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1 SIAM J. NUMER. ANAL. Vol. 46 No. pp c 008 Society for Indstrial and Applied Mathematics Downloaded 01/04/14 to Redistribtion sbject to SIAM license or copyright; see ON STABILITY OF STAGGERED SCHEMES AMY L. BAUER RAPHAËL LOUBÈRE AND BURTON WENDROFF Abstract. This paper investigates the theoretical stability bond of a Lagrangian staggered scheme sed to solve hydrodynamics eqations. We present the two-dimensional D wave eqation as a possible model for this stdy and by sing the nmerical radis of the amplification matrix we prove that the family of schemes defined with two time-centering parameters is limited by a nonclassical stability bond limit defined with an analytical crve. We frther show that D nmerical experiments agree with this theoretical reslt. Key words. staggered schemes stability wave eqations Eler eqations AMS sbject classifications. 65N1 65N 65M1 65J10 65J15 DOI / Introdction. The concept of stability was first introdced in the seminal paper [1] of Corant Friedrichs and Lewy in 198 where they discssed finite-difference methods for solving partial differential eqations. Amazingly in the very same paper they stdied one of the first staggered nmerical schemes for the one-dimensional 1D wave eqation v t = cw x w t = cv x which was frther developed in 1967 by Richtmyer and Morton in [6] and cast in its modern form as: v j v n j Δt = c wn j+1/ wn j 1/ Δx w j 1/ wn j 1/ Δt = c v j v j 1. Δx By considering these works as the genesis for nmerical analysis and comptational flid dynamics one wold expect the stability of staggered nmerical schemes to be at present well nderstood. However this is far from being the case. Althogh staggered schemes have since been widely employed comptationally not mch is known analytically abot the stability of staggered schemes. The se of staggered schemes also dates back to the inception of Los Alamos National Laboratory where the calclation of certain time-dependent flid flows played an important part in the wartime work of the laboratory preface to the first edition of [6]. Staggered schemes have freqently been sed to solve the compressible flid dynamics eqations in their Lagrangian formlation. In this case a staggered spatial placement of variables is sed where the position and velocity are defined at grid points and density internal energy and the pressre are defined at cell centers. Since this time other Lagrangian hydrodynamics staggered nmerical codes have been de- Received by the editors May ; accepted for pblication in revised form October ; pblished electronically March This work was performed nder the aspices of the U.S. Department of Energy at Los Alamos National Laboratory nder contract W-7405-ENG-36. The athors acknowledge the partial spport of the DOE/ASCR Program in the Applied Mathematical Sciences and the Laboratory Directed Research and Development program LDRD. The athors also acknowledge the partial spport of DOE s Accelerated Strategic Compting Initiative ASCI. Theoretical Division T-14 Los Alamos National Laboratory MS-B6 Los Alamos NM albaer@lanl.gov. Institt de Mathematiqes et CNRS UMR-519 Université Pal-Sabatier Tolose Tolose Cedex 9 France lobere@mip.ps-tlse.fr. Theoretical Division T-7 Los Alamos National Laboratory MS-B84 Los Alamos NM bbw@lanl.gov. 996

2 ON STABILITY OF STAGGERED SCHEMES 997 Downloaded 01/04/14 to Redistribtion sbject to SIAM license or copyright; see veloped arond the globe bt again little advancement has been made towards a deeper nderstanding and analytical proof of the stability of sch schemes. Some years ago Shashkov proposed a simple problem for the prpose of testing the stability of Lagrangian hydrodynamics codes. The initial data are given as zero velocity constant pressre and constant density. One then comptes the total kinetic energy which shold be zero for all time. Caramana recently tried this problem with his two-dimensional D staggered grid Lagrangian code and discovered that it was nstable for certain mesh ratios that folklore indicated shold be otherwise the instability manifested itself as an explosive growth of kinetic energy. The srprise here is not that a 1D symmetry-preserving D code cold be nstable for 1D data; the srprise lies in the contradiction of the folklore. The precise nmerical scheme we are referring to and which is the focs of or investigation is the staggered-grid predictor-corrector compatible Lagrangian hydrodynamics scheme. We refer the reader to [3] [4] for the details of this scheme which we omit here. Briefly there are two parameters α and β sch that if α = β =1/ the scheme resembles a time-centered Crank Nicolson scheme while if α = β = 1 it looks like backward Eler. For the Corant Friedrichs Lewy CFL condition CFL = 1 the scheme is stable for α = β =1/. However for α =1β=1/ the scheme is nstable for CFL > 0.71 indicating that as α increases the CFL limit decreases. Baer et al. then created a 1D problem by sing the same data; they seed their 1D staggered-grid predictor-corrector compatible hydrodynamics Lagrangian code with a small random pertrbation of the pressre. Moreover they created the mltidimension version of the problem and ran their D and Caramana s 3D code as well. They observed the same phenomenon in 1D D and 3D [3]. The stability limit was conjectred to be CFL = 1/ αβ and nmerical tests sing the fll nonlinear eqations show that this is tre provided α 1/ and β 1/ [3]. We prove the above conjectre in D which contains 1D as a special case. Of corse there is no possibility of doing this for the fll nonlinear problem the Eler eqations. Ultimately one applies a von Nemann analysis to the linearized system. Baer et al. [3] almost scceeded in doing this theoretical analysis in 1D resorting to nmerical sampling only at the very end of the proof. We have scceeded in D by sing the nmerical radis of the amplification matrix as a tool an idea apparently first applied in []. This proof is the goal of this paper which is articlated as follows: First we present the Lagrangian coordinates and jstify the se of the wave eqation as a possible linear model for the Eler eqations in 1D and D; second we introdce the notation; third we show that the staggered implicit scheme is nconditionally stable for α 1/ β 1/; forth we show that the predictorcorrector staggered scheme is stable for α 1/ β 1/ and CFL 1/ αβ. In a paragraph we show on a specific 1D example that the schemes for α<1/ or β<1/ cannot be stable as some Forier components are amplified. Finally we show D nmerical reslts for the wave and Eler eqations sing a compatible Lagrangian hydrodynamics code.. Lagrangian coordinates and wave eqation models..1. 1D. The hydrodynamic eqations in 1D Lagrangian coordinates expressed in mass variables m are d dt = p m dτ dt = m de = p dt m

3 998 A. L. BAUER R. LOUBÈRE AND B. WENDROFF Downloaded 01/04/14 to Redistribtion sbject to SIAM license or copyright; see where is the velocity τ is the specific volme e is the specific internal energy and p = pτe is the pressre. Total energy E and entropy S are conserved: and de dt = 1 d dt + de dt = p m p m = p m T ds dt = de dt + pdτ dt =0. Therefore for smooth flows the exact differential eqations for p are and taking the coefficients p dp dτ d dt = p m dp dt = dp dτ dτ dt + dp de dp de dt = dτ pdp de m and dp de constant coefficients. Now assme that dp dτ to be constant yields a linear system with dp < 0 and de > 0. As dp c = dτ + pdp de is the sond speed the linearization of the system with constant coefficients is obtained by freezing the sond speed; that is c is assmed to be constant in space and time. Conseqently the reslting system is jst the wave eqation d dt = p m dp = c dt m. By sing the new variables = /c and dm/c = da and dropping the prime notation this linearized system can be written as.1 d dt = p a. dp dt = a... D. The same linearized redction can be performed for Lagrangian coordinates in D bt it is not qite as straightforward. In D the Eler eqation system

4 ON STABILITY OF STAGGERED SCHEMES 999 Downloaded 01/04/14 to Redistribtion sbject to SIAM license or copyright; see is ρ d dt = p x ρ dv dt = p y ρ dτ dt = x + v y ρ de dt = p x + v y dx dt = dy dt = v. By following [7] [8] we make the change of variables: with the choice x = xa b t y = ya b t a = xa b 0 b = ya b 0. The Jacobian matrix of the transformation x y a b is given by x x a J = y a y the determinant of which is the Jacobian J = x y a y x a. For the two velocity eqations we now have Jρ d dt = y p a y p a Jρ dv dt = x p a x p a. Bt then by starting the linearization by freezing the entries of the Jacobian we obtain Jρ d x dt a + v y = J p a a Jρ d x dt + v y = J p. Ths by letting = x a + v y a v = x + v y

5 1000 A. L. BAUER R. LOUBÈRE AND B. WENDROFF Downloaded 01/04/14 to Redistribtion sbject to SIAM license or copyright; see we obtain We next need to evalate x + v y ρ d dt = p a ρ dv dt = p. in the energy eqation. First J = y y a v Jv = x x a v. Then taking the derivative of and v yields J x = y y a y a J v y = x x a x a Smming the previos eqations gives x J x + J v y = + x x a + y a y y a x a y a y v a + v a y a x v a x a v v. x x a + y y a x y + a a and in order to cancel the cross terms we are forced to assme here that.3 x x a + y y a =0 so that x + v y = d 1 a + d v x d 1 = 1 y J + x d = 1 y J +. a a So ρ d dt = p a ρ dv dt = p ρ dτ dt = d 1 a + d v ρ de d dt = p 1 a + d v. v

6 ON STABILITY OF STAGGERED SCHEMES 1001 Now ρ can be scaled into a b and as before a simple sbstittion yields Downloaded 01/04/14 to Redistribtion sbject to SIAM license or copyright; see Finally retrning to v a b set d dt = p a dv dt = p dp dt = d c 1 a + d v. = c d 1 v = c d v a = c d 1 ã b = c d b to obtain the D wave eqation as a model:.4 d dt = p ã dv dt = p b dp dt = ã + v b. However this formlation critically depends on.3 so ltimately we present this as only a potential model Notation. This section defines the notation that will be sed throghot the rest of this paper. The standard inner prodct on a complex vector space of dimension N is f g = N i=1 f i ḡ i for two complex vectors f =f i i=1...n and g =g i i=1...n. Time. We will assme eqal time steps; that is the temporal interval [0T > 0] is discretized into eqal intervals [t n t ] with t = t n +Δt. Space. In D we define niform rectangles with vertices x y where x i+1j = x +Δx and y +1 = y +Δy. Nodal qantities are indexed by i j while cell qantities are indexed by i + 1 j+ 1. Discretization. The ratio of time step to space step in each dimension is given by λ x = Δt Δx λ y = Δt Δy. Hence any qantity P at some point i j at time t n is represented as P n. Similarly any cell-based qantity C at time t is represented as C. i+ 1 j+ 1 For any variable w defined at two time levels t >t n on a point or in a cell we define its interpolated vale at an intermediate time n + κ as: w n+κ = κw +1 κ w n 0 κ 1. We frther define a vector w = v p t and w αβ = n+α v n+α p n+β t where { } = { : i j} v = {v : i j} and p = p i+ 1 j+ 1 : i j. 1 The Jacobian matrix J contains information relating to the volme shape and orientation of an element after the transformation a b x y. Λ = J T J is the associated metric tensor: x λ11 λ Λ = 1 = a + x x x a + y y a λ 1 λ x x a + y y y a a + y Λ is a symmetric matrix and moreover the constraint.3 states that λ 1 = 0: The metric tensor Λ is diagonal meaning that the implied system of coordinates is orthogonal.

7 100 A. L. BAUER R. LOUBÈRE AND B. WENDROFF Downloaded 01/04/14 to Redistribtion sbject to SIAM license or copyright; see Bondary conditions. We assme throghot that all sms exist and that Forier transforms can be taken. Also in the sal way it is spposed that effectively there are no bondaries and therefore that indices can be shifted in a sm withot changing its vale. For example for all integers k p i+ 1 j = i+kj p i+ 1 +kj. i i Operators. There are two sets of data: nodal qantities and cell qantities. Consider the operator Q x that maps cell data to nodal data given by Q x p = 1 p i+ 1 j+ 1 + p i+ 1 j 1 p i 1 j+ 1 p i 1 j 1. By sing the sal scalar prodct in both nodal and cell spaces the adjoint Q x maps nodal data to cell data and is defined by Q x p = Q x p = 1 p i+ 1 j+ 1 + p i+ 1 j 1 p i 1 j+ 1 p i 1 j 1 = 1 pi+ 1 j i+1j i+1j+1 ; that is Q x i+ 1 j+ = i+1j i+1j+1. Similar definitions exist for Q y and Q y. The notation for a staggered scheme on a rectanglar grid for the D wave eqation is depicted in Figre 3.1 where p is defined at the center of each cell and and v are defined at nodes. For this scheme p has fractional indices whereas and v have integer indices. Moreover one defines interpolated vales at each midedge point; for example for a midedge point defined with indices i + 1 j +1 one has p i+ 1 j+1 = 1 p i+ 1 j+ 3 + p i+ 1 j 1 and i+ 1 j+1 = i+1j+1 whereas for a midedge point i j + 1 one has p + 1 = 1 p i+ 1 j+ 1 + p i+ 3 j+ 1 and + 1 = i+1j. By sing this notation a flly implicit staggered scheme applied to system.4 can be written as 3.1 = n + λ x p n+α i+ 1 pn+α j i 1 j v = v n + λ y p n+α p n+α p = p n i+ 1 j+ 1 i+ 1 j+ + λ 1 x n+β n+β i+1j λ y The difference operator is given by the matrix 0 0 Q x M = 0 0 Q y Q x Q y 0 together with Λ = λ x λ y v n+β i+ 1 vn+β. j+1 i+ 1 j

8 ON STABILITY OF STAGGERED SCHEMES 1003 Downloaded 01/04/14 to Redistribtion sbject to SIAM license or copyright; see p i j 1/ i j 1/ v i j 1/ p p p p p i+1/ j+1 i+1/ j+1 v i+1/ j+1 i 1/ j+3/ i 1/ j+1/ i j+1 i j i+1/ j+3/ i+1 j+1 i+1 j p p p i 1/ j 1/ * p i+1/ j+1/ * * * i+1/ j 1/ i+1/ i i+1 i+3/ j+3/ p i+3/ j+1/ i+3/ j 1/ j+1 j+1/ j p i+1 j+1/ i+1 j+1/ v i+1 j+1/ Fig Rectanglar staggered scheme. p is defined at cell centers with fractional indices: p i+ 1 j+ 1 whereas v are defined at nodes with integer indices:. Moreover v p are interpolated at each midedge point. For example p i+ 1 j+1 = 1 p i+ 1 j+ 3 + p i+ 1 j 1 and i+ 1 j+1 = i+1j+1 and similarly for v. Ths the implicit difference scheme 3.1 has the form w = w n + ΛMΛw αβ. Theorem 3.1. The staggered implicit scheme is stable for any λ x λ y if α 1 and β 1. Proof. Throghot the proofs all sms are taken over both indices i and j. Applying the energy method yields Hα β = This follows from the fact that and therefore for real w n n+β + v v n v n+β + p p n i+ 1 j+ 1 i+ 1 j+ 1 M = M p n+α =0. i+ 1 j+ 1 Mw w =0.

9 1004 A. L. BAUER R. LOUBÈRE AND B. WENDROFF Now let s define UβVβPα sch that Hα β =Uβ+V β+p α and Downloaded 01/04/14 to Redistribtion sbject to SIAM license or copyright; see U β = = β V β = v = β v P α = p i+ 1 j+ 1 = α p i+ 1 j+ 1 n n+β 1 β n β 1 v n v n+β 1 β v n β 1 v p n+α i+ 1 j+ 1 p n i+ 1 j+ 1 1 α α 1 p i+ 1 j+ 1 p n i+ 1 j+ 1 p n i+ 1 j+ 1. n v n We detail the steps in the proof by sing U and comment that the steps in the proof for V and P are similarly obtained. Differentiating U with respect to β gives U β = + n n. Then by the Schwarz ineqality U β > 0 U = 0 only in the trivial case that the initial data are constant. Since 1 U = 1 n it follows that for β 1 while for β< 1 U β 1 U β < 1 n n. By applying the same reasoning to V and P we see that for β 1 and α 1 n + v v n + p i+ p n 1 j+ 1 i+ 1 j+ 0; 1 that is we have stability becase + v + p i+ 1 j+ 1 n + v n + p n i+ 1 j+ 1.

10 ON STABILITY OF STAGGERED SCHEMES 1005 Downloaded 01/04/14 to Redistribtion sbject to SIAM license or copyright; see 4. Rectanglar predictor-corrector scheme. The predictor-corrector scheme in D [4] for the wave eqation can now be described. Predictor step. ũ = n + λ x Q x p n ṽ = v n + λ y Q y p n p i+ 1 j+ = p n 1 i+ 1 j+ λ 1 x Q x n+β λ i+ 1 j+ 1 y Q y v n+β ; i+ 1 j+ 1 recall that w n+β = β w +1 βw n for w = v. Corrector step. = n + λ x Qx p n+α v = v n + λ y Qy p n+α p = p n i+ 1 j+ 1 i+ 1 j+ λ 1 x Q x n+β i+ 1 j+ λ 1 y Q y v n+β ; i+ 1 j+ 1 recall that p n+α = α p +1 αp n and now w n+β = βw +1 βw n for w = v. By sbstitting the eqation reslting from the predictor step into the corrector step we obtain the following difference scheme: w = Sw n. Rather than write this ot in terms of the difference operators we immediately move to the Forier transforms of the variables and operators. The Forier transform employs a sbstittion of variables; for instance for p we have p n i+ 1 j+ 1 p 0 e θnδt+iδi+ 1 Δx+γj+ 1 Δy where θ is complex and δ γ are real. After factoring we obtain: Qx p = p 0 e θnδt+iδiδx+γjδy 1 i sinδδx e iγδy + e iγδy = p 0 e θnδt+iδiδx+γjδy i sinδδx cosγδy with i = 1. Then if one denotes ξ = δδx and η = γδy the dimensionless wave nmbers in the x and y directions respectively on an niform mesh the Forier transforms of the operator Q x Q y are 4.1 Qx p = p 0 e θnδt+iiξ+jη i sin ξ cos η 4. Qy p = p 0 e θnδt+iiξ+jη i cos ξ sin η. The same formlas can be similarly obtained for Q. By setting Φ x =λ x sin ξ cos η and Φ y =λ y sin η cos ξ

11 1006 A. L. BAUER R. LOUBÈRE AND B. WENDROFF Downloaded 01/04/14 to Redistribtion sbject to SIAM license or copyright; see and dropping the hat notation we obtain 1 αφ x αφ xφ y iφ x 1 αβφ x +Φ y S = αφ xφ y 1 αφ y iφ y 1 αβ Φ x +Φ y iφ x 1 αβφ x +Φ y iφ y 1 αβφ x +Φ y 1+αβ Φ x +Φ y β. Φ x +Φ y The Lax Richtmyer stability theory [6] tells s that we mst show that the amplification factor S has niformly bonded powers. First observe that SS S Ssoitis not sfficient to show that the eigenvales of S are less than 1. We shall show instead that the nmerical radis of S is bonded by 1 since this implies that the powers of S are niformly bonded by. A srvey of reslts abot the nmerical radis is given in [5] along with a direct proof that if the nmerical radis is bonded by one the powers are bonded by. We remark that the Kreiss matrix theorem implies a weaker reslt namely that the powers are bonded by a constant that depends on the size of the matrix [6]. The nmerical radis was apparently first sed as a stability analysis tool in [] and here we se the basic ideas of that work. We henceforth se the same symbols for grid fnctions and their Forier transforms realizing that this is an abse of the notation bt hoping that no confsion will occr. The nmerical radis of S denoted RS is RS = sp Sw w with w w =1. w The matrix S can be split into real and imaginary parts as where Now let A = B = S = A +ib 1 αφ x αφ x Φ y 0 αφ x Φ y 1 αφ y αβ Φ x +Φ y β Φ x +Φ y 1 αβφ x +Φ y 0 0 Φ x 0 0 Φ y Φ x Φ y 0. r = Aw w j = Bw w. Since A and B are real and symmetric r and j are real. Then Sw w = Aw w +i Bw w = r +ij = r + j. Theorem 4.1. The D staggered rectanglar scheme is stable if α 1 β 1 and 4αβ maxλ xλ y 1. Proof. Sppose that α 1 β 1 and 4αβ maxλ xλ y 1. Let First note that x= αβ Φ x y= αβ Φ y. z = x + y 1

12 ON STABILITY OF STAGGERED SCHEMES 1007 Downloaded 01/04/14 to Redistribtion sbject to SIAM license or copyright; see since z = x + y = αβ Φ x +Φ y = αβ 4λ x sin ξ cos η +4λ y sin η cos ξ 4αβ max λ xλ [ y cos η sin ξ + cos ξ sin η ] }{{} 1 cos η sin ξ + cos ξ sin η 1. Recall that + v + p = 1 and note that 4.3 since 4.4 Also by setting x + yv 1 p x + y x + yv =x + yvx + yv =x + y vv + xyv + v x + y vv + x v + y a=1/α = + v x + y 1 p x + y. b=1/β we have that a b are positive and bonded by. Then r =1 b x + yv a x + y p + a x + y p j = ab R p x + yv 1 x + y where Rq is the real part of q and Now set r = 1 b x + yv az 1 z p j 4ab 1 z p x + yv. By γ 1. We ths have that Also note that for all 0 p 1 x + yv = z 1 p γ. r = 1 γbz 1 p az 1 z p j 4a γb z 1 z p 1 p. p 1 p 1 4. Since γb has the same range as b we can replace γb byb. What we need to show then is that g zpa b [ = 1 bz 1 p az 1 z p ] + abz 1 z 1 for 0 a b and 0 zp 1.

13 1008 A. L. BAUER R. LOUBÈRE AND B. WENDROFF Downloaded 01/04/14 to Redistribtion sbject to SIAM license or copyright; see Then Lemma 4.. The fnction gzpa b 1 for 0 a b and 0 zp 1. Proof: Let q = p t = z. g = [ 1 bt 1 q at 1 t q ] + abt 1 t. We first show that g 1 on the bondary of the t q domain. bondary vales: and since b1 q g t =0=1 g t =1 1. Consider the t For the q bondary vales: [ g q =0 1=bt +bt + a1 t ] [ ] bt +bt + a1 t [ ] bt +t +1 t =0 while g q =1 1=at 1 t [ ] [ ] +at 1 t+b1 t +at + b1 t 0. To finish the proof we need only to show that for each fixed t g a qadratic fnction in q does not have a maximm for 0 <q<1. This is achieved by showing that either g q q =1 0 in which case g is either monotone nondecreasing or has a minimm in 0 <q<1 or that g g q q =0 0 and q q =1 0 in which case g is monotone nonincreasing in 0 <q<1. We proceed as follows: g q =κ[ 1 bt + κq ] where Note that 4.5 κ = bt at + at. 1 bt + κ =1 at1 t 1 a Case a bα β: In this case κ 0 so by 4.5 g q q =1=κ 1 bt + κ κ 1 0. Case a bα β: Then κ 0ift t 0 =1 b a and κ 0ift t 0. If t t 0 k 0. Then as in the case above g q q =1> 0. If t t 0 k 0. Then by 4.5 both g g q q =1< 0 and q q =0< 0. Ths ends the proof of the lemma and completes the proof of the theorem. This case is the most important since for the compatible Lagrangian hydrodynamics scheme applied to the Eler eqations one mst necessarily have β = 1 for the scheme to conserve total energy.

14 ON STABILITY OF STAGGERED SCHEMES 1009 Downloaded 01/04/14 to Redistribtion sbject to SIAM license or copyright; see 5. Instability for α < 1/ or β < 1/. Both the flly implicit and the predictor-corrector schemes are nstable if either α<1/ orβ<1/ for any mesh ratios. We jst need to show this in 1D for the special case of p = for then the schemes are being applied to d dt = x with one set of difference eqations sed to advance at half-integer indexes i +1/ and a different set at integer indexes i. Instability occrs if some Forier component is amplified. So by setting Φ y = 0 in the amplification factor S for the predictor-corrector factor we need only to show first that 1 αφ x +iφ x 1 αβφ x > 1ifα<1/. This is clearly the case if Φ x is positive and sfficiently small since 1 αφ x +iφ x 1 αβφ x 1 =1+Φ x 1 α +Φ 4 x α β }{{} β +Φ x αβ >0 } {{ } oφ 4 x A similar argment works for the second row of S if β < 1/. The flly implicit scheme yields to the same kind of analysis; we omit the details. 6. Nmerical reslts. We solve the simple problem already described in [3] where on a nit domain sing a qadranglar mesh of velocity is set to zero density and sond speed are set to nity by sing a perfect eqation of state with γ the ratio of specific heats eqal to 5/3 and reflective bondary conditions. Since for this problem no velocity shold develop simlations are rn for a very large nmber of time cycles typically 10 5 and with varying vales for the CFL nmber λ = λ x = λ y and parameters α β. A sensitive gage of instability is to keep track of the total kinetic energy K λ t n = 1 [ n +v n ] for a given CFL nmber λ at a given time t n. Since the density and domain size are scaled to nity this nmber shold remain at the sqare of machine precision abot in or case. For an nstable scheme it is observed that K λ t n grows by several orders of magnitde long before the 10 5 cycle limit is reached. In this way one can accrately identify the stability bondary in the CFL nmber and α space for a given β. D wave eqations. We solve the D wave eqations with the rectanglar predictor-corrector scheme. From Theorem 4.1 we have that 4αβ max λ xλ 1 y 1 = λ αβ provided that λ = λ x = λ y. Ths the maximm CFL nmber achievable denoted CFL max therefore depends on α and β and is described by the following fnction: 6.1 CFL max = CFL max α β = 1 αβ. In Figre 6.1 we plot the maximal CFL as a fnction of α for a sample of five β vales: β = { }. The maximm CFL nmbers obtained nmerically with the code for a pair of α β are sperimposed; + for β =0.5 for β =0.6 for β =0.75 for β =0.9 and for β =1.0. As expected they perfectly match the theoretical CFL limit 6.1 given by Theorem 4.1. D Eler eqations. We present the reslts obtained with or D Lagrangian predictor-corrector code for the Eler eqations by sing Lagrangian coordinates. In this context see [3] for example the conservation of total energy is ensred provided that β = 1. In Figre 6. we plot seven data points symbol + obtained by the code which represent the maximm CFL nmber that can be sed for a given α for which.

15 1010 A. L. BAUER R. LOUBÈRE AND B. WENDROFF Downloaded 01/04/14 to Redistribtion sbject to SIAM license or copyright; see CFL Exact β= max CFL = Fig D wave eqations maximm CFL nmber λ as a fnction of α β fixed for the kinetic energy to remain on the order of machine precision for the test case. The vertical dashed line is the α =1/ limit and the horizontal one is the CFL =1limit. The theorem predicts the continos thick lines and the code prodces the data: + for β =0.5 for β =0.6 for β =0.75 for β =0.9 and for β =1.0. Any scheme defined by a vale α 1/ and β 1/ is stable with the following CFL nmber: λ 1/ αβ. CFL Exact D Lagrang code α Fig. 6.. Hydrodynamics eqations maximm CFL nmber λ as a fnction of α β =1/ for the kinetic energy to remain on the order of machine precision for the test case. The theorem predicts the continos line; the D code prodces the dashed line. Any scheme defined by a vale α 1/ is stable with the CFL nmber λ 1/ α and is nstable otherwise. α

16 ON STABILITY OF STAGGERED SCHEMES 1011 Downloaded 01/04/14 to Redistribtion sbject to SIAM license or copyright; see nmerical stability is preserved. The exact fnction CFL dashed line predicted by 6.1 is sperimposed. 3 We can easily see that any scheme defined by a vale α 1/ is stable with the CFL nmber λ 1/ α and is nstable otherwise. We comment that for α =1/ one exactly reaches the maximm CFL nmber λ =1. The scheme with α = β = 1/ is the only staggered predictor-corrector scheme that reaches the maximm CFL nmber. Therefore the choice of sing this scheme in several Lagrangian simlation codes at least for this optimal stability reason seems jstified. The reslts of or simlations perfectly match the predicted crve; this is also tre of or 1D and 3D codes. 7. Conclsion. We have proposed the D wave eqation as a linear constant coefficient model for D Lagrangian hydrodynamics and we have established a sfficient stability condition for the standard staggered-grid nmerical scheme. The 1D case is concrrently obtained as a special case of the D analysis. This stability bondary has been tested with the D compatible Lagrangian hydrodynamics scheme for the wave eqations and the Eler eqations in Lagrangian coordinates. The nmerical reslts fit the theoretical crve showing that the general belief that for sch nmerical schemes the more implicit the more stable is not always tre the classical choice made by several generations of Lagrangian code developers to se the α = β =1/ scheme is reasonable as it leads to optimal stability reslts at least for the artificial test case presented. For the Eler eqations these nmerical reslts have been obtained in 1D D and 3D. Acknowledgment. The athors thank Misha Shashkov for many fritfl discssions. REFERENCES [1] R. Corant K. O. Friedrichs and H. Lewy Uber die partiellen Differenzengleichngen der mathematischen Physik Math. Ann pp [] P. D. Lax and B. Wendroff Difference schemes for hyperbolic eqations with high order of accracy Comm. Pre Appl. Math. XVII 1964 pp [3] A. L. Baer D. E. Brton E. J. Caramana R. Lobère M. J. Shashkov and P. P. Whalen The internal consistency stability and accracy of the discrete compatible formlation of Lagrangian hydrodynamics J. Compt. Phys pp [4] E. J. Caramana D. E. Brton M. J. Shashkov and P. P. Whalen The constrction of compatible hydrodynamics algorithms tilizing conservation of total energy J. Compt. Phys pp [5] M. Goldberg and E. Tadmor On the nmerical radis and its applications Linear Algebra Appl pp [6] R. D. Richtmyer and K. W. Morton Difference Methods for Initial Vale Problems nd ed. Interscience [7] W. H. Hi P. Y. Li and Z. W. Li A nified coordinate system for solving the two-dimensional Eler eqations J. Compt. Phys pp [8] W. H. Hi The nified coordinate system in comptational flid dynamics Commn. Compt. Phys. 007 pp In this case CFL = 1/ αβ =1/ α becase β =1/.