Unstructured-Grid Third-Order Finite Volume Discretization Using a Multistep Quadratic Data-Reconstruction Method

Transcription

1 AIAA JOURNAL Vol. 48, No. 4, April 010 Unstructured-Grid Third-Order Finite Volume Discretization Using a Multistep Quadratic Data-Reconstruction Method D. Caraeni and D. C. Hill ANSYS, Inc., Lebanon, New Hampshire DOI: / Finite volume discretizations on unstructured grids that are more than second-order-accurate have not yet gained wide acceptance. This is due to the high computational cost and memory requirements of the proposed schemes and the technical difficulties in achieving a high-order-accurate solution. It is especially true when considering flows in complex geometries. In this paper, a third-order cell-centered finite volume discretization is detailed that is based on quadratic data recovery, which is computationally efficient and has optimal memory requirements. The densitybased flow-solver discretization and solution algorithm are described briefly, followed by a detailed presentation of the least-squares-based multistep quadratic data-reconstruction methodology proposed here. The paper is concluded with numerical examples to verify the accuracy and numerical efficiency of this high-order finite volume discretization. Nomenclature dx = coordinate vector displacement, m E = fluid total energy, C v T v v=, J=kg H = fluid total enthalpy, E p=, J=kg p = static fluid pressure, N=m T = static fluid temperature, K v = fluid velocity vector, m=s x = coordinate vector, m = pseudotime interval, s = fluid density, kg=m 3 = pseudotime, s I. Introduction THE aerospace industry continues to show an increased interest in the use of computational fluid dynamics (CFD) as a design tool to help improve product quality and decrease overall costs. This in turn has brought a renewed interest in fundamental research in accurate numerical discretizations and fast and reliable solution methods. Second-order-accurate finite volume method (FVM) solvers on unstructured grids are currently the commercial CFD industry standard. Although researchers have looked for and proposed veryhigh-order (higher than second-order) FVM schemes for complex applications, there are three main reasons that these methods have not developed a wide following: 1) The development of K-exact higher-order FVM on unstructured grids is technically difficult. ) The CPU time and memory requirements of these schemes can be prohibitive. 3) The parallel efficiency of these K-exact higher-order FVM schemes is inherently reduced by comparison with second-order FVM. Research on very-high-order FVM schemes originated in the 1990s, with the papers of Barth and Frederickson [1] and Presented as Paper 133 at the 47th AIAA Aerospace Sciences Meeting, Orlando, FL, 5 8 January 009; received 0 April 009; revision received 17 September 009; accepted for publication 1 November 009. Copyright 009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per copy fee to the Copyright Clearance Center, Inc., Rosewood Drive, Danvers, MA 0193; include the code /10 and $10.00 in correspondence with the CCC. 10 Cavendish Court, Centerra Park; DCaraeni@gmail.com. Senior Member AIAA. 10 Cavendish Court, Centerra Park; Chris.Hill@ansys.com, Senior Member AIAA. 808 Vankeirsbilck and Deconinck []. More recently, other researchers have reported on K-exact FVM schemes (see [3 6]). Barth [3] compares a version of a K-exact FVM scheme with the existing piecewise-linear (second-order) FVM scheme and with the fluctuation splitting schemes by Roe [7] and Deconinck et al. [8]. Barth [3] concludes that although very promising, the higher-order K-exact FVM approach requires further research to reduce computational cost. Recently, Ollivier-Gooch et al. [6] and Nejat et al. [9] presented an improved least-squares (LSQ)-based K-exact FVM scheme, with demonstrated third- and fourth-order accuracy for Euler solutions on unstructured grids. Although the computational cost of schemes proposed so far for higher-order FVM schemes is still too high, especially when compared with the classical second-order FVM, industrial clients, especially in the aerospace community, are longing for the potential benefits of using the higher-order schemes for complex flow simulations. In this paper, the details of an efficient quadratic reconstruction algorithm are presented, based on the least-squares method and the construction of a third-order FVM scheme for unstructured grids. The following requirements have been considered for the algorithm s design: 1) simplicity and ease of implementation in an existing unstructured-grid FV code, ) high numerical efficiency for both serial and parallel computations, and 3) minimum additional memory requirements. The CPU, memory, and accuracy of the original second-order cell-centered FVM scheme/code are used as a basis for comparison. First, the numerical discretization and the solution algorithm of the original second-order-accurate flow solver is presented. Then the details of the third-order-accurate quadratic reconstruction are presented and the higher-order flux integration method and boundary conditions are discussed. To make the presentation more concise, only a -D inviscid steady-state discretization is presented here. The paper is concluded with numerical examples to demonstrate the performance of the data-reconstruction algorithm and the numerical accuracy and computational efficiency of the resulting third-order FVM scheme for unstructured grids. Future work plans and remaining open questions are also outlined. II. Proposed Finite Volume Algorithm The research reported here has the algorithm of a new secondorder cell-centered FVM solver [10] as a starting point. Briefly, the key ingredients of this algorithm are as follows: 1) Solve the low-mach number preconditioned flow equations. ) Use state-of-the-art numerical flux discretizations, e.g., Roe flux-difference splitting, for high-fidelity results.

2 CARAENI AND HILL 809 3) Use least-squares-based linear-exact gradient reconstruction for guaranteed second-order accuracy. 4) For efficient inexact-newton solution advancement, use a combination of full-implicit or/and Runge Kutta enhanced pointimplicit relaxation steps to maximize numerical efficiency and optimize memory usage for very large applications. 5) Use full nonlinear multigrid [11] for quick solutions of large flow problems, as reported by Jameson and Caughey [1] and Caraeni et al. [10]. A. Flow Equations and Low-Mach Preconditioning The -D Euler flow equations may be written in integral form as ZZZ Q F da 0 (1) where Q f; u x ;u y ;Eg t is the vector of conserved variables and F fv;vu x ;vu y ;vhg t is the flux vector. The equation of state (ideal gas), p; T, completes the system. The variable W is defined as the working vector of primitive variables W fp; u x ;u y ;Tg t. The equations are rewritten in terms of W, where the time advancement is done in pseudotime toward convergence: FW da 0 () where the Jacobian p 0 0 p u x 0 T u x p u y 0 T u y 5 p H 1 u x u y T H C p (3) j j p The high-order spatial reconstruction is performed in terms of the primitive variable vector W. To precondition the system, the is replaced with the low-mach preconditioning matrix proposed by Weiss et al. [13]. This is called time-derivative preconditioning and results in the ZZZ ZZ W dv FW da 0 where the preconditioning matrix is given by T u x 0 T u x u y 0 T u y 5 (5) H 1 u x u y T H C p and 1 T u ref C p where u ref is a reference velocity for low-mach preconditioning, of the same magnitude as the local flow speed. As the flow becomes supersonic, the reference velocity approaches the local speed of sound. B. Numerical Discretization The preconditioned system of Eq. (4) is discretized using a cell-centered finite volume scheme, wherein the integral equations are applied to each cell. The current interest is in steady-state simulations. Thus, the time-derivative term in Eq. (4) can be approximated ZZZ W V 0 (6) where W 0 is the value of W at the reference point (cell centroid) of cell 0. This approximation of the time-derivative term does not affect the converged steady-state solution, as this term becomes machine-accuracy small at convergence. Hence, Eq. (4) W 0 V FW da 0 0 The surface integral is discretized as a sum of numerical fluxes that result from an exact or approximate solution of the discrete Riemann problem at the interface of the two neighbor cells. The discretized equation W 0 1 V 0 F num f W r 0 ;W r 1 da 0 (8) f; 0 ; 1 1 where W r 0 and W r 1 are reconstructed values of W from cells 0 and 1 reference points to (one of) the integration point(s) on the face f. The summation extends over all the faces of the cell 0, and 1 is a neighbor through face f of the cell 0. The numerical flux formulation used in this work is Roe s flux-difference-splitting scheme. When a very-high-order integration is performed, e.g., the integration points are the Gauss quadrature points, there is more than one integration point per face. Thus, for very-high-order schemes, the summation also extends over all integration points of the face f. The cell residual is defined as R ES0 F num f W r 0 ;W r 1 da (9) f; 0 ; 1 1 After a Taylor series expansion around W at pseudotime n n, i.e., W n, the implicit system that is solved can be written as V nw ; 0 1 nw1 k R n ES 0 (10) where W W n1 W n and superscript n refer to values computed for W n. Note that the linearization of the cell residual, ES =@W 0 ES =@W 1, is inexact. That is, only a linearization of the first-order approximation of the numerical fluxes is performed. Again, the choice of the linearization does not affect the high-order converged solution. More details on the solution algorithm are given by Caraeni and Hill [14]. The full solution coupling described above has been used for all - D simulations reported here, thus emphasizing maximum numerical efficiency. III. Efficient Quadratic Reconstruction Method Several strategies have been proposed (see [3,5,6]) for piecewise quadratic reconstruction. Here, we propose a new view of the basic high-order local LSQ reconstruction idea. To simplify the presentation, we will consider only the -D problem here. Consider a field variable. For quadratic third-order-accurate reconstruction for each cell (denoted as c 0 ), we need to compute a polynomial R x; y of the form: R x; x c0 0 rj c0 dx 1 dxt r j c0 dx (11)

3 810 CARAENI AND HILL where the underlined quantities 0, rj c0, and r j c0 are LSQcomputed values: rj c0 is a second-order-accurate gradient of in the cell centroid of cell c 0, x c0 x c0 ;y c0 is the cell c 0 centroid position, dx xx c0 ;y y c0 is the distance vector from x c0, and r j c0 is a first-order-accurate local Hessian of the field : r j c0 xx xy (1) xy yy c 0 Note that, in general, the LSQ-computed 0 will be different from c0 : e.g., the value of at cell centroid c 0. Once the gradient and the Hessian are constructed with the desired accuracy, we can use the relation (11) to quadratically reconstruct the value at any integration point x IPk. A simple and numerically efficient multistep algorithm computes these derivatives with the desired accuracy. The reconstruction algorithm has three main steps: 1) A least-squares method, similar to the one proposed by Kim et al. [15] and extended by Caraeni et al. [10], is used to compute and store a first-order-accurate gradient of, e.g., r I j c0. Thus, the gradient can be computed (see Sec. III.A) as a weighted average of neighbor cell c i values of i j 0i j c0 i (13) As proven in Sec. III.C, formula (13) has an error-cancellation property when its repetitive application is required (say, for computing high-order derivatives), as it satisfies the following identity: 0 (14)! j 0i which is evident from relation (13), as the gradient of a constant function equals zero. ) Using the same LSQ-computed weighting coefficients! j 0i,a first-order-accurate Hessian of, r I j c0 (as gradients of each component of the first-order-accurate gradient r I j c0 ) is computed and stored (for r I j c0, we used the notation 0;k j 0i i 0; k (15) j c0 A mathematical justification of the first-order accuracy for the computed Hessian, while recursively using formula (13), is given in Sec. III.B. 3) Equating the relation (11) for each neighbor cell c i, the value of, e.g., j c0 $ j 0i i 1 dxti r I j c0 dx i (17) where the LSQ weights $ j 0i in formula (17) are identical with the weights! j 0i used in Eq. (13). The derivation of relation (17) is trivial (see Caraeni and Hill [14]). The second-order-accurate gradients r II j c0 will be stored instead of the previously computed first-order-accurate gradients r I j c0. The LSQ-computed 0 is also stored and will be used in the third-order-accurate quadratic reconstruction formula (11). During the simulation, at each solver iteration, the values of 0, second-order-accurate gradient r II j c0, and first-order-accurate Hessian r I j c0 are computed and stored for each cell for all independent variables that are used in the flux computation. Then formula (11) is used at every integration point to compute the leftand right-side face values of for numerical flux estimates. A. First-Order-Accurate LSQ-Based Gradient Formula The problem of computing cell-centroid first-order-accurate gradients based on cell-centroid data using a least-squares method has been addressed in previous work [1,5,6,8,10,16] and references therein. Here, we briefly summarize the approach preferred for this work. Assume a discrete field i, with values specified at location x i.a local linear interpolation function based on the surrounding discrete data is defined as x; x 0 0 r x x 0 (18) where the unknowns r and 0 are computed using a least-squares technique. The derivations presented in this section are based on the normal-equations approach, but a more foolproof method (based on singular-value decomposition) was proposed in [6]. Either way, the final result of these derivations can be formally expressed as relation (13) above. The proof is simple and has been omitted here, for brevity. Here, we define a positive-definite functional F : F x i 0 x ; y x i x 0 ;y i y 0 T i (19) where i are appropriate positive weights and x, y, and 0 are found as solutions of the minimization problem. That is, x, y, and 0 are selected so that the functional F has a minimum. Then x, y, and 0 are the solutions of the coupled linear system: ci 0 r II j c0 dx i 1 dxt i :r I j c0 dx i 8 c i NB x 0 y (and eventually using other constraints, such as the conservation of mean suggested by Ollivier-Gooch et al. [6] to obtain a consistent third-order-accurate reconstruction scheme), we construct an overdetermined linear system to compute 0 and the second-orderaccurate gradient of in c 0, r II j c0, using a least-squares technique described in Sec. III.C. At this step, the Hessian r I j c0 is considered known, as computed with first-order accuracy in step. When not considering the conservation of mean, the j II j c0 can simply be computed with a formula similar to (13) above: Here, we define x i0 x i x 0 and y i0 y i y 0 and the geometrical moments: and I xx x i0 i I xy x i0 y i0 i I yy y i0 i (1)

4 CARAENI AND HILL 811 S x x i0 i S y y i0 i S 0 i () With these notations and after simple algebraic manipulations (see [14]), system (0) becomes S x S y S 0 x 4 I xx I xy S x 54 y 5 i i 4 x i0 i 5 (3) I xy I yy S y y i0 i 0 Solving the linear system yields 3 3 x 4 y 5 S x S y S 0 i 4 I xx I xy S x 5 I xy I yy S y 0 1 i 3 4 x i0 i 5 (4) y i0 i where the coefficient vector that multiplies i, 3 3 3! x i0 S x S y S 1 0 i 4! y 5 i0 4 I xx I xy S x 5 4 x i0 i 5 (5)! i0 I xy I yy S y y i0 i contains the needed LSQ-computed weights. Thus, the least-squares formula to compute x, y, and 0 is 3 3 x 4 y 5! x i0 i 4! y 5 i0 (6)! i0 0 which proves relation (13). B. First-Order Accuracy of LSQ-Computed Hessian Here, a mathematical justification is presented to support the assertion that a first-order-accurate Hessian is computed by applying formula (13) twice. Formula (13) produces a first-order-accurate gradient on general unstructured grids. Hence, we j! j i0 i x 0 Ox (7) where x is a constant local measure of the grid size. Here, it is assumed that the first-order error-term coefficient 0 is locally (or even globally) bounded, as the grid is refined and x goes to 0 (x! 0), e.g., k k, and is a finite positive number. It is also recognized that the LSQ weights are O1=x. j j i0 x i x 0 Ox (8) where x is a local constant. Hence, the identity! j i0 0 can be written as j i0 0 (replace! j i0 with j i0 =x and then take 1=x as a common factor and out of the summation sign). The successive application of formula (13) to compute the second derivatives of (e.g., the Hessian) leads k lnb c0 k l0 x inb cl j il x i x l Ox x 0 0 Ox (9) where 0;j 0 =@x j. Ignoring all Ox terms in j il x i x l Ox k i;l inb cl k j l0 il x x i lnb c0 k l0 0 r 0 x 0l Ox (30) where a local Taylor expansion in space, l 0 r 0 x 0l, has been used to express l. Once again, by grouping all Ox terms together and taking out the constant 0 from the summation, we 0;j k j l0 k x x k i l0 0 Ox (31) i;l Observe now that the second term vanishes, as lnb c0 lnb c0 lnb c0 k l0 0 k l0 0 for any k (k 1, ). Thus, by recursively applying formula (13), the actual computed Hessian is first-order-accurate. C. Mean-Conserving Third-Order-Accurate LSQ Reconstruction Here, it is shown that by using the precomputed first-orderaccurate local Hessian r j c0, we obtain an efficient third-orderaccurate LSQ-based reconstruction, which also enforces the preservation of mean. We consider the third-order-accurate quadratic interpolation formula (11) to formulate the LSQ problem, with the unknowns 0 and the second-order-accurate gradient r c0, while enforcing the condition for conservation of mean: c0 1 V c0 Z V c0 0 rj c0 dx 1 dxt r j c0 dx dv (3) where dx x x c0. Note that, in general, 0 0 ( 0 is the value of at cell centroid c 0 ). We equate the relation (11) for all neighbor values i x i, for i NB c0 : x i 0 rj c0 x i x 0 x i x 0 T r j c0 x i x 0 (33) Observe that we can write Eqs. (33) as where i0 i 0 i r x i x 0 i (34) i0 x i x i x 0 T r j c0 x i x 0 (35) and i are appropriate weights, as described above. The volume integrations in formula (3) must be performed with the appropriate

5 81 CARAENI AND HILL accuracy (exact for quadratic polynomials) (see Ollivier-Gooch et al. [6] and Nejat et al. [9]): Denote x 1 Z x x V 0 dv c0 V c0 y 1 Z y y V 0 dv c0 V c0 xx 1 Z x x V 0 dv c0 V c0 xy 1 Z x x V 0 y y 0 dv c0 V c0 yy 1 Z y y V 0 dv (36) c0 V c0 as the cell c 0 integral geometric moments. Define 00 as 00 c0 1 f xx xx xy xy yy yy g (37) which can be directly computed with third-order accuracy using the known first-order-accurate Hessian r j c0. Equations (3) and (34) form an overdetermined linear system 0 rj c0 x ; y 00 0 i rj c0 x i x 0 i i0 i with i NB c0 (38) in the unknowns 0 and rj c0, which must be solved using a LSQ technique while strictly enforcing its first equation, e.g., relation (3). Note that in the framework of cell-centered finite volume discretizations, it is common practice to compute the cell-centroid position as x 0 1 Z x dv (39) V c0 V c0 It is very easy to show that with the above definition of x 0, the integral geometric moments x and y are identically zero: x ; y 0; 0. This simplifies the above overdetermined linear system, which becomes 0 00 (40) 0 i rj c0 x i x 0 i i0 i with i NB c0 or 0 00 rj c0 x i x 0 i i0 00 i with i NB c0 (41) Note that, in general, for cell-vertex finite volume (and control volume finite element method) discretizations, the above simplifications are not possible: i.e., x ; y 0; 0. In the work described here, the cell-centroid position has been computed with relation (39), which simplifies the whole algorithm and makes it very efficient. After explicitly eliminating 0 from the system (the subsystem with unknowns) the second-order-accurate gradient rj c0, rj c0 x i x 0 i i0 00 i with i NB c0 (4) is solved either by using the normal-equations approach, as described in Sec. III.A, or by using a singular-value-decomposition technique, as suggested by Ollivier-Gooch et al. [6]. Either way, we obtain a formula similar to Eq. (6); for example, x y i0 00 ~! x i0 ~! y i0 (43) where ~! x i0 ; ~!y i0 are the LSQ-computed weights. The other unknown, 0, is computed explicitly as 0 00 (44) Note that in contrast with the formulation presented in [14], the mean-preserving third-order-accurate reconstruction presented here requires extra storage for the LSQ weights ~! x i0 ; ~!y i0 used in step 3 of our algorithm. The results presented in this paper are obtained using the meanconserving third-order-accurate quadratic reconstruction described above. D. Boundary Quadratic Reconstruction Method The reconstruction procedure described in the previous sections to compute 0, the second-order-accurate gradient rj c0, and firstorder-accurate Hessian of (r j c0 ) has been applied for all interior cells. The interior cells are cells that, by definition, do not have boundary faces. A complete set of data for the local LSQ-based reconstruction can then be used at any of the three steps of quadratic polynomial construction, as described above. For the purpose of the current algorithm, all of the other cells are considered to be boundary cells and require a special treatment. Here, we briefly describe the boundary-reconstruction procedure. To perform the first step of our polynomial reconstruction for all boundary cells, we use an algorithm that provides the gradients at boundary cells based only on interior-cell gradient values, while preserving the local values for the Hessian (to be computed in the second step). This can be achieved by local linear gradient extrapolation from interior cells toward the boundary cells. This extra information needed, which is a list of the closest interior cells (minimum of three cells in two dimensions) for a given boundary cell, is produced in a one-time preprocessing step and saved. After computing the first-order-accurate Hessians for all interior cells, linear extrapolation is used again to compute the Hessians for each boundary cell, using the closest-interior-cell list. Then 0 is computed from relation (44) and the second-order-accurate boundary-cell gradients are computed using the second-orderaccurate gradients and first-order-accurate Hessians of the corresponding interior/boundary cells, from the list discussed above. IV. Numerical Flux Integration With the third-order-accurate quadratic reconstruction available, performing the higher-order flux integration becomes straightforward. The higher-order computations require the use of a numerical quadrature scheme with appropriate accuracy of numerical integration: i.e., at least third-order accuracy for a quadratic reconstruction third-order FVM algorithm. The Gauss quadrature integration formula is preferred here, due to its known efficiency. Thus, Gauss quadrature with n integration points can exactly integrate polynomials of degree n 1. Thus, in two dimensions for a third-order of accuracy of integration we need a Gauss quadrature formula using two integration points. Consider an interior face f, with the two end-node coordinates x a and x b. Then the two Gauss quadrature points will be x 1; x a x b x b x a p (45) 3 and the computational weights are w 1; 1 (46) The third-order-accurate numerical flux through face f becomes Z ~FW d w ip ~FW L x ip ;W R x ip ja f j (47) A f ip1; where we denote W LR x ip as the left (right) of face f, third-orderaccurate quadratic reconstructed (primitive) solution variables at integration point ip, and ~F is the numerical flux formula, e.g., the approximate Riemann problem solver used (Roe flux-difference splitting).

6 CARAENI AND HILL 813 Here, we briefly describe our wall-boundary (curved) integration approach (for more details, see the work by Ollivier-Gooch et al. [6]). To achieve the expected higher-order results, integration of fluxes along curved boundaries must be done with extreme care; solution variables, boundary normals, and integration weights must accurately reflect the true shape of the boundary and have the required order of accuracy. When setting up the Gauss integration points and the local boundary normals, a true high-order (cubic spline) geometry representation has been used in the present work. Gauss quadrature weights are assigned based on arc length (-D) (see [6]). V. Monotonic Quadratic Reconstruction When computing flows with discontinuities (shocks) or even during the fast transient-phases toward convergence, local extrema can grow unbounded, especially for a higher-order scheme. Thus, special measures must be taken to make sure that the scheme satisfies the so-called maximum principle. Consider the local third-order-accurate quadratic reconstruction polynomial: 3 R x; x c0 c0 r II j c0 dx 1 dxt r j c0 dx Fig. 1 Close-up of the coarsest computational grid for Mach 0. inviscid flow around the circular cylinder. To enforce the monotonicity of the reconstruction (and of the overall scheme), a positive parameter is used to modify the above relation: 3lim R x; x c0 c0 r II j c0 dx 1 dxt r j c0 dx (48) Thus, one needs to find the maximum of, 0;...; 1, such that the values of the reconstructed function 3lim R x; x c0 at the integration points will not exceed the maxima and minima at the neighboring cells. Recent investigation results published by Michalak and Ollivier-Gooch [17] have consistently addressed the problem of gradient/solution limiting for very-high-order numerical discretizations. As all numerical example presented are for smooth-flow situations, these solutions have not used any gradient limiting, which could have affected the asymptotic accuracy of the results presented here. VI. Numerical Examples The accuracy assessment performed in this work follows the research by Nejat et al. [9]. Here, we give three numerical examples in two dimensions to show the level of accuracy and the computational efficiency of our third-order FVM discretization. A. Mach 0. Inviscid Flow Around a Circular Cylinder An inviscid low-speed airflow at Mach 0. is considered over a circular cylinder of 1 m radius, with the far field positioned at R 50 m. Air is modeled as an ideal gas and an inviscid simulation is performed for a series of four increasingly finer meshes of 637, 5,864, 10,05, and 40,17 triangular cells (see the cylinder closeup for the coarsest mesh in Fig. 1). The grid refinement has been done very carefully in order to achieve a quasi-self-similar refinement throughout the whole domain. Each mesh is about four times as dense as its immediate coarser mesh, with mesh length scales in all parts of the domain reduced by approximately a factor of. Figures and 3 display the velocity magnitude distribution around the cylinder as computed on the coarsest mesh (637 triangles) using the secondorder and third-order discretizations, respectively. The accuracy improvement provided by the third-order solution is evident, as profiles of iso-mach number become symmetric and the apparent wake behind the cylinder, present for the second-order solution and produced by the numerical dissipation of the scheme, practically disappears for the third-order solution. As suggested by Nejat et al. [9], we define an error function E P t =P 1 t 1, where P t is the local total pressure and P 1 t is the total pressure in the far field. For inviscid shock-free flows, the total Fig. Circular cylinder M 1 0:; second-order solution and velocity magnitude distribution. pressure should be preserved. Figures 4 and 5 display the distribution of error function E inside the domain for the second- and third-order solutions, respectively. The variations in total pressure are a good estimate of a scheme s numerical dissipation. The composite measure of the error is defined as an area-averaged L 1 norm of the error E, defined as L 1 n v i1 n v A i j E i j i1 A i (49) where A i is the -D cell area and E i is computed as Z E i Ex da (50) A i Figure 6 displays the asymptotic solution accuracy, expressed as the L 1 -norm of the error E for the new third-order discretization as compared with the original second-order discretization. Our estimates gave a 1.91 and.88 for the second- and third-order schemes, respectively.

7 814 CARAENI AND HILL Fig. 3 Circular cylinder M 1 0:; third-order solution and velocity magnitude distribution. Fig. 5 Circular cylinder M 1 0:; third-order solution and relative total pressure variation [P t =Pt 1 1]. Asymptotic accuracy 1.00E E-05 3-rd order scheme -nd order scheme nd order slope 3rd order slope L1-Norm of Error 1.00E E E-08 1.E+03 1.E+04 1.E+05 1.E+06 Control Volumes Fig. 4 Circular cylinder M 1 0:; second-order solution and relative total pressure variation [P t =Pt 1 1]. Fig. 6 Asymptotic accuracy comparison between the original secondorder (triangles) and the third-order (squares) finite volume discretization. We provide direct timing results for this test case, done with the instrumented code, for the different code segments. Results are presented in Table 1 as the average over the timing results for all grids. Although it took 68 to 14 iterations on the first three meshes for the second-order scheme to converge residuals eight orders of magnitude, it took 15 45% more iterations to converge the thirdorder scheme (45% more iterations on the fine mesh). Note that the measured computational time per iteration was only 35 40% larger for the third-order scheme than for the second-order scheme, on the same mesh. Table 1 shows the computational effort for the different parts of the respective algorithms, based on per-iteration timing results. Based on the table data and also considering error levels shown in Fig. 6, we see that although it takes about % more CPU time to compute a third-order solution on the medium mesh (5,864 cells) using our scheme, the solution obtained is about one order of magnitude more accurate than the second-order solution on that same mesh. This clearly shows the numerical efficiency of the third-order scheme starting from rather coarse meshes. B. Mach 0.3 Inviscid Flow Around NACA001 Profile at a Zero-Degree Angle of Attack An inviscid airflow at Mach 0.3 is simulated over a NACA001 profile at a zero degree angle of attack (AOA). The airfoil chord is 1 m and the far field is positioned at R 30 m. Air is modeled as an ideal gas and the simulation is performed for a series of four increasingly finer meshes of 4980, 16,99, 6,764, and 40,73 triangular cells. A Table 1 Test-case timing results Discretization Normalized time/iter Linear system solution Flux computation Linearization LSQ reconstruction LSQ setup Second-order 100% 59% 1% 16.4% 3.% 0.4% Third-order ~ % 44.6% 8.0% 1.6% 14.4% 0.4%

8 CARAENI AND HILL close-up of the airfoil for the coarsest mesh is shown in Fig. 7. Similar to the observations made in the previous test case on grid refinement, each mesh here is about four times denser than its immediate coarser mesh. Figures 8 and 9 display the Mach number distribution around the airfoil as computed on the coarse mesh using the second-order and third-order discretizations, respectively. The accuracy improvements provided by the high-order solution are evident; e.g., see the maximum Mach number value (on the profile s suction side), etc. Figures 10 and 11 display the distribution of error function E inside the domain for the second- and third-order solutions, respectively, on the coarse NACA001 mesh. The asymptotic accuracy study based on the L1 -norm of error E (see the detailed description above) gave a computed order of accuracy for this test case (based on the relative error reduction as a function of grid size) of 1.89 and.7 for the second- and third-order schemes, respectively. Note that as in the previous case, we have not obtained exactly the nominal accuracy that we have expected ( and 3, respectively). This is most probably due to the fact that when creating the four successively finer meshes, we could not exactly guarantee the required reduction (1:) in grid size or a perfect similarity of the grid topology for the entire series. The relative computational efficiency study of the second- and third-order schemes in this test case showed very similar results with the previous test case. For brevity, we will not reproduce those results here. C. Fig. 7 Close-up of coarse grid for M1 0:3 inviscid flow around NACA 001 airfoil. 815 Mach 0.3 Inviscid Flow Around RAE8 Profile at 1.5 of AOA An inviscid airflow at Mach 0.3 is simulated over the asymmetric RAE8 profile at a 1.5 AOA. The airfoil chord is 1 m and the far field is positioned at R 30 m. The simulation setup is similar to the test described above. A series of four increasingly finer meshes of 380, 13,76, 50,80, and 94,937 triangular cells is used (see the Fig. 8 NACA 001 M1 0:3; second-order solution and Mach number distribution. Fig. 10 NACA 001 profile at M1 0:3; second-order solution and relative total pressure variation [ Pt =Pt1 1]. Fig. 9 NACA 001 M1 0:3; third-order solution and Mach number distribution. Fig. 11 NACA 001 profile at M1 0:3; third-order solution and relative total pressure variation [ Pt =Pt1 1].

9 816 CARAENI AND HILL Fig. 1 Close-up of coarse grid for M1 0:3 inviscid flow around an RAE8 airfoil. Fig. 15 RAE8 profile at M1 0:3 AOA of 1.5 ; second-order solution and relative total pressure variation [ Pt =Pt1 1]. airfoil close-up for the coarsest mesh in Fig. 1). Figures 13 and 14 display the Mach number distribution around the airfoil as computed on the coarse mesh using the second-order and third-order discretizations, respectively. Similar observations as were made Fig. 16 RAE8 profile at M1 0:3 AOA of 1.5 ; third-order solution and relative total pressure variation [ Pt =Pt1 1]. Fig. 13 RAE8 M1 0:3, AOA of 1.5 ; second-order solution and Mach number distribution. above can be made about the accuracy improvements provided by the high-order solution. Figures 15 and 16 display the distribution of error function E inside the domain for the second- and third-order solutions, respectively, on the coarse RAE8 mesh. The asymptotic accuracy study based on the L1 -norm of error E gave a computed order of accuracy for this test case (based on the relative error reduction as a function of grid size) of 1.86 and.83 for the second- and third-order schemes, respectively. The relative computational efficiency study of the second- and third-order schemes in this test case showed very similar results with the previous test cases; for brevity, these results will not be presented here. VII. Fig. 14 RAE8 M1 0:3, AOA of 1.5 ; third-order solution and Mach number distribution. Conclusions The numerical examples presented here prove that third-orderaccurate solutions computed using the present multistep quadratic reconstruction algorithm are both very accurate and computationally efficient. This third-order discretization, which is simple enough to be implemented in existing unstructured cell-centered FVM CFD codes, provides the right ingredients for application in mainstream CFD. Future work will include extension of the present scheme for transonic turbulent-flow simulations. Extension to numerical accuracy higher than third-order is possible using a similar multistep reconstruction idea. But these extensions may not be as numerically

10 CARAENI AND HILL 817 efficient for industrial CFD application today, as observed for the third-order scheme proposed here. More investigations are required to clarify this issue. Acknowledgments The first author acknowledges Laszlo Fuchs from Lund Institute of Technology and Royal Institute of Technology in Sweden for supporting his original research in the field of high-order numerics for practical large eddy simulation of turbulent flows and combustion. The authors would like to acknowledge F. Boysan and D. Choudhoury from ANSYS, Inc., for their continuous support and encouragements. Thanks also to V. Ivanov from ANSYS, Inc., for reading the original manuscript. References [1] Barth, T. J., and Frederickson, P. O., Higher Order Solution of the Euler Equations on Unstructured Grids Using Quadratic Reconstruction, AIAA Paper , [] Vankeirsbilck, P., and Deconinck, H., Higher Order Upwind Finite Volume Schemes with ENO-Properties for General Unstructured Meshes, AGARD Rept. R-787, Neuilly-sur-Seine, France, 199. [3] Barth, T., Recent Developments in High-Order K-Exact Reconstruction on Unstructured Meshes, AIAA Paper , [4] Agarwal, R. K., and Halt, D. W., A Compact High-Order Unstructured Grids Method for Solution of Euler Equations, International Journal for Numerical Methods in Fluids, Vol. 31, No. 1, 1999, pp doi:10.100/(sici) ( )31:1<11::aid-fld959>3.0. CO;-S [5] Delanaye, M., and Essers, J. A., Quadratic-Reconstruction Finite Volume Scheme for Compressible Flows on Unstructured Adaptive Grids, AIAA Journal, Vol. 35, No. 4, 1997, pp doi:10.514/.183 [6] Ollivier-Gooch, C., Nejat, A., and Michalack, K., On Obtaining High- Order Finite Volume Solutions to the Euler Equations on Unstructured Meshes, 18th AIAA CFD Conference, AIAA Paper , 007. [7] Roe, P. L., Linear Advection Schemes on Triangular Meshes, Cranfield Inst., Rept. 870, Cranfield, England, U.K., [8] Deconinck, H., Struijs, R., Bourgois, G., Paillere, H., and Roe, P. L., Multidimensional Upwind Methods for Unstructured Grids, AGARD Rept. R787, Neuilly-sur-Seine, France, 199. [9] Nejat, A., Ollivier-Gooch, C., and Michalak, K., Accuracy Assessment Methodology for a High-Order Unstructured Finite Volume Solver, 18th AIAA CFD Conference, AIAA Paper , 007. [10] Caraeni, D., Mathur, S., and Zori, L., A Next-Generation DBNS Solver: Accurate Solutions and Fast Multigrid Driven Convergence, AIAA Paper , 007. [11] Jameson, A., Solution of the Euler Equations for Two-Dimensional Transonic Flow by a Multigrid Method, Applied Mathematics and Computation, Vol. 13, Nos. 3 4, 1983, pp doi: / (83) [1] Jameson, A., and Caughey, D., How Many Steps Are Required to Solve the Euler Equations of Steady Compressible Flow: In Search of a Fast Solution Algorithm, AIAA Paper , 001. [13] Weiss, J., Maruszewski, J., and Smith, W., Implicit Solution of Preconditioned Navier-Stokes Equations Using Algebraic Multigrid, AIAA Journal, Vol. 37, No. 1, 1999, pp doi:10.514/.661 [14] Caraeni, D., and Hill, D. C., Efficient Third-Order Finite Volume Discretization Using Iterative Quadratic Data Reconstruction on Unstructured Grids, AIAA Paper , 009. [15] Kim, S.-E., Makarov, B., and Caraeni, D., A Multidimensional Linear Reconstruction Scheme for Arbitrary Unstructured Grids, AIAA Paper , 003. [16] Hill, D. C., A Variant of Least-Squares Reconstruction Based on Properties of the Laplacian Operator, AIAA Paper , 005. [17] Michalak, C., and Ollivier-Gooch, C., Accuracy Preserving Limiter for the High-Order Accurate Solution of the Euler Equations, Journal of Computational Physics, Vol. 8, No. 3, 009, pp doi: /j.jcp Z. Wang Associate Editor