FRANCIS FILBET, CLAUDIA NEGULESCU AND CHANG YANG

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1 NUMERICAL STUDY OF A NONLINEAR HEAT EQUATION FOR PLASMA PHYSICS FRANCIS FILBET, CLAUDIA NEGULESCU AND CHANG YANG Abstract. Ths paper s devoted to the numercal approxmaton of a nonlnear parabolc balance equaton, whch descrbes the heat evoluton of a magnetcally confned plasma n the edge regon of a tokamak. The nonlnearty mples some numercal dffcultes, n partcular for the long tme behavor approxmaton, when solved wth standard methods. An effcent numercal scheme s presented n ths paper, based on a combnaton of a drectonal splttng scheme and the IMEX scheme ntroduced n [6]. Keywords. Nonlnear heat equaton, IMEX scheme, fnte volume method. Introducton The descrpton and smulaton of the transport of charged partcles confned n a tokamak s nowadays one of the man problems for fuson generated energy producton. A tokamak s a devce usng a magnetc feld to confne a plasma n the shape of a torus [] as shown n Fgure -(a), where the magnetc feld lnes move around the torus n a helcal shape. In the plasma tokamak, parallel drecton s along the helcal feld lne, whle the radal drecton or perpendcular drecton s perpendcular to the magnetc surfaces. The crcle orthogonal to the magnetc feld lnes s stratfed nto core and edge layers. In ths paper, we are nterested n magnetcally confned fuson plasmas n the edge regon called scrape off layer (SOL) of a tokamak (see Fgure -(a)). The understandng of the physcs n ths edge regon s fundamental for the performances of the tokamak, n partcular the plasma-wall nteractons as well as the occurrng turbulence have an mportant mpact on the confnement propertes of the plasma. For example, an obstacle as a lmter s nstalled n SOL to cool down the devce [7]. From a modelng pont of vew, an accurate approxmaton of the plasma evoluton n the edge regon s essental snce energy fluxes as well as partcle fluxes at the boundary are used as boundary condtons to descrbe the plasma evoluton n the center regon (core) of the tokamak. The physcal propertes of these two regons (core/edge) are rather dfferent, so that several models are used for the respectve plasma-evoluton modelng: the gyroknetc approach for the collsonless core-plasma and the flud approach for the collsonal edge-plasma. A large varety of models can be found n the lterature, we refer to [7, 8] for the descrpton of the SOL, based on varous assumptons and amed to descrbe dfferent physcal phenomena. We shall concentrate n ths paper on a smplfed model, ntroduced n [7]. The am of ths model s the nvestgaton of the nstabltes occurrng n ths plasma edge regon, as for example the Kevn-Helmholtz nstablty, the electron-temperature-gradent (ETG), ontemperature-gradent nstabltes (ITG), etc. Ths model s based on a two-flud descrpton (ons, electrons) and conssts of the usual contnuty equaton, equaton of moton and energy F. Flbet s partally supported by the European Research Councl ERC Startng Grant 9, project NuSKMo, C. Negulescu s partally supported by the ANR project ESPOIR.

2 FRANCIS FILBET, CLAUDIA NEGULESCU AND CHANG YANG Polodal drecton Torodal drecton Axs of symmetry Radal drecton r= r=/ Lmter radal drecton r Wall Scrape off layer Separatrx Lmter Core SOL perodc BC Transton layer perodc BC Magnetc feld lne (a) Lmter Torodal angle Wall r= s= CORE (b) parallel drecton s s= Fgure : (a) The Tokamak devce and ts structure n polodal secton, (b) the D computatonal doman. balance equaton, closed by the so-called Bragnsk closure. These equatons are t n α + (n α u α ) = S nα, (.) m α n α [ t u α + (u α )u α ] = p α + n α e α (E + u α B) + R α, 3 n α [ t T α + (u α )T α ] + p α u α = q α + Q α, where the unknowns are the partcle densty n α where α = e for electrons and α = for ons, u α the velocty and T α the temperature, the parameters m α s the partcle mass, e α s the partcle charge wth e e = for electrons and e = for ons. Moreover, the pressure s p α := n α T α (perfect gas assumpton), S nα s a partcle source term comng from the core plasma, R α the frcton force due to collsons, q α s the energy flux, whch s supposed to have a dffusve form q α := κ α T α comng from the Fourer law wth κ α the thermal conductvty coeffcent. Fnally, Q α s the partcle exchange energy term, due to collsons Q α := ±3 m e m n α τ e (T e T ), where τ e s the electron-on collson tme and (E,B) represents the electro-magnetc felds. Fnally, boundary condtons have to be mposed, whch s a rather delcate task from a physcal, mathematcal and numercal pont of vew. Several dffcultes arse when tryng to solve numercally the system (.). We shall concentrate n ths paper only on the temperature equaton, whch requres at the moment stll a lot of effort, due to ts nherent numercal burden. The resoluton of the two other equatons was the am of the PhD thess [7]. The numercal dffcultes n solvng the temperature equaton are frstly related to the thermal conductvty coeffcents, whch depend on the temperature tself, leadng thus to a nonlnear problem. Secondly, the strong magnetc feld whch confnes the tokamak plasma ntroduces a sharp ansotropy nto the problem. Indeed, the charged partcles gyrate around the magnetc feld lnes, movng thus freely along the feld lnes, but ther dynamcs n the perpendcular drectons s rather restrcted. Quanttes as for example the resstvty or the conductvty, dffer thus n several orders of magntude when regarded n the parallel or perpendcular drectons. Let us now present n more detals the model we are nterested n. In ths paper, we shall study a smplfed verson of the temperature evoluton equaton n (.), and propose an effcent numercal method to solve t. We shall focus on how to handle wth the nonlnear terms and the boundary condtons, the hgh ansotropy beng the am of a forthcomng work [, 5].

3 NUMERICAL STUDY OF A NONLINEAR HEAT EQUATION FOR PLASMAS 3 Here we assume that the magnetc feld to confne the plasma s strong enough such that all quanttes are nvarant wth respect to the polodal drecton, that s the short way around the torus, thus by followng the helcal feld we reduce the energy equaton of (.) to two-dmensonal problem. The smulaton doman = (, ) (, ) wth boundary s presented n Fgure -(b). It conssts of a perodc core regon, separated by a Separatrx from the non-perodc SOL regon. Its axes represent the drecton parallel to the magnetc feld lnes (s) and the radal drecton (r). The parallel thermal conductvtes κ,α depend on Tα 5/ whereas the perpendcular ones κ,α, governed by the turbulence, are ndependent of the temperature []. Therefore, by neglectng the conductve terms n the energy equaton of (.), we obtan the followng temperature evoluton equaton (.) t T α s (K,α T 5/ α s T α ) r (K,α r T α ) = ±β α (T e T ), (s,r), completed wth the boundary condtons [9] (.3) r T α = Q,α, r =, s (,), r T α =, r =, s (,), K,α T 5/ α s T α = +γ α T α, r (/,), s =, K,α T 5/ α s T α = γ α T α, r (/,), s =, T(t,,r) = T(t,,r), r (,/), and the ntal condton (.4) T α () = T α. The dffuson parameters < K,α K,α and the core-heat flux Q,α > are consdered as gven. Accordng to [9], Bohm boundary condtons descrbe the contnuty of the heat fluxes at the lmter, whch gve rse to the thrd and fourth boundary condtons n (.3). The constant γ α s dfferent for electrons and ons, n partcular γ for ons and γ e for electrons. In the case of ons, we have thus homogeneous Neumann boundary condtons at the lmter. From the energy estmate gven later n (.), we ndeed observe that these boundary condtons gve an addtonal contrbuton to the temperature dsspaton at the SOL. The outlne of ths paper s the followng. In Secton, we wll focus on the D nonlnear parabolc problem t T s (K T 5/ s T) =, completed wth the nonlnear boundary condtons n s =,. Explct, mplct and IMEX schemes are compared for the resoluton of ths D problem, wth respect to accuracy and computatonal cost. In Secton 3 we consder the complete D problem for one speces (wthout the source term). A drectonal Le splttng method s used n order to transform the D problem n two D problems and to apply the results of the prevous secton. Fnally, n Secton 4 we solve the complete D on-electron coupled problem. The shapes of the dfferent electron/on temperatures are compared.

4 4 FRANCIS FILBET, CLAUDIA NEGULESCU AND CHANG YANG. The D nonlnear problem Let us consder n ths secton the D nonlnear problem, correspondng to the temperature balance equaton n the parallel drecton,.e. t T s (K T 5/ s T) =, (t,s) R + (,), (.) K T 5/ s T = +γt, s =, K T 5/ s T = γt, s =, T(, ) = T, where γ s a gven constant, K L (), T L (), wth T and K almost everywhere. Let us denote n ths secton the doman by = (,). The am of ths secton s to ntroduce an effcent numercal scheme for ts resoluton. From a physcal pont of vew, problem (.) descrbes the rapd dffuson process of the ntal temperature T and the outflow through the boundary. Concernng the exstence of soluton to (.), we refer, for nstance, to [, 3, 4, 6] for a general parabolc equaton wth a homogeneous Neumann boundary condton. Ths proof can be easly adapted to nonlnear boundary condton case of (.) thanks to the followng energy estmate (.) + γ t T(t,s) ds + t K s (T 7/ ) dsdτ [ T(τ,) + T(τ,) ] dτ = T (s) ds. Indeed, from ths latter estmate, we deduce that T s bounded n L () and we obtan also the followng estmaton of nonlnear operator t K s (T 7/ ) dsdτ T L (). Now, we present several numercal schemes for (.)... A fnte volume approxmaton. In ths secton, we propose to derve a numercal scheme for (.) n whch we apply a fnte volume approach for the dscretzaton n the space varable. Let us consder a set of ponts (s / ) ns of the nterval (,) wth s / =, s ns / = and n s + represents the number of dscrete ponts. For n s, we defne the control cell C by the space nterval C = (s /,s +/ ). We also denote by s the mddle of C and by s the space step s = s +/ s / where we suppose that there exsts ξ (,) such that (.3) ξ s s s, {,...,n s }, wth s = max s. We shall construct a set of approxmatons T (t) of the average of the soluton to (.) on the control volume C and frst set T = s C T (s)ds.

5 NUMERICAL STUDY OF A NONLINEAR HEAT EQUATION FOR PLASMAS 5 Applyng a fnte volume dscretzaton to (.), T s soluton to a system of ODEs, whch can be wrtten as dt (.4) dt (t) = F +/ F /, n s, s where the numercal flux s gven by (.5) F +/ = 4K 7 T (t = ) = T, n s, (T + ) 7/ (T ) 7/ s + + s, =,...,n s. Moreover, at the boundary s = and s =, we apply the boundary condtons, +γ T, f =, (.6) F +/ = γ T ns, f = n s. Note that the above dscretzaton on space s frst order due to the loss of precson at the boundary. To complete the dscretzaton to the system (.), the fnte volume scheme (.4)- (.6) has to be supplemented wth a stable and consstent tme dscretzaton step. In the followng we present dfferent tme dscretzatons startng from classcal explct and mplct schemes and then propose a new stable and accurate numercal approxmaton based on the work [6]... Tme explct dscretzaton. We denote by > the tme step, t n = n for any n N and T n s an approxmaton of the soluton T to (.) at tme t n. Then, we apply a backward Euler scheme to (.4)-(.6), whch yelds (.7) T n+ T n = Fn +/ Fn / s, n s, wth gven T, n s, wth F n +/ the flux (.5)-(.6) computed from the approxmaton at tme T n. Classcally, to guarantee the stablty of the scheme (.7), the tme step s restrcted by a CFL condton. Proposton.. Consder that the ntal datum T s nonnegatve and T L (,) and assume the stablty condton (.8) ξ s ( ) 4 K, max 7 T 5/,γ s where ξ s gven n (.3). Then, the numercal soluton (T n ),n obtaned by the explct scheme (.7) s stable and converges to the exact soluton to (.). We refer to [5] or to [] for the proof of a smlar result. Unfortunately, ths smple scheme s not really effcent snce t becomes costly when the mesh s very fne, the constrant on the tme step s too restrctve..3. Tme mplct dscretzaton. To avod the restrctve constrant on the tme step (.8), an mplct scheme s more sutable. Therefore, we consder the fnte volume scheme (.4)-(.6) to the system of equatons (.), but apply a forward Euler tme dscretzaton.

6 6 FRANCIS FILBET, CLAUDIA NEGULESCU AND CHANG YANG Ths yelds, (.9) T n+ T n = Fn+ +/ Fn+ / s, n s, wth gven T, n s, wth F n+ +/ the flux (.5) computed from the approxmaton at tme T n+. Hence, a fully nonlnear system has to be solved at each tme step. The scheme (.9) coupled wth (.5)-(.6) s unformly stable and leads to a numercal approxmaton whch converges to the exact soluton to (.). We refer to [4, 5] for a smlar result or to [] for a complete proof of convergence. Here, we only perform a stablty analyss, whch s the key pont of the convergence proof of the numercal soluton to the unque weak soluton. Proposton.. Consder that the ntal datum T s nonnegatve and T L (,). Then the numercal soluton gven by the mplct scheme (.9) coupled wth (.5)-(.6) s uncondtonally stable,.e. (.) T n T L, and (.) s = s T n+ s = s T. Moreover, the followng dscrete sem-norm s unformly bounded [ (T ) N t s n+ 7/ ( ) ] + T n+ 7/ (.) C, s + s + n= = where the constant C > only depends on the ntal datum T. Proof. Let us consder a convex functon φ C (R, R), then we have (.3) φ(t n+ ) φ(t n ) φ (T n+ )(T n+ T n ). Thus, we multply the scheme (.9) by s φ (T n+ ) and sum over {,...,n s }, t gves s = s s φ(t n+ ) = s φ(t n ) s = s = ( ) φ (T n+ ) F n+ +/ Fn+ /, F n+ ( +/ φ (T+ n+ ) φ (T n+ ) ) F n+ / φ (T n+ ) + F n+ n s / φ (T n+ n s ). Usng the defnton of the numercal flux (.5) and the dscrete boundary condtons (.6), we get s = (.4) s s φ(t n+ ) = 4K 7 s φ(t n ) γ φ (T n+ )T n+ γ φ (Tn n+ n+ s )Tn s s = [ φ (T n+ + ) φ (T n+ ) ] ( T+ n+ ) 7/ ( ) T n+ 7/. s + s +

7 NUMERICAL STUDY OF A NONLINEAR HEAT EQUATION FOR PLASMAS 7 Observng that a smlar nequalty holds true when φ(x) s only Lpschtzan, we take φ(x) = x, where x = x for x < and x = for x, and prove the nonnegatvty of the approxmaton T n, that s, s = s (T n+ ) s = s (T ) =. Therefore, assumng that T, for all n s, we obtan that T n for all n s and n N. Moreover, takng φ(x) = (x M) +, wth M = T L, where x + = x for x > and x + = for x, we have s = s (T n+ M) + s = s (T n M) + Hence we deduce that T n M, for all n s. Then we take φ(x) = x / n (.4), whch yelds that s = s s n+ (T ) s (T n ) C = s = s = [ T n+ + T n+ s (T M) + =. ] ( T n+ ) 7/ ( T n+ + s + s + and snce x x 7/ s a nondecreasng functon on R +, we have s n s s n n+ (T ) s (T s n ) C T n+ + T n+ ( ) T n+ 7/ ( + T n+ s = = = + s + Therefore, we use the fact that T n s unformly bounded to observe that T 7/ + T 7/ C T + T and get the followng nequalty n s = s (T n+ ) s = C s (T n ) s = [ (T ) n+ 7/ ( ) ] + T n+ 7/. s + s + Fnally we sum over n {,...,N t } and mmedately deduce that there exsts a constant C > only dependng on the ntal datum T such that [ (T ) N t s n+ 7/ ( ) ] T n+ 7/ C. s + s + n= = The mplct scheme (.9) s uncondtonally stable, but t requres the numercal resoluton of a nonlnear system. For ths purpose, a Newton s method s appled. By rewrtng the mplct scheme n the form of F(T n+ ) =, where F s a nonlnear functon, and by denotng T n,k, k N, a sequence of approxmaton of T n+ calculated by the Newton s method, the crteron step s that the dscrete L norm of F(T n,k ) s smaller than a threshold ε = 6. However, ths Newton s method may ncrease consderably the computatonal cost and would make ths mplct scheme neffcent. Therefore, another strategy would consst n applyng a sem-mplct scheme for the tme dscretzaton, but t stll requres the mplementaton of a new lnear system at each tme teraton and the computatonal cost remans too mportant. In the followng we propose a numercal scheme nspred by the work of F. Flbet & S. Jn [6] to handle wth ths problem. ) 7/ ) 7/.

8 8 FRANCIS FILBET, CLAUDIA NEGULESCU AND CHANG YANG.4. An mplct-explct (IMEX) scheme. In [6], the authors proposed to handle wth a stff and nonlnear problem. The man pont s to wrte the nonlnear problem n a dfferent form n order to splt the nonlnear operator n the sum of a dsspatve lnear part, whch can be solved n an mplct way and a non dsspatve and nonlnear part whch wll be solved wth a tme explct solver. The man dffculty s to fnd an adequate decomposton of the operator. For nstance the nonlnear dffusve operator can be wrtten as ) ) ) K s (T 5/ s T = ν sst + s ((K T 5/ ν s T and the tme dscretzaton to (.) becomes T n+ T n ( s ν s T n+) = s ((K (T n ) 5/ ν (.5) ν s T n+ () + γt n+ () = ν s T n+ () γt n+ () = ) s T n), ( ) K (T n ()) 5/ ν s T n (), ( ) K (T n ()) 5/ ν s T n (). To choose an approprate ν for the scheme (.5), we perform an energy estmate of the numercal approxmaton. Proposton.3. Assume that the vscosty term ν s such that (.6) K T n 5/ ν, n N. Then the numercal soluton satsfes the followng (.7) ( T n+ ) ds + ν s T n+ ds (T n ) ds + ν s T n ds. Proof. We multply (.5) by T n+ and ntegrate on s (,), hence we have T n+ ds ( (T T n n+ ds ) T n+ T n) ds (( ν K T n 5/) s T n s T n+ ν ( s T n+) ) ds γ ( (T n+ () ) + ( T n+ () ) ). Usng the assumpton that K T n 5/ ν and applyng the Young s nequalty, we obtan ( (ν ν K T n 5/) s T n s T n+ ε K T n 5/) ( st n ) ( + s T n+) ε ε ( st n ) + ν ( s T n+). ε Therefore wth the choce ε = ν, we have ( T n+ ) ν ds + s T n+ ds (T n ) ds + ν s T n ds. Hence, the scheme (.5) s stable when K T n 5/ ν. Now, we can gve the fully dscrete scheme, called n the sequel IMEX, as follows T n+ T n = Fn+/ +/ Fn+/ /, n s, (.8) s wth gven T, n s,

9 NUMERICAL STUDY OF A NONLINEAR HEAT EQUATION FOR PLASMAS 9 wth the numercal flux F n+/ +/ s gven for {,...,n s } by ( (( ) (.9) F n+/ T n 5/ ) ) +/ = K + + (T n ) 5/ T n ν + T n (.) F n+/ +/ = s + + s + ν T n+ + T n+, s + + s whereas at the boundary s = and s =, we apply the boundary condtons wrtten n the form (.5), +γ T n+, f =, γ T n+ n s, f = n s. Moreover, the vscosty ν > s ntally chosen as an upper bound of K T 5/ readjusted along teratons n N n order to satsfy the condton (.6): Algorthm to compute ν ν := K T 5/ and n = whle n N Tend do compute the numercal soluton T n+ and s then f ν 5 4 K T n+ 5/ ν K T n+ 5/ end f then f ν 4K T n+ 5/ then ν K T n+ 5/ / end f n n + end whle.5. Numercal results. To compare the numercal results obtaned wth the dfferent schemes, we take γ =, K = and the ntal temperature s T = 5, whereas the fnal tme of the numercal smulaton s equal to T end =. We do not consder a degenerate case, for nstance an ntal condton whch vanshes on some nterval, snce n physc the temperature could not reach zero kelvn. On the one hand a reference soluton s computed usng the fnte volume method wth an explct scheme (.7) on a unform grd wth n s = 45. On the other hand, we bascally compare both mplct (.9) and IMEX (.8)-(.) schemes wth dfferent unform grds wth n s = 5, 5. Furthermore, we choose the tme step equal to =, 3, 4 and 5 respectvely. Implct scheme (.9) IMEX scheme (.8)-(.) n s = n s = n s = n s = Table : Computatonal tme for the mplct scheme (.9) and the IMEX scheme (.8)-(.) n seconds at the fnal tme of the numercal smulaton T end =.

10 FRANCIS FILBET, CLAUDIA NEGULESCU AND CHANG YANG In the mplementaton of mplct scheme (.9), the mean number of teratons by tme step for Newton s method s between and 4, and when 4 we need only teratons to acheve the convergence. We observe from Table that the IMEX scheme s much more effcent than the mplct scheme n terms of computatonal cost snce the lnear system correspondng to the mplct part does not depend on the teraton n when the vscosty ν > s large enough. For n s = 5, the computatonal tme of the IMEX scheme (.8)- (.) s less than one fourth of the one correspondng to the mplct scheme (.9) whereas for n s = 5, the mplct scheme s ten tmes more consumng than the IMEX scheme. Implct scheme (.9) IMEX scheme (.8)-(.) n s = n s = n s = n s = Table : Relatve errors obtaned usng an mplct scheme, IMEX scheme at tme T end =. Relatve error logscale t= t= 3 t= 4 t= 5 S Relatve error logscale t= t= 3 t= 4 t= 5 S 3 3 s logscale (a) Implct scheme 3 3 s logscale (b) IMEX scheme Fgure : Convergence speed of mplct and IMEX schemes. We compare the relatve errors on the mesh szes 6, 3, 64, 8, 9, 56 et 3 respectvely at fnal tme T end =. Concernng the accuracy and stablty, Table shows that the numercal soluton computed wth both mplct and IMEX schemes s stable for any tme step and the numercal errors are of the same order. Moreover, we get smlar results when tme step s smaller than 4. Of course, when we ncrease the number of ponts n s, the numercal error decreases and the IMEX scheme (.8)-(.) seems to be more accurate for small tme steps. In Fgure, we observe that these two schemes have frst order convergence speed n the case =. However n the case 3, the convergence speed approaches second order n a fne mesh. Ths s because frst order approxmaton s used at two boundares, thus n a coarse mesh large errors appear around the boundares. Fnally, Fgure 3 llustrates the temperature evoluton n log scale. We note that the temperature has a fast decay at the begnnng, then the decay rate slows down when t approaches the fnal tme T end =. Furthermore we observe that the temperature develops steep gradents at the boundary modelng the coolng of the plasma due to the lmter effects. Indeed, on the one hand the thermal dffuson depends on the term T 5/ whch s large at the begnnng and then becomes smaller and smaller. On the

11 NUMERICAL STUDY OF A NONLINEAR HEAT EQUATION FOR PLASMAS other hand, due to the nonlnear flux at the boundary when the temperature becomes small, the temperature gradent becomes larger and larger. Fgure 3: Temperature evoluton of problem (.) n log scale. We use the IMEX scheme to approxmate (.) on a mesh wth 5 ponts and choose tme step of = The D problem In ths secton, we consder the two dmensonal problem where the temperature T depends on tme t and two space varables (s,r) = (,) (,) wth approprate boundary condtons (3.) t T s (K T 5/ s T) r (K r T) =, t, (s,r), where K and K are nonnegatve constants wth K K. For the boundary condtons we mpose a boundary flux n r = and assume that for r = the flux of temperature s zero, that s, r T(t,s,) = Q, s (,), r =, t, (3.) r T(t,s,) =, s (,), r =, t, and at the boundary s = and s = we consder ether perodc boundary condtons or of modelng descrbng the effects of the lmter whch allows to decrease the temperature n the devce. Thus, n the transton layer, we have (3.3) T(t,,r) = T(t,,r), r (,/), t, and n the scrape layer, we have K T 5/ (t,,r) s T(t,,r) = γ T(t,,r), r (/,), t, (3.4) K T 5/ (t,,r) s T(t,,r) = γ T(t,,r), r (/,), t.

12 FRANCIS FILBET, CLAUDIA NEGULESCU AND CHANG YANG Ths model also satsfes an energy estmate gven by d T(t,s,r) dsdr = K dt T 5/ s T dsdr K r T dsdr ( γ T(t,,r) + T(t,,r) ) dr + K Q T(t,s,)ds. / To dscretze the system (3.)-(3.4), we apply a fnte volume method n space coupled wth a tme splttng scheme for the tme dscretzaton. We frst present the numercal scheme and descrbe precsely the dscretzaton of the boundary condtons. Fnally we compare our numercal results wth those obtaned by standard explct and mplct tme dscretzatons. 3.. Tme splttng scheme. We apply a tme splttng scheme n both drectons. As for the one dmensonal case, we apply an IMEX scheme to treat the nonlnear equaton and fnd a condton on the vscosty ν > to get a unformly stable scheme. We frst consder the nonlnear problem n the s drecton, T T n ) (3.5) s ((K (T n ) 5/ ν s T n) ν sst =, (s,r), wth the boundary condton T (,r) = T (,r), r (,/), (3.6) ( ) K (T n (,r)) 5/ ν ( ) K (T n (,r)) 5/ ν s T n (,r) = γt (,r) ν s T (,r), r (/,), s T n (,r) = γt (,r) ν s T (,r), r (/,), whch allows to compute a frst approxmaton T. Then we compute a numercal approxmaton of the lnear heat equaton, (3.7) T n+ T r (K r T n+ ) =, (s,r), wth non homogeneous Neumann boundary condtons r T n+ (s,) = Q, s (,), r =, (3.8) r T n+ (s,) =, s (,), r =. For the sake of clarty we present a stablty estmate on ths sem-dscrete scheme (dscrete n tme and contnuous n space), but the proof can be easly adapted to the fully dscrete case. Proposton 3.. Assume that the vscosty term ν s such that for any r (,), K T n 5/ ν, n N. Then the numercal soluton satsfes the followng ( T n+ ) ν dr ds + s T n+ dr ds ( T ) ν dr ds + s T dr ds n+ [ ] K r T k dr ds Q T k (s,)ds. k=

13 NUMERICAL STUDY OF A NONLINEAR HEAT EQUATION FOR PLASMAS 3 Proof. Multplyng (3.5) by T and ntegratng n, we obtan ( (T ) (T n ) ) ( ) dr ds K (T n ) 5/ ν s T n s T dr ds ν s T dr ds γ / T (,r) + T (,r) dr. Then, applyng the Young nequalty and takng ν such that for all r (,), we have (3.9) (T ) dr ds + ν K T n (s,r) 5/ ν, n N, s T dr ds (T n ) dr ds + ν s T n dr ds. Smlarly, we multply (3.7) by T n+ and ntegrate wth respect to (s,r), we get ( (T n+ ) (T ) ) ( drds K r T n+) drds (3.) + Q K T n+ (s,)ds. Furthermore, we derve (3.7) wth respect to s and get s T n+ s T K ( rr st n+ ) =. Then we multply ths latter equalty by ν s T n+ and ntegrate over (s,r), [ ( s ν T n+) ( s T ) ] dr ds ν K rs T n+ dr ds + ν [ s ( r T n+ ) s T n+] r= r= (. Hence usng that s r T n+ (s,r) ) =, r {,}, t yelds [ ( s (3.) ν T n+) ( s T ) ] dr ds. Then, gatherng (3.) and (3.), we get ( T n+ ) ν [ dr ds + s T n+ + K r T n+ ] dr ds K Q T n+ (s,)ds (T ) dr ds + ν s T dr ds. Fnally, the latter nequalty together wth (3.9), t gves ( T n+ ) ν [ dr ds + s T n+ + K r T n+ ] dr ds K Q T n+ (s,)ds (T n ) dr ds + ν s T n dr ds. By nducton and summng over k =,..., n, we get the result ( T n+ ) ν dr ds + s T n+ dr ds ( T ) ν dr ds + s T dr ds n+ [ ] K r T k dr ds Q T k (s,)ds. k=

14 4 FRANCIS FILBET, CLAUDIA NEGULESCU AND CHANG YANG 3.. A fnte volume approxmaton. For the space dscretzaton, we consder a set of ponts (s / ) ns a set of ponts of the nterval (,) wth s / =, s ns / = and n s + represents the number of dscrete ponts n the drecton s and (r j / ) j nr a set of ponts of the nterval (,) wth r / =, r nr / = and n r + represents the number of dscrete ponts n the drecton r. For n s, j n r, we defne the control cell C,j by C,j = (s /,s +/ ) (r j /,r j+/ ). We also denote by (r,s ) the center of C,j, the space step s = s +/ s / and the space step r j = r j+/ r j / where we assume that there exsts ξ (,) such that (3.) ξ h s, r j h, (,j) {,...,n s } {,...,n r }, wth h = max,j { s, r j }. We shall construct a set of approxmatons T,j (t) of the average of the soluton to (.)- (.3) on the control volume C,j and set T,j = C,j C,j T (s,r)ds dr. Hence, the fnte volume dscretzaton to (3.5) can be wrtten as T,j T n,j = Fn+/ +/,j Fn+/ /,j, (,j) {,...,n s } {,...,n r }, s where the flux F +/,j corresponds to the one dmensonal flux gven by (.9) and perodc boundary condtons are appled for r j (,/) and condtons (.) for r j (/,). Then, the fnte volume dscretzaton to (3.5) can be wrtten as T n+,j T,j where G,j+/ s gven by = Gn+,j+/ Gn+,j / r j, (,j) {,...,n s } {,...,n r } T,j+ n+ (3.3) G,j+/ = K T,j n+, j =,...,n r. r j+ + r j Moreover, at the boundary r = and r =, we apply the boundary condtons, K Q, f j =, (3.4) G,j+/ =, f j = n r Numercal results. In ths secton we compare the dfferent numercal results related to the D problem (3.)-(3.4) obtaned usng a tme splttng scheme wth an explct, mplct and IMEX treatment of each step. As before, we frst compute a reference solutons obtaned from an explct scheme wth a small tme step satsfyng a CFL condton h. In the followng numercal smulatons, we choose the dfferent physcal parameters as K =, K =, γ =, Q =. Moreover, the ntal temperature s gven by (3.5) T (s,r) = 3, and the fnal tme of the smulaton s T end =. To compute the reference soluton, we have chosen n s = 3 and n r = 3, whereas the numercal results usng mplct and IMEX schemes are obtaned wth n s = and n r = wth several tme steps =,, 3, and 4. Frst, concernng the computatonal tme we observe n Table 3, that the IMEX scheme s much faster than the mplct scheme. Furthermore, the numercal error presented n Table 4 for both scheme s of the same order of

15 NUMERICAL STUDY OF A NONLINEAR HEAT EQUATION FOR PLASMAS 5 magntude and thus the IMEX scheme s clearly much more effcent than the fully mplct scheme. 3 4 Implct scheme IMEX scheme Table 3: Computatonal tme for the D problem (3.)-(3.4) usng mplct and IMEX schemes at tme T end =. 3 4 Implct scheme e-4 IMEX scheme e-4 Table 4: The relatve errors for the mplct and IMEX schemes compared wth a reference soluton for the D problem (3.)-(3.4) at tme T end =. Now we want to nvestgate the effect of the splttng scheme on the numercal error and the computatonal cost. Therefore, we also propose a comparson between the dfferent schemes. We frst compare the computatonal tme applyng the IMEX scheme wth and wthout the splttng method wth a tme step = 3, (n s,n r ) = (5,5), (,), (3,3) and (5,5) respectvely. On the one hand, we observe n Table 5 that the splttng method s much faster than the non-splttng method when the number of dscrete ponts ncreases. n s n r IMEX Non-splttng scheme 6 55 IMEX splttng scheme Table 5: Computatonal tme of IMEX wth and wthout splttng scheme at tme T end =. On the other hand, we compare the numercal errors correspondng to the two strateges wth (n s,n r ) = (,), = 3 n Table 6, n partcular the fully mplct scheme wth and wthout splttng and the IMEX scheme wth and wthout splttng. We observe that the method wthout splttng s always more accurate than the one wth the splttng method. Scheme Splttng mplct Splttng IMEX Implct IMEX Numercal error Table 6: Relatve errors for dfferent numercal schemes compared wth a reference soluton for (n s,n r ) = (,), = 3 at tme T end =. In Fgure 4, we present the evoluton of the approxmaton of the temperature (3.)-(3.4) n computatonal doman, whch s dvded nto two regons : the transton layer and the scrape-off layer (SOL) as llustrated n Fgure -(b). We frst ntalze the temperature to a constant and then observe mmedately that temperature decreases rapdly n the scrape-off

16 6 FRANCIS FILBET, CLAUDIA NEGULESCU AND CHANG YANG layer and becomes sngular around the lmter (whch corresponds to the boundary s = and wth r /). On the other hand, n the transton layer, the temperature converges to a steady state whch s homogeneous n s (, ). The dfferent numercal schemes gve the same qualtatve behavor of the soluton. (a) t = (b) t =. (c) t =.5 (d) t =.5 (e) t = (f) t = Fgure 4: Temperature evoluton of problem (3.).

17 NUMERICAL STUDY OF A NONLINEAR HEAT EQUATION FOR PLASMAS 7 In Fgure 5, we plot the temperature evoluton at the secton r =.5, r =.75 and s = and s =.5 respectvely. Accordng to Kočan et al. [, ], the parallel thermal dffusvty s much larger than the perpendcular one,.e. K K. Therefore, the temperature becomes constant along the magnetc feld lnes, that s for s (,). We observe n Fgures 5 that the temperature s constant at all tme n transton regon whereas steep gradents develop at the boundary layer s = and s = n the SOL regon. In the perpendcular drecton r, the stuaton s dfferent. We also observe that at tme t = the temperature decreases lnearly wth respect to r n the transton layer ( r.5), accordng to the heat flux Q at edge r =, and then decreases exponentally n the scrape-off layer (.5 r ). These numercal results correspond to the retardng feld analyzer (RFA) [9,, ] Temperature Tme=. Tme=.5 Tme= Tme= Temperature.5 Tme=. Tme=.5 Tme= Tme= s axs s axs (a) r = /4 (b) r = 3/4 Tme=. Tme=.5 Tme= Tme= Tme=. Tme=.5 Tme= Tme= Temperature 3.5 Temperature r axs r axs (c) s = (d) s = / Fgure 5: Temperature evoluton at secton r = /4, r = 3/4, s = and s = / at tme t =.,.5, and respectvely. Fnally, we present the evoluton of the energy dsspaton wth respect to tme: d T(t,s,r) dsdr = E + E + E 3, dt

18 8 FRANCIS FILBET, CLAUDIA NEGULESCU AND CHANG YANG wth ( ) E := K T 5/ s T + K r T dr ds, E := γ / T(t,,r) + T(t,,r) dr, E 3 := +Q K T(t,s,)ds. The Fgure 6 states the terms E, E, E 3 as functon of t. We plot these terms obtaned from mplct and IMEX schemes. Note that these two fgures are almost the same. In fact, at the begnnng of smulaton, there s a fast decay of the temperature, thus the quantty E representng the total energy exchange rato n the doman, s ncreasng for t <.. Then, t converges to an equlbrum state for larger tme. On the other hand, the quantty E decreases wth respect to tme, t s due to the ansotropy between K and K. Indeed, the energy s transferred to the lmters n the scrape-off layer regon whereas n the perpendcular drecton r, the thermal dffusvty s small. Fnally, as we have seen n Fgure 5 on the edge of the core, the temperature does not vary sgnfcantly, thus the quantty E 3 ncreases slghtly wth respect to tme. E E E E E 3 E 3 Energy dsspaton Energy dsspaton Tme (a) Implct scheme Tme (b) IMEX scheme Fgure 6: Evoluton of the energy dsspaton wth respect to tme for problem (3.), wth =.. 4. The couplng problem In ths secton, we consder the full D model (.) composed of two dfferent partcle speces,.e. ons and electrons. We denote by T (resp. T e ) the temperature of ons (resp. electrons) whch depends on tme t and two space varables (s,r). The two equatons are coupled by a non-zero source term whch balances the temperature between the two partcle speces, (4.) t T s (K, T 5/ s T ) r (K, r T ) = +β(t T e ), for (s,r), t T e s (K,e T 5/ e s T e ) r (K,e r T e ) = β(t T e ), for (s,r),

19 NUMERICAL STUDY OF A NONLINEAR HEAT EQUATION FOR PLASMAS 9 where K, K,, K,e K,e and β s a negatve constant. These two equatons are completed wth the boundary condtons (.3) and the ntal condton (.4). 4.. Tme splttng scheme. Now we dscretze the full system (4.) usng a splttng scheme n three steps. We assume that an approxmaton of the soluton (T e,t ) at tme t n s known and denote t by (Te n,t n ). Therefore, we frst approxmate the source part couplng the two temperatures T e and T usng an mplct scheme, whch yelds T = ( ) Te n + ( ) + T n, β β (4.) Te = ( + β )T e n + ( β )T n. It s clear that (4.) guarantees the postvty of the temperature. Then we apply the same tme splttng steps as before n drecton s and n drecton r as follows. On the one hand we compute Tα for α {, e} by solvng (3.5)-(3.6). On the other hand we apply the last step (3.7)-(3.8) n the drecton r. Furthermore, for the scheme (3.5)-(3.6), (3.7)-(3.8) and (4.), we also prove an energy estmate Proposton 4.. Consder that the ntal datum T α s nonnegatve and T α L (,). Assume that for α {, e}, the vscosty term ν s such that for any r (,), max K,α Tα n α {,e} 5/ ν, n N. Then the numercal soluton, gven by (4.), satsfes the followng [ T n+ α + ν s Tα n+ ] dr ds α {,e} [ T α + ν s Tα ] dr ds + α {,e} α {,e} k= α {,e} k= n+ K,α r T k α dr ds n+ K,α Q,α Tα k (s,)ds. Proof. We frst observe that the energy estmate of the two last steps n the drecton s and r are the same as the one proved n Proposton 3., hence we have [ T n+ α + ν ( s Tα n+ + K,α r Tα n+ ) ] dr ds [ T α + ν s Tα ] dr ds + K,α Q,α Tα n+ (s,)ds. (4.3) Therefore, to acheve the proof on the energy estmate, we only observe that (4.) can be wrtten as follows T T n = + β (T Te ), T e T n e = β (T T e ).

20 FRANCIS FILBET, CLAUDIA NEGULESCU AND CHANG YANG Multplyng the frst equaton (4.3) by T and the second by Te and ntegratng on (r,s), t yelds T + Te drds T n + Te n drds Moreover, dfferentatng (4.3) wth respect to s and multplyng the frst equaton by ν s T and the second one by ν s Te, we get ν s T + s Te drds ν s T n + s Te n drds. Fnally, we have α {,e} α {,e} α {,e} [ T n+ α + ( ν s Tα n+ + K,α r T n+ ) ] dr ds [ T n α + ( ν s T n α + K,α r T n α ) ] dr ds K,α Q,α Tα n+ (s, )ds. Summng over k =,...,n, we complete the proof. Fnally space dscretzaton s performed usng the fnte volume scheme presented n Secton Numercal results. In ths secton, we compare the numercal results obtaned from the mplct scheme and the IMEX scheme for (4.). We choose K, =., K,e =, K, =., K,e =., γ =, γ e =.5, Q, = Q,e = and β =.. The ntal temperature s such that T (s,r) = 3, and T e (s,r) = 3, (s,r). The fnal tme of the smulaton s T end = and the mesh sze s chosen as n s =, n r =. We plot the electron and on temperature and compare ther rato at dfferent tme. The am s to compare the dfferent behavors between electron and on temperatures at the edges and n the scrape-off layer of a Tokamak [8]. On the one hand, we propose n Fgure 7, the temperature evoluton. On the left hand sde, we present the electron temperature, whereas on the rght hand sde we gve the on temperature. We frst notce that the electron parallel thermal dffusvty s about tmes larger than the one for ons [, ], and the electron energy exchange rato at the edge r (.5,) depends on O(Te 3/ ), thus the temperature has a fast decay when t s small n the scrape-off layer. However, the boundary condtons for ons n the scrape-off layer s gven by the homogeneous Neumann condton s T =, whch means that there s no energy exchange at the lmters. Thus the on temperature does not vary sgnfcantly at scrape-off layer. α On the other hand, the rato between electron temperature and on temperature s presented n Fgure 8. The Fgure 8 llustrates that n the transton layer, the on and electron temperatures are almost dentcal. However, n the scrape-off layer, at the fnal tme T end = the rato τ becomes large around the lmters due to the boundary condton s T e Te 3/. The evoluton of the rato τ n the radal drecton s gven n Fgures 8. We observe that n the transton layer the rato τ s almost equal to, whereas n the scrape-off layer ths rato becomes large. For example, at tme t = the rato τ = 6 for s = / whle t s τ = 45 for

21 NUMERICAL STUDY OF A NONLINEAR HEAT EQUATION FOR PLASMAS (a) T e at t =.5 (b) T at t =.5 (c) T e at t =.5 (d) T at t =.5 (d) T e at t = (e) T at t = Fgure 7: Temperature evoluton of problem (4.). s =. These behavors correspond to the experment results n Kočan et al. [, ]. At last we vary the parameter β to study the equlbrum source term n Fgure 9 and observe that when the parameter β s large, the rato τ decreases.

22 FRANCIS FILBET, CLAUDIA NEGULESCU AND CHANG YANG Tme=. Tme=.5 Tme= τ. Tme=. Tme=.5 Tme= τ s axs Tme=. Tme=.5 Tme= s axs (a) r = /4 (b) r = 3/4 6 5 Tme=. Tme=.5 Tme= τ τ r axs r axs (c) s = (d) s = / Fgure 8: Rato τ = T /T e at secton r = /4, r = 3/4, s = and s = / at tme t =.,.5 and respectvely. 5. Concluson We have presented varous numercal approxmatons for a nonlnear temperature balance equaton descrbng the heat evoluton of a magnetcally confned plasma n the edge regon of a tokamak. Numercal comparsons show that an IMEX scheme based on a smart decomposton of the nonlnear dffusve operator coupled wth a splttng strategy gves an effcent numercal scheme n terms of accuracy, stablty and reasonable computatonal cost. The next step would conssts to couple the present model wth the transport equatons for the plasma densty and momentum. References [] M. Bostan, A. Mentrell, C. Negulescu, Asymptotc Preservng scheme for hghly ansotropc, nonlnear dffuson equatons. Applcaton: SOL plasmas, n preparaton. [] S.I. Bragnsk, Transport processes n a plasma, Revews n Plasma Physcs, New York Consultant Bureau Edton, (965). [3] J.M. Brzard, T.S. Hahm Foundatons of nonlnear gyroknetc theory, Rev. Modern Phys. 79 (7), pp

23 NUMERICAL STUDY OF A NONLINEAR HEAT EQUATION FOR PLASMAS β=. β=. β= 6 5 β=. β=. β= τ 5 τ r axs r axs (a) s = (b) s = / Fgure 9: Rato τ = T /T e at secton s = and s = / for dfferent parameters β =, and at tme t =. [4] C. Chanas-Hllaret, Y.-J. Peng, Convergence of a fnte volume scheme for the drft-dffuson equaton n D, IMA Journal of Numercal Analyss, 3, (3), pp [5] R. Eymard, Th. Gallouët, R. Herbn, Fnte volume methods, Handbook of numercal analyss, Vol. VII. Amsterdam: North-Holland, (). [6] F. Flbet, S. Jn, A class of asymptotc preservng schemes for knetc equatons and related problems wth stff sources J. Comput. Phys. 9 (), pp [7] L. Isoard, Modélsaton du transport dans le plasma de bord d un tokamak, PhD thess (), Unversté Paul Cézanne. [8] L. Isoard, H. Bufferand, G. Chavassa, G. Craolo, F. Schwander, E. Serre, S. Vazzo, N. Fedorczak, Ph. Ghendrh, J. Gunn, Y. Sarazn, P. Taman, D modelng of electron and on temperature n the plasma edge and SOL, Journal of Nuclear Materals, (). [9] M. Kočan, J.P. Gunn, M. Komm, J-Y Pascal, E. Gauther and G. Bonhomme, On the relablty of Scrape-off layer on temperature measurements by retardng feld analyzers, Revew of Scentfc Instruments, 79:735, (8) [] M. Kočan, J.P. Gunn, T. Gerbaud, J-Y Pascal, G. Bonhomme, C. Fenz, E. Gauther and J-L. Segu, Edge on-to-electron temperature rato n Tore Supra tokamak, Plasma Physcs and Controlled Fuson, 5:59, (8). [] M. Kočana, J.P. Gunn, J.-Y. Pascal, G. Bonhomme, P. Devynck, I. Ďuran, E. Gauther, P. Ghendrh, Y. Marandet, B. Pegoure and J.-C. Vallet, Measurements of scrape-off layer on-toelectron temperature rato n Tore Supra ohmc plasmas, Journal of Nuclear Materals, 39-39, (9), pp [] Kamenomostskaja, S.L., On the Stefan problem, Mat. Sb. 53, (995), pp [3] O. A. Ladyženskaja, V.A. Solonnkov and N.N. Ural ceva, Lnear and quaslnear equatons of parabolc type, Transl. of Math. Monographs 3, (988). [4] A.M. Mermanov, The Stefan problem, (99) [5] J. Narsk, C. Negulescu, Asymptotc Preservng scheme based on mcro-macro decomposton for nonlnear degenerate, ansotropc parabolc equatons, n preparaton. [6] O. A. Olenk, A method of soluton of the general Stefan problem, Sov. Math. Dokl.,, (96), pp [7] P. Taman, Etude des flux de matère dans le plasma de bord des tokamak: almentaton, transport et turbulence, PhD thess (7), Unversté de Provence. [8] P. Taman, Ph. Ghendrh, E. Trstone, V. Grandgrard, X. Garbet, Y. Sarazn, E. Serre, G. Craolo, G. Chavassa, TOKAM-3D: a 3D flud code for transport and turbulence n the edge plasma of tokamaks, J.Comp. Phys. 9 (), pp [9] Stangeby P.C., The plasma boundary of magnetc fuson devces, Chapter. IOP, Unversty of Toronto Canada, [] J. Wesson, Tokamaks, Oxford Unversty Press 9, thrd edton.

24 4 FRANCIS FILBET, CLAUDIA NEGULESCU AND CHANG YANG [] J. L. Vazquez, The Porous Medum Equaton: Mathematcal Theory, Clarendon press, Oxford Mathematcal press (6). [] C. Yang, Analyse et mse en œuvre des schémas numérques pour la physque des plasmas onosphérques et de tokamaks, PhD thess (), Unversté Llle.