u u dv. (8) If the energ can be shown to deca in time, this is proof of global stabilit. It generall means proving that the last term on the right-hand side of Eqn. (8), i. e.,the volume integral, which is negative and of magnitude O(), dominates the other terms. For p = 0 the volume integral vanishes, and a separate proof is needed, We assume that V and S are both of magnitude O(), and h is a measure of the average element size. Furthermore, we assume u lies in a polnomial space of order p and the solution is smooth to begin with (discontinuous initial values are excluded). This allows us to estimate the boundar- and interior-face terms. The integral over the interior faces is larger than the integral over the domain boundar, because the area of E is /O(h), as compared to O() for S. Using angled brackets to indicate the interface average of a function, the integrant of the former integral can be reduced as follows: [u f + (u f) u] ˆn = {[u] f + [u f] u + u f [ u]} ˆn V = {[u]( f + u ) + ( u f)[ u]} ˆn. (9) Here [u] and u f are both of order O(h p+ ), while [ u] = O(h p ) for p and 0 for p = 0. Hence, the two terms on the final right-hand side of (9) are O(h p+ ) and O(h p+ ), 4

5 respectivel, for p. The first term dominates, so the integral over E, with total area /h, becomes O(h p ) for p. On a sufficientl fine grid this integral therefore becomes negligible compared to the negative volume integral. Now consider the domain-boundar terms. Let us first assume that the boundar is adiabatic, i. e., the diffusive flux D f ˆn vanishes at the boundar. This removes the first term of the boundar integral in (8). B realizing that u = f + (u f) we ma reduce the integral to (u f) D ˆn ds, = O(h p+ ), (0) S which is small on a sufficientl fine grid, for all p. Thus, on a sufficientl fine grid the negative volume integral dominates both boundar- and interior-face integrals for p ; the solution energ decreases in time, impling stabilit. For a Dirichlet boundar condition, e. g., f = 0 on S, the integrant in the domainboundar term reduces to u f + (u f) u = (u f) (u + f) = O(h p+ ), () which vanishes with h for all p. Again, for p the negative volume integral dominates both boundar- and interior-face integrals on a sufficientl fine grid, causing the solution energ to deca. For p = 0 we have u = 0, so the volume integral in (8) vanishes; the integral over E now takes over its role. The integrant (9) reduces to [u] f ˆn; in this case [u] = O(h) and the recover procedure ields f ˆn = [u] h cn, () where h cn is the distance between the centroids of neighboring elements measured along their interface-normal. Therefore, [u] f ˆn = [u] h cn = O(h), (3) and the integral over E becomes negative of order O(). This dominates the domainboundar integral in both Neumann and Dirichlet cases, so stabilit is ensured. Note that (3) ma be interpreted as stemming from the Baker[0]-Arnold[] penalt term µd[v][u]/h, with positive µ = O(); cf. Sec. 3.. For p = 0 this term has to provide the phsical content of the diffusion operator, so the value of µ must be chosen correctl with regard to consistenc []. The recover procedure automaticall generates the term with the proper value. This completes the stabilit proof for RDG-x; the proof for the naive form of RDG-x (see Sec. 4.) is obtained b omitting the terms proportional to u f in the integrals over E and S in Eqn. (8). The global RDG form (8) is not just useful in a stabilit proof, but, as pointed out b Marshall Galbraith (Universit of Cincinnati), it actuall suggests an excellent local 5

6 implementation formula that is equivalent to RDX-x: (v k ) j u t dω = D {(v k ) j f + (u f) (v k ) j } ˆn d Ω D (v k ) j u dω, k =,..., K. (4) The formula looks like a hbrid of RDG-x and RDG-x: it has the volume integral of RDG-x but shows two boundar terms as in RDG-x. The great advantage is that one does not have to use recovered information in evaluating u in the volume integral; one ma use u as is. The second boundar term looks somewhat like a penalt term; it contains the information to elevate the accurac of naive RDG-x (without recovered volume integral) to that of RDG-x, most noticeable on a Cartesian grid. See further Secs. 4. and Penalt-term expansion in one dimension We have succeeded in writing the -D form of the RDG-x scheme of arbitrar order as an expansion in penalt-like terms. On a uniform grid it reads: vu t dx = ( u x [v] D j+ Ω j + u x [v] + R ( ) 0 [u] [v] j+ + [u] [v] x +R x ( ) [u x ] [v x ] j+ + [u x ] [v x ] +R x ( ) [u xx ] [v] j+ + [u xx ] [v] ) ( u [vx ] j+ + u [v x ] +R 3 x ( ) 3 [u xxx ] [v x ] j+ + [u xxx ] [v x ] +R 4 x ( ) 3 [u xxxx ] [v] j+ + [u xxxx ] [v] +R 5 x ( ) 5 [u xxxxx ] [v x ] j+ + [u xxxxx ] [v x ] +R 6 x ( ) 5 [u xxxxxx ] [v] j+ + [u xxxxxx ] [v] +... ; ) v x u x dx here the jump of a function at an interface is alwas computed as right value minus left value. It is seen that the i-th bilinear penalt term, with coefficient R i, contains the jump in the i-th derivative of u and the jump in either v or v x, depending on whether the index i is even or odd, respectivel. For an RDG scheme based on a polnomial space of degree p, the index runs from 0 to p. The coefficients R i are given in Table for schemes up to p = 5. Note that for p > the spatial operators are not smmetric; however, the real part of their eigenvalues is found (b numerical evaluation) to be negative, allowing for stable update schemes. These discretizations are also highl accurate: the truncation error is of the order 3p + or 3p + for p even or odd, respectivel. (5) 6

7 p µ = R 0 R R R 3 R 4 R Table : Coefficients of the penalt-like terms in -D RDG for p 5. 4 Progress in implementation 4. RDG schemes in standard form As reported in [9], it is possible to insert the information obtained in the recover procedure into the standard DG form (6) so as to arrive at an RDG-x discretization that competes in accurac with the RDG-x discretization with equal p. In this section we review a hierarch of three RDG-x implementations based on (6), and one based on the quasi-standard form (4), which is reall equivalent to RDG-x. The most naive form of RDG-x has u replaced b f in the boundar term, but u in the volume integral computed as is; this leads to an accurac that is considerabl lower than that of RDG-x with equal p. For instance, for p = on a -D Cartesian grid, naive RDG-x is onl second-order accurate, while RDG-x is fourth-order accurate. The loss of accurac is solel caused b the absence from Eqn. (6) of the O(h p+3, h p+ ) second boundar term present in Eqn. (4). Next, we consider replacing u in the volume integral b f, where f is the average of the solutions recovered on cell and each of its neighbors. The accurac resulting from this procedure is expected to be better than that of naive RDG-x, but onl b a factor. The reason is that the recovered solutions are particularl accurate onl in their dependence on the coordinate normal to the interface of the two participating cells []; their representation of the solution in the direction parallel to the interface is not better than order p, as for u itself. Aware of the anisotrop in the accurac of the recovered solutions, we recommended in [9] to evaluate u using onl the interface-normal derivatives f j,l / n j,l,of the solutions recovered on and Ω l, where l ccles through all neighbors of element j. Since there are more neighbors than dimensions, even for a simplex cell, this leads to a small least-squares problem that is alwas well conditioned; note that its size is independent of p. With the 7

8 gradient thus recovered, which we shall call ˆf j, the accurac of RDG-x competes with that of RDG-x on a Cartesian grid. But the implementation is more involved than the quasistandard implementation (4), which we expect to become the standard of implementation of RDG for diffusion. It is worth reiterating from [9] that in our first paper on RDG [] we derived, b truncation-error minimization, a remarkabl accurate member of the classical DG famil [] vu t dx = D ( ) < u x > [v] j+ + < u x > [v] I D v x u x dx j I j + σd ( ) < v x > [u] j+ + < v x > [u] µd ( ) j+ [v][u] + [v][u] x, (6) with σ = 4, µ = 9 4. (7) This scheme turns out to be an RDG-x scheme with u in the volume integral evaluated as ˆf or f (these are equal in one dimension). In Sec. 6 the above discretizations are numericall tested and compared on a Cartesian grid. Note that even on a regular triangular grid, let alone on an on an irregular grid, all RDG-x and RDG-x schemes reduce to the order of accurac p + [9]; differences in error level and in eigenvalue magnitude ma still remain. In particular, the eigenvalues of RDG-x with recover of u appear to be more than 50% larger than those of RDG-x for p = and. 4. RDG for a nonlinear diffusion sstem While the twice-integrated form () was essential for the development of the RDG approach, and leads to the simplest discretization of a linear scalar diffusion equation, it ma not necessaril be the preferred choice when considering the DG discretization of a nonlinear sstem of diffusion operators, such as contained in the Navier-Stokes equations. Below the two weak nonlinear forms are displaed as the would appear in DG discretizations of the Navier-Stokes equations in conservation form (Euler terms suppressed); bold face indicates a vector or matrix in the 5-dimensional space of state variables, wheras an arrow indicates a vector in 3-D phsical space. For instance, D indicates an 3 3 arra of 5 5 matrix-valued diffusion coefficients corresponding to diffusive fluxes of all state quantities in all coordinate directions. Both the dot product and the gradient operator refer to phsical space. Differential form: U t = D U; (8) Weak form once partiall integrated (RDG-x): (v k ) j u t dω = (v k ) j ( D f) ˆn d Ω ( D ˆf) (vk ) j dω, k =,..., K; (9) 8

9 Weak form twice partiall integrated (RDG-x): (v k ) j u t dω = + [(v k ) jd f { D (vk ) j }f] ˆn d Ω [ { D (vk ) j }]u dω, k =,..., K. (0) The first weak form has the advantage that we ma use the discrete solution u j as is to compute the volume integral, rather than the set of recovered solutions f j,.. as recommended for the first form. This advantage, though, disappears when computing on irregular, unstructured grids, where all DG methods with a basis of order p have their accurac reduced to the order p +, so ˆf in the volume integral of Eqn. (9) ma as well be simplified to u. Moreover, the first form includes u or f on the right-hand side solel in the combination D u or D f; this expression greatl simplifies when switching to the primitive variables w = (ρ, u, v, w, T ) T favored for the Navier-Stokes equations. We have D u = M w, () where the blocks of M are ver sparse. In contrast, the second weak form also contains the combinations { D (vk ) j }u and { D (vk ) j }f, which do not benefit from simplification b switching to primitive variables. Even the quasi-standard form (4) extended to a nonlinear sstem, retains the expression in the second boundar term: (v k ) j u t dω = [(v k ) jd f + { D (vk ) j }(u f)] ˆn d Ω ( D u) (vk ) j dω, k =,..., K; () Summarizing: RDG-x, with the simplest possible evaluation of the volume integral, ma be the most effective choice for use with the Navier-Stokes equations. However, there is no bod of practical numerical evidence et to support this presumption; furthermore, the aspect of a possibl reduced stabilit range associated with larger eigenvalues needs to be weighed in. The quasi-standard scheme (4) remains a strong contender. 4.3 Use of the recover basis As discussed at length in [3] and [9], the recover procedure on Ωj+ can be first applied to the discontinuous basis functions (v k ) j, k =,.., K, and (v k ) j+, k =,.., K, ielding a set of smooth basis functions (w l ) j,j+, l =,.., K. The principal recover theorem sas that the expansion of u on Ωj+ in the discontinuous basis functions and the expansion of f in the corresponding smooth basis functions have identical coefficients. Since the recover process is onl influenced b the geometr of and +, the recover basis will not change if the grid stas fixed. Hence, for calculations on a fixed grid it suffices to compute the smooth recover bases for all pairs of neighboring elements at the start of a calculation, and 9

10 store these; no further recover needs to be done at an later time. Recover thus reduces to a pre-processing step for an entire calculation, ielding large computational savings. Our numerical experiments so far have alwas been based on fixed grids; for unstructured grids our RDG schemes have indeed been coded using pre-computed recover bases. 5 Isotropic recover and cell-centered recover So far we have limited ourselves to recovering a solution on the union of two neighboring, interface-sharing cells; we shall refer to this as a binar recover. As pointed out before, such a recover is anisotropic in its accurac []: its representation of the solution in the direction normal to the interface is accurate to the order p +, while the representation in an direction parallel to the interface is onl of the order p. The anisotrop could be remedied, if so desired, b adding some information from neighboring cells that span a distance along the interface. Figure (a) shows, in two dimensions, the obvious extended Cartesian stencil, while the butterfl in Figure (b) is a possible extended stencil for an unstructured triangular grid. When recovering the solution on these stencils it makes sense to match fewer data in the more remote cells than in the principal pair. C D C A B E A B D F E F (a) (b) Figure : Extended Cartesian (a) and triangular butterfl (b) stencils for more isotropic interface-based recover. For instance, with p = the principal cells A and B each carr three independent data (defining a linear function) that must be matched b the recovered solution; in each of the four remote cells C-F we shall onl match the cell-average. Thus, the number of recover equations increases from 6 to 0. The latter is the correct number of data for a complete cubic basis, but the collection of basis functions representable on the rectangle or butterfl is not quite right. Calling the coordinate normal to the principal interface r and the coordinate along the interface s, it is seen that we will fail to obtain a coefficient for the function r s; instead, we obtain a coefficient for the function rs 3, not needed for the cubic basis. In the end, the recover is onl quadraticall correct in all directions. To reduce the waste we might constrain the recovered function to have the correct average value in C D and E F, rather than in each of the four cells separatel. This 0

11 reduces the number of recover equations to 8; the coefficients of rs and rs 3 no longer can be found. Left over is a quadratic basis augmented with r 3 and s 3. Initiall we asked ourselves if it would be possible to obtain the detailed s-dependence at the interface between A and B purel from binar recoveries, i. e., from the cell pairs (A, C), (A, E), (B, D) and (B, F). This is in the same vein as the gradient reconstruction (Sec. 4.) for computing the RDG-x volume integral, sa, in cell A; there we intentionall avoided performing a new, cell-centered recover on the union of A and its neighbors B, C and E. As it turned out, the binar-recover information increases the accurac of RDG-x onl on a Cartesian grid; even on a regular triangular grid [9] the accurac falls back to the usual order p +. So we are pessimistic about re-using binar-recover information to enhance RDG accurac. x x x x x x (a) (b) Figure : Cartesian stencil (a) used b Park et al. [5] for crdg, and an especiall thrift stencil (b) for cubic crdg. Nourgaliev et al. [4, 5], though, have begun to explore cell-centered recover to enhance the accurac of DG; we shall refer too this as crdg, to contrast it with interface-centered recover-based DG or irdg. Rather than removing an interface discontinuit, the goal of crdg is to increase the order of the solution basis in the central cell b weak interpolation. The increase could of course simpl be achieved b starting out with a higher-order DG basis; the advantage of increasing the order b recover is that the number of unknowns does not increase. For Nourgaliev et al. this is the prime motivation, as their temporal discretization is implicit and treated with a Newton-Krlov solver. The computational expense of cellcentered recover is small change compared to the expense of the implicit solver further down the line. The -D numerical example of crdg given in [5] is based on a manufactured travelingwave Navier-Stokes solution; it starts from a p = basis and uses the stencil and basis functions displaed in Figure (a) for the cell-centered recover. This recover-enhanced solution provides the input for the Riemann solver used to obtain the inviscid interface fluxes. Diffusive interface fluxes are computed on the stencil shown in Figure (a); not mentioned in the paper is whether crdg and irdg are applied in parallel, or crdg pre-processes the data used in irdg. The crdg-enhanced solution has 5 parameters but is not full quartic; two mixed terms, x 3 and 3 x, are missing, while x 5 and 5 are included. The numerical results show an increase from nd-order accurac without crdg to 6th-order accurac with crdg. The

12 former result is puzzling, as a Navier-Stokes DG calculation with upwind Euler fluxes and diffusive fluxes from binar recover should ield third-order accurac; the latter result ma be flattered because the test problem could be too simple. Nevertheless, the increase in accurac is impressive and suggests irdg was applied on top of crdg. With regard to cutting waste we offer the observation that the combination of the Cartesian stencil and basis functions shown in Figure (b) is about as thrift as it gets: from a DG basis with p = it recovers a complete cubic basis, with onl one parameter to spare. The extra parameter results from the fact that there are two was to obtain the coefficient of the function x in the central cell: b using horizontal neighbors or b using vertical neighbors. The difference between the two is a combination of 4th-order terms. Based on our experience with irdg, though, we expect the gains on an unstructured triangular grid to be rather modest. For instance, when enhancing a p = basis in cell A, using all the information from its neighbors B, C and E ( data in total), still onl a quadratic basis can be recovered without overl constraining the grid. In particular, if the centroids of 3 triangles line up, as in Figure 3, there is insufficient coverage of -D space. In the displaed stencil no coefficient can be found for the function x. Similarl, if onl the solution-averages are matched in the neighboring cells (6 data in total), the same degenerate geometr even prohibits going beond a p = basis. In this case no coefficient of x can be found. x x x x Figure 3: Degenerate triangular stencil: three centroids are aligned along the x-axis, spoiling the potential for cubic recover. We ma regard crdg as a cross between a finite-volume and a finite-element method; in fact, a finite-volume Euler method such as MUSCL ma be interpreted as a crdg method with p = 0, where the recover aims at increasing the order of the solution representation to p =. The use of cell-centered recover as a pre-processing step to enhance a DG solution offers a wealth of possibilities, especiall when combined with interface-based recover. We are bound to see more of this approach.

13 6 Numerical results In order to demonstrate the differences in accurac among the various versions of RDG-x, and with respect to RDG-x, we solved a Dirichlet problem on the square [0, ] [0, ] with exact stead solution U(x, ) = cos(πx) + cos(π). (3) For our purpose it sufficed to stud the case p =. At the boundar, recover in the normal direction involved the next cell inward to make it cubic [], as needed to match the accurac at interior interfaces. We tested three RDG-x schemes, with the following features (cf. Sec. 4.): (A) volume integral computed with u, (B) volume integral computed with f, (C) volume integral computed with ˆf; for comparison, results were also obtained with RDG-x cast in (D) quasi-standard form (4). 0 0 L Error in u Slope= 0 5 Slope= h RDG Version A RDG Version B RDG Version C RDG Version D Figure 4: Grid-convergence stud: L error in cell-average versus mesh width for three RDG-x schemes and the RDG-x scheme (4). Dirichlet problem on Cartesian grid; p =. The convergence results displaed in Figure 4 (L -error in cell-averages versus mesh-width h) confirm the theor-based prognosis made in Sec. 4.. Versions (A) and (B) both are 3

14 second-order accurate, but (B) has about half the error of (A) owing to the averaging of accurate with inaccurate information in f. Version (C) is fourth-order accurate, though its error coefficient is more than twice that of the RDG-x version (D). Not shown is that for (C) and (D) the other solution coefficients, u x and u, are converging faster than u, possibl heading toward 5th-order accurac on still finer grids. 7 Conclusions During the past ear Recover-based Discontinuous Galerkin methods have been put on a firm footing. There now is a sufficientl general stabilit proof based on an energ argument, and in one dimension we also know the precise string of penalt terms and coefficients that generate RDG for an order of the DG basis. Users can choose between various implementations in the standard, once-partiall-integrated form RDG-x, and the twice-partiallintegrated form RDG-x, which is more accurate and stable. The latter scheme can again be rewritten in the standard form, with a small but significant extra boundar term remaining. When implemented on a Cartesian grid, RDG-x is the most accurate and most stable among all known DG diffusion methods, for a given order p of the basis (Hunh, 009). Nevertheless, for a sstem with nonlinear diffusion terms like the Navier-Stokes equations, RDG-x has the advantage of being somewhat easier to implement than RDG-x. Besides interface-based recover (irdg) for DG diffusion fluxes, there is now also cellcentered recover (crdg), developed b Nourgaliev et al. to enhance the DG basis; it is still restricted to Cartesian grids. This technique increases the accurac of upwind advection fluxes and ma also be used as a preprocessor for irdg. One ma regard crdg as a cross between DG and the Finite-Volume method. We expect to see much more of this, once a robust formulation for unstructured grids has been established. Altogether, RDG appears to have a bright future. 8 Acknowledgement This work was supported b AFOSR Grant Nr. FA References [] B. van Leer and S. Nomura, Discontinuous Galerkin for diffusion, AIAA Paper , 005. [] B. van Leer, M. Lo, and M. van Raalte, A Discontinuous Galerkin method for diffusion based on recover, AIAA Paper , 007. [3] H. T. Hunh, A reconstruction approach to high-order schemes including Discontinuous Galerkin for diffusion, AIAA Paper ,

15 [4] R. Nourgaliev, T. Theofanous, H. Park, V. Mousseau, and D. Knoll, Direct numerical simulation of interfacial flows, AIAA Paper AIAA , 008. [5] H. Park, R. Nourgaliev, V. Mousseau, and D. Knoll, Recover Discontinuous Galerkin Jacobian-free Newton Krlov method for all-speed flows, Tech. Rep. INL/CON-08-38, Idaho National Laborator, 008. [6] K. W. Morton and M. A. Rudgard, Shock recover and the cell-vertex scheme for the stead Euler equations, Lecture Notes in Phsics, vol. 33, pp , 989. [7] F. Bassi and S. Reba, High-order accurate discontinuous finite element solution of the D Euler equations, Journal of Computational Phsics, vol. 38, pp. 5 85, 997. [8] J. Peraire and P. O. Persson, The compact discontinuous Galerkin method for elliptic problems, SIAM Journal on Scientific Computing, vol. 30, pp , 008. [9] B. van Leer and M. Lo, Unification of Discontinuous Galerkin methods for advection and diffusion, AIAA Paper , 009. [0] G. A. Baker, Finite element methods for elliptic equations using nonconforming elements, Mathematics of Computation, vol. 3, pp , 977. [] D. N. Arnold, An interior penalt finite element method with discontinuous elements, SIAM Journal on Numerical Analsis, vol. 9, pp , 98. [] M. H. van Raalte, Multigrid Analsis and Embedded Boundar Conditions for Discontinuous Galerkin Discretization. PhD thesis, Universit of Amsterdam, 004. [3] M. van Raalte and B. van Leer, Bilinear forms for the recover-based discontinuous Galerkin method for diffusion, Communications in Computational Phsics, vol. 5, pp ,