Positivity-preserving time discretizations for production-destruction equations. with applications to non-equilibrium flows.

Transcription

1 Postvty-preservng tme dscretzatons for producton-destructon equatons wth applcatons to non-equlbrum flows Juntao Huang and Ch-Wang Shu Abstract In ths paper, we construct a famly of modfed Patankar Runge-Kutta (MPRK methods, whch s conservatve and uncondtonally postvty-preservng, for producton-destructon equatons, and derve necessary and suffcent condtons to obtan second-order accuracy. Ths ordnary dfferental equaton solver s then extended to solve a class of sem-dscrete schemes for PDEs. Combnng ths tme ntegraton method wth the postvty-preservng fnte dfference weghted essentally non-oscllatory (WENO schemes, we successfully obtan a postvty-preservng WENO scheme for non-equlbrum flows. Varous numercal tests are reported to demonstrate the effectveness of the methods. Keywords: Compressble Euler equatons; postvty-preservng; chemcal reactons; productondestructon equatons; fnte dfference Zhou Pe-Yuan Center for Appled Mathematcs, Tsnghua Unversty, Beng 00084, Chna. E-mal: huangt3@mals.tsnghua.edu.cn Dvson of Appled Mathematcs, Brown Unversty, Provdence, RI 09, USA. E-mal: shu@dam.brown.edu. Research supported by ARO grant W9NF and NSF grant DMS-7940.

2 Introducton We consder the model for non-equlbrum flows wthout conducton or radaton [0], a system of hyperbolc conservaton laws wth source terms U t + F(U x = S(U. (. Here U, F(U and S(U are column vectors wth m = n s + components where n s s the number of speces: U = (ρ,, ρ ns, ρu, ρe 0 T, F(U = (ρ u,, ρ ns u, ρu + p, ρue 0 + up T, S(U = (s,, s ns, 0, 0 T, where ρ s the densty of the -th speces, u s the velocty and e 0 s the total energy per unt mass of mxture. The total densty s defned as ρ = n s = ρ and the pressure p s gven by p = RT n s where R s the unversal gas constant and M s the molar mass of the -th speces. The total energy has the expresson: = ρ M, n s n s ρe 0 = ρ e n, (T + ρ h 0 + ρu, = where e n, (T = C T s the nternal energy of the -th speces wth C = 3R/M and 5R/M for monoatomc speces and datomc speces, respectvely, and the enthalpes h 0 are constants. The source term S(U descrbes the chemcal reactons occurrng n gas flows whch result n changes n the amount of mass of each chemcal speces. We assume that there are J reactons of the form = ν, X + ν, X + + ν n s, X n s ν, X + ν, X + + ν n s, X n s, =,, J,

3 where ν, and ν, are respectvely the stochometrc coeffcents of the reactants and products of the -th speces n the -th reacton. For non-equlbrum chemstry, the rate of producton of speces due to chemcal reacton, may be wrtten as ( J s = M (ν, ν, n s k f, ( ρ n s s ν s, kb, ( ρ s ν s,, =,, n s. M s M s = s= For each reacton, the forward and backward reacton rates, k f, and k b, are known functons of the temperature. Notce that the reactve Euler equatons, whch are often used to model detonaton waves [8], can also be wrtten n ths form (.. We wll turn back to ths ssue n secton 3.3. The man obect of ths work s to develop postvty-preservng schemes for ths class of hyperbolc equatons. A general framework for constructng hgh-order postvty-preservng dscontnuous Galerkn (DG, fnte volume and fnte dfference schemes was proposed by Zhang and Shu n a seres of works [4, 5, 7]. It has been successfully appled to shallow water equatons [, ], convecton-dffuson equatons [8], Naver-Stokes equatons [3] and so on. Ths framework was also generalzed to compressble Euler equatons of gas dynamcs wth several knds of source terms n [6]. We remark that almost all these works rely on the dea of convex combnatons: one frst proves the postvty-preservng property for the frst-order explct Euler forward scheme and then obtan a scheme hgh-order n tme by usng strong-stablty-preservng (SSP Runge-Kutta (RK tme dscretzaton [5]. However, for equatons wth stff source terms, e.g., the non-equlbrum flows (. consdered n ths work, the explct tme ntegraton may suffer from very severe restrcton on the tme step and thus cause large computatonal cost. Recently, there are a few works on postvty-preservng schemes for PDEs wth stff source terms. Chertock et al. proposed a class of sem-mplct RK schemes wth sgn-preservng and well-balanced propertes for a partcular class of ODEs wth stff terms [], and appled t to shallow water equatons wth stff frcton terms [3]. In [8], we constructed a class of second-order postvty-preservng mplct-explct (IMEX RK methods for the system of ODEs arsng from the sem-dscretzaton of the Kerr-Debye model (a specal relaxaton 3 s=

4 system. In [9], we developed a class of exponental SSP hgh order tme ntegratons wth bound-preservng property, and appled t to scalar hyperbolc equatons wth stff source terms. Most recently, Hu et al. proposed a famly of second-order IMEX schemes for the stff BGK knetc equatons [7], whch shares both the asymptotc-preservng and postvtypreservng propertes. For chemcal reactve flows, the chemcal source terms are mostly stff and one usually apples the tme-splttng technques (e.g. Strang s splttng [7]. However, ths approach has several lmtatons. Frst, the splttng technques are generally only up to second-order accuracy. It s far more dffcult to desgn splttng schemes wth hgh-order accuracy. Second, after splttng, the ODEs wth only the chemcal reactons are not easy to solve, due to the stffness, especally takng the postvty of the numercal solutons nto consderaton. In ths work, we frst gnore the convecton terms n (. and only consder a system of ODEs whch descrbes the chemcal reactons, ( dρ J dt = M n s (ν, ν, k f, ( ρ n s s ν s, kb, ( ρ s ν s,, =,, n s. (. M s M s = s= Instead of solvng (. drectly, we move to a larger class of ODEs and concentrate on s= producton-destructon equatons whch have the form: dc dt = P (c D (c, =,,, N, (.3 wth P (c = N p (c, D (c = = N d (c, = and p (c = d (c 0. Here c = (c, c,, c N T and c denotes the concentraton of the -th component. The producton functon p (c denotes the rate at whch the -th component transforms nto the -th component, whle the destructon functon d (c denotes the rate at whch the -th component transforms nto the -th component. The exact solutons to (.3 share 4

5 the conservaton property,.e., N = c (t remans unchanged wth respect to tme t. Also, the postvty of the soluton s guaranteed as long as the ntal condton s postve and d, (c = 0 for c = 0 []. We remark that the chemcal reacton system (. s a specal class of producton-destructon equatons (.3, see e.g. [4]. There have been many works on constructng numercal schemes for (.3 whch uncondtonally preserve the conservaton and postvty of the solutons. The famous Patankar trck [4] s to modfy the explct Euler forward scheme for (.3 nto c n+ = c n + t(p (c n D (c n cn+, (.4 c n to reach the uncondtonal postvty of the numercal soluton. However, ths trck loses the conservaton property and may fal for some stff ODEs. For preservng the conservaton, n [], the authors modfed the classcal Patankar scheme (.4 and obtaned the modfed Patankar scheme c n+ = c n + t ( p (c n cn+ c n d (c n cn+. (.5 c n The postvty-preservng property of ths scheme can be easly shown by wrtng (.5 as a lnear system for c n+ = (c n+, c n+,, c n+ of whch the coeffcent matrx s an M-matrx. N It was also generalzed to a second-order modfed Patankar method [] ( c ( = c n + t p (c n c( d c n (c n c(, (.6a c n ( c n+ = c n + t (p (c n + p (c ( cn+ (d c ( (c n + d (c ( cn+. (.6b c ( Ths tme ntegraton method was then appled to shallow water equatons wth wettng and dryng n [3]. Recently, Kopecz and Mester formulated a class of modfed Patankar Runge-Kutta scheme n a more general form, and obtaned second-order [0] and thrd-order schemes []. The second-order modfed Patankar Runge-Kutta (MPRK scheme reads as [0] c (0 = c n, (.7a 5

6 c ( = c n + ta c n+ = c n + t ( ( p (c (0 c( π (b p (c (0 + b p (c ( cn+ σ d (c (0 c(, (.7b π (b d (c (0 + b d (c ( cn+. σ (.7c wth parameters a = α, b = /(α, b = /(α, and π = c n, σ = c n (c( /c n /α, and α /. Ths class of schemes (.7 works well for the producton-destructon equatons (.3. However, t would fal, f we would lke to solve the chemcal reacton flow (. wthout usng the tme-splttng approach, as the convecton terms n (. cannot be coupled nto the tme dscretzaton n (.7. Ths s the startng pont of our work. Instead of usng the RK schemes n the classcal form, we apply the RK schemes of the Shu-Osher form [6] and develop another class of modfed Patankar Runge-Kutta scheme for (.3. In order to cover the chemcal reactve flows, the convecton term s then ncluded n the ODEs solver and the accuracy s analyzed. Combnng these tme ntegraton methods wth the sem-dscrete fnte dfference weghted essentally non-oscllatory (WENO schemes [], we successfully obtan the postvty-preservng WENO schemes for (.. The paper s organzed as follows. In secton, we formulate another class of modfed Patankar RK scheme (MPRK wth the Shu-Osher form, and then derve the necessary and suffcent condtons for ths ODE solver to be second-order accurate. The solver s also generalzed to solve sem-dscrete schemes for PDEs and the accuracy s analyzed. In secton 3, we combne the tme ntegraton method wth the postvty-preservng fnte dfference WENO schemes, and dscuss the postvty-preservng property by usng the lmter. Numercal results for the ODEs and PDEs are presented n secton 4. Some concludng remarks are gven n secton 5. 6

7 The ODE solver In ths part, we frst formulate a class of modfed Patankar RK scheme wth the Shu-Osher form. Followng the procedures n [0], the necessary and suffcent condtons for ths ODE solver to be second-order accurate are derved. Afterwards, the solver s generalzed to solve sem-dscrete schemes for PDEs and the accuracy s analyzed. The explct SSP RK scheme for (.3 wrtten n the Shu-Osher form [6] reads as: c (0 = c n, (.8a c ( = α 0 c (0 + tβ 0 ( P (c (0 D (c (0, (.8b c n+ = α 0 c (0 ( + tβ 0 P (c (0 D (c (0 + α c ( ( + tβ P (c ( D (c (. (.8c For (.8 to be second-order accurate, the condtons on the coeffcents are α 0 =, α 0 + α =, β 0 + β + α β 0 =, β 0 β =. (.9 Followng [0], for preservng conservaton and postvty uncondtonally, we modfy the producton and destructon terms n (.8 and obtan the modfed Patankar RK scheme (MPRK c (0 = c n, ( c ( = α 0 c (0 + tβ 0 p (c (0 c( π (.0a d (c (0 c(, (.0b π c n+ = α 0 c (0 + α c ( ( + t (β 0 p (c (0 + β p (c ( cn+ σ (β 0 d (c (0 + β d (c ( cn+. σ (.0c Here π, σ 0 are functons of c (0 and c ( and wll be determned later. Actually, for avodng solvng nonlnear equatons n each stage of (.0, t s requred that π and σ are ndependent of (c (, c n+ and c n+, respectvely. Furthermore, snce (.0 s only a modfcaton of (.8, t s natural to assume that the constrans (.9 on the 7

8 coeffcents also hold n (.0, and π and σ are some approxmatons of c n,.e., π = c n + O( t, σ = c n + O( t. For smplcty, we choose π = c n. Remark.. Although the RK methods n the classcal form and the Shu-Osher form are equvalent, they are dfferent after modfcaton (see (.7 and (.0. The scheme (.0 reduces to (.7 by takng α = 0.. Necessary condton In ths part, we wll derve the necessary condton for (.0 to be of second-order. We consder a specfc class of ODEs: In (.3, we take, for fxed I, J {,,, N} and I J, µc I, = J, = I, p (c = 0, otherwse, and µc I, = I, = J, d (c = 0, otherwse, wth constant µ > 0. In the followng, we wll assume that the MPRK scheme (.0 s of second-order, and apply t to ths specfc class of ODEs. By Taylor expanson, the exact soluton has the form Takng dervatve on (.3 yelds c (t n+ = c (t n + dcn dt t + d c n dt t + O( t 3. d c dt = d dt (P (c D (c = (P (c D (c dc c dt = (P (c D (c (P(c D(c, c Here P(c := (P (c, P (c,, P N (c T and D(c := (D (c, D (c,, D N (c T. Hence, we have c (t n+ = c n + t(p n D n + t (P n D n (P n D n + O( t 3. c 8

9 Snce the solver s second-order accurate, we obtan ( cn+ α 0 c n + α c ( + t (β 0 p n + β p ( (β 0 d n σ + β d ( cn+ σ = c n + t(p n D n + t (P n D n (P n D n + O( t 3, c and subsequently α (c ( c n + t ( t(p n D n t cn+ σ cn+ σ (β 0 p n + β p ( (β 0 d n + β d ( (P n D n (P n D n = O( t 3, c by the order condton α 0 + α =. Set = I, and wrte I as, α (c ( c n t(β 0 D n + β D ( cn+ σ + td n t D n c Dn = O( t 3. (. Note that (. holds for any =,, N, snce I s chosen arbtrary. Usng (.0b, we have c ( = c n tβ 0 D n c ( π = c n tβ 0 µc n c ( π. Substtute t nto the left hand sde of (.: α β 0 tµc n = α β 0 tµc n = µ tc n ( c ( t(β 0 µc n + β µc ( π ( c ( t π cn+ σ β 0 µc n + β µ(c n tβ 0µc n c ( α β 0 + (β 0 + β cn+ π σ + tµc n t µ c n, c ( c n+ σ + µ t c n π ( + tµc n t µ c n, c n+, π σ where we have used the order condton β β 0 =. Now we have ( ( µ tc n c ( α β 0 + (β 0 + β cn+ + π σ µ t c n c ( c n+ = O( t 3. π σ Snce the constant µ > 0 s arbtrary, we reach the followng two constrants wth the ad of c ( Lemma 3. n [0]: c ( α β 0 + (β 0 + β cn+ = O( t, (. π σ 9

10 and c ( c n+ = O( t. (.3 π σ The second constran (.3 s automatcally fulflled snce π = c n and σ = c n + O( t. Substtute π = c n nto the frst stage (.0b: ( c ( = α 0 c n + tβ c ( 0 p n c n ( and thus = c n + tβ 0 d n c n p n + O( t c n = c n + tβ 0 (P n D n + O( t. c ( π = c( c n = + tβ 0 P n D n c n c (, c n d n c n + O( t, c n + O( t. The frst constran (. s smplfed: ( P n D n α β 0 + tβ 0 + (β c n 0 + β cn+ = O( t, σ α β 0 t P n D n c n + (β 0 + β cn+ = β 0 + β + O( t, σ and we solve out: Thus, the expresson of σ s derved: c n+ = σ α β0 t P n D n β 0 + β c n + O( t. σ = cn+, c n+ σ = We fnally reach the relaton c n+ α β 0 β 0 +β t P n Dn c n + O( t, =(c n + t(p n D n ( + α β0 t P n D n + O( t, β 0 + β c n =c n + ( + α β 0 β 0 + β t(p n D n + O( t. σ = c n + ( + α β 0 β 0 + β t(p n D n + O( t. (.4 0

11 . Suffcent condton In ths part, we prove that the necessary condton (.4 s also suffcent. Frst, we do expanson for c ( n the frst stage (.0b c ( =c n + tβ 0( c ( p n c n d n =c n + tβ 0 (P n D n + O( t. c (, c n Iterate once, c ( =c n + tβ 0 ( c n p n + tβ 0(P n Dn c n =c n + tβ 0(P n D n + t β 0 ( P p n n Dn c n d n c n + tβ 0(P n D n + O( t 3, c n d n P n D n + O( t 3. c n By Taylor expanson for φ ( := φ(c ( wth φ = p or d, we have and thus, φ ( = φ(c ( = φ(c n + φn c (c( c n + O( t, (.5 β 0 φ n + β φ ( =(β 0 + β φ n φ n + β c (c( c n + O( t =(β 0 + β φ n φ n + tβ β 0 c (P n D n + O( t. Subsequently, we have expanson for c n+ : c n+ Iterate once, =α 0 c n + α (c n + tβ 0(P n D n + t(β 0 + β (P n D n + O( t, =(α 0 + α c n + (α β 0 + β 0 + β t(p n D n + O( t, =c n + t(p n D n + O( t. c n+ σ = c n + t(p n D n + O( t c n + ( + α β0 β 0 +β t(p n D n + O( t, = t α β0 P n D n β 0 + β c n + O( t.

12 and thus, (β 0 p (c (0 + β p (c ( cn+ (β 0 d (c (0 + β d (c ( cn+ σ σ = ( (β 0 + β p n + tβ p n ( β 0 c (P n D n t α β P 0 n Dn β 0 + β c n ( (β 0 + β d n + tβ d n ( β 0 c (P n D n t α β0 P n D n β 0 + β c n =(β 0 + β (P n D n (P n D n + tβ β 0 (P n D n ( c P tα β0 p n n Dn d n P n D n c n + O( t. c n + O( t, Substtutng the above relaton nto the fnal stage (.0c yelds: ( c n+ = α 0 c (0 + α c n + tβ 0(P n D n + t β0 ( P p n n Dn c n + t((β 0 + β (P n D n + tβ (P n D n β 0 (P n D n c tα β0 ( P p n n D n d n P n D n c n + O( t 3, c n = c n + t(p n D n + t Now the second-order accuracy has been proved. (P n D n (P n D n + O( t 3. c d n P n D n c n.3 MPRK schemes We have shown that the relaton (.4 s a necessary and suffcent condton for (.0 to be second-order accurate. In the followng, we wll derve an explct expresson of σ for (.4 to be satsfed. We try to take σ = (c ( s (c n s (.6 wth s a constant to be undetermned. Thanks to the expanson of c (, we have σ = (c n + tβ 0(P n D n + O( t s (c n s, = c n ( + tβ 0 P n D n c n + O( t s,

13 = c n ( + tβ 0 s P n D n + O( t, c n = c n + tβ 0 s(p n D n + O( t. To satsfy the constran (.4, t s only requred that Now we have a famly of schemes s = + αβ 0 β 0 +β = β 0 + β + α β0. (.7 β 0 β 0 (β 0 + β c (0 = c n, ( c ( = α 0 c (0 + tβ 0 p (c (0 c( c n (.8a d (c (0 c(, (.8b c n c n+ = α 0 c (0 + α c ( ( + t (β 0 p (c (0 + β p (c ( c n+ (c ( s (c (0 s wth the coeffcents satsfyng the condtons (.9 and s = β 0+β +α β 0 β 0 (β 0 +β. (β 0 d (c (0 + β d (c ( c n+ (c ( s (c (0 (.8c Note that there are four constrants n (.9 and sx parameters (α 0, α 0, α, β 0, β 0, β. Hence, there are two free parameters. Set α = α, β 0 = β, and express other parameters n terms of α and β by (.9: and subsequently, α 0 =, α 0 = α, α = α, β 0 = β, β 0 = β αβ, β = β. s = αβ + αβ. β( αβ For guaranteeng the non-negatvty of these parameters (α 0, α 0, α, β 0, β 0, β, t s requred that 0 α, β > 0, αβ + β. We summarze the above results n the followng theorem: 3 s.

14 Theorem. (MPRK schemes. Wth the parameters satsfyng and α 0 =, α 0 = α, α = α, β 0 = β, β 0 = β αβ, β = β, αβ + αβ s =, β( αβ 0 α, β > 0, αβ + β, the MPRK scheme (.8 s second-order accurate. Moreover, t s conservatve: N c n = = N = c ( = N = c n+, and uncondtonally postvty-preservng: f c n 0 for =,,, N, then c n+ =,,, N and all t > 0. 0 for Remark.. Our scheme (.8 are generalzatons of the schemes n [, 0]. By takng α = 0 n (.8, t reduces to the scheme n [0]. If we further set β =, t reduces to the scheme n []. Remark.3. If we take α = and β =, the coeffcents of the optmal SSP RK method are recovered: and accordngly s =. α 0 = β 0 =, α 0 = α = β =, β 0 = 0, Remark.4. The form of σ n (.8 s not the only possble choce. Followng [0], we can also take a convex combnaton of (c ( s (c n s : σ = λ(c ( s (c n s + ( λ(c ( s (c n s. wth 0 λ. To satsfy the condton (.4, the relatons of parameters λ, s, s can be derved n the same approach. For smplcty, we do not nvestgate ths ssue n detal and wll only focus on (.8 n the followng. 4

15 .4 Extenson to sem-dscrete schemes To cover the sem-dscrete scheme for the PDEs, we formulate a system of ODEs n the followng form: dc k, dt = F k, (c + P k, (c D k, (c, k =,, M, =,, N. (.9 Here c k, = c k, (t denotes the concentraton of the -th speces at the k-th grd pont, N and M denote the number of speces and nodes, respectvely. The vector c s the collecton of all unknown varables wth the form c := (c, c,, c N,, c M, c M,, c MN T and s of length M N. F k, = F k, (c denotes the contrbutons of the convecton terms after spatal dscretzatons n the PDEs. The producton and destructon terms are P k, = P k, (c = N = p k,,(c and D k, = D k, (c = N = d k,,(c whch satsfy p k,, (c = d k,, (c,,, k and c 0. We make the followng assumpton on (.9: Assumpton.. The Euler forward method for the convecton term satsfes the postvtypreservng property: f c n k, 0 for all k,, then c n k, + tf k,(c n 0, for all k, and t t 0. Remark.5. Ths famly of ODEs (.9 covers the sem-dscrete fnte dfference WENO scheme for chemcal reactng flows (. by settng the producton and destructon terms to be zero n the momentum and energy equatons. The varable c k, does not necessarly denote the concentraton or densty n ths case. We generalze the MPRK scheme (.8 by ncorporatng the convecton term F k, usng the Euler forward dscretzatons n each stage and arrve at c (0 k, = cn k,, (.0a 5

16 c ( k, = α 0c (0 k, + tβ 0F k, (c (0 + tβ 0 ( p k,, (c (0 c( k, c (0 k, c n+ k, = α 0 c (0 k, + α c ( k, + t(β 0F k, (c (0 + β F k, (c ( + t( (β 0 p k,, (c (0 + β p k,, (c ( (β 0 d k,, (c (0 + β d k,, (c ( c n+ k, (c ( k, s (c (0 k, s c n+ k, (c ( k, s (c (0 k, s. d k,, (c (0 c( k,, (.0b c (0 k, (.0c where the parameters are the same wth those n Theorem., Obvously, the scheme (.0 satsfes the postvty-preservng property f the tme step satsfes for c ( k, t mn{ α 0 β 0, α 0 β 0, α β } t 0. Next, we show that the scheme (.0 s second-order accurate. Frst, we do expanson n the frst stage (.0b: c ( k, = cn k, + tβ 0F n k, + tβ 0( c ( p n k, k,, c n k, c ( d n k, k,, c n k,, = c n k, + tβ 0(F n k, + P n k, Dn k, + O( t. Iterate once, c ( k, = cn k, + tβ 0Fk, n + tβ 0 ( c n p n k, + tβ 0(Fk, n + P k, n Dn k, k,, c n k, d n k,, c n k, + tβ 0(Fk, n + P k, n Dn k, + O( t 3, c n k, = c n k, + tβ 0 (F n k, + P n k, D n k, + t β 0( + O( t 3. F p n k, n + P k, n Dn k, k,, c n k, d n k,, Fk, n + P k, n Dn k, c n k, By dong expanson for c n+ up to O( t n the second stage (.0c, we have c n+ k, = α 0 c n k, + α (c n k, + tβ 0(F n k, + P n k, Dn k, + t(β 0 + β (F n k, + P n k, Dn k, + O( t, = c n + t(f n k, + P n k, Dn k, + O( t, and then c n+ k, (c ( k, s (c (0 k, s = cn k, + t(f n k, + P n k, Dn k, + O( t (c n k, + tβ 0(F n k, + P n k, Dn k, + O( t s (c n k, s, 6

17 = cn k, + t(f n k, + P n k, Dn k, + O( t c n k, + tsβ 0(F n k, + P n k, Dn k, + O( t, = c n k, + t(f k, n + P k, n Dn k, + O( t c n k, + ( + α β0 β 0 +β t(fk, n + P k, n Dn k, + O( t, = t α β0 Fk, n + P k, n Dn k, + O( t. β 0 + β c n k, Defne the vector ψ := (ψ, ψ,, ψ N,, ψ M, ψ M,, ψ MN for ψ = F, P, D. Expanson for φ ( = φ(c ( wth φ = p k,,, d k,, or F k, : β 0 φ n + β φ ( = β 0 φ n + β (φ n + φn c (c( c n + O( t = (β 0 + β φ n φ n + β c (c( c n + O( t = (β 0 + β φ n φ n + tβ β 0 c (F n + P n D n + O( t. Substtutng the above relaton nto the man part of the rght hand sde of the second stage (.0c yelds (β 0 p k,, (c (0 + β p k,, (c ( cn+ k, (β 0 d k,, (c (0 + β d k,, (c ( cn+ k, σ k, σ k, = ( p n ( (β 0 + β p n k,, k,, + tβ β 0 c (F n + P n D n t α β0 ( (β 0 + β d n k,, + tβ d n ( k,, β 0 (F n + P n D n t α β0 c + O( t, β 0 + β F n k, + P n k, Dn k, c n k, =(β 0 + β (Pk, n Dn k, + tβ (Pk, n β Dn k, 0 (F n + P n D n ( c F tα β0 p n k, n + P k, n Dn k, k,, F d n k, n + P k, n Dn k, c n k,, + O( t. k, c n k, Fnally, we smplfy the second stage (.0c: β 0 + β F n k, + P n k, Dn k, c n k, c n+ k, = α 0 c (0 k, + α (c n k, + tβ 0(Fk, n + P k, n Dn k, + t β 0 ( p n k,, F n k, + P n k, Dn k, c n k, d n k,, Fk, n + P k, n Dn k, c n k, + t((β 0 + β F n k, + β β 0 F n k, c (F n + P n D n 7

18 (P + t((β 0 + β (Pk, n Dk, n k, n + tβ β Dn k, 0 (F n + P n D n c tα β0 ( F p n k, n + P k, n Dn k, k,, F d n k, n + P k, n Dn k, c n + O( t 3, k, c n k, = c n + t(fk, n + Pk, n Dk, n + t (Fk, n + P k, n Dn k, (F n + P n D n + O( t 3. c The second-order accuracy has been proved. We have the followng results: Theorem.. The MPRK scheme (.0 s second-order accurate for the system of ODEs (.9. It s conservatve: k, k, c ( k, = k, = α 0 c n+ k, c n k, + tβ 0 F k, (c n, k, c n k, + α k, k, c ( k, + t (β 0 F k, (c n + β F k, (c (, k, where the summaton of the convecton terms wll vansh f perodc boundares are mposed. It s postvty-preservng: f c n k, t mn{ α 0 β 0, α 0 β 0, α β } t 0. 0 for all k,, then c( k, 0 and cn+ k, 0 for all k, and Remark.6. Note that the condton for the MPRK scheme (.0 to be postvty-preservng does not depend on the producton and destructon terms P k, and D k,, and only rely on the explct SSP RK solver for the convecton part. Ths s hghly desrable for solvng Euler equatons wth stff chemcal reacton sources. The CFL condtons for postvty-preservng schemes developed n [6, 8] all depend on the stffness of the source terms. 3 Postvty-preservng fnte dfference WENO schemes In ths secton, we frst revew the postvty-preservng fnte dfference WENO scheme [7]. Then we apply our ODE solver (.0 to reactve Euler equatons [8] and Euler equatons wth three speces reactons [0] and dscuss the postvty-preservng propertes. 8

19 3. The fnte dfference WENO scheme for hyperbolc conservaton laws We frst revew the fnte dfference WENO scheme for hyperbolc conservaton laws wthout source terms [5] U t + F(U x = 0, (3. where U s the unknown varable vector and F(U s the physcal flux. Consder a unform mesh wth node x. Defne x + = (x +x +, x = x + x and I = [x, x + ]. Denote the pont values at x and tme level n by W n. The fnte dfference scheme wth hgh order spatal dscretzaton and Euler forward tme dscretzaton solvng (3. has the form where x ( ˆF + W n+ = W n t x ( ˆF + ˆF, (3. ˆF should be a hgh order approxmaton to F(W x at x = x. If there exsts a functon H(x dependng on the mesh sze x such that F(x = x x+ x/ x x/ H(ξdξ, the we call F and H a reconstructon par and denote them by H = R x (F, F = R x (H. Then F x = x x (H(x+ H(x. Thus, f ˆF x + of H(x +, then ( ˆF x + We use the global Lax-Fredrchs splttng, s a hgh order accurate approxmaton ˆF wll be a hgh order approxmaton of F(W x at x = x. F ± (W = W ± F(W α, Here α s the maxmum egenvalue of the Jacoban matrx F(U, and the maxmum s taken U over all the grd ponts at tme level n. For Euler equatons of gas dynamcs, α = max( u +c where u and c are the velocty and the speed of sound. For clarty, the notaton ± on the subscrpts denotes the postve and negatve parts of the splttng flux, and that on the 9

20 superscrpts denotes dfferent stencls n the WENO reconstructons for cell I and I +. We also ntroduce the notaton L + of F(U U takng value at W + and R + and satsfy L + W and W +. For smplcty, we take W + scheme are formulated as follows: At each fxed x +, whch denote the left and rght egenvector matrx = R. Here W + + s some average of = (W + W +. The fnte dfference WENO Denote H ± = R x (F ±, then we have the cell averages ( H ± n = F ± (W n. Transform all the cell averages ( H ± n for n a neghborhood of to the local characterstc feld by settng ( V ± n = L + ( H ± n. Perform the WENO reconstructon for each component of ( V + n to obtan approxmatons of the pont value of the functon L + H + at the pont x + on the left sde and denote them as (V +. Also, perform the WENO reconstructon for each component + of ( V n to obtan approxmatons of the pont value of the functon L + H at the pont x + on the rght sde and denote them as (V +. + Transform back nto physcal space by (H + + = R + (V +, (H = R + (V +. + Form the flux by ˆF + = α ((H + + (H + +. Fnally, we get the conservatve scheme W n+ = W n λ( ˆF + ˆF, (3.3 wth λ = t/ x. 0

21 3. Postvty-preservng fnte dfference WENO schemes In ths part, we brefly revew the postvty-preservng fnte dfference WENO schemes n [7]. For (., we defne the set of admssble states G: G = {U = (ρ,, ρ ns, m, E T ρ > 0, =,, n s, p > 0}. (3.4 Defne ẽ := n s = ρ e n, (T. Then ẽ as a functon of U s convex. The convexty of G s obtaned by notcng G = {U ρ > 0, =,, n s, ẽ > 0}. See also Lemma. n [6] for detals. Next, we present the suffcent condton n [7] for the scheme to keep W n+ W n G. G provded W n+ = W n λ( ˆF + ˆF, = (( H + n + ( H n λα ((H + + (H + + (H + + (H +, := T + + T, where T + = ( H + n λα((h + + T = ( H n + λα((h + + (H +, (H +. Then, the cell averages H + and H are splt nto a convex combnaton of values at quadrature ponts and the followng theorem can be proved: Theorem 3. (Zhang and Shu [7]. Under the CFL condton αλ ŵ, f q +,, (H + +, (H +, q,, (H +, (H + + G, then the fnte dfference scheme wll be postvtypreservng,.e., W n+ q +, = G, where (( ŵ H + n ŵ N (H +, q, + = N (( ŵ H n ŵ (H +. (3.5 ŵ = ŵ N are the quadrature weghts for the two end ponts n Gauss-Lobatto quadrature rules.

22 Note that ŵ = ŵ N = /6 for the thrd-order WENO schemes and / for the ffth-order WENO schemes. To enforce the suffcent condtons n Theorem 3., the postvty-preservng lmter s appled. Here, we slghtly modfy the smple and robust lmter n [8] snce the pressure s no longer a convex functon of W n our model. Let ( H + n = (( ρ,, ( ρ ns, m, Ē T, (H + = ((ρ +,, (ρ + ns, m, E T, q +, = ((ρ,, (ρ n s, m, E. The lmter s presented as follows: Set up small parameters ǫ s = mn {0 3, ( ρ s } for s =,, n s. For each cell I, modfy the denstes: for s =,, n s, ( ˆρ s = θ + ((ρ s ( ρ + s + ( ρ s, θ = mn{ ( ρ s ǫ s ( ρ s (ρ s mn, }, (3.6 where (ρ s mn := mn{(ρ s, (ρ + s }. Then denote (Ĥ+ + and ˆq +, := ŵ N (( H + n ŵ N(Ĥ +. Then we modfy ẽ. For convenence, let q = (Ĥ+ + ẽ(q m 0, set t m = ; otherwse, set Then modfy ( H + + t m = := (( ˆρ,, ( ρˆ + ns, m, E T and q = ˆq +,. For m =,, f ẽ(( H + n ẽ(( H + n ẽ(qm. (3.7 = θ (( ˆ H + + ( H + n + ( H + n, θ = mn{t, t }. (3.8 Smlarly, we get the revsed pont value ( H +. Then we have the modfed WENO scheme wth the numercal flux replaced by ˆF + = α(( H + + ( H + +. (3.9 It s straghtforward to extend the postvty-preservng fnte dfference scheme for onedmenson to mult-space dmensons. The CFL condton for preservng the postvty for D case s replaced by t( a x + a y ŵ, (3.30

23 where a = max{ u +c} and a = max{ v +c}, wth u, v are veloctes n x and y drectons, c the speed of sound. For the tme ntegraton, the SSP hgh order RK tme dscretzaton [5] wll keep the valdty of Theorem 3. snce G s convex. 3.3 Reactve Euler equatons We consder the reactve Euler equatons whch are often used to model the detonaton waves [8] n D case: U t + F(U x + G(U y = S(U, (3.3 wth U = (ρ, m, n, E, ρy, F(U = (m, ρu + p, ρuv, (E + pu, ρuy, G(U = (n, ρuv, ρv + p, (E + pu, ρvy, S(U = (0, 0, 0, 0, ω, and m = ρu, n = ρv, E = ρ(u + v + p + ρqy. (3.3 γ Here q > 0 s the heat release of reacton, γ s the specfc heat rato and 0 Y denotes the reactant mass fracton. The source term s assumed to be n an Arrhenus form ω = KρY e T/T, (3.33 where T = p/ρ s the temperature, T > 0 s the actvaton constant temperature and K > 0 s a constant. To ft nto our framework, we rewrte (3.3 n an equvalent form. The unknown varables and the correspondng physcal fluxes and source terms are replaced by U = (ρy, ρz, m, n, E, F(U = (ρuy, ρuz, ρu + p, ρuv, (E + pu, 3

24 G(U = (ρvy, ρvz, ρuv, ρv + p, (E + pu, S(U = (ω, ω, 0, 0, 0, where Z denotes the unreacted mass fracton. The admssble set G s defned as G = {U = (ρy, ρz, m, n, E ρy 0, ρz 0, p > 0}. The sem-dscrete fnte dfference WENO scheme for the equvalent PDEs reads as dw, dt = x ( ˆF +, ˆF, y (Ĝ,+ Ĝ, + S(W,, (3.34 wth W, := (W,,, W,,, W,,3, W,,4, W,,5 = ((ρy,, (ρz,, m,, n,, E,. Ths system of ODEs can be wrtten n the form (.9 by splttng the source terms nto S (W = P (W D (W, (3.35 wth the producton and destructon terms p, (W = d, (W = ω 0, (3.36 and p m,n (W, d m,n (W vansh for other set of m, n 5. Then we apply our ODE solver (.0 and obtan ( W ( t, =α 0 W, n β 0 x ( ˆF n n t ˆF +,, + y (Ĝn,+ Ĝn, W n+,,, W,, n + tβ 0 (ω nw( =α 0 W, n + α W (,,, W,, n, ω nw( ( t β 0 x ( ˆF n n t ˆF +,, + y (Ĝn,+ Ĝn, ( t ( ( t β ( ˆF ˆF x +,, + y (Ĝ(,+ Ĝ(, + t((β 0 ω n + β ω (, 0, 0, 0 T, (3.37a W n+,, (W (,, s (W n,, s, (β 0ω n + β ω ( W n+,, (W (,, s (W n,, s, 0, 0, 0T. (3.37b 4

25 By ntroducng auxlary varables V (, and V, n+, we rewrte (3.37 nto an equvalent form V (, ( t =α 0 W, n β 0 x ( ˆF n n t ˆF +,, + y (Ĝn,+ Ĝn, (3.38a, W (, =V ( V n+,,, W,, n, + tβ 0 (ω nw(,, W,, n, ω nw( ( =α 0 W, n + α W ( t, β 0 x ( ˆF n n t ˆF +,, + W n+, =V n+ ( t ( ( t β ( ˆF ˆF x +,, + y (Ĝ(,+ Ĝ(,, + t((β 0 ω n + β ω (, 0, 0, 0 T, (3.38b W n+,, (W (,, s (W n y (Ĝn,+ Ĝn,, (3.38c,, s, (β 0ω n + β ω ( W n+,, (W (,, s (W n,, s, 0, 0, 0T. (3.38d At tme level n, gven W n, G for all,, then the postvty of denstes and pressure of V (, n the frst stage (3.38a s guaranteed by the lmter n secton 3.. In the second stage (3.38b, the postvty of denstes of W (, s trval. In addton, note that the densty ρy of W (, s no greater than that of V ( (,,.e., W,, V (,,, due to the fact that ωn 0. Then the postvty of pressure of W (, s guaranteed by notcng that p = (γ (E m qρy. ρ In the thrd and the fourth stage, the postvty of denstes and pressure of V, n+ and W, n+ can be preserved by the same argument. 3.4 Euler equatons wth three speces reactons We consder the three speces model wth a more general equaton of state n [0] U = (ρ, ρ, ρ 3, ρu, E T, F(U = (ρ u, ρ u, ρ 3 u, ρu + p, (E + pu T, S(U = (M ω, M ω, 0, 0, 0. and ρ = 3 ρ s, s= p = RT 3 s= ρ s M s, E = 3 ρ s e s (T + ρ h 0 + ρu. (3.39 s= 5

26 the nternal energy e s (T = 3RT/M s and 5RT/M s for monoatomc and datomc speces respectvely. The rate of chemcal reacton s gven by ω = ( k f (T ρ k b (T( ρ 3 ρ s (3.40 M M M s k f = CT e E/T, k b = k f / exp(b + b log z + b 3 z + b 4 z + b 5 z 3, z = 0000/T (3.4 where b, C and E are constants whch can be found n [6]. The sem-dscrete fnte dfference WENO scheme for the equvalent PDEs reads as s= dw dt = x ( ˆF + ˆF + S(W, (3.4 wth W := (W,, W,, W,3, W,4, W,5 = ((ρ, (ρ, (ρ 3, m, E. For ths model, we splt the source terms M ω nto two parts: M ω = ω + ω, wth the postve part and the negatve part ω + = M k f (T ρ M ω = M k b (T( ρ M 3 s= 3 s= ρ s M s 0, ρ s M s 0. Ths system of ODEs can be wrtten n the form (.9 by splttng the source terms nto S (W = P (W D (W, (3.43 wth the producton and destructon terms p, (W = d, (W = ω 0, p, (W = d, (W = ω + 0, (3.44 and p m,n (W, d m,n (W vansh for other set of m, n 5. Then we apply our ODE solver (.0 and obtan W ( =α 0 W n t β 0 x ( ˆF n ˆF n + 6

27 W n+ W ( + tβ 0 (ω+ n, ω n W, n W (,, ω n W, n + W (, W n, =α 0 W n + α W ( t β 0 x ( ˆF n ˆF n + + t((β 0 ω n + + β ω ( (β 0 ω n + + β ω ( + + W n+, + ω n W (,, 0, 0, 0 T, (3.45a W, n t ( β ( ˆF x + (W (, s (W n, s (β 0ω n + β ω ( W n+, (W (, s (W n, s + (β 0ω n + β ω ( ( ˆF W n+, (W (, s (W n, s, W n+, (W (, s (W n, s, 0, 0, 0T. (3.45b For ths model, our scheme (3.45 could only preserve the postvty of the nternal energy (E m, but could not guarantee the postvty of the pressure. Actually, for ths chemcal ρ reactng flow, the pressure s p = ρ M + ρ M + ρ 3 M 3 3ρ M + 5ρ M + 5ρ 3 (E ρ h 0 m, (3.46 ρ M 3 whch may become negatve due to the exstence of ρ h 0. We also remark that ths dffculty seems to be essental, snce even the frst-order splttng wth the exact evoluton n tme for the ODEs part does not necessarly preserve the postvty of the pressure. In our numercal experments, f there exsts negatve pressure n the calculaton, we ust put absolute values n the pressure when calculatng the sound of speed. Moreover, we do not enforce the postvty of pressure n the lmter n secton 3.. To conclude ths secton, we gve several remarks below: Remark 3.. Here we use the fnte dfference as the spatal dscretzaton and not the fnte volume and DG schemes. The reason s that, for the fnte volume and DG schemes, the sem-dscrete scheme n general could not preserve the orgnal form of the source. Remark 3.. We also remark that our ODE solver can be appled to convecton-dffusonreacton equatons by treatng the convecton and dffuson terms explctly n the tme ntegraton, as long as the reacton terms can be wrtten n the producton-destructon form. 7

28 4 Numercal tests In ths part, the numercal results wll be presented. We frst dscuss the convergence order for the non-stff problems and the performance on the stff problems n solvng the ODEs. Then we move to reactve Euler equatons and Euler equatons wth three speces reactons and general equaton of state. Snce the second-order SSP RK method s lnearly unstable when coupled wth the ffth-order WENO spatal dscretzaton [9], we adopt the thrd-order fnte dfference WENO scheme proposed n []. 4. ODEs Example 4. (lnear case. The lnear test case s wth constant a > 0, and ntal value The exact soluton s dc dt = c ac, dc dt = ac c, c (0 = c 0, c (0 = c 0. (4.47 c (t = ( + b exp( (a + tc, c (t = c 0 + c0 c (t, (4.48 wth the parameters c and b determned by c = c0 + c0 a +, b = c0 c. (4.49 In the numercal experment, we take c 0 = 4.5, c 0 = 3., a =.7 and the fnal tme t =. The errors between numercal solutons and exact solutons at the fnal tme are lsted n Table 4.. The second-order accuracy s clearly observed. Example 4. (nonlnear case. To test the accuracy of our solve (.0, we make up a non-stff nonlnear problem: dc dt = F (c c c c +, 8

29 dc dt = F (c + c c c + ac, dc 3 dt = F 3(c + ac. where (F (c, F (c, F 3 (c denotes convecton terms. To express ths system of ODEs n the form of producton-destructon equatons, we set p (c = d (c = c c c +, p 3(c = d 3 (c = ac, (4.50 and p = d = 0 for other sets of,. In the numercal examples, the ntal condtons are set as c 0 = 9.98, c = 0.0 and c 3 = 0.0. The convecton terms are (F (c, F (c, F 3 (c = (c c c 3, c 3, c c c 3. (4.5 c The fnal tme s t = and the parameter a =. The errors are lsted n Table 4., whch shows second-order accuracy. Example 4.3 (stff case. One of the most promnent examples of the stff ODEs s the Robertson test case, whch descrbes the chemcal reactons [6]: dc dt = 04 c c c, Table 4.: Example 4.: Error table for lnear ODEs. t error order /0.0e-03 - / e / e /60.96e /30 4.9e Table 4.: Example 4.: Error table for nonlnear ODEs. t error order /0.35e-03 - /40 3.8e / e /60.89e / e

30 c exa c num 0 4 c exa 0 4 c num c3 exa c3 num t Fgure 4.: Example 4.3: Tme evoluton of c, =,, 3. wth ntal values c (0 = c 0 for =,, 3. dc dt = 4 0 c 0 4 c c c, dc 3 dt = 3 07 c, For ths problem, the producton and destructon terms are p (c = d (c = 0 4 c c 3, p (c = d (c = 4 0 c, p 3 (c = d 3 (c = c, (4.5 and p = d = 0 for other sets of,. In the numercal smulatons, we take c 0 = and c 0 = c 0 3 = 0. Followng [0], the tme step sze n the k-th tme step s chosen as t k = k t wth the ntal tme step sze t = 0 6. The small ntal tme step was set to obtan an adequate resoluton of the component c n the startng tme nterval. To vsualze the evoluton of c, t s multpled by 0 4 n Fg. 4.. From Fg. 4., we observe the excellent accuracy of our scheme n the case of a hghly stff problem. 30

31 y 6 y x x y 5 y x x (a (b Fgure 4.: Example 4.4: Detonaton dffracton problems. Top: colored contour map and contour lne of densty; bottom: colored contour map and contour lne of pressure. 4. Reactve Euler equatons Example 4.4 (Detonaton dffracton problems. We test the detonaton dffracton n ths example. The same parameters and ntal condtons wth [8] are appled here. The ntal condtons are, f x < 0.5, then (ρ, u, v, E, Y = (, 6.8, 0, 970, ; otherwse, (ρ, u, v, E, Y = (, 0, 0, 55,. The boundary condtons are reflectve except that at x = 0, (ρ, u, v, E, Y = (, 6.8, 0, 970,. The termnal tme s t = 0.6. The parameters are γ =., q = 50, T = 50 and K = The numercal results wth x = y = /48 are shown n Fgure 4., whch are comparable to the results n [8]. Example 4.5 (Multple obstacles. Followng Example 4.6 n [8], we desgn a numercal test wth multple obstacles. The locatons of the obstacles are dfferent from those n [8]. In 3

32 y x y y x y x x (a (b Fgure 4.3: Example 4.5: Multple obstacles. Top: colored contour map and contour lne of densty; bottom: colored contour map and contour lne of pressure. our test, the locaton of the frst obstacle s [, 3] [0, 3] and the second one s [5, 0] [0, 5]. The unform mesh can be easly appled. The ntal condton s, f x + y 0.36, then (ρ, u, v, E, Y = (7, 0, 0, 00, 0; otherwse, (ρ, u, v, E, Y = (, 0, 0, 55,. The boundary condtons are reflectve for any boundares. The parameters are set as γ =., q = 50, T = 0, K = 40.. The colored contour map and the contour lne of the densty and pressure wth the mesh sze x = y = /0 are presented n Fgure Euler equatons wth three speces reactons and general equaton of state Example 4.6. For the three speces model of the one-dmensonal Euler system wth a more general equaton of state n [0, 7]. We take the same parameters as n [0, 7]. The 3

33 parameters are M = 0.06, M = 0.03, M 3 = 0.08, h 0 = , R = , C 0 = m 3, E 0 = K, and b =.855, b = 0.988, b 3 = 6.8, b 4 = 0.03, b 5 = The egenvalues of the Jacoban are (u, u, u, u c, u + c where c = wth γ = + T P 3 p γp ρ s= ρse s (T. The ntal condtons are: the denstes ρ, ρ and ρ 3 are , , on the left, and , , on the rght. The veloctes are zero. The pressures are 000 on the left and on the rght. The fnal tme s t = The profles of densty, velocty and pressure are presented n Fgure. 4.4, where the converged solutons are observed. 5 Concludng remarks In ths paper, we have developed a class of tme ntegraton methods for the productondestructon equatons, whch s conservatve and uncondtonally postvty-preservng. The necessary and suffcent condtons for the methods to be second-order accurate are derved. Ths ODE solver s then extended to cover a class of sem-dscrete schemes for PDEs and successfully appled to fnte dfference WENO schemes for non-equlbrum flows. We have tested the thrd-order WENO scheme wth the postvty lmter coupled wth the tme ntegraton method on a varety of numercal examples. Generalzatons to thrd-order schemes consttute our ongong work. Acknowledgements We would lke to thank Xangxong Zhang from Purdue Unversty and Tao Xong from Xamen Unversty for many frutful dscussons. References [] H. Burchard, E. Deleersnder, and A. Mester. A hgh-order conservatve Patankar-type dscretsaton for stff systems of producton destructon equatons. Appled Numercal 33

34 ρ (num ρ (exa ρ (num ρ (exa ρ3 (num ρ3 (exa x x u (num u (exa p (num p (exa x Fgure 4.4: Example 4.6: Three speces reacton problem at t = The sold lnes are the reference solutons wth x = /8000. Symbols are the numercal solutons wth x = /

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