The PPM Compressible Gas Dynamics Scheme (This is a Draft. References especially are incomplete.)

Transcription

1 The PPM Compressible Gas Dnamics Scheme (This is a Draft. References especiall are incomplete.) Pal R. Woodward Universit of Minnesota, CSE Feb. 0, 005. Introdction. The development of the PPM gas dnamics scheme grew ot of earlier work in the mid 970s with Bram van eer on the MUSC scheme [-3]. At ivermore, in the late 970s, the se of moments of the internal cell distribtions which makes MUSC resemble finite element schemes was abandoned for reasons of compatibilit with ivermore prodction codes. In order to recaptre the accrac of MUSC while sing onl cell averages as the fndamental data, PPM, the Piecewise-Parabolic Method, was developed in collaboration with Phil Colella [4-7]. Over the last 0 ears, PPM has evolved considerabl in order to address shortcomings of the scheme as it was laid ot in [6] and [7]. PPM has also been etended to MHD [8-8] in collaboration with Wenlong Dai, applied etensivel to the std of jets and spersonic shear laers [9-5] in collaboration with Karl-Hein Winkler, Steve Hodson, orm Zabsk, Jeffre Pedelt, and Gene Bassett, sed to simlate flow in disk galaies and arond stationar and moving objects [6-8] in collaboration with B. Kevin Edgar, applied etensivel to convection and trblence problems [9-56] in collaboration with David Porter, Annick Poqet, and Igor Stine (see later article in this volme), and most recentl etended to mltiflid gas dnamics problems. Implicit versions of the method have also been worked on [59-66] in collaboration with Brce Frell and Karl-Hein Winkler and, later, Wenlong Dai. A cell-based AMR version of PPM has been nder development [see 67], in collaboration with Dennis Dinge. The PPM code has been implemented on man parallel compting sstems, inclding some with thosands of processors [68-69; 54-55]. Versions of the PPM gas dnamics scheme have become incorporated into 3 commnit codes aimed at astrophsics applications: FASH [refs], EZO [refs], and VH- [refs]. Becase the several improvements and modifications to the 984 or 986 versions of PPM in the literatre have not been pblished, we here la ot the present scheme for simple, single-flid, ideal gas dnamics from start to finish. Then selected reslts on homogeneos, isotropic, compressible trblence in a cbical, periodic domain are briefl reviewed.. Design Constraints. Before lanching into a sstematic description of the PPM algorithm, it is worthwhile to first eplain the constraints that have inflenced its design. These are:. Directional operator splitting.. Robstness for problems involving ver strong shocks. 3. Contact discontinit steepening. 4. Minimal dissipation. 5. merical errors dominated b dissipation, as opposed to dispersion. 6. Preservation of signals, if possible, even if their shapes are modified, so long as the travel at roghl the right speeds. 7. Minimal degradation of accrac as the Corant nmber decreases toward 0. The first of these design constraints was to garantee ver high processing speeds on all CPU designs. It has the side benefit of allowing the PPM scheme to emplo complicated techniqes to achieve high accrac that wold be impractical in a trl mlti-dimensional implementation, especiall in 3D. Accrac is still limited in principle b the se of -D sweeps, bt in practical problems errors prodced b the se of -D sweeps have rarel, if ever, appeared to be important. The demand that the scheme be robst in the presence of ver strong shocks has prodced a general orientation that dissipation, in the right places, times, and amonts is good rather than bad. Earl eperience with Glimm s random choice scheme for mltidimensional compressible gas dnamics in the 970s illstrated the error of letting the formal dissipation of the scheme vanish for flows containing shocks. In low-speed trblent flows, the /0/05

2 need for dissipation in the nmerical scheme is less obvios bt no less real. Contact discontinit steepening is inclded in PPM in order to allow mltiflid problems to be treated simpl throgh the addition of passivel advected constittive properties, sch as the constants of an analtic representation of the eqation of state. The view of dissipation in PPM has alread been mentioned. atrall, a design goal has alwas been to minimie the dissipation consistent with an accrate soltion. evertheless, it has also been a design goal that it is better to dissipate a signal than to significantl falsif its speed. This has led to an intolerance for signals with ver short wavelengths, sch as, for eample, 4 cell widths. However, a frther design goal was to preserve those signals that cold be propagated accratel, even if this were to reqire a falsification of the signal shape. Sch signals tend to move throgh the grid at the flid velocit, since ver short wavelength sond waves, with the eception of shocks, are difficlt to propagate at the right speeds. The final design goal, that the accrac of the scheme shold be maintained in the limit of ver man ver small time steps is achieved throgh special featres of the interpolation discssed below. The attitde that has been taken in the PPM design toward nmerical dissipation is the most relevant of these design constraints for the se of PPM in simlating trblent flows. This point will be discssed after the scheme has been laid ot, sing simlations of decaing compressible trblence as illstrations. 3. PPM Interpolation. The heart of an nmerical scheme is its approach to interpolation. PPM tilies the cell averages of varios qantities in a fairl broad stencil srronding a cell in 3D in order to constrct a pictre of the internal strctre of the cell. Becase the eqations of hdrodnamics are copled, the PPM interpolation of the relevant qantities is also copled. However, to see how the interpolation process works, it is easiest to consider it first for the simple case of a single variable, which we will represent b the smbol a. We will represent the cell averages of a b a. We write the vale of a at the left- and righthand interfaces of the cells as a i, i and a R, i. We will determine the coefficients of a parabola a ( ~ ) a ~ ~ 0 a a representing the distribtion of the variable a within the cell in terms of a cellcentered local coordinate ~ ( M ) /. Here M is the cell center and is the cell width. For simplicit, we first assme a niform grid. (In modern AMR codes, interpolation on non-niform grids is nnecessar.) Or first task is to determine whether or not the fnction a () is smooth in the region near or grid cell. To do this, we will essentiall compare the first and third derivatives of the fnction in or cell. This process can be formlated in several different was, all of which boil down to essentiall the same criterion. We begin b determining the niqe parabola that has the prescribed cell averages in or cell of interest and its two nearest neighbors. If we write then we have a a i a i a i ( a a ) ; a ( a a ) / R / R Here we have dropped the sbscript i b writing ar for a, i. Where possible, we will tr to eliminate confsing sbscripts b sch devices. However, all or sbscript indices will refer to cells, and none to interfaces (no half-indices will be sed). If no sbscript inde is given, the sbscript i for or cell of interest will alwas be implied. In PPM, interpolated variables are discontinos at interfaces, which makes the se of inde sbscripts sch as i / ambigos. Ths, a R, i, or simpl a R, is not eqivalent to a., i /0/05

3 3 We will consider a fnction to be smooth near or grid cell if the above niqe parabola, when etrapolated to the net nearest neighbor cells, gives a reasonable approimation to the behavior of the fnction there. This will not be tre, of corse, if the third derivative of the fnction is sfficientl large. We will consider the fnction to be smooth in or cell of interest if the parabola in its neighbor on the left, when etrapolated into or cell s neighbor on the right gives an accrate estimate of that cell s average, and if also that cell s parabola gives a good estimate of the average in or cell s neighbor on the left. Writing the average of the etrapolated parabola in the neighbor cell as a and in the net nearest neighbor cell as a i i±, we find that i i± a i a i i i i i i i ( a, a ) ; a a ( a a, ) Here we have sed the fact that, b constrction of these parabolae, fractional error in this etrapolation b f err, given b f /0/05 a a i i± i± {( a a )/( a sign( α, a ), ( a a )/( a sign( a ) } err ma, i R R, i α,. We denote the Here sign is the Fortran sign transfer fnction, which applies the sign of its second argment to the absolte vale of its first argment. We also denote a trivial vale of the qantit a b α. Ths the term with the sign fnction is simpl sed to protect the divide operation in a Fortran program. Of corse, we ignore this error estimate when this difference is ver small, since in that case it is of no conseqence whether we get a good etrapolation or not. To get some perspective on this measre, f err, of the smoothness of the fnction a on or grid, it is instrctive to evalate f err for sine waves of wavelengths n. For the case where a( ) sin(π /( n )), we have a s a0, where a 0 is the vale at the center of the cell and s ( / ) sin( / ). The higher-order coefficients of the interpolation parabola are given b a s sin( ) cos(π M /( n )), where M is the coordinate of the cell center, and b a s (cos( ) ) a0. We therefore find that for vales of n ranging from 4 to 5, or fractional error, eqal simpl to cos( ), assmes the vales:.00, 0.69, 0.50, 0.38, 0.9, 0.3, 0.9, 0.6, 0.3, 0., 0.0, and If we assert that a sine wave over 4 cells is smooth and one over 0 cells is not, then we ma conclde that it is reasonable to assert that a fnction is smooth if f err 0. 0 and it is not smooth if f err In between these two vales, we ma treat the fnction as partiall smooth, b taking a linear combination of a smooth and an nsmooth interpolation parabola. The choice of which fnctions we will treat as smooth and which not comes from eperience. Using onl cell averages, it takes five cell vales to determine, for eample, whether one is reall in a shock or not, as we will later see, or, as we have jst seen, whether the fnction is locall smooth or not. Under sch conditions, it is simpl nreasonable to believe that one can properl treat a sine wave with a wavelength of onl 6 cells. There are those in the commnit who believe this can be done, bt PPM assmes this is impossible withot frther independent information provided pon which the nmerical scheme can operate. Eperience also shows that nmerical noise, which can originate from a nmber of sorces, some of which will be identified later, tends to show p principall at wavelengths between 4 and 8 cells. Signals that appear at these wavelengths are ths qite possibl noise, and therefore deserve to be treated as if the might actall be noise. PPM takes the view that a little noise is not bad, bt too mch is intolerable. This view is motivated b mch eperience indicating that elimination of noise is sall accompanied b significant collateral elimination of the signal. In PPM, we interpolate several different fnctions. Some of them, like the Riemann invariants, are defined onl as differences (the Riemann invariants are defined b ineact differentials). Other fnctions, like the transverse velocities, are defined b cell averages. Some ma have sharp transitions, associated with contact discontinities, that deserve special treatment. Below, we describe the most general algorithm, for the special case of a niform grid. This is the algorithm that PPM ses to interpolate a sb-

4 4 grid strctre for the Riemann invariant associated with the entrop, for which the differential form is given b da dρ dp / c From this most complicated interpolation algorithm, all other, simpler forms sed in PPM can be derived as special cases. We will therefore describe the interpolation for the entrop Riemann invariant in detail, and later briefl note in which respects the other interpolations are degenerate cases of this one. We begin the interpolation of da b defining a variable A derived from it b determining the arbitrar constant of integration sch that the cell average of A vanishes. Ths each cell has its own local scaling for A in which the cell average vanishes bt the derivatives are correct. This local scaling is mch like the local, cell-centered coordinate, ~, that we se to define the interpolation parabola in a cell. There is a discontinos jmp in the integration constant when we go from one cell to its neighbor. This jmp is A ρ p For the gamma-law eqation of state, we estimate c ( ρ ρ ) ( p p ) / c / c i i c via [( γ p / ρ ) ( γ p / ρ )] i i Here we see a common practice sed in PPM, namel the evalation of an estimate of the cell average of a fnction of primitive flid state variables as the fnction of the cell averages. Eperience shows that attempts to do better than this are sall nrewarded b compensating increases in simlation accrac. In the specific case of the kinetic energ, PPM does do a better job, bt the effect on the flow accrac is onl marginal in this case, and the cost is not insignificant. Using the above definitions, we evalate a fractional error estimate as follows: f err A, i AR A AR A A A dα R A, i d α represents a trivial Riemann invariant difference that is sed to protect the divide in the Fortran Here program. We note that this error estimate is ver similar to bt not eactl the same as the one discssed earlier. Using this error estimate, we constrct a measre, Ω, varing between 0 and, of the roghness of the fnction near this grid cell as follows: Ω min {, ma{ 0,0( f 0.) } We now constrct estimates A s and A m of the first-order parabola coefficient, which gives the change of the variable across the cell. The first, A s, applies to the parabola that has the 3 prescribed average vales in this and the nearest neighbor cells. The second, A m, applies to this parabola after the application of a monotonicit constraint. These qantities will be sed in constrcting frther qantities below. We have alread seen that As ( A AR )/. Defining a sign variable, s, of absolte vale nit to have the sign of A s, we ma then compte A m in the following seqence of steps: ~ A m ~ ~ min{ s A, s A R }, A m s ma 0, min{ s As, Am } A m Ω A s err ~ ( Ω) Am { } /0/05

5 5 (The speriorit of Fortran over mathematical notation for the description of nmerical algorithms is here becoming apparent.) We now define two estimates, A s and A m, of the vale at the left-hand cell interface. The first of these, A s, corresponds to an nconstrained interpolation polnomial, while A m corresponds to one that is constrained. A s 6 ( As, i As ) A, Am ( Am, i Am ) A We note that these interface vale estimates involve the cell averages in two cells on either side of the interface. The nconstrained estimate is the vale of the niqe cbic crve that has the prescribed average vales in these 4 cells. The constrained vale is garanteed to lie within the range defined b the cell averages adjacent to the interface. Of corse, both these vales refer to the choice of integration constant that gives a vanishing cell average to the right of this interface (in the cell of interest, i ). We desire a linear combination of these two interface vale estimates, which we will regard as a provisional estimate Ap ( Ω) As Ω Am. Rather than bild this vale A p b blending the nconstrained and constrained estimates, we find it more sefl to bild blended parabola coefficients, Ap ( Ω) As Ω Am, and to form A p from these. This procedre allows s to make se of the coefficients A p later in or contact discontinit detection and steepening algorithm. We now proceed to modif or provisional interface vales A p in cells where the fnction is not smooth. First, we set or interpolation fnction to a constant in cells where etrema occr, nless of corse the fnction is smooth there. Ths, if A 0, we set new provisional vales, indicated b a sbscript q, as follows: A ( Ω) p A pr A ( Ω) q A p, qr ApR. Otherwise we simpl set A q Ap and A q Ap. ow, in cells where the fnction is not smooth, we constrain the internal strctre to be monotone. We mst make sre in this process not to revise the cell strctre frther in the cells containing etrema. We therefore constrct limiting vales, A and A, which we mst be carefl to set to 0 in cells l lr containing etrema. In the remaining cells, these vales are: l ApR and lr Ap. This reflects the fact that a parabola having ero slope at one side of the cell will assme at the opposite interface the negative of twice this vale, so long as the cell average is ero. If these limiting vales are eceeded in cells where the fnction is not smooth, we mst reset them to these limits. Therefore we constrct new, bt still provisional, vales A r and A rr (in Fortran, of corse, no new names are reqired), which reflect this additional constraint: 6 A A ( AqR Aq )( Aq Al ) < 0, then Ar ( Ω) Aq Ω Al, otherwise Ar Aq ( AqR Aq )( AlR AqR ) < 0, then ArR ( Ω) AqR Ω AlR, otherwise ArR AqR if if From these provisional interface vales, we constrct provisional interpolation parabola coefficients: A A A ( A A ), A A /, Ar 3 r rr r0 r rr r r For interpolation of sond-wave Riemann invariant differences, for which we do no contact discontinit steepening, the above vales are or final reslts. However, for the entrop Riemann invariant differences, we contine as described below. We also note that interpolation of a variable, sch as the transverse velocit component, which is not epressed as a differential form, we ma first constrct differences of the cell averages and then proceed as otlined above. We wish to detect cells that are inside sharp jmps in the entrop Riemann invariant that are associated with contact discontinities. In sch cells, it is inappropriate to tr to fit smooth crves to determine the sbgrid strctre, since it is actall discontinos (PPM is solving the Eler eqations). For the prpose of comptation, we will nevertheless reqire that the distribtion inside the grid cell be a /0/05

6 6 parabola; however, we ma se the knowledge that the actal strctre is discontinos to constrct an appropriate choice for this parabola. The idea here is qite simple. We bild a test that detects cells that are within contact discontinit strctres. Then we obtain estimates of the interface vales for sch cells b etrapolating to the cell interfaces presmabl smoother strctres from otside the cell and hence, hopefll, from otside the discontinit. Together with the prescribed cell averages, these more appropriate cell interface vales allow s to bild an improved, steeper parabola to describe the distribtion within the cell. Obviosl, the fnction in sch cells is not smooth, so we appl monotonicit constraints to the new edge vales and also to this improved parabola. This procedre works ver well in practice. A similar procedre can be adapted for se with other nmerical schemes that emplo linear interpolation fnctions for the cell strctres, bt it is less effective. Apparentl, the crvatre provided b the se of parabolae allows the dimensionless constants that will be introdced below to make the steepening process effective for contact discontinities withot reqiring that the steepening process be applied for marginal cases, which can occasionall trn ot not to actall be contact discontinities. That is, sing parabolae we are able to appl the steepening process onl in neqivocal circmstances and ths to avoid steepening strctres b mistake that shold not actall be steepened. We begin, as with or test for fnction smoothness, with a measrement of the sie of the third derivative of the fnction relative to its first derivative. Resing the smbol s for a different sign variable, we bild a steepness measre S in a seqence of steps as follows: S ( A A A ) ( A s dα ) sr s s / s 0 Here s has absolte vale nit and has the same sign as A s in order to protect the divide withot altering the sign of the overall epression. As before, d α is a trivial Riemann invariant difference. We mst take care to realie that the nmerator in this epression is not necessaril ero, since the edge vales that appear there come from cbic crves rather than the parabolae that define the A s vales. We mst also take care to realie that becase each cell has its own vale of the integration constant for this Riemann invariant s differential form, we mst set AsR As, i AR. With those cations, we see that this epression for S is ver similar to or measre of the lack of fnction smoothness, ecept that no absolte vale signs appear in it. The first epression in the nmerator for the variable difference across the cell involves cbic interpolation, which acconts for the third derivative of the fnction. From this we sbtract A s, which removes the part that is de to the first fnction derivatives. To obtain S, we divide b an estimate of the first derivative. The lack of absolte vale operations in this formla for S is cased b or desire to detect sharp jmps in the fnction, and to reject sdden flat spots. Flat spots will correspond to negative vales of S. We now scale S and limit it to the range from 0 to : { 0, min{, 0 ( 0.05) } S ma S To gard against appling or contact discontinit steepening anwhere bt at tre sharp jmps in the entrop, we reset S to ero where the second derivative of the fnction does not change sign and also where the amplitde of the jmp is not sfficient to warrant this special treatment (we do not want to preserve, or worse, to amplif small nmerical glitches). Ths: if A s, i As, i, then S3 0, otherwise S3 0 S Here we se the earlier definition of the coefficients of the parabola that has the prescribed 3 cell averages, writing As ( AR A )/. To eliminate trivial jmps from consideration, we form S 4 : if A α, S AR < d then S4 0, otherwise S4 3 Before proceeding with the steepening algorithm, we make one frther test to make certain that steepening is appropriate. This test makes sense for contact discontinities, bt for other variables, sch /0/05

7 7 as a fractional volme of a second flid, it might not. Therefore this last test is optional, bt we do se it for the entrop. The calling program to the interpolation rotine provides two variable differences along with the differences A. If the jmp is one of the tpe we seek to detect, the differences B will be ver small compared with the differences D. Both these sets of differences are cell centered and have the same nits. We enforce this demand b constrcting or final steepness measre, S, as follows: if D 0 or B / D 0., then S 0, otherwise S S In previos versions of PPM, the c ρ B were pressre differences and the D were estimates of, while the densit was the qantit being interpolated with potential contact discontinit steepening. We still perform this additional test, althogh it is somewhat redndant when interpolating the entrop Riemann invariant, for which the differential form is jst ρ. /0/05 A p / c With or contact discontinit detection process complete, and all cells inside sch strctres marked b S, varing from 0 to, we are at last read to begin the steepening operation. Mch earlier, we compted constrained variable differences A m across the cells. In principle, these variable differences cold be onl partiall constrained, bt it is safe to assme that if or cell is in a contact discontinit, the parameter Ω that measred the fnction roghness assmes the vale nit in this and the neighboring cells. We define cell interface vales, A c and A cr, appropriate for a contact discontinit strctre b etrapolating the constrained slopes of the fnction in neighboring cells to the interfaces of or cell of interest: A c A Am, i / and A cr AR Am, i /. We then constrct or nearl final estimates for the interface vales as: t ( S Ar S Ac, A tr ( S) ArR S AcR A ) We mst once again appl the constraints on the implied parabola and then compte the coefficients of that parabola, jst as before (however, hardl an cells are steepened, and hence this etra work occrs in a scalar loop for onl these cells, which incrs hardl an additional comptational cost). These operations will not be repeated here. The are jst like those we performed to arrive at first A q and then A r beginning with A p. It is worthwhile to note the was in which the above interpolation algorithm addresses the previosl listed design constraints for PPM. Some of the points are obvios, bt others are less so. In particlar, it is not obvios that dissipation errors will dominate dispersion errors, althogh this is in fact the case. The reason for this is the action of the monotonicit constraints on all short wavelength distrbances, which alwas fail or test for fnction smoothness. The monotonicit constraints are nonlinear, in that the alter the shape of a sine wave b introdcing higher freqenc components throgh effects sch as the clipping of etrema. Their overall effect is strongl dissipative for ver short wavelength signals, which of corse are the ones for which the nmerical scheme wold otherwise introdce significant dispersion errors. This dominance of dissipation over dispersion error as a design goal ma seem incompatible with the goal of minimal dissipation. Or task is to deliver as little dissipation as possible while still having dissipation overwhelm dispersion. PPM s se of parabolae rather than the more commonl sed linear interpolation fnctions is meant to address this design goal. The se of cbic interpolation fnctions to help define these parabolae throgh interpolation of cell interface vales redces dissipation errors still frther, since not all parabolae provide eqall good fits. However, even more importantl this se of cbic interpolation in PPM preserves the accrac of the scheme in the limit of vanishing Corant nmber. Since, as is common practice, we will determine a single time step vale for the entire grid based pon Corant nmber limitations for the single most demanding cell, the blk of the cells we pdate will have ver small Corant nmbers. When AMR is added to the scheme, the Corant nmbers sed for the blk of the cells are likel to become even smaller. Therefore it is ver important for the accrac of the calclation to hold p in this limit. The se of cbic interpolation of cell interface vales is or wa of 4

8 8 addressing this isse in PPM. Over man ears of appling PPM to a wide variet of flow problems, ver little Corant nmber sensitivit has ever appeared. The monotonicit constraints and contact discontinit detection and steepening algorithms incorporated in PPM have important conseqences for the propagation of barel resolved or effectivel nresolved signals. Or design goal is to preserve these signals, so long as we can propagate them at roghl the proper speeds, rather than to destro them throgh nmerical diffsion processes. The monotonicit constraints and, to a far greater degree, the contact discontinit steepening in PPM have this effect. The act pon passivel advected signals, sch as entrop variations, that can easil be propagated at the correct speeds. The appl an artificial compression along with their other effects, sch as maintaining positivit where appropriate. This tends to maintain signal amplitde while altering signal shape when the signal is not adeqatel resolved. As the description of the contact discontinit detection algorithm shows, we take great care to appl the steepening method onl where appropriate, since some small signals, better described as nmerical glitches of varios tpes and cases, deserve to be dissipated. 4. Using the Interpolation Operators to Bild a Sbgrid-Scale Model for a Cell. The above interpolation process is qite elaborate. evertheless, it is not the entire stor of PPM interpolation. As we remarked earlier, the eqations of gas dnamics are copled, and therefore the PPM interpolations of the flid state variables are also copled. The goal is to come p with a complete and consistent pictre of the internal strctre of a grid cell that is, to come p with a sbgrid-scale model. This is not a trblence model, bt it is a sbgrid-scale model. This model mst satisf, for consistenc, certain primar constraints. First, the integrated mass, momenta, and total energ mst be consistent with the prescribed cell averages of these qantities. This reflects the need for the nmerical scheme to be in strict conservation form in order to correctl captre and propagate shocks. PPM regards the fndamental cell averages on which it operates to be the cell averages of densit and pressre (both volme-weighted) and those of the 3 velocit components (all mass-weighted). From these, PPM defines the mass, momenta, and total energ of a cell to be: m ρ, m, m, m, p γ the transverse directions, and ( ) m and. Here we do not bother to inclde the cell widths in, becase the cancel ot of all or eqations for the -pass. The last epression, for the cell s total energ, is misleading. At the beginning of the grid cell pdate for a -D pass, we compte the total energ of the cell in jst this wa. However, once we have done this, we make a more accrate estimate of the cell s kinetic energ and then revise its thermal energ, keeping this total constant. We do this b sing nearest neighbor cells to compte cell-centered slope estimates for all 3 velocit components. We then appl the standard monotonicit constraints to these slopes. All these comptations are similar to the following steps, for the slope of in the direction: ( ), ( ) min{ s ( ), s ( )} k k ma Here s has absolte vale nit and the same sign as kinetic energ to be given b m s ma k { 0, min{ s, ( ) } ma k. We now estimate the cell s average /0/05

9 9 E kin [( ) ( ) ( ) ] m [( ) ( ) ( ) ] m [( ) ( ) ( ) ] m This process captres the first correction to the average kinetic energ that arises from the internal strctre of the velocit within the grid cell. The method wold still be second-order accrate withot this correction, bt this correction proves to be of some marginal vale (relative to its implementation cost) in flows involving strong shear laers. Terms of this sort make p part of some trblence closre models, althogh one might arge that this is inappropriate, since the have nothing specificall to do with trblence. In PPM we mst compte estimates for cell averages of several other qantities, and inclding terms like these for all those comptations wold roghl doble the cost of the scheme. This has been tried, and little noticeable benefit is delivered in practical problems. Therefore onl these velocit terms are inclded in PPM. One can see wh these terms are important as follows. The velocit slopes can be ver large in a cell, becase shear laers that are stretching de to the local flow or de to their own instabilit natrall become ever thinner. The contribtion to the kinetic energ from the velocit slope terms can therefore be significant, and can ths have a significant effect pon the pressre (since the total energ is prescribed), and hence pon the dnamics. evertheless, this effect pon the dnamics is strongl localied, so that omitting the terms, as was done in PPM for man ears, onl tends to increase b roghl 50% the nmerical friction in thin, strong (roghl sonic or spersonic) shear laers. In order to propagate signals, we need to interpolate their sbgrid strctres. We therefore appl the previosl described interpolation algorithm to the 5 Riemann invariants of the Eler eqations for 3-D compressible flow. For the two transverse components of the velocit, and, which are advected passivel with the flid velocit in the -pass, we form differences and then appl the interpolation scheme described above. We do no contact discontinit detection or steepening for these variables, even thogh the ma in fact jmp at sch discontinities. We have alread discssed in detail the treatment for the entrop Riemann invariant. The sond wave Riemann invariants, which we denote b R ±, are defined b the following differential forms and their corresponding nmerical approimations: R p dr d ± ± dp C m m m m m m ( p p ) ( C C ) ± ± ± / i i i C Here we introdce the agrangian sond speed C ρ c, and we approimate its cell average as C ρ c γ p ρ. The pressre, of corse, is p, and onl the -component of velocit appears becase we are describing the -pass of the directionall split PPM algorithm. It is distrbances in R that propagate to the right at speed s c and distrbances in R that propagate to the left at speed s c. We interpolate parabolae to describe the strctre of these sond wave signals in the grid cells, sing R ± in place of A in the algorithm previosl described, and of corse performing no contact discontinit detection or steepening. Performing interpolations for the Riemann invariants attempts to ncople the gas dnamic eqations as mch as possible. However, we will not work in terms of these variables directl. Instead, we will immediatel go abot constrcting interpolation parabolae for the primar flid state variables the densit, pressre, and 3 components of velocit in a manner that attempts to achieve as mch consistenc as possible with the interpolated /0/05

10 0 strctres of the Riemann invariant signals. To perform or hdrodnamical cell pdate, we will reqire internal cell strctres for all these variables as well as the total energ. An choice of strctres for 5 variables will impl strctres for all the others, bt those implied strctres ma not satisf reasonable constraints, sch as monotonicit or positivit when appropriate. PPM attempts to achieve consistenc between interpolated and implied strctres b first interpolating constrained parabolae for the Riemann invariant signals, constrcting the implied parabolae for the primar state variables, and then constraining those implied parabolae where appropriate. In a sense this is an attempt to have it both was, which is of corse impossible. However, eperience has shown that the etra labor involved in this process delivers additional accrac and robstness of a vale worth of its comptational and programming cost. From the interpolated parabolae for the sond wave Riemann invariants, we ma compte the cell interface vales of the pressre and -velocit as follows: p p C ( R R )/, p p C ( R R )/ R R R R ( R R )/, ( R R )/ R R R We also compte a measre, Ω, of the lack of smoothness of the associated fnctions as the maimm of the vales fond for the two sond wave Riemann invariants separatel dring the process of their interpolation: Ω ma { ΩR, ΩR }. We will se this measre Ω to control the application of constraints to or interpolation parabolae for p and. We first appl monotonicit constraints, controlled b Ω, to the cell interface pressres and -velocities obtained above. When Ω, we demand that the interface vales lie within the ranges defined b the averages in adjacent cells. We then, again when Ω, demand that the parabolae defined b these constrained interface vales and the cell averages are monotone. These constraints are essentiall the same as those described earlier for the interpolation of the entrop Riemann invariant, and hence the will not be stated in detail here. Once the interpolation parabola for the pressre has been so derfined, we ma se the interpolated interface vales for the entrop to determine interface vales for the densit. Using the maimm of the Ω measres for the pressre and for the entrop, we ma then constrain the interface densities and ltimatel the parabolae that the and the cell averages define. At the end of this length process, we have interpolation parabolae defined for the densit, pressre, and all 3 velocit components. The formlae presented assme niform cell sies, althogh more general formlae are easil derived and were presented in the description of PPM in [whatever]. We can think of these interpolations as occrring in a cell nmber variable, so that the are valid, althogh potentiall somewhat less accrate, even if the cell sie is smoothl varing. evertheless, we mst take care in interpreting these parabolae. Some of the variables described b them, sch as the velocit components, mst be considered to var according to mass fraction across each cell, since the (mass-weighted) cell average is associated directl with the conserved cell momentm. For the velocities, therefore, we take the interpolation variable ~ within the cell to have its origin at the center of mass and to describe fractions of the cell mass rather than of the cell volme. For the pressre or the densit, whose volme integrals are directl associated with the mass and internal energ, we mst interpret ~ as having its origin at the center of the cell in a volme coordinate and to describe volme fractions within the cell. Or se of the niform grid formlae therefore reflects an assmption that all interpolated qantities are smoothl varing in a cell fraction variable, be it a volme or a mass fraction. We know this assmption to be false at contact discontinities and slip srfaces (for some of the variables) and at shocks, bt we have agmented or interpolation procedre to deal with these discontinities. The interpolation is therefore valid, so long as we interpret it properl. The process described above has enabled s to constrct a sbgrid-scale model for each cell. Since each flid state variable varies as a simple parabola inside a cell, this sbgrid-scale model cannot possibl describe sbgrid-scale trblent motions. To perform that fnction, the model wold have to be agmented b the addition of one or more new state variables, sch as a sbgrid-scale trblent kinetic energ variable, which cold also be given an interpolation parabola within the cell. evertheless, or demand that the strctre of a velocit component within a cell be no more complicated than a parabola, /0/05

11 and or constraints on that parabola, provide a powerfl mechanism to dissipate kinetic energ of smallscale motions nresolvable or marginall resolvable on or grid into heat. This is one of the important fnctions of trblence closre models, especiall when incorporated into difference schemes that lack an other method to accomplish this ver necessar dissipation. In this respect, trblence models can be sed to stabilie otherwise nonlinearl nstable nmerical schemes, a role that has nothing to do with trblence itself and that confses the prpose and fnction of a trblence model. As we will see later, with PPM we ma compte trblent flow directl from the Eler eqations that govern it, and we accomplish the dissipation of small-scale motions throgh the trncation errors of or nmerical scheme rather than from differencing an eplicit viscos diffsion term. We take the view that the details of this dissipation are nimportant so long as kinetic energ trning p on the smallest possible scales in or comptation is dissipated into heat with no nphsical side effects. We presme that in a far more epensive and carefl simlation of all relevant phsics, this same kinetic energ wold be dissipated in an event, and the same amont of heat generated (since total energ is conserved), bt the (hopefll nimportant) details on tin length and time scales wold differ. The assmption here is that the kinetic energ that appears on scales where the nmerical dissipation comes into pla has arrived there de to a phsical process, sch as a trblent cascade, that is correctl simlated in or calclation becase it has nothing to do with moleclar viscosit. 5. The PPM Approimate Riemann Solver We will pdate the cell averages in time b appling the laws of conservation of mass, momentm, and total energ. To do this, for a -D -pass of the algorithm, we mst compte time-averaged fles at the left- and right-hand cell interfaces of these 5 conserved qantities. This is done with the help of an approimate Riemann solver. A fndamental approimation that we make is that the soltion to the Riemann problem for left and right states representing spatial averages over the appropriate domains of dependence is roghl eqivalent to the average in time of the man separate Riemann problem soltions that each epress the interaction of the specific Riemann invariant vales arriving at the interface at that particlar time. The Riemann solver is a nonlinear operator, and we assme that its otpt when operating on the averaged inpts is eqivalent to the average of the otpts from the separate, timevaring inpts. This assmption makes the Riemann solver relativel simple and efficient. Or elaborate constrction of sbgrid-scale strctres allows s to easil arrive at the spatial averages in the appropriate domains of dependence. In the linear regime, when we have simple advection of the Riemann invariants, we get the fll advantage of or interpolation parabolae for the Riemann invariant signals. Eperience shows that when variations in R ± are large, shocks rapidl develop, and these demand techniqes onl loosel related to concepts of formal order of accrac. Here we will discss how we handle the domain of dependence for the Riemann invariant R. The treatment for R is similar, and will not be described. We will treat the characteristic speed s c to be constant both in space and time within each cell for this single time step. Acconting for the space and time variation of s is demanded onl if we desire formal third-order accrac. We will focs or discssion on the left-hand interface of or cell of interest. We calclate a Corant nmber, σ, which applies to one or the other of the adjacent cells, depending pon the signs of s in them: σ t /, if s, i 0 ; t /, s, i otherwise s if s We note that the two possibilities listed above are not ehastive. In the case where s is directed awa from the interface on both sides, we will have a centered rarefaction there. We will handle this case later. For the moment it will be sfficient to set σ to ero in this rare event. The Corant condition demands that we control the time step t so that averages of the densit and pressre, σ σ does not eceed nit. We now estimate the spatial, in the domain of dependence via: ρ and p 0 /0/05

12 We obtain nmber, m ρ p ρ σ ρ σ 3 ρ,,,, R i i i if s i ρ σ σ ρ, otherwise. 3 ρ ρ sing similar formlae. From ρ we ma calclate a mass fraction Corant s, we have σ σ ρ ρ, and otherwise we have σ. When 0, i /0/05 m / i σ m σ ρ / ρ. We can then obtain sing the same formlae as for the densit or pressre, bt with σ replaced b σ m. ote that we choose the pwind domain of dependence in the case when R Riemann invariant signals reach the interface from both adjacent cells. This can happen if the interface is inside a shock strctre, in which case the R signals from the cell on the right will become lost in the shock transition. Hence we take the signals from the cell on the left in this case, which will correspond to the proper post-shock state. For the R signals, we prefer signals from the cell on the right, if sch signals can reach the interface from both adjacent cells. In the case when s, i < 0 and also s > 0, the left-hand interface of or cell is located inside a centered rarefaction fan. We desire Riemann invariant information corresponding to a vanishing characteristic speed. We interpolate ρ, p, and linearl between the two edge vales, as in: p f pr i f, ( ) p, where s ( s s ) /, i f. A simple, linearied soltion to the Riemann problem at the cell interface can be obtained from the demands that the time-averaged state at the interface have the same vales of R ± as the domains of dependence which we have jst determined. At this point we desire onl the time-averaged pressre and -velocit at the interface, which we denote b p and. Using the differential forms that define the Riemann invariants, we ths have: ± ( p p ) / C 0 p ± ±. Therefore: ( ) ( p p ) C p C ( ) Becase this is a nmerical scheme, and anthing can happen, we sbseqentl make sre that p is at least as big as some trivial, floor vale. This simple, linear Riemann solver sffices for almost all the cell interfaces, bt it is not good enogh for interfaces that are in strong shock strctres. For sch cells, which are ver small in nmber, we can do mch better. However, first we mst determine which cells are in shock strctres. In PPM, we go to considerable lengths to identif these cells, and we will se the information we glean in the process to bild a smart diffsion velocit that we will se to greatl diminish noise that can be generated at shocks in nmerical comptations. Eperience has shown that shock detection mst be done sing information from all 3 dimensions. This fact means that the difference stencil of the PPM -D pass etends into the transverse dimensions. This eplicit acknowledgement that the -D pass is actall part of a mlti-d comptation allows PPM to avoid a nmber of nmerical pathologies that wold otherwise occr. Several of these have been discssed in the original paper laing ot the design philosoph and major techniqes of PPM [W&C]. (These pathologies were dbbed Cra instabilities, since at the time one needed to have access to a Cra- compter in order to afford grids fine enogh to observe them.) As with the detection of fnction, 0 ;

13 3 roghness and sdden jmps associated with contact discontinities, detection of shocks reqires the inspection of data from cells on either side of the cell of interest. For the case of shock detection, we se cells on either side in each of the 3 dimensions. Taking differences over 4 cell widths allows s to get a good estimate of the entire shock transition, and hence of its propagation speed. We will need to know the propagation speed relative to the grid as well as the shock orientation relative to the grid in order to detect those sitations which reqire diffsion at the shock, and hence to be able to constrct or smart diffsion velocit. Or approach is to first compte in vectoriable loops qantities that allow s to eliminate most candidate cells, and then to do the more involved comptations determining shock orientation and propagation speed onl for the ver few cells that reqire this. We will demand that to be considered as within a shock strctre a cell mst be in a region of compression and in a sdden pressre jmp of at least 5%. These demands will allow s to eliminate from consideration the great blk of or cells. First, in each of the 3 dimensions, we evalate the jmp between the net nearest neighbor cell averages of the pressre on either side of or cell. In this process, we also determine which of these net nearest neighbor cells has the lower pressre vale. We then find the largest of these pressre jmps and divide it b the lower pressre vale in that dimension. For or cell to be considered inside a shock, this ratio mst eceed 0.5, althogh this particlar vale is not critical to the sccessfl operation of the scheme (a vale of 0.33, for eample, works well too). The choice of 0.5 indicates a Mach nmber, which one cold derive, for a shock below which no special treatment b PPM is necessar. Setting this vale too low is harmless, bt it wold case the scheme to devote considerable nnecessar labor to weak distrbances. Setting the vale too high will let small, incorrect signals be generated at some shocks nder pathological circmstances. The choice 0.5 has served well for decades in a wide variet of problems. In the same vectoriable loop, we evalate simple difference approimations to the divergence of the velocit. One of these ses differences between velocit component averages in nearest neighbors to or cell and the other ses differences of sch averages in net nearest neighbors. If either of these estimates of the velocit divergence is positive, we eliminate or cell as a candidate for being inside a shock strctre. We mark or candidate cells, those with negative velocit divergence estimates and with 5% or greater pressre jmps, with a flag vector that we call shocked. This flag arra vanishes everwhere bt in or candidate cells, where it has the vale nit. For the cells marked b shocked, we now embark pon a complicated calclation described below. First we se averages in net nearest neighbor cells to evalate the contribtions from each dimension to the velocit divergence estimate. We will se weight factors, f, f, and f, for each dimension according to the relative sie of these contribtions. Becase we desire no negative weight min 0, factors, we define the contribtions to the velocit divergence as {, ( )} j with similar formlae for the and dimensions. We then define the weight factors as follows: f ( ) ( ) ( ) ( ), f ( ) ( ) ( ) ( ) We define the remaining weight factor b sbtraction: nearest cells in each dimension, we select the pre- and post-shock cells in each. We then blend these 3 pre- and 3 post-shock cell averages sing the above weight factors to obtain estimates of the pre- and post-shock densities, pressres, and 3 velocit components. Using the jmps across the shock in the 3 velocit components compted from these blended vales, we get the -, -, and -components of a nit vector pointing in the direction of the velocit jmp, hence in the direction normal to the shock front. We now project the pre- and post-shock velocities onto this normal direction b taking the dot prodcts of those velocities with the nit vector. We also compte the pre- and post-shock agrangian sond speeds. The nonlinear agrangian wave speed, W, for the shock is compted via: post pre f j f f. ooking at vales in net ( V V 0. V ) ~ W p p / pre post post /0/05

14 4 This formla clearl involves some defenses against potential pathologies. In mltidimensional flows net to walls where Mach reflections of shocks can occr, some rather wild cases can arise to which or formlae mst be insensitive. The tilde in the formla indicates that this is a provisional vale. To force this wave speed estimate to be reasonable, we additionall eecte the following constraints, reslting ltimatel in or final estimate, W : ~ ~ ~ min{ W, ma{ }, W ma W, min{, } W C pre, C post { C pre C post } We now compte the Elerian shock speed, w, b sing the post-shock densit and normal velocit: w W / ρ post One might wonder wh we need all this information abot the shock. We will se it to compte a diffsion speed that will determine the amont of additional dissipation that we introdce here. As discssed in the original articles on PPM [W&C, C&W, W], shock representations on or grid tend to become too thin when the shocks move slowl relative to the grid. Sch a shock can linger near a grid line, and then the action of the focsing characteristics can case the shock jmp to occr mostl across that single cell interface. As the shock moves so that it becomes centered on a grid cell instead, it is forced to have a qite different and broader nmerical representation. The slow oscillation of the shock strctre from thicker to thinner nmerical representation cases small distrbances to be emitted in all the characteristic families that cross the shock (i.e. in or case in the entrop and in the oppositel propagating sond wave Riemann invariants). This happens in -D problems, bt in - or 3-D problems we can get glitches cased b oversteepening of the shock strctre as it crosses grid lines obliqel. At each graing intersection, a tin shock and contact discontinit pair can be emitted. The soltion to all these problems is to broaden the nmerical shock representation so that it is roghl independent of phase relative to the grid. The glitches and noise emission can be redced dramaticall b onl a modest broadening of the shock. We do this b means of a diffsion operation that will be described later. The diffsion coefficient, which we evalate as a diffsion velocit, mst relate to the potential severit of the nmerical problems for a given shock in a given cell. We find that the diffsion coefficient needs to reflect two independent factors the shock strength and the wavelength of the fndamental noise signals that it tends to emit. Or diffsion velocit, which we denote b diff, has a factor eqal to the negative of the velocit divergence in order to reflect the strength of the shock. It also has a factor arising from the fndamental wavelength of noise emission. The fndamental period for noise emission b the shock is the interval between grid line crossings. The wavelength of an sch signals, measred in grid cell widths, is then dependent pon the Riemann invariant famil involved and the post-shock flid velocit and sond speeds. The longest fndamental wavelength of signal emission, measred in grid cell widths, will appl to the sond wave Riemann invariant, and it can be estimated as follows: λ noise post ( W / ρ post ) c post ( W / ρ post ) post c pre When the shock moves rapidl across the grid, this wavelength is ver short, and an noise signals are immediatel damped b the nmerical scheme, if the can propagate at all. It is onl as this wavelength eceeds that we need become concerned. We therefore compte or diffsion velocit in the following steps: Θ ~ diff 0.3 ( shockd) i i j j k k ma { 0, ( ) } 3 3 λ, Θ / ( Θ ) noise Ξ, diff 0. ( 9Ξ) diff ~ Some aspects of these formlae ma seem a bit arbitrar, and no dobt other variations might work eqall well. evertheless, the above formlation is qite serviceable, and has kept nmerical glitches and noise at ba in PPM simlations for at least 5 ears. Changing the 0.3 to 0.5 in the formla for /0/05

15 5 ~ diff will roghl doble the thicknesses of shocks, almost completel eliminate nmerical noise, and redce the overall qalit of the soltion b an amont comparable to coarsening the grid b a factor of in ever spatial dimension and time. Setting this constant to 0. will case the noise to appear and to mar the soltion. The recommended vale of 0.3 is ths a compromise, giving ecellent reslts with thin shock strctres and crispl resolved details bt allowing ver low level signals of nmerical origin into the simlation at levels almost alwas below abot.5% of the amplitde of the relevant, emitting shock jmp. The smart diffsion described here, and applied at the end of the grid cell pdate, spersedes all previos formlations in the literatre on PPM. Earlier formlations went to significant lengths to avoid the se of an constrct that cold directl be interpreted as an artificial viscosit. This was a conseqence of a bet made with Bill oh at ivermore. Since his death man ears ago, there has been no satisfaction from doing withot eplicit diffsion, and the present formlation, in se for abot 5 ears, is easil made compatible with other code packages. ow that we have identified the cells inside shock strctres and compted the diffsion velocities to be sed there, we proceed, in a scalar loop for onl these cells, to appl a nonlinear Riemann solver to compte the fles at their interfaces. A fndamental approimation we will make is to se the shock formlae for pressre and velocit differences across rarefaction fans. For weak rarefactions, this approimation is eas to jstif. Strong rarefactions at single cell interfaces are alwas transient phenomena, with the eception of centered rarefactions attached to featres in walls or solid objects. We take the view that it is impossible to do a good job for these eceptional cases in an event, save to refine the grid there sing AMR or other techniqes. We will describe the nonlinear Riemann solver for gammalaw gases, bt a generaliation of it to arbitrar eqations of state has been given in [W86]. We will perform the following operations onl at those interfaces for which one of the adjacent cells is marked b the flag arra shocked. We will do the first step of what cold be sed as an iteration. Eperience has shown that there is essentiall no vale in performing additional steps of sch an iteration. The se of a nonlinear Riemann solver allows s to compte a portion of the entrop jmp in the shock from shock jmp conditions rather than obtaining it from a diffsion process that smears the shock strctre. However, the shock jmp conditions that appl at a single cell interface are essentiall never the proper ones for the shock as a whole, and as a reslt it is not worthwhile to take them too seriosl. A single nonlinear iteration therefore sffices. We begin b compting positive-definite agrangian wave speeds in the gas immediatel to the left, W, and to the right, W R, of the contact discontinit that initiall forms at the interface. To improve or estimate, p, of the pressre at this contact discontinit, we also evalate the slopes, ( p / ) and ( p / ) R, of the tangent lines at the pressre p to the Hgoniot crves in the velocit-pressre plane: ( p / ) f ( ) p ( γ ) p f f ( ) p ( γ ) p γ, R γ W f ( γ ρ /, W ρ / f W ) ( p p ) R f R, ( p / ) R f R f R ( γ W ) R ( p p ) We now compte the point of intersection of the two tangents to the Hgoniot crves, which gives s an improved estimate, p shk, for p. The tangent lines pass throgh the points (, p ) and, p ) in the p plane. The intersect at the point, p ). ( R ( shk shk ( p p ) W, R ( p p ) / WR / Ψ ( R ) ( p / ) R ( p / ) ( p / ) R /0/05

16 6 p shk {, [ p Ψ( p ) ]} ma φ /, Ψ shk We mst now take care to handle slowl moving strong shocks properl. From the above reslts for the pressre and -velocit at the contact discontinit in or Riemann problem, we can compte the Elerian wave speeds, w and w R, for both the leftward and rightward facing waves: w W ρ /, w R W / ρ R Using this information, we now select from the 3 possibilities or improved estimates for which we will adorn with primes: if shk < 0 and w < 0, then, R p if shk > 0 and w > 0, then, p p otherwise, p p shk shk p and p, Getting good estimates of the nonlinear wave speeds is crcial here, althogh in principle the fles of mass, momentm, and total energ on either side of a stationar strong shock are identical. Up to this point, we have worked onl with the time-averaged -velocities and pressres at the cell interfaces. ow that we know the time-averaged interface velocities, we can trace back along the flid streamlines from the interfaces into the adjoining cells in order to estimate what fraction of which cell crosses the interface dring the time step. We ma also estimate the time-averaged densit at the interface b taking the spatial average at the beginning of the time step of this material that crosses the interface and compressing or epanding it sing the shock jmp conditions to the time-averaged pressre at the interface, which has jst been compted. In the absence of shocks, this compression or epansion shold be adiabatic, bt if a strong shock is involved, the behavior cold be etremel different. Using the shock jmp conditions will give s an acceptable reslt in either case. We begin b compting the spatial averages, ρ, p 0 0,, and 0 0, in essentiall the same wa that we compted similar spatial averages earlier over the domains of dependence of the R ± Riemann invariant characteristics. Here we se the sbscript 0 to denote the streamline characteristics. The timeaveraged densit at the cell interface is now: ρ ρ ( γ ) p ( γ ) 0 ( γ ) p ( γ ) 6. Constrction of the Time-Averaged Interface Fles and Application of the Conservation aws. ow that we have evalated the time averages of the flid state variables at the cell interfaces, it is a simple matter to constrct from them the time-averaged fles of the conserved qantities. At each interface, we compte an advected volme, d, an advected mass, dm, 3 advected momenta, dµ, dµ, dµ, and an advected total energ, de. These are all signed qantities, with positive signs appling to advection to the right: d t, dm ρ d, d dm p t p p 0 0 µ, d µ dm [ ] d µ dm, de ( ) ( ) ( ) γ dm p d γ /0/05

17 7 To gard against the prodction of negative densities, we also demand that dm not eceed advection of 95% of the pstream cell s mass. Or time step controls shold prevent this sitation, bt this constraint provides additional code robstness. We now appl the conservation laws for mass, momentm, and total energ in order to arrive at provisional new vales for the cell averages at the end of the time step for this -pass. We se the sbscript to denote the new time level and the sperscript ( ) to denote this first, provisional vale. Despite or application of monotonicit constraints it is still possible in this Elerian treatment for the new cell mass to be negative. We therefore demand that the new cell mass not fall below a floor vale, given b ρ min. The vale of ρ min is of corse problem dependent, and it mst be spplied b the ser for an given rn, along with trivial vales for other positive definite qantities sch as pressre and energ, p min and E min. To prevent problems of negative cell masses from occrring, we will compte a cell mass Corant nmber that we will se to help control the time step. This time step control is the Elerian eqivalent of the one sed in agrangian calclations to prevent the tangling of grid cells. m () {( m dm dm ), } () ma ρ, ρ / () () The cell mass Corant nmber we se is ( ) R min 0. m m / m t / t Corant nmber eqal to. ( ) [ ( ) ] R R () m, and we se a similar cell volme 0. These Corant nmbers assert that we do not want either the mass of the Elerian cell or the volme of its agrangian conterpart to decrease b more than 80% in a single time step. We sall find that if the Corant nmber eceeds nit on an time step, even thogh we have taken measres to see that no disasters, sch as sqare roots of negative nmbers, will ense, it is best to have the comptation halve the time step and tr again. This featre is bilt into or PPM code, inclding its massivel parallel versions, and occrs atomaticall, althogh, gratefll, ver rarel. The provisional new cell velocities are now obtained from the momentm conservation laws: () () ( m dµ d ) m µ R / () () with similar eqations for and. Althogh it is optional, we sall choose to reset trivial cell interface and cell average velocities to ero. This prevents the generation of nderflows, and it can allow s in some problems to avoid doing an comptational labor in cells where nothing is happening et. () From these new velocities, we obtain the provisional new cell kinetic energ, E. E kin () () () () ( ) ( ) ( ) ote that at this point we make onl a simple estimate of the cell average of the kinetic energ. It wold make the scheme a great deal more comple and epensive to attempt to determine the new internal cell velocit strctres at this stage of the calclation. That will be done at the otset of the net -D pass, where the total energ in the cell will first be compted according to the above prescription, so that the total energ conservation law, stated below, will be obeed. E () () () ( E m de der )/ m, ε ma{ E } () () min, E Ekin Here we se the smbol ε to represent the internal energ. We now compte the new provisional cell pressre from the gamma-law eqation of state: kin p () () () ( γ ) ρ ε /0/05

18 8 We now have a complete set of provisional new cell averages. These will actall be or final reslts for the overwhelming majorit of the cells. However, near slowl moving strong shocks we still need to appl or smart diffsion. 7. Smart Diffsion and Reapplication of the Conservation aws. The final step of the PPM -D pass is applied onl to those cells with interfaces for which a smart diffsion velocit was compted earlier. There are ver few of these cells, and therefore the comptational cost of this final step is small. However, this step increases the nmerical difference stencil b one cell on either side of or cell of interest. Hence when the scheme is implemented on mltiprocessor or clster sstems, the principal cost of this additional step of the algorithm is the increased qantit of data that mst be commnicated from one processor to another. We reiterate that the smart diffsion is mltidimensional. The detection of shocks does not favor an single dimension, sch as the dimension of the present -D pass, and the diffsion is applied in all directional passes, not jst those in which the shock jmp is seen as sdden. This mltidimensional character of the smart shock dissipation in PPM avoids mltiple nmerical pathologies, and the reqired sie of the difference stencil in the 3 dimensions is therefore a small price to pa for this benefit. We begin b compting diffsive fles at those cell interfaces adjacent to cells in which the diffsion velocit, diff, does not vanish. We also compte a diffsion Corant nmber eqal to doble the advected cell fraction for this diffsion step. diff { } ma,, diff, i diff () diff t diff, i dm ρ, dm t diffr diff ρ () dm diff dm dm, diff diffr dµ diff () () dmdiff dm, i diffr diff diff () dµ dm dm, (), i diffr dµ () () diff dmdiff dm, i diffr de () () diff dmdiff E dm, i diffr E With these fles, for the affected cells onl, we now go throgh the same steps described earlier sing the non-diffsive fles, appling the conservation laws to obtain the new cell averages: ρ, p,,, and. This completes the -pass of the PPM single-flid gas dnamics algorithm. The fll 3-D algorithm consists of si, smmetried -D sweeps, in an seqence and with a constant vale of the time step. 8. PPM Code Performance and Scalabilit on Modern Mltiprocessing Sstems. PPM follows sond wave signals eplicitl, so that all its comptations, save the calclation of the time step vale, are local. The scheme is ver comptationall intensive, becase it epends so mch effort to ensre that the varios special operations sch as monotonicit constraints, contact discontinit steepening, and special treatment of strong shocks are applied onl when and where the shold be. On all CPUs from the Cra- to the modern Intel-based laptop machine or ASCI spercompter, PPM has achieved ecellent performance. This is becase almost all of the arithmetic can be performed in vector mode, and the code has alwas been bilt to perform the entire algorithm for each strip of grid cells at once, thoroghl eploiting modern cache-based memor sstems. On mltiprocessor sstems, each CPU pdates an entire 3-D grid brick, agmented b enogh etra grid cells to make either 3 or 6 -D passes possible withot an frther information. Dring this brick pdate, the reslts of the previos brick pdate are written back to global memor and a new brick of data is fetched. Ths, even when the global memor is implemented on disks, all CPUs are alwas kept bs. The above description has been long and involved, and it ma therefore seem that an nde amont of comptation is reqired b the PPM scheme. Hence it is sefl to qantif jst how mch comptation is actall involved. On a difficlt flow problem involving man strong shocks, on the average each cell pdate for a -D pass involves 9 flops /0/05

19 9 and 49 vectoriable logical operations (which do not cont as flops). This work is performed at a speed of 55 Mflop/s on a.7 GH Intel Pentim-M CPU working from its cache memor in a laptop machine. This laptop performance degrades to 954 Mflop/s for a fll D grid brick pdate, so that fll 8 3 test rns can be performed overnight at a hotel. These speeds are for 3-bit arithmetic, which is all that PPM ever reqires. On a 3. GH Intel Pentim-4 CPU, these 3-bit rates become 607 Mflop/s and 73 Mflop/s, respectivel, while the corresponding rates for 64-bit arithmetic, which is nnecessar bt qoted for comparison prposes with other applications, are 3 Mflop/s and 938 Mflop/s, respectivel. PPM performance has scaled essentiall linearl on ever mltiprocessing machine on which the code has ever been implemented, inclding ASCI machines with over 6000 CPUs. Below, we review some reslts from appling the PPM gas dnamics scheme to compressible trblent flow problems. However, one can download a Windows PC version of the code from the CSE Web site at the Universit of Minnesota and tr it ot for oneself. From links on the main page at one can download -D, -D, and 3-D test versions to pla with that have been sed in teaching and code development at the Universit of Minnesota. User gides are even online for the -D and -D versions. All versions online can perform either single flid or -flid simlations. A librar of compiled PPM code modles, PPMIB, is also available from the CSE Web site, althogh these modles are at present not compiled for the latest machines. These codes, distribted in binar form from the CSE Web site, are designed to rn a variet of test problems that have proven sefl in or code development at the CSE. The -D and -D codes rn, among other things, test problems introdced in the original PPM papers in 980 and 984. The 3-D code rns a 3-D shear laer instabilit problem. Tr them; the re fn. Versions of PPM are incorporated in the commnit codes FASH, EZO, and VH- for the more serios ser. /0/05

20 0 At the left we see the ser interface to the D PPM gas dnamics test program available at the CSE Web site. It is set p to perform b defalt the dal blast wave problem from the earl PPM comparison with other difference schemes in 984. The space-time wave diagram at the bottom right can be enlarged to fill the entire window area b doble clicking on it. Displa in this window of an of a whole list of variables ma be generated. Below at the left is this same application, bt the ser has clicked on the Sond Wave btton to set p a sond wave steepening test problem. For this tpe of problem, the Riemann invariants are shown in the plotting windows, since these variables are the most sensitive indicators of shock-emitted noise, some of which can be seen at ver low amplitde in the lower left-hand plotting window. This tin distrbance is not noticeable at all in the space-time plot. Rns of this tpe were sed to help determine the optimal vales of the dimensionless constants in the PPM smart shock diffsion algorithm described in the tet. The bttons at the bottom of the ser interface form set p a series of test problems from a comparison of gas dnamics schemes carried ot b Brton Wendroff at os Alamos /0/05

21 The ser interface to the -D PPM gas dnamics code is shown here in the process of compting a tpe of wind tnnel test problem originall introdced b Emer in 967. This test code can perform mltiflid variations on this problem, and it can do single-flid comptations that inject smoke streams, set sing the little tet windows at the right of the form, and track them in order to visalie the flow. An of a large nmber of possible displas can be selected sing the list-bo at the bottom center of the form. 9. References.. van eer, B., J. Compt. Phs., 3, 0 (979).. van eer, B., and P. R. Woodward, The MUSC Code for Compressible Flow: Philosoph and Reslts, Proc. of the TICOM Conf., March, van eer, B., and P. R. Woodward, Proc. Internat. Conf. Compt. Meth. onlinear Mech., edited b J. T. Oden (Amsterdam:orth-Holland), p Woodward, P. R., and P. Colella, High-Resoltion Difference Schemes for Compressible Gas Dnamics, ectre otes in Phs. 4, 434 (98). 5. Woodward, P. R., and P. Colella, The merical Simlation of Two-Dimensional Flid Flow with Strong Shocks, J. Compt. Phs. 54, 5-73 (984). 6. Colella, P., and P. R. Woodward, The Piecewise-Parabolic Method (PPM) for Gas Dnamical Simlations, J. Compt. Phs. 54, 74-0 (984). 7. Woodward, P. R., merical Methods for Astrophsicists, in Astrophsical Radiation Hdrodnamics, eds. K.-H. Winkler and M.. orman, Reidel, 986, pp Wenlong Dai and P. R. Woodward, An Approimate Riemann Solver for Ideal Magnetohdrodnamics, Jornal Comptational Phsics, (994). /0/05