Robust MUSCL Schemes for Ten-Moment Gaussian Closure Equations with Source Terms

Transcription

1 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms Asha Meena, Harsh Kumar To cte ths verson: Asha Meena, Harsh Kumar. Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms. Internatonal Journal on Fnte Volumes, Insttut de Mathématques de Marselle, AMU, 7. <hal-69> HAL Id: hal-69 Submtted on 8 Oct 7 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms Asha Kumar Meena Dept. of Mathematcs, IIT Delh, Inda 6 ashameena@gmal.com Harsh Kumar Dept. of Mathematcs, IIT Delh, Inda 6 hkumar@maths.td.ac.n Abstract In ths artcle, we present postvty preservng second-order numercal schemes to appromate Ten-Moment Gaussan closure equatons wth source terms. The challenge here s to preserve the postvty of the densty and the symmetrc pressure tensor. We propose MUSCL type numercal schemes to overcome these dffcultes. The prncpal components of the proposed schemes are a Strang splttng of the source terms, postvty preservng frst order scheme and sutable lnear reconstructon process whch ensures the postvty of the reconstructed varables. To acheve postvty of reconstructed varables, we mpose the addtonal restrctons on the slopes of the lnear reconstructons. Addtonally, the source s dscretzed usng both eplct and mplct methods. In the case of eplct source dscretzaton, we derve the approprate condton on the tme step for dscretzaton to be postvty preservng. Implct dscretzaton of the source terms s shown to be uncondtonally postvty preservng. Numercal eamples are presented to demonstrate the superor robustness and stablty of the proposed numercal schemes. Key words : Ten-Moment equatons, Fnte Volume Methods, MUSCL Scheme, Postvty preservng schemes.

3 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms Introducton Flud components of the plasma flows are often modeled by Euler equatons of compressble flows. These equatons are derved by takng moments of Boltzmann equaton wth respect to the velocty. The resultng set of equatons s then closed by assumng local thermodynamc equlbrum. However, for many applcatons especally related to the plasma flows, See [3, 4, 6, 7, 9,, 5, 7, 8]), the local thermodynamc equlbrum assumpton s not vald, and one need to take the ansotropc nature of the pressure nto account. To acheve ths, Levermore et al. proposed Ten- Moment equatons model See [3, 4]), whch s derved by the Gaussan closure of the knetc model. Ths results n a hyperbolc system of conservaton laws where the pressure s descrbed usng symmetrc tensor. Due to hyperbolcty of the system, we need to consder the weak solutons, as solutons may develop dscontnutes lke shock and contact waves. However, the class of weak soluton s too large. An addtonal crteron n the form of entropy nequalty s mposed to choose the physcally relevant soluton. Numercal methods to dscretze hyperbolc conservaton laws are often based on fnte volume schemes or fnte dfference schemes) See [5] [8],[],[]). The ntal soluton s dscretzed usng cell averages, whch are then evolved by usng numercal flu over the cell edges. The numercal flu s ether based on the eact soluton of the local Remann problem or some appromaton of t. The hgher-order schemes are desgned by reconstructng the solutons nsde the cells usng ENO, WENO or TVD lmters based procedures. For the tme update, SSP-RK schemes are used. A prmary concern for any numercal scheme for hyperbolc conservaton laws s to be robust,.e., gven the cell averages n the physcally admssble set, we would lke numercal schemes to produce updated cell averages n the same set. It s essental; otherwse, we may lose the hyperbolcty of the system. For the Euler equatons of compressble flows, ths s equvalent of preservng the postvty of the densty and the pressure. There are several works on desgn of postvty preservng schemes for Euler equatons See [],[],[5], [6] and []). For the Ten-Moment equatons, the robustness translates nto ensurng postvty of the densty and the symmetrc pressure tensor unlke Euler equatons, where the pressure s a scalar). In the case of frst-order schemes, ths s based on the constructon of the sutable numercal flu. For Ten-Moment equatons ), a relaaton based scheme s proposed n [3]. In addton to ensurng postvty, the scheme s also shown to be entropy stable. In [7], a Ten-Moment equaton based plasma flow model s consdered, whch s dscretzed usng HLLC numercal flu solver and shown to be postvty preservng. For hgher order schemes, a wave propagaton based dscretzaton s proposed for a plasma flow model based on Ten-Moment equaton n [9]. However, the scheme does not guarantee postvty. More recently, n [4] a Ten-Moment based plasma flow model wth source terms s dscretzed to smulate laser effects on matter. The dscretzaton s based on an equvalent relaaton model, whch also takes source terms nto consderaton. Ths results n a frst-order scheme, whch s shown to be postvty preservng and entropy stable. In ths artcle, we consder the Ten-Moment equatons wth source terms model consdered n [4]. We propose second-order dscretzatons whch ensure the postv- Internatonal Journal on Fnte Volumes

4 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms ty of the densty and the pressure tensor. We acheve ths as follows: For the numercal flu, we use postvty preservng appromated Remann solvers based numercal flues. There are several eamples of them, lke La- Fredrchs, HLLE and HLLC solvers See [7]). To obtan second-order postvty preservng schemes, we follow [] and [], and propose robust MUSCL See []) reconstructon processes. We propose two slope lmters based on the reconstructon of the prmtve varables, namely: Generalzed slope lmter and Conservatve slope lmters. We prescrbe eact condtons for both the cases, for the schemes to be postvty preservng. The source s dscretzed usng both eplct and mplct schemes. In the case of eplct dscretzaton, a condton on tme step s derved, whch ensures postvty of the soluton. The mplct treatment of the source s shown to be uncondtonally postvty preservng. Furthermore, we demonstrate that we do not need to solve any system of algebrac equatons to mplement mplct source update. Both source and flu dscretzatons are then combned usng Strang splttng, whch ensures postvty of the whole scheme. The rest of the artcle s organzed as follows: In the followng Secton, we frst present the HLLC solvers for Ten-Moment model wthout source terms, whch s smplfed from [7]. In Secton 4, we present the MUSCL based reconstructon process. In Secton 5, we present the analyss of the source dscretzaton, followed by tme dscretzaton n Secton 6. Numercal smulatons to demonstrate the superor robustness of the proposed schemes are presented n Secton 7. Ten-Moment Equatons wth Source Terms Followng [4], we consder the Ten-Moment equatons wth source terms whch model nhomogeneous heatng of the electrons n dense plasmas usng lasers. In two dmensons, these equatons can be wrtten as t ρ + ρv) =, a) t ρv) + ρv v + p) = ρ W, b) t E + E + p) v) = ρ W v + v W ). c) 4 Here, ρ s the densty, v = v, v ) s the velocty vector and E s the symmetrc energy tensor wth components E, E and E. The set of equatons s closed by the equaton of state, E = p + ρv v), ) Internatonal Journal on Fnte Volumes 3

5 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms where p s the symmetrc pressure tensor wth components p, p and p. The gven functon W, y, t) s the electron quver energy due to laser lght. The equaton a) represent balance of mass, followed by equaton b) for momentum balance and c) s the balance of energy tensor. The system ) can also be wrtten as, t u + f u) + y f y u) = su), 3) wth conservatve state varable u = {ρ, ρv, ρv, E, E, E }. The flu components are gven by, f u) = and f y u) = ρv ρv + p ρv v + p E + p )v E v + p v + p v ) E v + p v ρv ρv v + p ρv + p E v + p v E v + p v + p v ) E + p )v, 4a). 4b) The source terms can be wrtten component wse as follows: ρ W su) = ρv + ρ yw. 5) W 4 ρv W 4 ρv y W ρv y W For solutons to be physcally meanngful, state varable u needs to be n the conve set of physcally admssble states, { } Ω = u R 6 ρ >, and p >, R /{}, 6).e the densty and the pressure tensor has to be postve. For the soluton n Ω, we have the followng result from [3]: Lemma. The system ) wthout source terms, s hyperbolc for u Ω and admts the egenvalues, 3p.n).n p.n).n v.n, v.n ±, v.n ±, ρ ρ Internatonal Journal on Fnte Volumes 4

6 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms along the untary vector n. The egenvalue v.n has two order of multplcty whle other egenvalues have one order of multplcty. The egenvalue v.n s assocated to a lnearly degenerate feld. The egenvalues v.n ± genunely nonlnear feld whle egenvalues v.n± degenerate feld. p.n).n ρ 3p.n).n ρ In addton, we observe that the source terms are Ω-nvarant,.e.: are assocated to a are assocated to a lnearly Lemma. The soluton of the source ODE du dt = su) are n Ω f the ntal condtons are n Ω. Proof To smplfy, we consder the case of W as a functon of and t only. Dependance on y can be consdered n smlar way. Then we have dρ dt =, 7) for the densty component. For the momentum, usng 7) we get, dρv dt = ρ dv dt = ρ W, whch mples dv dt = W and dv =. 8) dt Consderng energy tensor, we get, So, for p usng 8), d p + ρv ) d = ρv W, dt dt p + ρv v ) = ρv W, d p + ρv ) =, dt dp dt Smlarly, we can also show that, Combnng, = ρv dv dt ρv W = ρv dp dp dt = and dp dt dv dt + ) W =. = and d dt dt p p p ) =. 9) So, the evolutons of densty and the pressure tensor are not affected by source term. Hence source s Ω-nvarant. For the system of Ten-Moments equatons ), followng [3], we ntroduce entropy e and entropy flu q as follows: ) det p sp, ρ) = ln, e = ρs and q = ev. ) ρ 4 =. Then we can deduce the followng entropy stablty result from [4]. Internatonal Journal on Fnte Volumes 5

7 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms Lemma.3 The weak solutons of ) satsfes followng entropy stablty condton: t e + q. ) In addton, the model above can be shown to be rotatonal nvarant See [4]). As we have seen that at the contnuous level, solutons are n Ω, t s mportant to desgn numercal schemes whch are Ω-nvarant. 3 Frst Order Schemes for Homogeneous Model We wll consder the homogeneous equatons n one dmensons to smplfy the dscusson.e. we wll consder the followng equatons: t u + f u) =, ) nstead of 3). The etenson of the numercal schemes proposed here to the two dmensons s straght forward. Let us consder a computatonal mesh, wth ) Z as the cell centres of the cells I =, + ) wth + = + + )/. We wll consder unform mesh for smplcty.e. we assume = + for Z. The cell averages of the state varable u at the tme t n are defned as, w n = u, t n )d. 3) I We wll assume that the cell averages w n) Z belongs to set of physcally admssble states.e. w n Ω for all Z. Our am s to propose numercal schemes whch ensures that the evolved cell average w n+ Ω. We defne t formally as follows: Defnton 3. Ω-nvarance: An update U of the soluton w n ) Ω, for all Z, s called Ω-nvarant or postvty preservng), f updated soluton w n+ = Uw n) s also n Ω. A frst order fnte volume scheme for the dscretzaton of ) can be wrtten n the form, w n+ = w n t F w n, w n +) F w, n w n ) ), 4) where F s the numercal flu, whch s conservatve and consstent wth contnuous flu f. For the frst order schemes, the Ω-nvarance of the scheme depends on the choce of numercal flu. Several numercal flues ests for the -moment equatons, whch are Ω nvarant, e.g. HLLE, Rusanov, HLLC See [7]), relaaton solver See [3]). Here, we wll now present HLLC solvers for Ten-Moment equatons, whch s smplfed from the HLLC solver presented n See [7]) for a more general system. Internatonal Journal on Fnte Volumes 6

8 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms 3. HLLC Numercal Flu for Ten-Moment Equatons The HLLC numercal flu s based on the appromated Remann solver consstng of two ntermedate states, namely, wl and wr.e we are lookng at the solutons of the form: w hllc = w l, f s l t w l, f s l < t v w r, f v < t s r w r, f s r t. 5) Here s l, s r are appromatons of the fastest left and rght wave, respectvely see []) and ntermedate wave speed s v. The correspondng numercal flu s, F hllc = F l, f s l t F l, f s l < t < v F r, f v < t < s r F r, f s r t. 6) Assumng that the epressons for s l and s r s known, we can determne states wl and wr, usng Rankne-Hugonot RH) condtons across the three waves. In addton, followng [7], we also mpose followng condtons across the contacts: v l = v r = v, v l = v r = v, p l = p r = p, p l = p r = p. 7a) 7b) Then we have the epressons for veloctes as, v = p l p r + ρ l v l v l s l ) ρ r v r v r s r ), 8a) ρ l v l s l ) ρ r v r s r ) v = p l p r + ρ l v l v l s l ) ρ r v r v r s r ). 8b) ρ l v l s l ) ρ r v r s r ) Epresson for the denstes are then derved usng RH condton across s l and s r waves and we get, ρ l = ρ lv l s l ) v s l Now the pressures can be derved as follows:, and ρ r = ρ rv r s r ) v s. 9) r p = p l + ρ l v s l v ) ρ l v l s l v l ) a) = p r + ρ rv s r v ) ρ r v r s r v r ), p = p l + ρ l v s l v ) ρ l v l s l v l ) b) = p r + ρ rv s r v ) ρ r v r s r v r ). Now, we can derve epressons for the energy components. For left states we get, E l = v l p l v p + E l v l s l ) v s, a) l Internatonal Journal on Fnte Volumes 7

9 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms E l = v l p l + v l p l v p v p v s l) + E l v l s l ) v s, b) l E l = v l p l v p + E l v l s l ) v s. c) l Smlarly, epresson for E r, E r and E r can also be derved. To prove postvty of the solver we need condtons on s l and s r. We have the followng result from [7]. Proposton 3. The HLLC Remann solver for Ten-Moment equatons s postvty preservng.e. w hllc Ω for {w l, w r } Ω, f the fastest left and rght wave speeds s l and s r, satsfes the followng bounds: where, s l mnv l ĉ l, v l c l ), and s r mav r + ĉ r, v r + c r ), ) pl ĉ l =, c l = p pr l ρl, ĉ r =, and c r = p r ρr. ρ l p l ρ r p r There are several choces of wavespeeds s l and s r are possble See []), whch satsfes these bounds. We follow [7] and consder followng speeds: where s l = mnλ Roe mn, v l c l, v l c l ), s r = maλ Roe ma, v r + c r, v r + c r ), 3pl 3pr c l =, c r =, c l = p l ρl, and c r = p r ρr. ρ l ρ r p l p r 3a) 3b) and λ Roe mn, λroe ma are the left and rght speed for Roe averaged state,.e.: wth, H l ρl + H r c = ρl + ρ r λ Roe mn = v c, and λ Roe ma = v + c, ρr v, v = v l ρl + v r ρr ρl + ρ r, and H l = vl + 3p l, H r = vr + 3p r. ρ l ρ r 4 Second Order Schemes for Homogeneous Model To acheve the second order of accuracy n space, we use MUSCL procedure. Ths s based on the lnear reconstructon usng the cell averages. Consder a scalar functon u and ts cell averages w over the computatonal cells I. A pecewse lnear reconstructon based the cell averages s defned as, p) = w + Dw ), I j, 4) Internatonal Journal on Fnte Volumes 8

10 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms where the slope Dw s defned usng a TVD lmter See [,, 8]). One such commonly used lmter s MnMod lmter gven by, w w Dw = mnmod, w ) + w, 5) where mnmoda, b) = ma, mna, b)) + mn, maa, b)). The traces on the reconstructed functon p) on the edges of the cell I are denoted as follows: w n,± = w n ± w, w = Dw. 6) Ths reconstructon process s performed on prmtve or conservatve varables, component wse. In ths work, we shall restrct ourself to reconstructon of the prmtve varables, as ths s the most commonly used reconstructon process n MUSCL scheme. However, note that ths s not equal to drect reconstructon of the conservatve varables. Let us assume that w n,± s the reconstructed conservatve varable obtaned by reconstructed conservatve varables component wse and convertng them to conservatve varable. The reconstructon process s called conservatve f, w n,+ + w n, = w n, 7) hold. Note that ths s not true for every reconstructon. Followng the reconstructon, a second order scheme can be wrtten as, w n+ = w n t ) F w n,+, w n, + ) F w n,+, wn, ), 8) The scheme s second order accurate n space, however there s no guarantee that t s Ω-nvarant. We wll now descrbe the process whch ensures that. Followng [], we ntroduce the ntermedate state w n,, whch satsfes, w n = αw n, + α)w n, for some α, 3 ]. Then we can rewrte 8) as follows: w n+ = w n t ) F w n,+, w n, + ) F wn,+, wn, ) = αw n, = αw n, = α + α)w n, + α)w n, F w n,, w n,+ w n, t α + α) +α ) F w n, w n, w n,+ t α F w n, + αw n,+ + αw n,+, w n, t + αw n,+, 9) F w n,+ t )) F w n,+ F w n,, w n, ) F w n,+, wn, ) ), w n, + ) F wn,+, wn, ), w n, ) F w n,+, wn, )) ), w n, + ) F wn,, w n,+ )) ) ) t ) ) F w n,, w n,+ ) F w n,, w n, ) α) F w n,+, w n, + ) F wn,, w n,+ )) ). 3) Internatonal Journal on Fnte Volumes 9

11 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms We observe that w n+ s a conve combnaton of frst-order schemes wth tme steps. Ths results n followng See []): t α and t α) Theorem 4. Consder a frst-order Ω-nvarant postvty preservng) scheme under a standard CFL condton wth CFL number C and α, 3 ]. Then the MUSCL scheme 8) s Ω-nvarant wth CFL number αc f the followng condtons hold: a) w n Ω, for all Z. b) w n,± Ω, for all Z. The standard reconstructon process descrbed above doesn t ensure second condton of the Theorem 4.. We wll now present two reconstructon processes whch ensures these condtons. Frst lmter s a general slope lmter and second lmter s conservatve slope lmter. Both are based on the reconstructon of the prmtve varables. 4. General Slope lmtng of Prmtve Varable ) Consder the prmtve varables ŵ = ρ, v, v, p, p, p ). We defne cell edge values ŵ n,±, n cell I as, ρ n,± = ρ n ± ρ, v n,±, = v, n ± v,, v n,±, = v n, ± v,, p n,±, = pn, ± p,, p n,±, = pn, ± p,, p n,±, = pn, ± p,. To ensure the Ω-nvarant of the scheme, we am that the conservatve varable w n,± correspondng to ŵ n,± and the state w are n Ω. Ths leads to the addtonal condtons on the slopes. We proceed as follows: 3) Postvty of ρ n,±, p n,±, and pn,±, can be easly establshed f, ρ < ρ n, p, < p n,, p, < p,. 3) To ensure the postvty of the pressure tensor we need that the determnant of tensors, p n,± s postve. Ths results n, p n, + p, < p n, pn, + pn, p, + p n, p, + p, p,, p n, p, < p n, pn, pn, p, p n, p, + p, p,. We now consder the state w n,. We take α = 3 and observe that See Equaton 9)), ρ n, = ρ n. 34) Internatonal Journal on Fnte Volumes 33)

12 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms So, the densty component of w n, s postve. The pressure components can be wrtten as, ) ) p n,, = ρ pn, + ρ n v,, 35a) Also detp n, ) can be wrtten as, ) ) p n,, = ρ pn, + ρ n ρ n v, v,, ) ) p n,, = ρ pn, + ρ n v,. p n,, pn,, pn,, = pn,p n, p n,) ρ n The postvty of p n, slopes, v, < ρ n and pn, p n, ρ n ρ n ) ) ρ + ρ n 35b) 35c) p n, v, + p n, v, p n, v, v, ). 36) + ρ ρ n I s guaranteed wth the followng condtons on velocty ) ), v, < To ensure the postvty of detp n, ), we assume, v, pn, v, < p n, ρ n p n, pn, pn, ) p n, ρn v, ρ n pn, + ρ ρ n p n, + ρ ρ n ) ). 37) ) )) + ρ ρ n ) ). 38) whch mples, ) )) ) p n p n, v, pn, pn, pn, p ) n, ρn v, + ρ ρ, n p n v, < ) ),39), ρ n pn, + ρ ρ n p n, v, pn, p n, v, ) < ρ n pn, pn, pn ) ) v, + ρ ρ n ρ n, ) ) p n, pn, p n., So, we get, ) ) ρ n ρ p n +, v, + p n, v, p n ), v, v, < p n,p n, p n,, whch ensure the postvty of the detp n, ). We can now state the followng result: Internatonal Journal on Fnte Volumes

13 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms Proposton 4. Assume that the ntal data w n s n the set Ω for all Z. Then, the -reconstructon satsfes the suffcent condtons of the Theorem 4.) wth α = 3 f 3), 33), 37) and 38) hold. To mplement condtons 3), 33), 37) and 38) numercally, we proceed as follows: Frst calculate the slopes of prmtve varables usng MnMod lmters. We can use any other lmter also. Then we update the slopes ρ, p, and p, to satsfy 3), by takng, ρ = ma ρ n, mnρ n, ρ )), p, = ma p n,, mnp n,, p, )), p, = ma p n,, mnp n,, p, )). To ensure 33), we restrct the slopes of p as follows: where, wth 4a) 4b) 4c) p, = mamap l,, p l,), mnmnp u,, p u,), p, )), 4) p, l = p u, = p n,+, pn,+, =, pn,, p l p n,+, pn,+, pn,, p u, = Condton 37) and 38), s satsfed f we consder: v, l = ρ n p n,, pn,, + pn,, p n,, pn,, + pn,. v, = mav l,, mnv u,, v, )), 4) p n, + ρ ρ n ) ), v, u = ρ n p n, + ρ ρ n ) ), and wth v l v, = mamav l,, v l,), mnmnv u,, v u,), v, )), 43), = ρ n p n, + ρ ρ n ) ), v u, = ρ n p n, + ρ ρ n ) ), and v l, = v, p n p n, v u, = v, p n, p n, + p n, pn, pn p n, pn, pn,) ) )) p n, ρn v, + ρ ρ n ) ), ρ n pn, + ρ ρ n,) ) )) p n, ρn v, + ρ ρ n ) ). ρ n pn, + ρ ρ n Internatonal Journal on Fnte Volumes

14 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms 4. Conservatve Slope lmtng of Prmtve Varable ) If the slopes are conservatve we can rewrte Equaton 3) as, w n+ = w n, t F w n, / + w n,+ t F w n,+, w n, + / ) F wn, Ths result n followng result See []):, w n,+ ) F w n,+, wn, ) ) ), w n,+ )) ). Theorem 4.3 Consder a frst-order doman nvarant scheme under a standard CFL condton wth CFL number C, then the MUSCL scheme s Ω-nvarant wth CFL number C under the followng condtons: a) w n Ω for all Z, b) w n,± Ω, for all Z. Let us consder the followng cell edge values of prmtve varables: ρ n,± = ρ n ± ρ, v n,+, = v, n + ρn, ρ v,, v n,, = v, n ρn,+ ρ v,, v n,+, = v, n + ρn, ρ v,, v n,, = v, n ρn,+ ρ v,, p n,± ρ n,+, = pn, ± p, ρn, ρ v, ), p n,± ρ n,+, = pn, ± p, ρn, ρ v, v,, p n,± ρ n,+, = pn, ± p, ρn, ρ v, ). One can easly check that ths reconstructon s conservatve. Now we wll derve the suffcent condtons for Theorem 4.3 by mplementng restrctons on the slopes. For postvty of densty we have, The postvty of p n,±, p, < p n,, Smlarly for p n,±,, we need, p, < p n,, can be acheved f, 44) ρ < ρ n. 45) v, ) < ρn pn, + p,), and v, ) < ρn pn, p,). ρ n, ρ n,+ ρ n, ρ n,+ 46) v, ) < ρn pn, + p,), and v, ) < ρn pn, p,). ρ n, ρ n,+ 47) and fnally for detp n,± ) to be postve, t s suffcent f, pn, p, ρn, ρ n,+ ρ n v, v, < p n,, pn,,, 48a) and pn, + p, ρn, ρ n,+ v, v, < p n,+, pn,+,. 48b) ρ n ρ n, ρ n,+ Internatonal Journal on Fnte Volumes 3

15 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms Now we have the followng result: Proposton 4.4 Assume that the ntal data w n s n the set Ω for all Z. Then, the -reconstructon satsfes the suffcent condtons of the Theorem 4.3) f the condtons 45)-48) holds. For the mplementaton of these condtons we proceeds as follows: Frst calculate the slopes of prmtve varables usng MnMod lmter. Agan we can use any other lmter for reconstructon. We now modfy the slope of densty as ρ = ma ρ n, mn ρ n, ρ )). 49) Condtons on slope of pressure component p s followng: p, = ma p n,, mnp n,, p, )). 5) Smlarly condtons on slope of pressure component p s acheved by: p, = ma p n,, mnp n,, p, )). 5) Condtons on the slope of the velocty v, s the followng: v, = mamav l,, v l,), mnmnv u,, v u,), v, )), 5) wth, v l, = ρ n p n, + p,) ρ n,+ ρ n,, and v u, = ρ n p n, + p,), ρ n,+ ρ n, v l, = Smlarly, for v,, we take, ρ n p n, p,) ρ n,+ ρ n, ρ n, and v, u p n, = p,) ρ n,+ ρ n,, v, = mamav l,, v l,), mnmnv u,, v u,), v, )), 53) wth, v l, = ρ n p n, + p,) ρ n,+ ρ n,, and v u, = ρ n p n, + p,), ρ n,+ ρ n, v l, = ρ n p n, p,) ρ n,+ ρ n, ρ n, and v, u p n, = p,) ρ n,+ ρ n,, Desred condtons on slope of p can now be acheved by takng, p, = mamap l,, p l,), mnmnp u,, p u,), p, )), 54) where, p l, = p n, ρn,+ ) ρ n, v, v, ρ n p n,, pn,,, Internatonal Journal on Fnte Volumes 4

16 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms and p l, = p u, = p u, = p n, + ρn,+ p n, ρn,+ p n, + ρn,+ ) ρ n, v, v, ρ n p n,+, pn,+,, ) ρ n, v, v, ρ n + p n,, pn,, ) ρ n, v, v, ρ n + p n,+, pn,+,., 5 Dscretzaton of the Source Terms Let us consder the source ordnary dfferental equatons, dw dt = sw ), wth su) = ρ W ρv W 4 ρv W. 55) Here, we have consdered the one dmensonal case of the source term. Etenson to hgher dmensons s straght forward. Furthermore, let us assume that the gven ntal data w n Ω. We want to dscretze 55), so that w n+ Ω. We now present eplct and mplct Euler dscretzatons of 55). 5. Eplct Euler Source Update An Euler eplct update of 55) s gven by, Let us denote ths wth w n+ = S e, to densty s zero, we note that, w n+ = w n + tsw n ). 56) t wn ). As the source component correspondng ρ n+ = ρ n. Smlarly, we observe that v, n+ = v, n. From the frst momentum component we get, ρ n+ v, n+ = ρ n v, n t ρn W,, n where W, n = W, t n ) for the gven functon W, t). Ths mples, v n+, = v n, t W n,. Consderng energy tensor component E, we observe that, p n+, + ρn+ v, n+ ) = p n, + ρ n v,) n tρ n v,w n,, n Internatonal Journal on Fnte Volumes 5

17 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms So, fnally we get, whch s postve f, = pn+, pn, ρ n Smlarly, we can show that, = v,) n v, n+ ) tv,w n, n 57) ) t = W n,). p n+, = p n, t 4 ρn W,, t pn, ρ n W, n p n+, = p n, and p n+, = p n,. To show postvty of the tensor we consder, p n+, pn+, whch s postve, f p n+, ) = p n, t 4 ρn W, ). 58) ) p n, p n ), = p n,p n, p n ) t, 4 ρn p n,w,. ) t pn, pn, p n, ρ n pn, W, n. 59) ) Combnng the dscusson above, we have the followng result: Lemma 5. The eplct Euler source update S e, t s Ω nvarant under the tme step restrctons 58) and 59). 5. Implct Euler Source Update We wll now consder mplct dscretzaton of the source terms. A frst order Euler mplct scheme for the source ODE 55) s wrtten by, w n+ = w n + tsw n+ ). 6) Let us denote ths wth w n+ = S, t wn ). Assumng wn Ω, smlar to the case of eplct dscretzaton, ρ n+ = ρ n, v n+, = v n, and p n+, = p n,. Note that we do not have to solve a system of equatons to mplement ths update. We frst update momentum by, ρ n+ v n+, = ρ n v n, t ρn W n+,. Internatonal Journal on Fnte Volumes 6

18 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms whch s then used to update energy tensor. Usng ths, from the momentum equaton, we get, v, n+ = v, n t W, n+. The evoluton of energy component E s, p n+, + ρn+ v, n+ ) = p n, + ρ n v,) n tρ n+ v, n+ W, n+, Rearrangng terms, we get, p n+, pn, ρ n = v,) n v, n+ ) tv, n+ W, n+ ) t ) = W, n+ + tv n, v, n+ )W, n+ = So, fnally we have, t ) W n+, ) + t t p n+, = p n, + t 4 ρn ) W n+, W n+, ). ) = t ) ) W, n+. Hence, p n+, s postve uncondtonally. Smlarly, we can check that p n+, = p n, )., Ths ensures the postvty of p n+, pn+, p n+, uncondtonally. We have the followng result: Lemma 5. The mplct Euler source update S, t s uncondtonally Ω nvarant. 6 Tme Dscretzaton For the tme dscretzaton we use second order SSP-Runge Kutta methods See [9]). Let us denote flu update wth second order SSP-RK scheme wth H t, where each nternal Euler update s gven by 8). Smlarly, let us denote S e, t second order SSP-RK tme update wth each nternal Euler update s gven by 56) and S, t second order SSP-RK tme update wth each nternal update s gven by 6). Usng Strang splttng, we propose two second order schemes: O-ep O-me w n+ = S e, t w n+ = S, t H ts e, t H ts, t w n. 6) w n. 6) The source dscretzaton consst of evaluaton of dervatves W, n n+ and W, of gven functon W. As n all our test cases the functon W s smooth, we calculate the dervatves eactly and then evaluate them at the grd ponts. Internatonal Journal on Fnte Volumes 7

19 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms Both of the above schemes are Ω-nvarant f each nternal eplct tme step satsfes correspondng tme restrcton. However, t s not possble to ensure ths when calculatng the tme step usng w n. So, we reduce t slghtly usng t = t t 5 and check at each nternal step f the correspondng stablty condton s satsfed. If not, then we reduce ntal tme step more and repeat the process. In practce, we note that the soluton wll not change drastcally over a tme step. So, t chosen above s suffcent. 7 Numercal Results In ths secton, we wll present the numercal eperments to ehbt the accuracy and robustness of the proposed algorthms. For the results presented here, we use HLLC flu as the numercal flu. Ths s because HLLC s more accurate less dsspatve) solvers compared to La-Fredrchs, Rusanov and HLLE solvers. Also, the focus of the work here s second-order reconstructon process, not frst order solver. In the followng Secton 7., we present the test cases for the Ten-Moment equatons wthout source terms. Ths s to show the accuracy and robustness of the proposed reconstructon procedure. In Secton 7., we present computatonal results for the complete model, to show the robustness of source dscretzaton. ] 7. Numercal Results for Homogeneous Case 7.. Smooth Solutons: Rate of Convergence To check the formal order of accuracy of the proposed lmters and ther comparson wth the standard MnMod lmters, we have desgned a smooth soluton of Ten Moment equatons ) wthout source terms, n one dmenson case. We consder the doman [.5,.5] wth ntal densty profle of ρ, ) = + snπ). Ths s assumed to be movng wth velocty v =, ). The pressure components are taken to be p = p = and p =. Assumng perodc boundary condtons, the eact soluton s advecton of densty profle n -drecton,.e. ρ, t) = +snπ t)). All other varables remans the same. We have plotted the L -errors of densty n Fgure for conservatve ), general ) and standard MnMod slope lmters usng HLLC solver. The errors are calculated usng, 4, 8, 6, 3, 64 and 8 cells. We have also plotted reference slope for second order convergence for comparson. All the schemes converge wth the second-order of accuracy. Furthermore, errors of the and solutons are same as of standard MnMod lmter. Ths s because the soluton does not contan any low densty or low-pressure area. So, and reduce to standard MnMod based reconstructon. Hence, all the schemes have almost same errors n ths case. 7.. Sod Shock Tube Problem We consder the nterval [.5,.5] to be doman and assume that the ntal dscontnuty s at =.. The ntal condtons for the Sod s shock tube Remann Internatonal Journal on Fnte Volumes 8

20 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms Order of Convergence: L Error plots for densty MnMod Reference Slope L Error of densty No. of cells Fgure : Convergence Rate: L-error of densty for, and MnMod solutons usng HLLC solver. All the schemes acheve second order of accuracy. State ρ v v p p p Left.5.6 Rght.5... Table : Intal Condtons for Sod s Shock Tube Remann Problem problem are two constant states, gven n the Table. Solutons are computed tll tme t =.5 and outflow boundary condtons are used. The eact soluton of the Remann problem conssts of a shock wave and a rarefacton wave separated by contact dscontnuty. So, numercal schemes are tested for the performance on all knds of possble waves n soluton. Numercal results for the problem s presented n Fgures. The soluton s computed usng cells for MnMod, and slope lmter. We agan observe that there s no dfference n the performance of and lmters compare to the standard MnMod lmter. Ths holds for all the state varables and det p See Fgures a)-f)). Also, all the waves are resolved by the three schemes. We note that the addtonal waves are present n v and p components See Fgure c) and e)). The smulaton tmes for, and MnMod schemes were.8838,.8457 and.558 seconds, respectvely Two Shock Waves Remann Problem We consder the same doman as the prevous Remann problem wth Remann problem agan centered at =.. The ntal left and rght states for the Two shock wave Remann problem are gven n Table. We assume outflow boundary condtons and solutons are computed tll tme t =.5. The eact soluton of the problem s two shock waves movng away from each other and separated by a contact dscontnuty. Internatonal Journal on Fnte Volumes 9

21 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms.9 Mnmod Eact.9.8 Mnmod Eact Densty.6.5 v a) Densty b) Velocty n -drecton.5.4 MnMod Eact.8 MnMod Eact v p c) Velocty n y-drecton d) Pressure component: p.65.6 MnMod Eact.4. MnMod Eact.55.5 p.45.4 p p p e) Pressure component: p f) Determnant of pressure tensor Fgure : Sod Shock Tube Problem: Numercal Solutons of MnMod, and lmters wth HLLC flu usng cells at tme t =.5. HLLC solver s used as numercal flu. Internatonal Journal on Fnte Volumes

22 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms.7.6 Mnmod Eact.8 Mnmod Eact Densty v a) Densty b) Velocty n -drecton MnMod Eact MnMod Eact v p c) Velocty n y-drecton d) Pressure component: p 3.5 MnMod Eact 8 7 MnMod Eact 3 6 p.5 p p p e) Pressure component: p f) Determnant of pressure tensor Fgure 3: Two Shock Waves Problem: Numercal Solutons of MnMod, and lmters wth HLLC flu usng cells. HLLC solver s used as numercal flu. Internatonal Journal on Fnte Volumes

23 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms Mnmod Eact Mnmod Eact.5.5 Densty v a) Densty b) Velocty n -drecton MnMod Eact.5 MnMod Eact.5 v p c) Velocty n y-drecton d) Pressure component: p.5.4 MnMod Eact.8 MnMod Eact p p p p e) Pressure component: p f) Determnant of pressure tensor Fgure 4: Two Rarefacton Waves Remann Problem: Numercal Solutons of Mn- Mod, and lmters wth HLLC flu usng cells. HLLC solver s used as numercal flu. Internatonal Journal on Fnte Volumes

24 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms Densty.6 v a) Densty b) Velocy n -drecton p.5 p p p c) Pressure component: p d) Determnant of pressure tensor Fgure 5: Two Rarefacton Waves wth Near Vacuum State: Numercal Solutons of and lmters wth HLLC flu usng cells. HLLC solver s used as numercal flu. State ρ v v p p p Left Rght - - Table : Intal condtons for Two Shock Waves Remann Problem for Homogenous Case Internatonal Journal on Fnte Volumes 3

25 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms ρ v v p p p Left Rght Table 3: Intal condtons for Two Rarefacton Waves Remann Problem ρ v v p p p Left -5 Rght 5 Table 4: Intal condtons for Two Rarefacton Waves Remann Problem: Near Vacuum Test Case The numercal solutons are presented n Fgures 3 whch are computed usng cells. Solutons are compared for MnMod, and lmters. As we do not have any low densty or low pressure areas, all the three schemes produce smlar results and resolve all the waves wth smlar accuracy. Furthermore, addtonal waves are present n v and p components. The computatonal tme of and schemes were.454 and.3775 seconds, respectvely. For the standard MnMod lmter t was.967 seconds Two Rarefacton Waves Problem We now consder the Remann problem for whch soluton conssts of two rarefacton waves separated by a contact. The doman and the pont of the ntal dscontnuty are same as two above Remann problems. The ntal left and rght states are gven n Table 3. The solutons are computed tll tme t =.5 wth outflow boundary condtons. Computed solutons are presented n Fgures 4. They are computed usng cells. Smlar to prevous Remann problems we observe that the performance of the all the three schemes s smlar. We also note that all the fve waves are present n p components. The computatonal tme of the and schemes were.894 and.45 seconds. The MnMod scheme has the computatonal tme of.3458 seconds Two Rarefacton Waves Problem wth Near Vacuum State To demonstrate the superor robustness of the presented schemes, we now consder a Remann problem, for whch the solutons contans low-densty and low-pressure area See [7]). The doman of the Remann problem s the same as n the prevous cases. The ntal states are gven n Table 4. The soluton of the Remann problem contans two Rarefacton waves movng from each other and creatng a low densty, low-pressure area n the center. Internatonal Journal on Fnte Volumes 4

26 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms Computatonal results are presented at tme t =.5 and usng cells n Fgures 5. We note that standard MnMod lmter based scheme s not stable n ths case and breaks down mmedately. So, we are not able to present the results for the scheme. Sgnfcantly though, both and schemes are stable, and both schemes capture low-densty and low-pressure areas. Also, the performance of both and schemes s comparable n ths case. In addton, the computatonal tme of scheme was.4978 seconds whereas took.3367 seconds Two Dmensonal Near Vacuum Test Case In ths Secton, we present a two-dmensonal test case whch contans low densty and pressure areas. The test s the generalzaton of the one-dmensonal case presented n Secton We consder the doman [.5,.5] [.5.5] wth outflow boundary condtons. Ths doman s flled wth flud at constant unt densty wth pressure p =, p = and p =. The velocty s taken to be 5 n, where n s the unt normal at the pont drected outwards. So, the flud s pushed outsde the doman. The numercal solutons are presented n Fgures 6 at tme t =.5 computed usng cells. In Fgures 6a), we have plotted the densty for. We note that, as n the one-dmensonal case, a low-densty area has appeared n the center of the doman and both the lmters capture t. We also note that standard MnMod lmter based scheme fals n ths case. Smlarly, n Fgure 6b), we have plotted detp) for lmters. To compare both lmters more accurately, In Fgures 6c)-6d) we have compared the cuts of two dmensonal plots at y = for densty, pressure components p, p, and detp). We observe that both schemes produce smlar results. The smulaton tme of scheme was seconds whch was slghtly more than the scheme whch took 58.6 seconds. 7. Numercal Results: Non-Homogeneous Case 7.. Two Rarefacton Waves wth Gaussan Source Terms To test the effect of the source terms we consder the one-dmensonal test case from [4]. We consder the doman [, 4] wth ntal dscontnuty at =. The ntal states are gven n Table 5. The soluton wthout source terms conssts of two rarefacton waves leavng behnd a low-densty area n the mddle, smlar to the case of Secton For the one-dmensonal test case wth source terms, we consdered Gaussan profle gven by, W, t) = 5 ep ) ). The numercal results are presented usng 5 cells at fnal tme t =. n Fgure 7). At ths tme wthout source term Homogeneous Case) the low densty area s not completely developed n the mddle. So, proposed lmter and standard MnMod lmter produce smlar results. However, when source terms are added, a near vacuum area has developed around pont =. So, one need a robust lmtng Internatonal Journal on Fnte Volumes 5

27 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms Densty detp y as as y as as a) Densty b) Determnant of Pressure Tensor Densty.5 p p p as c) Densty as d) Determnant of Pressure Tensor Fgure 6: Two Dmensonal Near Vacuum State: Numercal Solutons of and lmters wth HLLC flu usng cells. HLLC solver s used as numercal flu. Internatonal Journal on Fnte Volumes 6

28 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms.4. MnMod Homogeneous O ep O ep.4. MnMod Homogeneous O me O me.8.8 Densty Densty a) Densty b) Densty.4. Homogeneous O ep O me.4. Homogeneous O ep O me.8.8 Densty Densty c) Densty d) Densty Fgure 7: Two Rarefacton Waves wth Gaussan Source Terms: Numercal Solutons of and lmters wth HLLC flu usng 5 cells. HLLC solver s used as numercal flu. Internatonal Journal on Fnte Volumes 7

29 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms ρ v v p p p Left Rght Table 5: Intal condtons for Two Rarefacton Waves wth Gaussan Source Terms Lmter Scheme Ep Ime Ep Ime Tmes) Table 6: Smulaton tme for Two Rarefacton Waves wth Gaussan Source Terms process n addton to postvty preservng source dscretzaton. For ths reason we use and lmters when source terms are consdered. In Fgure 7a)), we have compared standard MnMod lmter for the homogeneous case wth and lmter for the non-homogeneous case usng eplct source dscretzaton O-ep). Here, the soluton usng contans more detals compared wth. Smlar observaton s made for the case mplct source O-me) n Fgure 7b)). In Fgure 7c)) we have compared lmter for the homogeneous case wth n O-ep and O-me. We fnd that there s no vsble dfference n eplct and IMEX scheme. Ths s because the source s not stff and tme step s governed by flu dscretzaton. Smlar observaton can be made for lmter usng Fgure 7d)). The smulaton tmes of lmter were smaller than the lmter for both eplct and IMEX schemes See Table 6). 7.. Unform Plasma State wth Gaussan Source n Two Dmenson To demonstrate two dmensonal effects of the source term we consder a unform plasma state wth ntal condtons gven n Table 7. n the doman [, 3] [, 3] wth outflow boundary condtons. The Gaussan source term consdered s W, y, t) = 5 ep ) + y ) )). Numercal results are presented usng cells at fnal tme t =. n Fgure 8). In Fgure 8a)), we have plotted densty usng CPSV lmter wth O-ep ρ v v p p p Table 7: Intal condtons for Unform Plasma wth Gaussan Source n Two Dmenson Internatonal Journal on Fnte Volumes 8

30 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms Densty Densty O ep O ep O me O me y as as as a) Densty: Lmter b) Densty Fgure 8: Two Dmensonal Near Vacuum State: Numercal Solutons of and lmters wth HLLC flu usng cells. HLLC solver s used as numercal flu. Lmter Scheme Ep Ime Ep Ime Tmes) Table 8: Smulaton tme for Unform Plasma State wth Gaussan Source Terms scheme. we note that source term has created a low densty area at the centre. Furthermore, we observe that the effects are ansotropc. In Fgure 8b)), we have plotted cut along lne y = + 4 for both lmters and ) and both schemes O-ep and O-me). We agan note that soluton for both IMEX and eplct schemes are smlar and lmter provde more detals compared to lmter. On the other hand, the computatonal tme of lmter were slghtly smaller than the lmter See Table 8) Realstc Smulaton n Two Dmensons In ths eample we consder a problem smlar to the Eample 7.4 from [4]. We consder a doman of [, ] [, ] flled wth plasma wth densty.9885, ntally at rest wth pressure p = p = and p =. Ths s ected wth source term only n -drecton wth ) 5 ) ) y 5 W, y) ep We consder outflow boundary condtons. In addton, we also consder addtonal source term v T ρw for the energy part, two smulate nverse bremsstrahlung n Internatonal Journal on Fnte Volumes 9

31 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms. cells v T =. cells v T = cells v T = cells v T = cells 44 v T = cells 44 v T = Densty Densty as as a) Densty b) Densty Fgure 9: Two Dmensonal Realstc Smulaton wth Absorpton Coeffcent v T = : Numercal Solutons of and lmters wth HLLC flu usng, and 4 4 cells along lne y = Densty.98 Densty cells v T = cells v T = cells 44 v T =.97 cells v T = cells v T = cells 44 v T = as as a) Densty b) Densty Fgure : Two Dmensonal Realstc Smulaton wth Absorpton Coeffcent v T = : Numercal Solutons of and lmters wth HLLC flu usng, and 4 4 cells along lne y = 5 Internatonal Journal on Fnte Volumes 3

32 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms. cells 44 v T =.4 cells 44 v T = cells 44 v T = cells 44 v T =. cells 44 v T = cells 44 v T =. cells 44 v T = cells 44 v = T Densty.98 p as as a) Densty b) Densty Fgure : Two Dmensonal Realstc Smulaton Comparson for Absorpton Coeffcent v T = and v T = : Numercal Solutons of and lmters wth HLLC flu usng 4 4 cells. Densty v T = Densty v T = y as.98 y as as as a) Densty b) Densty Fgure : Two Dmensonal Realstc Smulaton wth Absorpton Coeffcent v T = and v T = : Numercal Solutons of lmters wth HLLC flu usng 4 4 cells. Internatonal Journal on Fnte Volumes 3

33 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms plasma. We consder two cases, one wth absorpton coeffcent v T to be and another wth no absorpton.e. v T =. In prevous eamples, we notce no vsble dfference n solutons for eplct and IMEX schemes. So, here we present results for IMEX schemes only. Solutons are evolved tll tme.5. The numercal results are presented n Fgures 9), ), ) and ). In Fgure 9), we gnore the absorpton by settng coeffcent v T =. We plot the densty along the lne y = 5 usng, and 4 4 cells for See Fgure 9a)) and See Fgure 9b)) lmters. In both the cases we observe convergence of the results when we refne the mesh. To observe effect of the absorpton coeffcent we now consder case wth v T =. In Fgure we plot densty and p for, and 4 4 for both and along lne y = 5. We observe that both densty and pressure p have converged. Furthermore, we have compared the results wth v T = and v T = for 4 4 mesh n Fgure ). We observe that densty has decreased sgnfcantly at the centre of laser and pressure p has changed ts shape and now hghest at that pont. In Fgure we have compared densty contours wth 4 4 mesh usng lmter. Agan we observed that densty has decreased n the center when we take v T =.. 8 Concluson In ths work, we have presented postvty preservng second-order MUSCL scheme for Ten-Moment Gaussan closure model wth source terms. Ths s acheved by enforcng sutable restrctons on the slopes of the reconstructed varables. We prove that under the presented restrctons on the slopes, schemes are postvty preservng. Furthermore, we have presented robust treatment of the source terms. We have presented numercal eperments for several test cases and compared the presented schemes wth the standard second-order scheme. We note that the proposed restrctons on the slopes result n comparable results to the standard scheme for the cases wthout low densty and pressure areas. For the cases, where we have low densty or pressure areas, the presented schemes are shown to have superor robustness. Acknowledgement: H. Kumar has been funded n part by SERB, DST grant wth fle no. YSS/5/663. References [] C. Berthon, Stablty of the MUSCL schemes for the Euler equatons, Commun. Math. Sc., 3 ), 33-57, 5. [] C. Berthon, Robustness of MUSCL schemes for D unstructured meshes, J. Comput. Physcs, 8), , 6. [3] C. Berthon, Numercal appromatons of the -moment Gaussan-closure, Mathematcs of Computaton, 75 56), 89-83, 6. Internatonal Journal on Fnte Volumes 3

34 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms [4] C. Berthon, B. Dubroca, A. Sangam, An entropy preservng relaaton scheme for the Ten-Moments equatons wth source terms, Comm. Math. Sc., 3 8), 9-54, 5. [5] F. Bouchut, Nonlnear stablty of fnte volume methods for hyperbolc conservaton laws, and well-balanced schemes for sources, Fronters n Mathematcs Seres, Brkhauser, 4. [6] S. L. Brown, P. L. Roe and C. P. Groth, Numercal Soluton of a -Moment Model for Nonequlbrum Gasdynamcs, th AIAA Computatonal Flud Dynamcs Conference, 995. [7] B. Dubroca, M. Tchong, P. Charrer, V. T. Tkhonchuk and J. P. Morreeuw, Magnetc feld generaton n plasmas due to ansotropc laser heatng, Physcs of Plasmas,, , 4. [8] E. Godlewsk and P. A. Ravart, Numercal Appromaton of Hyperbolc Systems of Conservaton Laws, Appled Mathematcal Scences,8, Sprnger, New York, 996. [9] A. Hakm,Etended MHD Modellng wth Ten-Moment Equatons, J. Fuson Energy 7, 36-43, 8. [] A. Harten, P. D. La, B. Van Leer, On Upstream Dfferencng and Godunov- Type Schemes for Hyperbolc Conservaton Laws, SIAM Revew, 5), 35-6, 983. [] E. A. Johnson, Gaussan-Moment Relaaton Closures for Verfable Numercal Smulaton of Fast Magnetc Reconnecton n Plasma, Ph.D. Thess, UW- Madson,. [] R. J. Leveque, Fnte Volume Methods for Hyperbolc Problems, Cambrdge Tets n Appled Mathematcs, Cambrdge Unversty Press, 3. [3] C. D. Levermore, Moment Closure Herarches for Knetc Theores, Journal of Statstcal Physcs, 83, -65, 996. [4] C. D. Levermore and W. J. Morokoff, The Gaussan Moment Closure for Gas Dynamcs, SIAM Journal of Appled Mathematcs, 59), 7-96, 996. [5] P. Morreeuw, A. Sangam, B. Dubroca, P. Charrer and V. T. Tkhonchuk, Electron temperature ansotropy modelng and ts effect on ansotropy-magnetc feld couplng n an underdense laser heated plasma, Journal de Physque IV, 33, 95-3, 6. [6] B. Perthame, C. W. Shu, On postvty preservng fnte volume schemes for Euler equatons, Numer. Math., 73 ), 9-3, 996. [7] A. Sangam, An HLLC scheme for Ten-Moments appromaton coupled wth magnetc feld, Int. J. Computng Scence and Mathematcs, ),/, 73-9, 8. Internatonal Journal on Fnte Volumes 33

35 Robust MUSCL Schemes for Ten-Moment Gaussan Closure Equatons wth Source Terms [8] A. Sangam, J. P. Morreeuw, V. T. Tkhonchuk, Ansotropc nstablty n a laser heated plasma, Physcs of Plasmas, 4, 53, 7. [9] C. W. Shu, TVD tme dscretzatons, SIAM J. Math. Anal. 4, 73-84, 988. [] E. F. Toro, Remann Solvers and Numercal Methods for Fluds dynamcs, A Practcal Introducton, Thrd Edton, Sprnger, Berln, 9. [] K. Waagan, A postve MUSCL-Hancock scheme for deal magnetohydrodynamcs, Journal of Computatonal Physcs, 8, , 9. [] X. Zhang, C. W. Shu, On postvty preservng hgh order dscontnuous Galerkn schemes for compressble Euler equatons on rectangular meshes, J. Comput. Phys., 9, ,. Internatonal Journal on Fnte Volumes 34