SECOND order computational methods commonly employed in production codes, while sufficient for many applications,

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1 th AIAA Comptational Flid Dynamics Conference 7-3 Jne, Honoll, Hawaii AIAA -384 Active Fl Schemes for Systems Timothy A. Eymann DoD HPCMP/CREATE Kestrel Team, Eglin AFB, FL 354 Philip L. Roe Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI, 489 We introdce a new formlation of active fl schemes and etend the idea to systems of hyperbolic eqations. Active fl schemes treat the edge vales, and hence the fles, as independent variables, dobling the degrees of freedom available to describe the soltion withot enlarging the stencil. Schemes p to third order accrate are eplored. The limiter employed ses soltion characteristics to set the bonds for the edge pdates. The process redces to simply accessing the soltion history from memory and ensring that the pdates stay within the bonded range. The limited edge vales are then sed to constrct the fles that conservatively pdate the centroid vale. Using data that most closely follows the soltion characteristics allows the limiter to better maintain tre etrema in the soltion. The scheme is sed to generate -D soltions for the linear advection, Brgers, and Eler eqations. I. Introdction SECOND order comptational methods commonly employed in prodction codes, while sfficient for many applications, lack the accracy reqired to perform certain simlations sch as those that reqire high-fidelity vorte tracking and comptational aeroacostics. One method for increasing the order of accracy of a finite-volme scheme is to epand the comptational stencil, generating higher-order reconstrctions from which the soltion may be interpolated. An early third order scheme sing this idea was developed by Warming, Ktler, and Loma (WKL). Later, essentially non-oscillatory (ENO) and weighted essentially non-oscillatory 3 (WENO) schemes were developed to obtain higher-order accracy withot generating sprios oscillations in the soltion. WENO and ENO schemes perform the higher-order reconstrctions by searching throgh a set of candidate stencils and either selecting the smoothest reconstrction or sing a weighted average of the possible choices. This approach is most straightforward to apply on a strctred mesh, bt it is also applicable to nstrctred meshes. 4, 5 One drawback of increasing accracy by enlarging the comptational stencil is that the solver attempts to bild a high order reconstrction sing data from regions of the flow that may not be physically relevant. Another is that synchronizing large stencils dring parallel comptations reqires etremely carefl implementation to minimize the commnication overhead between processes. The combined need for higher accracy and increased parallelization has driven massive interest in methods that achieve higher-order accracy by increasing the internal degrees of freedom within a cell. Poplar methods in this class inclde discontinos Galerkin 6 (DG), spectral difference 7 (SD), and spectral volme methods 8 (SV). These schemes are well sited for parallel comptations on nstrctred meshes since they all are capable of generating higher-order reconstrctions withot etending the comptational stencil beyond the neighboring cells. While a great deal of research has been concentrated on these schemes, they still sffer from two primary isses: a lack of robstness 9 /sfficient limiters for nonlinear problems and a high memory reqirement. The isse of generating higher-order soltions from compact stencils was recognized and addressed very early by van Leer. In his 977 paper, he presented a series of linear and nonlinear advection schemes that evolve not only the cell-average vale bt also a soltion gradient or edge vale. He then sed the etra degrees of freedom to increase the accracy of the soltion. Other novel attempts at increasing the accracy of schemes focsed not on the formal order, bt on redcing the dissipation throgh a carefl representation and interpolation of data on a space-time mesh. Iserles and Roe 3 investigated Upwind Leapfrog (ULF) schemes, which are two-level schemes with zero dissipation de to their symmetric stencils in space and time. The CABARET scheme presented by Karabasov 4 collapses the Research Engineer, Ph.D. Candidate, Member AIAA Professor, Fellow AIAA of This material is declared a work of the U.S. Government and is not sbect to copyright protection in the United States.

2 two-level ULF to a single-step scheme by independently pdating cell-averages and fl vales. The scheme retains all the advantages of the ULF scheme while only sing data from one time level. Active fl schemes encompass some of the advection schemes proposed by van Leer bt also se information from the larger space-time mesh to limit sprios oscillations. With these schemes, we attempt to retain the compactness of DG/SD/SV schemes, while lowering the memory overhead by basing the internal cell reconstrctions on shared data. The schemes have great potential for the soltion of hyperbolic conservation laws. The net section of this paper introdces the main concepts of active fl schemes by concentrating on their application to scalar eqations. After introdcing families of second and third order schemes, we discss the limiting strategy before moving on to the etension of the scheme to systems of eqations. Finally, key properties of the scheme and limiter are highlighted by presenting soltions from the one-dimensional linear advection eqation, Brgers eqation, and the Eler eqations. II. Scalar active fles Begin by interpreting the data contained within a given cell as a qadratic fnction defined by Eq. (), where is the cell centroid coordinate for cell. ( ) [ ( ) ] () = ū n + s + c () This fnction is a combination of Legendre polynomials, ensring that we recover the cell-average ū when () is integrated over each cell and divided by the cell width. Since the terms are orthogonal, each may be mltiplied by an independent coefficient. We interpret the coefficients s and c as the average slope and crvatre, which are fnctions of the verte vales ±, cell-average, and Corant nmber. The slope and crvatre definitions determine the reconstrction shape and hence the behavior of the scheme. n + - / n + n + +/! n +/ - / n +/ +/ n - / n n +/ Figre. Storage strctre for -D element One of the advantages of active fl schemes is that they store the data reqired to define and pdate a given reconstrction very economically. Figre illstrates the data storage for a -D element. Cell-averages are stored at the centroid of each cell, edge-averages at the face centroids and the vertices hold reconstrction point vales. Becase cells share data between faces and vertices, the scheme is able to achieve savings over other higher-order approaches. These savings become even more apparent in higher dimensions. In one dimension, active fl schemes reqire only two degrees of freedom per cell for third order accracy, bt they only reqire approimately 3.5 degrees of freedom per cell for the same level of accracy in either two or three dimensions. II.A. Linear eqations The key concept of active fl schemes is that the fles at the cell edges are evolved as part of the soltion rather than simply being calclated from the srronding data. This does not significantly add to the cost since some fl calclation is reqired with any choice of scheme. We define the active fl system by specifying an evoltion eqation for both the verte vale and the edge-average vale. The edge-average ū n+ is a time average of the velocity at the + interface and is therefore directly related to the fl. Consider a linear advection case with positive wave speed a. As illstrated in Fig., the new verte vale is the reconstrction vale at a point ν pwind from the edge, and the edge-average is proportional to the integral of the of

3 !! n - / n n +/ n + +/!! Figre. Update qantities reconstrction fnction over this interval. The evoltion eqations as a fnction of the slope, crvatre, and Corant nmber are: [ ] n+ = ū n + + s ( ν) + c ν ( ν) 6 () ū n+ + = ū n + s ( ν) + c ( ν) ( ν) 6 The edge-averages can then be sed to conservatively pdate the centroid vale, sing a standard time-integration scheme sch as the Eler eplicit form below. ū n+ = ū n t [ ( ) ( )] f ū n+ f ū n+ + (3) Parameterizing the eqations sing differences between the verte vales and cell-average vales greatly simplifies the identification of second and third order accrate schemes within the active fl family. The state differences can be considered flctations which generate signals that travel to a target interface. This is a similar approach to Roe s flctation-signal approach, 5 bt here, the flctations are calclated on half-cell intervals rather than between centroids. If we assme the wave speed is positive everywhere, the target interface is the downwind node +. The sbscripts of the parameterization coefficients indicate whether the flctation is adacent to the target node (near) or a half-cell width away (far). Eqation () and Eq. (4) can be compared to derive epressions for the slope and crvatre as a fnction of the signal coefficients. n+ + ū n+ + = n + = n + t ( ) ( )] [aη far ū n n + aη near n + ū n t ( ) ( )] [aφ far ū n n + aφ near n + ū n The pdates defined by Eq. (4) are eact by constrction for constant data. A family of second order schemes can be defined by reqiring that the epressions in Eq. (4) are eact when the cell contains linear data. This is eqivalent to specifying () =, leading to: (4) n+ + ū n+ + = ν = (ν ) (5) where the epressions have been simplified by transforming to a coordinate frame in which +/ sbstittion into Eq. (4), we arrive at the coefficient definitions for a family of second order schemes. =. After φ far + φ near = η far + η near = (6) Third order schemes are obtained by repeating the procedre for qadratic cell data () =. For third and higher order schemes, it is important to be mindfl that the flctation-signal formlation contains cell-averages, not the 3 of

4 reconstrction vale at the cell centroid. The verte pdate and edge-average with a qadratic internal reconstrction are: n+ + ū n+ + = (ν ) = 3 (ν ) (7) reslting in the signal coefficients valid for a family of third order schemes. φ far + φ near = ν η far + η near = 3ν (8) There are a variety of methods that can be sed to determine the signal coefficients η and φ, some of which were discssed in a previos paper by the athors. 6 By solving the system defined by Eq. (6) and Eq. (8) we obtain the third order scheme introdced by van Leer as Scheme V. Its signal coefficients are listed in Table along with those for the first order pwind (FUP) scheme. Scheme V contains many desirable properties that distingish it from other members of the active fl family we have investigated. The individal reconstrction within each cell has a clear physical interpretation and is C continos with reconstrctions from neighboring cells. Additionally, the scheme has a stability range from ν, qickly damps sprios modes, and has relatively simple epressions for the signal, slope, and crvatre coefficients. For these reasons, mch of or testing has centered on Scheme V. Table. Selected signal coefficients FUP Scheme V η near / ν 4 3 ν η far 3 ν φ near / ν ν φ far ν II.B. Nonlinear eqations For simplicity, the discssion above assmes a constant, positive wave speed bt it is not difficlt to formlate the scheme to work in more general cases. When the wave speed is negative, the signal coefficients shold be defined as fnctions of the absolte vale of ν. The signal target changes with the sign of the wave speed, so if a is negative everywhere, the flctation ( n ū n + ) mst now be mltiplied by the far vales and (ūn n ) by the near vales. Active fles may also be sed for nonlinear eqations, where the wave speeds can be a mitre of positive and negative vales, by simply linearizing the characteristic speed within each cell. This cold be done in varios ways; at present we are sing the linearized speed λ, defined in Eq. (9), to replace constant speed a in Eq. (4). In cases sch as Brgers eqation, the average wave speed epression may be simplified to avoid division by zero in constant regions of the flow. f λ = ( ) ( ) n f n + n + n = ( ) n + + n (9) The verte vales and edge-averages are pdated by sweeping throgh each cell, sending the flctations to + if λ and when λ <. After all the signals traveling to each node have been acconted for, the interface fles can be calclated from the appropriate analytical fl fnction and the cell centroid pdated according to Eq. (3). II.C. Limiting Godnov showed that any linear scheme capable of second or higher order accracy will generate oscillations. 7 Traditional limiting approaches introdce nonlinearity by eamining spatial neighbors and performing some type of activity sch as redcing the local gradient. A form of nonlinearity that we have fond to be effective eamines near 4 of

5 neighbors in both space and time. The strategy makes se of a characteristic coordinate concept illstrated in Fig. 3. If one chooses the pdate point as the origin, the coordinates of each point in the characteristic frame λt are the fnctions of ν shown in the figre.!!t v - v! v - v Figre 3. Characteristic coordinates Once again, momentarily assme λ is positive. Begin by finding those points in the stencil that are closest to the pdate point ( +, n + ) in the characteristic coordinate frame. When ν < these are the points at ( +, n) and (, n ). When ν >, the characteristic neighbors are (, n) and (, n ). If the edge pdate at n + is within the bonds of the appropriate set of characteristic neighbors, it is accepted, otherwise the vale closest to this interval is sed. By sing the vale at (, n ) rather than the vale at (, n), the limiter is made mch milder in its operation. The limiter ensres a version of the TVD property for the nodal pdates, bt does not strictly enforce it pon the cell-averages. Ths, the method tends to avoid the ecessive clipping of legitimate etrema in the soltion. The epression in Eq. () ensres that the node pdate does not eceed the maimm of its characteristic neighbors. Similar epressions keep the verte pdate above the minimm of the characteristic neighbors. [ ( )] min n+, ma n, n if ν + + = () [ ( )] min n+, ma n, n if ν > + When one relaes the reqirement that the wave speed is positive, the terms in Eq. () are simply mirrored across +. For linear advection cases, a straightforward implementation of the limiting strategy applied to the verte and edge-average vales sfficiently damped the oscillations. Nonlinear eqations introdce options to the strategy that we are crrently eploring. For instance, we determine the primary characteristic direction based average of the left and right cell-average speeds, bt other methods are certainly possible. Tests have shown that a limiting strategy that prodces good reslts with linear eqations does not necessarily work for nonlinear problems. This is a new approach to limiting and we acknowledge that there are still a nmber of details left to work ot. III. Active fles applied to systems As with the scalar case, we begin with the simple, linear case and etend the ideas to the more general nonlinear system. The governing eqations can be written compactly as Eq. (). t + A = () Diagonalizing the Jacobian matri A = RΛR = RΛL leads to a system of independent, linear advection eqations in the characteristic variables w = L. Eqation (4) can be recast as: w t + Λw = () n+ + ū n+ + = n + = n + t (δ R + δ L ) t (δū R + δū L ) (3) 5 of

6 where the flctation term δ L,R or δū L,R is decomposed as a sm over each of the k characteristic fields. For eample, δ R = = = k i= k i= k i= ηλ i δw (i) R r i ( ) ηλ i w n + w n r i (4) i ( ) ηλ i l i n + ū n r i The prodct of the row vector l i with the state difference is the strength of the i th wave generated by the flctation. To pdate the interface vales, the wave strength is modified by the appropriate signal coefficient, chosen based on the sign of λ i, and the qantity from Eq. (4) is sent to the target interface. Once the average state at each interface is determined, the vales can be sed in the analytical fl fnction associated with the governing eqations, st as in the scalar case. F + = R(Λ wn+ ) = R(ΛLū n+ + ) = Aū n+ + = f(ū n+ + ) (5) + Nonlinear systems may be compted by linearizing abot an average state. A natral choice is to se Roe s linearization 8 within each cell to find an average state in the cell, which can then be sed to calclate a local approimation for the eigenvectors and characteristic speeds (eigenvales). We crrently se verte vales for the linearization, bt once again there are other valid options. Once a linearization choice has been made, the limiter from section II.C may be applied to the system provided the bonds checks are performed one field at a time sing characteristic variables, not conserved or primitive variables. IV. Performance IV.A. IV.A.. Linear scalar performance No limiter, randomized grid We tested Scheme V on the linear advection eqation with a site of waveforms selected by Zaleszak 9 and others. The cases were rn sing a mesh of 3 eqally spaced cells as well as a mesh with randomized cell spacing. Figre 4 compares Scheme V to the standard WKL cbic-interpolating scheme. The more compact spatial stencil of Scheme V improves the accracy compared to the cbic-interpolating scheme. On both the niform and random meshes, Scheme V performs noticeably better on the narrow peak, and generally prodces a soltion closer to the eact soltion. Table highlights Scheme V s accracy when compting on a random mesh. Scheme V loses only 6.6% of the Gassian peak after one cycle throgh the mesh, while WKL redces the peak by 6.4%. On the smooth cosine crve, Scheme V drops the peak vale by st.5% compared to a 4.7% loss with the WKL scheme. (a) Uniform mesh Table. Peak vales withot limiting (b) Random mesh Wave WKL Scheme V Sqare.5.4 Cosine Gassian Ellipse.. Wave WKL Scheme V Sqare.55.6 Cosine Gassian Ellipse IV.A.. Performance with limiting Figre 5 compares a limited soltion, an nlimited soltion and a La-Wendroff (LW) soltion that employed the sperbee limiter. The reslts for the limited LW scheme and Scheme V are comparable for the sqare wave, bt for 6 of

7 Warming/Ktler/Loma Scheme V Eact (a) Uniform mesh, ν = (b) Random mesh Figre 4. Performance of third order schemes with compact and non-compact stencil the Gassian profile, the characteristic-based limiter does not sffer from either the ecessive clipping or the sqaring of the waveform cased by the compressive sperbee limiter. Table 3 smmarizes the peak vales predicted by Scheme V on a niform mesh sing the limiter. The vales show that the limiter preserves tre etrema in the soltion since only the Gassian peak is redced by more than %. Even this is not significantly higher than the redction of the peak seen in the nlimited soltion (3.7% redction with limiting verss.5% withot) La-Wendroff w/ Sperbee SchemeV(nolimter) SchemeV(limited) (a) Sqare wave (b) Gassian Figre 5. Limiter comparison after one cycle; ν =.44 7 of

8 Table 3. Scheme V peak vales with limiting Wave Scheme V (limited) Sqare. Cosine.993 Gassian.968 Ellipse.9987 IV.B. Nonlinear scalar performance We se Brgers eqation to demonstrate the scheme s performance on nonlinear eqations. Scheme V was tested on two cases; a Gassian plse and a centered compression wave. The Gassian plse starts with the following initial data: () = + 9 e 5( ) (6) Straightforward analysis shows that a shock forms at time t s =.74 and position s =.79. The initial speed of the shock is s =.63, bt it slows as it interacts with the epansion. Or previos implementation of Scheme V sed a flctation-signal approach based arond the cell-average vale and a linearization that inclded vales from neighboring cells. This performed well p to the shock formation, bt after that, the scheme began prodcing ecessive overshoots in the conserved variables. A shock detector and eplicit call to a Riemann solver were reqired to remove the overshoots. The new implementation removes the ambigity associated with basing the edge pdates off a cellaverage vale by basing the pdates on the verte vale. Additionally, the linearization is performed entirely within a single cell. As a reslt, the method no longer reqires eplicit shock detection to properly captre these featres. Figre 6 shows the soltion at t s =.74 (shock formation) and t =.375. Both the limited and nlimited schemes prodce similar reslts prior to the shock forming, bt differentiate after that. The reslts in Fig. 6(b) were prodced applying limiting to the st the verte vales. Also limiting edge-averages frther redces the overshoot, bt does not eliminate it. Both approaches captre the shock itself well and closely match the eact soltion in smooth regions..8.8 Limited No limiter (a) t = (b) t =.375 Figre 6. Soltion of Gassian plse at shock formation (left) and post shock formation (right). Eact soltion shown with dotted line. A steady shock case was also compted to evalate the scheme s ability to reach machine zero. The soltion was initialized with a compression wave in Eq. (7), sch that a standing shock forms at t =.5 between the cell centroid 8 of

9 and cell edge. Final shock locations directly at the cell edge and centroid were also investigated bt did not significantly change the convergence behavior. Figre 7 compares the norm of the residal with and withot the limiter. Here, only verte vales are being limited. The figre shows that the residal norm steadily drops to approimately 3 once the shock strctre is established. While there is some chatter toward the end of the nlimited residal history, activating the limiter eliminates the oscillations () = < (7) > Initial Data Scheme V Limited No limiter R (a) Soltion at t = Time (b) Convergence with and withot limiter Figre 7. Brgers steady shock case IV.C. Eler reslts To test the scheme on a more complicated problem, Scheme V was sed to generate soltions for Sod s shocktbe problem. The mesh sed cells with a niform spacing of =.. The time step was set at t =.. Figre 8 shows the compted density and Fig. 9 illstrates the velocity profile. The eact soltion is shown within the figres by the dotted line and eact cell-averages are indicated by filled circles. Each figre compares two limiting options: limiting only the verte vales and limiting both the verte and edge-average vales. Both soltions captre the shock and contact qite sharply, bt each also has its drawbacks. The verte-only soltion is more symmetric arond the shock and does a better ob of limiting the epansion; however, there appears to be some error near the original discontinity location of =.5. Limiting the edge-average vales along with the verte vales appears to clean p errors near the epansion, bt, as with the Brgers case, this limiting strategy appears to indce a new etremm at the shock that is not present in the nlimited soltion. The new velocity peak vale is relatively small thogh, measring only.77% above the eact soltion. V. Conclsion Active fl schemes have been demonstrated in one dimension and generated accrate soltions for the linear advection eqation, nonlinear Brgers eqation, and the Eler eqations. It has been shown that randomizing the grid has very little effect on the accracy. This is becase the reconstrctions are performed entirely within each cell, and do not depend on srronding geometry. Additionally, the limiting approach redces sprios oscillations in the soltion withot clipping legitimate etrema. The higher accracy does increase the cost of the schemes over traditional finitevolme approaches; however, the memory savings gained throgh the data storage strategy and simple, effective 9 of

10 .8.8 ρ.6 ρ (a) Verte only (b) Verte and edge-average Figre 8. Density at t =. sing two limiting options (a) Verte only (b) Verte and edge-average Figre 9. Velocity at t =. sing two limiting options limiter sggest that the scheme will be less comptationally epensive than other high order methods sch as DG or WENO schemes, even in higher dimensions. Ftre work on active fles will concentrate on improving the characteristic-based limiter and etending the approach to mlti-dimensional systems. The nonlinear tests condcted so far indicate that the limiting approach is sond, bt it reqires improvement to more accrately simlate the flow conditions, especially arond shocks. While the etension to mlti-dimensional systems will not be trivial, there are not any foreseeable limitations with the approach that make it impossible. Si degrees of freedom are reqired for third order accracy in -D. A seventh may be added as a symmetrical bbble fnction to ensre conservation. A natral location for the conserved variable is at the triangle centroid, leaving si locations at which to store soltion data that does not need to be conserved (i.e. fles). An obvios choice is to store fl data at the three cell vertices and three edge midpoints. For 3-D tetrahedra, the vertices and edges also provide the correct degrees of freedom. In either the two or three-dimensional case, once a location for the variables is determined, all the -D concepts of careflly conserving centroid data while pdating fl of

11 data sing appropriate pwinding may be directly applied. There are, of corse, other second and third order members of the active fl family that may be robst, sefl schemes. An additional area of research will be eploring the varios schemes in search of favorable properties. Active fl schemes are targeted for se within the Kestrel flow solver, bt other codes are epected to benefit from their generality. Becase active fl schemes are an offshoot of the finite-volme approach that forms the basis for many of the CFD codes crrently sed throghot indstry, we anticipate that integrating these new schemes within eisting solvers that sers are familiar with will be a mch simpler task than inserting other high order methods. References Warming, R., Ktler, P., and Loma, H., Second- and Third-Order Noncentered Differnce Schemes for Nonlinear Hyperbolic Eqations, AIAA Jornal, Vol., No., 973, pp Harten, A., Engqist, B., Osher, S., and Charkravarthy, S. R., Uniformly high order accrate essentially non-oscillatory schemes, III, Jornal of Comptational Physics, Vol. 7, 987, pp Li, X.-D., Osher, S., and Chan, T., Weighted Essentially Non-oscillatory Schemes, Jornal of Compatational Physics, Vol. 5, 994, pp.. 4 Ollivier-Gooch, C. F., Qasi-ENO Schemes for Unstrctred Meshes Based on Unlimited Data-Dependent Least-Sqares Reconstrction, Jornal of Comptational Physics, Vol. 33, 997, pp Friedrich, O., Weighted Essentially Non-Oscillatory Schemes for the Interpolation of Mean Vales on Unstrctred Grids, Jornal of Comptational Physics, Vol. 44, 998, pp Cockbrn, B., Lin, S.-Y., and Sh, C.-W., TVB Rnge-Ktta local proection discontinos Galerkin finite element method for conservation laws III: one-dimensional systems, J. Compt. Phys., Vol. 84, 989, pp Wang, Z. J. and Li, Y., The Spectral Difference Method for the D Eler Eqations on Unstrctred Grids, 7th AIAA Comptational Flid Dynamics Conference, 5, AIAA Paper Wang, Z. J., Spectral (Finite) Volme Method for Conservation Laws on Unstrctred Grids: Basic Formlation, Jornal of Comptational Physics, Vol. 78,, pp Allmaras, S. R., Bssoletti, J. E., Hilmes, C. L., Johnson, F. T., Melvin, R. G., Tinoco, E. N., Venkatakrishnan, V., Wigdon, L. B., and Yong, D. P., Algorithm Isses and Challenges Associated with the Development of Robst CFD Codes, Variational Analysis and Aerospace Engineering, Springer New York, 9, pp. 9. Kzmin, D., A verte-based hierarchical slope limiter for p-adaptive discontinos Galerkin methods, Jornal of Comptational and Applied Mathematics, Vol. 33,, pp van Leer, B., Towards the Ultimate Conservative Difference Scheme. IV. A New Approach to Nmerical Convection, Jornal of Comptational Physics, Vol. 3, No. 3, 977, pp Iserles, A., Generalized Leapfrog Methods, IMA Jornal of Nmerical Analysis, Vol. 6, 986, pp Roe, P., Linear Bicharacteristic Schemes Withot Dissipation, SIAM Jornal of Scientific Compting, Vol. 9, No. 5, 998, pp Karabasov, S. and Goloviznin, V., Compact Accrately Bondary-Adsting high-resoltion Techniqe for flid dynamics, Jornal of Comptational Physics, Vol. 8, 9, pp Roe, P. L., Flctations and Signals - A Framework for Nmerical Evoltion Problems, Nmerical Methods for Flid Dynamics, 98, pp Eymann, T. A. and Roe, P., Active Fl Schemes, 49th AIAA Aerospace Sciences Meeting,, AIAA Paper Godnov, S. K., A Difference Scheme for Nmerical Comptation of Discontinos Soltion of Hydrodynamic Eqations, Matematicheskii Sbornik, Vol. 47, 959, pp Roe, P. L., Approimate Riemann solvers, parameter vectors, and difference schemes, Jornal of Comptational Physics, Vol. 43, 98, pp Zalesak, S. T., Flly Mltidimensional Fl-Corrected Transport Algorithms for Flids, Jornal of Comptational Physics, Vol. 3, No. 3, 979, pp Sod, G., A srvey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, Jornal of Comptational Physics, Vol. 7, 978, pp. 3. of