Numerical Solution of Quenching Problems Using Mesh-Dependent Variable Temporal Steps

Transcription

1 Numercal Soluon of Quenchng Problems Usng Mesh-Dependen Varable Temporal Seps K.W. LIANG, P. LIN and R.C.E. TAN Deparmen of Mahemacs Naonal Unversy of Sngapore Sngapore 7543 Absrac In hs paper, we nroduce a new adapve mehod for compung he numercal soluons of a class of quenchng parabolc equaons whch ehb a soluon wh one sngulary. Our mehod sysemacally generaes an rregular mesh wh mesh-dependen emporal ncremens based on he soluon behavor from whch an mplc fne dfference scheme assocaed wh he rregular mesh s consruced. The convergence and sably of he fne dfference scheme s analyzed for he soluon before quenchng. An equvalen lnearzed model s used o jusfy he sably of he mehod near quenchng as well. A numercal eample s provded o demonsrae he vably of he proposed mehod. Keywords. nonlnear reacon-dffuson equaons, quenchng problems, sngulary, adapve mehod, rregular grds, mplc fne dfference scheme AMS subjec classfcaons. 65M5, 8A, 8A5 Inroducon Many mahemacal problems derved from mporan physcal processes have he properes ha hey may develop sngulares n fne me. A ypcal eample of hs ype s he quenchng problem whose soluon remans bounded, whle he frs order me dervave of he soluon may blow up a ceran me []. The quenchng phenomena appear n varous applcaon areas [3,, 7]. In hs paper, we focus on he classcal quenchng problem malkw@nus.edu.sg malnp@nus.edu.sg scance@nus.edu.sg

2 n [7], whch descrbes he combuson of wo gases meeng n a gap beween porous walls [6]. I gves he followng paral dfferenal equaon (PDE) wh a nonlnear sngular source funcon, u = u yy + f(u), < y < a, < < T, () and nal-boundary condons u(y, ) = u, y (, a), u(, ) =, u(a, ) =, (, T ), () where f(u) = ( u) θ (θ > ) and u <. The source funcon f(u) s monooncally ncreasng for u < wh f() = >, lm f(u) =. u Dscusson of he esence and unqueness of s soluon can be found n [8,, ] and references heren. I s poned ou n Kawarada [5] ha he boundary dsance a plays a crucal role n he quenchng problems. There ess a crcal lengh a > such ha for a < a, he soluon of ()-() ess globally, whle for a a, he soluon quenches a a fne me T a,.e., lm u( a, ) =, T a lm u ( a T a, ) = +. (3) The quenchng defned n (3) only consders he values of u and u a = a because he soluon of ()-() s symmerc abou he lne = a and u(a, ) s he mama of he funcon u wh respec o. Classcal numercal mehods wh unform spaal meshes and emporal seps may no be effcen o reproduce quenchng phenomena. Is accuracy may dmnsh when he lengh scale of he sngulary s less han he sep sze. I has been demonsraed ha sgnfcan mprovemens n accuracy and effcency can be acheved by adapve mehod n numercally solvng problems wh hs ype of sngulares [4, 9, 5,,, 6], snce he mesh pons can be concenraed locally n he regons wh rapd varaon of he soluon. In pracce, here are hree ypes of adapve mehods,.e. local refnemen (or he h mehod), order enrchmen (or he p mehod) and mesh moon (or he r mehod) summarzed n [5]. Neverheless, for me dependen problems, he emporal sep sze s kep unform a all spaal nodes n each me-level as depced n Fgure..

3 Fgure.: Mos adapve mehods are mplemened on he same emporal ncremens for all spaal nodes. Space-me fne elemen mehod n heory allows dfferen emporal sep szes a dfferen space locaons, s ypcally mplemened as usng unform emporal seps (cf. []). I s desrable o assgn dfferen emporal ncremen a dfferen spaal nodes (we call hs meshdependen emporal ncremen). Our am of hs paper s o eplore a mesh-dependen varable emporal seps mehod. From he vewpon of adapve numercal mehods, hs sudy s a naural eenson of he usual unform emporal ncremen. If he soluon a some spaal pon s smooher we may march furher wh a larger emporal sep. So he resulng mehod may be more effcen n general. Anoher neresng problem abou equaon ()-() s wha happens beyond he quenchng. A recen survey conduced n [6] proposes mporan basc quesons n quenchng research, ncludng quenchng me, naure of quenchng (does occur a a sngle pon) and wha happens beyond quenchng. There are conjecures proposed by Levne []. Bandle and Brauner [] also presen some resul abou he beyond-quenchng soluon. Phllps [5] consders a slghly dfferen model and show he esence of he soluon away from he quenchng pon when he me evolves beyond he quenchng me. Chan [6, 7] sudes he model as well and under he assumpon ha f() >, f and f he shows ha he soluon wll evenually reach he value one (sngulary) a every spaal pon, whch s called complee quenchng n []. Beyond quenchng has physcal meanng as well, for eample, paral premed combuson (See [9]). Ths, from praccal pon of vew, movaes our consderaon of mesh dependen varable emporal sep mehod snce unform emporal ncremens can no go beyond he quenchng me when u goes o nfny. Whou addng any era mesh pons, we can move boh emporal seps and spaal meshes n mplemenaon. However, for smplcy of llusrang our mesh dependen emporal sep mehod we wll consder only a unform spaal mesh n hs paper. As depced n Fgure., 3

4 he mesh-dependen emporal sep mehod auomacally generaes an adapve rregular mesh based on he soluon behavor. If he varaon of he soluon a node A s smaller han ha a node B, hen he emporal ncremen τ A s chosen o be bgger han τ B. As a resul, he me lne wll be replaced by a curve (or pecewse lnes) a a me-level. Repeaedly, he ne emporal ncremen τ A and τ B wll be deermned by he varaons of he soluon n he same manner. Hence, an rregular mesh wll be auomacally generaed based on he soluon behavor. The man advanage of rregular mesh s ha can ge he soluon n he smooh regon faser snce he emporal ncremen s bgger. In he case of quenchng problem ()-() where only one quenchng pon s usually nvolved, all unform emporal ncremen mehods, ncludng he mehod n [3], have o sop a he quenchng me T a. Whle n our rregular mesh, he nodes afer ceran emporal ncremens locae on a curve nsead of a horzonal lne. Due o hs feaure of he rregular mesh, he mehod s able o provde useful nformaon of he soluon beyond quenchng. Fgure.: The emporal ncremen τ A s chosen o be bgger han τ B snce he varaon of he soluon a node A s smaller han ha a node B. As a resul, he dashed lne wll be replaced by a curve a a me-level. To he bes of our knowledge, no sudes have been performed on he numercal mehod for a soluon beyond he quenchng, or he blow-up for hs parcular problem. Our mehod seems o be a frs aemp n hs drecon. We have o adm ha such an aemp s sll raher premaure. I canno generally go very far beyond he quenchng me. Also he mehod so far only works for he case where he sngulary akes place a one pon. Boh of above dffcules are due o sably resrcon of he rregular mesh. Neverheless, s rue ha our model equaon ()-() has one quenchng (sngulary) pon and our mehod can produce beyond-quenchng soluon and can provde evdence for he complee quenchng resul gven n [6] (See numercal epermens laer). Our goal of hs paper s no o provde a mehod coverng general blow-up problem bu o focus on he model equaon and o undersand dffcules assocaed wh such ype of mehods. We hope ha hs work could movae 4

5 furher developmen o mprove or even avod sably resrcon of rregular meshes. The paper s organzed as follows. In secon, we nroduce he rregular mesh n - plane based on he soluon behavor of problem ()-(). Furher, n secon 3, we consruc an mplc fne dfference scheme, assocaed wh he rregular mesh, for compung he soluon and he quenchng me of he problem ()-(). In secon 4, necessary crera are obaned o guaranee he effcency n handlng he sysem of algebrac equaons resulng from he dscrezaon. The sably properes are dscussed for he dfference scheme and used as crera of rregular mesh consrucon. Fnally, n secon 5, compuaonal resuls are gven o demonsrae he vably of he mehod. Dscrezaons Leng y/a =, problem ()-() can be convenenly reformulaed no he followng form u = a u + u(, ) = u, u(, ) =, u(, ) =,, < <, < < T, (4) ( u) θ < <, < < T. (5) Le { } N = be he spaal nodes on [, ], where = < < < N < N = and h = +, =,, N. We denoe j, as he j-h dscree me level a spaal node and τ j, as he j-h emporal ncremen a spaal node, where j+, = j, + τ j,. Our emporal coordnaes and emporal ncremens wll be deermned n erm of u snce blow-up akes place n u. We adop he followng arc-lengh monor funcon for u [9, 4, 8, 6], µ(u ) = + u, (, ) (, ) (, T ). Then, wh τ,, τ, and τ, gven, τ j, can be deermned hrough τ j, = τ j, + ((u ) j, (u ) j 3, ) ((u ) j, (u ) j, ), (6) where ndces, N, correspond o he spaal node. Remark. In he case of unform emporal ncremens, j, = j and τ j, = τ j for =,,, N. Insead of usng sophscaed smoohng as n many adapve algorhms, we choose a mnmal and a mamal emporal sep sze conrollers τ mn and τ ma, < τ mn τ < τ ma, o avod sudden changes n grd movemen or any unnecessarly large number of compuaons mmedaely before he blow-up of u. 5

6 The conrollers can also help o provde necessary nformaon for a proper soppng creron durng he compuaon. Under such crcumsance, he acual emporal sep sze used can be obaned unquely hrough τ mn f τ j, < τ mn, ˆτ j, = τ j, f τ mn τ j, τ ma, τ ma f τ j, > τ ma. Furhermore, based on (3), we know ha he funcon u may blow up wh respec o me as T and he me adapaon echnque should be mplemened durng he numercal soluon process. For he convenence of llusraon, we wll consder a unform paron n space nerval [, ],.e., = h, =,, N and h = /N. 3 The Dfference Scheme The dea of consrucng dfference schemes on an rregular mesh can be found n [4, 8]. For any suffcenly dfferenable funcon u(, ), we may oban where u(, ) u h u + k u + h u. (7) u = u(, ), h =, k =. Based on equaon (7) hree ndependen equaons are needed o oban hree dfferen dervaves u, u and u a pon (, ). For eample, n order o oban an mplc scheme, he value of u a hree neghborng pons (, ), (, ) and ( 3, 3 ) are used as depced n Fgure 3.. Afer wrng equaon (7) for each of hree neghbors of pon (, ) and pung hem n a mar form, we mmedaely have h h l u(, ) u τ u( 3, 3 ) u. (8) h h l u(, ) u u u u Snce he 3 3 mar n he equaon (8) s non-sngular, hen 6

7 Fgure 3.: The pon (, ) has hree neghborng pons (, ), (, ) and ( 3, 3 ). u u u h l l τh h τ l + l h τh h u(, ) u u( 3, 3 ) u u(, ) u. (9) Obvously, he rregular mesh obaned n secon make he sysem (8) unquely solvable and an rregular dfference scheme can be consruced based on appromae dervaves obaned n (9). We subsue appromae paral dervaves (9) no (4) and ge an mplc dfference scheme τ j, b j+, a h U j+ ( + + ) τ j, b j+, a h U j+ τ j, b j+, a h U j+ + = U j + τ j, b j+, ( U j )θ. () where U j s an appromaon o u(, j ), b j+, = + l j+, + l j+,+ a h and l j+, (or l j+,+ ) s he dsance n emporal drecon beween j+, and j+, (or j+,+ and j+,, respecvely),.e., l j+, = j+, j+, and l j+,+ = j+,+ j+,. Remark 3. For a regular recangular mesh, he rregular dfference scheme () reduces o he usual mplc fne dfference formula wh b =. Remark 3. For hs model we can adop a specal echnque o oban he funcon values of u n (6) wh beer accuracy. Takng he me dervave of equaon (4), we have 7

8 wh he nal-boundary condons (u ) = a (u θ ) + ( u) θ+ (u ), () u (, ) =, (, ), u (, ) =, u (, ) =, (, T ). Smlarly o (), () can be appromaed by he mplc dfference scheme ) τ j, b j+, a h V j+ ( + τ j, + b j+, a h V j+ τ j, b j+, a h V j+ + = V j + θτ j, b j+, ( U j )θ V j, () where V j s an appromaon o u (, j ). We can solve boh () and () smulaneously o oban u and u whou any era compuaonal cos snce () and () have he same rdagonal coeffcen mar. 4 Accuracy and Sably Analyss In hs secon, we wll analyse he mesh-dependen varable emporal seps mehod descrbed n Secon 3. Followng he dea n [3], we dvde he analyss no wo pars. One s he soluon away from he quenchng pon. The oher s he soluon near he quenchng pon,.e. u approaches. For he laer case, we do no nend o gve a rgorous proof bu only provde an nuve jusfcaon of he sably near he quenchng pon hrough a lnearzed model. 4. Sably away from quenchng Leng u j = u(, j, ) be he eac soluon of (4) he local runcaon error of scheme () a pon (, j+, ) s E j+ = uj+ τ j, u j Then usng Taylor s seres epanson, we have uj+ uj+ + u j+ + b j+, a h f(uj ). (3) b j+, 8

9 E j+ = ( τ j, + l j+, + ) l j+,+ u j+ b j+, a h l j+, l j+,+ u j+ + b j+, a h + f(u j+ ) f(u j ). b j+, We assume ha u and u are sasfed n mesh consrucon are connuous n [, ] [, T ) and he followng condons l = ma l j, < Ch and b j+,, (4),j where C s a posve consan. Noe ha b j+, mples ha l j+, + l j+,+ s nonnegave. Then here es consans K and K such ha E j+ K h + K E, (5) where = ma τ j,.,j In fac, (5) s obvous f l = O(h ). If l = O(h r )( r ), hen l j+, + l j+,+ b j+, a h = l j+, + l j+,+ a h + l j+, + l j+,+ = O(h r ), and l j+, l j+,+ b j+, a h f(u j+ = (l j+, l j+,+ )h a h + l j+, + l j+,+ = O(h), ) f(u j ) = O( h r ). b j+, Remark 4. Alhough he consan C n he condon (4) can be an arbrary posve number s generally gven a small value o oban a beer accuracy. In our algorhm C = or 3. Remark 4. The followng resrcon s necessary o guaranee he condon b j, n (4). If l j, l j,+ < hen l j, + l j,+ mus be nonnegave. Ths can be ensured n he mesh consrucon for he one-pon sngulary case. Rearrangng he erm n (3), we can deduce ha 9

10 ( ) τ j, + b j+, a h u j+ = τ ( ) j, b j+, a h u j+ + uj+ + + u j + f(uj ) b j+, τ j, + E j+ τ j,. Le e j = uj U j, where U j s he numercal soluon of he mplc dfference scheme (). We subrac () from (6) and assume he condon (4) sasfed. The followng nequaly s hen obaned afer akng modulus of boh sdes of he resulng equaon. ( + τ j, b j+, a h ) e j+ where K 3 s he mamum magnude of f (u). If we le b j+, e j = ma e j, hen he above nequaly becomes τ ( ) j, e j+ b j+, a h + e j+ + + e j +K 3 + E j+, e j (6) (7) e j ( + K 3 )e j + E ( + K 3 ) j e + ( + K 3 ) j E K 3 ep(k 3 T )e + ep(k 3T ) E. K 3 Snce e = and (5), we have e j as h and. Sably can be obaned smlarly. Defne ) LU j+ τ j, = b j+, a h U j+ ( + τ j, + b j+, a h U j+ τ j, b j+, a h U j+ + U j and U j = ma U j. We can have U j ep(k 3 T )U + ep(k 3T ) K 3 Ths mples sably of he dfference scheme. We now summarze he above resuls and oban he followng heorem. ma LU j. (8) Theorem 4.3 If he soluon s away from he quenchng, he mplc dfference scheme () s lnearly sable f b j, for all and j. Furhermore, f (4) holds, he error e j = ma e j can be bounded by e j K h + K.

11 4. Sably near quenchng We frs rewre he quenchng problem (4) no he followng form u = a u + u, (9) ( u e ) θ+ where u e (, ) s he eac soluon of (4)-(5). So he soluon of (9) wh (5) s he same as ha of (4)-(5). I can also be shown from mamum prncple (ref. [3]) ha u e < (In our programme, we defne a small posve consan δ as a olerance, e.g., we could choose δ relaed o τ mn. And we wll say u quenches when u reaches δ). Now we nroduce a perurbaon z j = U j Ū j where U j s he soluon of (9) and Ū j s he perurbed soluon due o he nal perurbaon. Then he perurbaon z j sasfes he followng perurbaon equaon ( + ) τ j, b j+, a h z j+ = τ ( ) [ j, b j+, a h z j+ + zj+ + + ] τ j, b j+, ( u e ) θ+ z j. () Hence, f he emporal sep sze τ j, δ θ+ and he condon (4) holds, hen, smlarly o he argumen n (7), we have z j+ ma z j+ z j, for he soluon near he quenchng. 5 Numercal Epermens We apply he mehod nroduced n secons and 3 o solve he problem (4) and repor he case wh θ =. Several oher cases wh θ > were esed as well and resuls are smlar. Whou loss of generaly, he nal value u s se o be zero. The spaal sep sze h vares from. o., whle he nal emporal sep sze τ s chosen o be... The reason of choosng smaller spaal and nal emporal sep szes s no for he sably of numercal scheme, bu for observng he quenchng and pos-quenchng behavors more accuraely. In Fgure 5., we show he evoluon of he funcon u and s dervave u. The parameer value a = π s adoped ogeher wh h =. and τ =.5. The s dfferen sages of he soluon a =.38,.3637,.4639, 37, 637 and 73 are depced n Fgure 5.. The locaon of he blow-up peak appears beween =.4 and =.6. Ths can be observed from he conour maps of he numercal soluon n Fgure 5.5.

12 u(,).4. u (,).5.6 u(,).4. u (,).6 u(,).4. u (,) u(,).6.4. u (,) 3 Fgure 5.

13 .8 8 u(,).6.4 u (,) 6 4. u(,) u (,) Fgure 5.: The evoluon profles of he funcon u (n he lef sde) and s dervave u (n he rgh sde) wh a = π, h =., τ =.5 and θ =. From he frs o he las, =.38,.3637,.4639, 37, 637, 73, respecvely. Furhermore, n Fgure 5. and 5.3, we plo evoluon profles of u and u for a = π n deal. Saus of u(, ) and u (, ) a fve dfferen spaal locaons ( =.,.,.3,.4 and ) are dsplayed respecvely. We observe ha he funcon u (, ) a =.4 (or.6) grows rapdly and he value of he peak ges very large, smlar o s behavor a =. I mples ha he soluon u a =.4 (or.6) blows up followng he node =. Alhough he values of he soluon u are below a some nodes, he dervave funcons u have been ncreasng as shown n Fgure 5.3. The rapd ncrease of u s also observed o sar a =.3 and s value has reached as shown n he Fgure 5.3. We can epec ha u of problem (4) blows up fnally for he whole spaal doman ecep wo boundary nodes. The conour maps n Fgure 5.5 also ndcae hs concluson, snce he conours ge flaer and flaer as ncreases. Ths resul provdes some numercal evdence ha complee quenchng as suded n [6] may ndeed es. Fgure 5.4 dsplays he curve of spaal nodes a one me-level rgh before he quenchng. I s observed ha he nodes near he boundares march furher han hose nodes locaed n he mddle. The soluon a nodes ecep he mddle node can move furher n he emporal drecon. I hen possbly provdes nformaon of he soluon beyond quenchng. In able 5., we ls he compued quenchng me T a and he mamal emporal coordnaes ma j, for varous values of a and h. The mamal emporal marchng dfference ma j, T a,j,j s near.8 n he case of a = and h =.. However, we can also noce ha larger emporal 3

14 u(,) =. =. =.3 =.4 = u (,) 5 5 =. =. =.3 =.4 = Fg Fg Fg Fg. 5.5 Fgure 5.: Profles of he funcon u correspondng o dfferen spaal locaon wh a = π, θ =. Form op o boom, =,.4,.3,.,., respecvely. Fgure 5.3: Profles of he dervave funcon u correspondng o dfferen spaal locaon wh a = π, θ =. Form op o boom, =,.4,.3,.,., respecvely. Fgure 5.4: The curve of spaal nodes a one me-level mmedaely before quenchng (a = π, θ = ). Fgure 5.5: The conour map of numercal soluon for a = π, θ =. marchng dfference leads o he delay of he quenchng me, or n oher words, reduces he accuracy of he compuaon of he quenchng me. The reason s, as menoned n secon, he appromaon error of fne dfference scheme depends no only on he dervaves of soluon and szes of spaal and emporal meshes, bu also on he spacng and he shape of mesh cells. The shape of cells becomes oo narrow and he angles beween mesh lnes become oo small f he emporal marchng s enlarged oo much, whch causes he reducon of accuracy. Such a phenomenon has been well known n he fne elemen cone where he angles of each elemen canno be oo small. From he resuls lsed n able 5., we can mprove he accuracy of he numercal soluon hrough decreasng he spaal sep sze h and he nal emporal sep sze τ. For eample, he values of he quenchng me T a for h =. s close o he resuls obaned n [6]. Thus, wh he help of he adapaon n space, he quenchng phenomenon can be reproduced more accuraely. On he oher hand, we also noce ha he delay of quenchng me does no seem o affec he concluson for he qualave behavor of he soluon. 4

15 Table 5. Quenchng me T (a) and mamal emporal coordnaes ma,j ma,j h =. h =.5 h =. h =. j, T a ma j, T a ma j, T a ma j,,j,j,j a = a = π a = a = j, T a 6 Conclusons In hs paper we have presened a mesh-dependen varable emporal sep mehod for he quenchng problem ()-() n one dmenson under a movng mesh seng n he emporal drecon. A each me level, dfferen emporal ncremen s assgned a dfferen space locaons based on he soluon profle. We have analyzed and mplemened he mehod. The correspondng rregular mesh s consruced o sasfy he sably crera and o manan he numercal accuracy. Usng he mehod, we can provde evdence o a fac ha a complee quenchng soluon may es. The accuracy of he mehod on an rregular mesh may be lower han usual mehod on regular mesh snce he accuracy depends on shape of mesh cell as well [7]. If necessary some mesh refnemen dea (cf [3]) may be combned o mprove he accuracy. Acknowledgemens The auhors would lke o hank he anonymous referees for her valuable commens and suggesons. References [] A. Acker and W. waler, The quenchng problem for nonlnear parabolc dfferenal equaons, Lecure Noes n Mah., Sprnger-Verlag, New York, 564(976), -. [] C. Bandle and C. M. Brauner, Sngular perurbaon mehod n a parabolc problem wh free boundary, n Proc. BAIL IVh Conference, Boole Press Conf. Ser. 8, Novosbrsk, 986, 7-4. [3] Marsha J. Berger and Randall J. Leveque, Adapve mesh refnemen usng wavepropagaon algorhms for hyperbolc sysems, SIAM J. Numer. Anal., 35(998),

16 [4] C. J. Budd, Wezhang Huang and R. D. Russell, Movng mesh mehods for problems wh blow-up, SIAM J. Sc. Compu., 7(996), [5] Wemng Cao, Wezhang Huang and R. D. Russell, An r-adapve fne elemen mehod based upon movng mesh PDEs, J. Compu. Phys., 49(999), -44. [6] C. Y. Chan and Lan Ke, Beyond quenchng for sngular reacon-dffuson problem, Mahemacal Mehods n he Appled Scences, 7(994), -9. [7] C. Y. Chan, New resuls n quenchng, n Proc. s World Congress Nonlnear Anal., de Gruyer, Berln, 996, [8] C. Y. Chan, Recen advances n quenchng phenomena, Pro. Dynamc Sysems & Appl., (996), 7-3. [9] H. Cheng, P. Ln, Q. Sheng and R. C. E. Tan, Solvng degenerae reacon-dffuson equaons va adapve Peaceman-Rachford splng, SIAM J. Sc. Compu., 5 (3), [] S. F. Davs and J. E. Flahery, An adapve fne elemen mehod for nal boundary value problems for paral dfferenal equaons, SIAM J. Sc. Sas. Compu., 3(98), 6-7. [] K. Deng and H. A. Levne, On he blow-up of u a quenchng, Proc. Amer. Mah. Soc., 6(989), [] Rchard S. Falk and Gerard R. Rcher, Eplc fne elemen mehods for lnear hyperbolc sysems, Dsconnuous Galerkn Mehods, Lec. Noes Compu. Sc. Eng., Sprnger Berln,, 9-9. [3] M. Fremond, K. L. Kuler, B. Nedjar and M. Shllor, One-dmensonal models of damage, Advances n Mah. Scences and Applcaons, 8(998), no., [4] R. M. Furzeland, J. G. Verwer and P. A. Zegelng, A numercal sudy of hree movng-grd mehods for one-dmensonal paral dfferenal equaons whch are based on he mehod of lnes, J. Compu. Phys., 89(99), [5] H. Kawarada, On soluons of nal-boundary problem for u = u +/( u), Pul. Res. Ins. Mah. Sc., (975), [6] C. M. Krk and Caherne A. Robers, A revew of quenchng resuls n he cone of nonlnear volerra equaons, Dyn. Conn. Dscree Impuls. Sys. Ser A Mah. Anal., (3), [7] Park M. Knupp and Sanly Senberg, Fundamenals of Grd Generaon, CRC Press, Inc., Florda, 994. [8] J. Lang and A. Waler, An adapve Rohe mehod for nonlnear reacon-dffuson sysems, Appl. Numer. Mah., 3(993),

17 [9] Bors Lasdrager, Barry Koren and Jan Verwer, Soluon of me-dependen advecondffuson problems wh he sparse-grd combnaon echnque and a Rosenrock solver, Compuaonal Mehods n Appled Mahemacs, (), [] H. A. Levne, The phenomenon of quenchng: a survey, n Trends In The Theory And Pracce Of Nonlnear Analyss, Norh-Holland, Amserdam, 985, [] H. A. Levne, Quenchng, nonquenchng and beyond quenchng for soluon of some parabolc equaons, Ann. Mah. Pura. Appl., 55(989), [] Ruo L, Tao Tang and Pngwen Zhang, Movng mesh mehods n mulple dmensons based on harmonc maps, J. Compu. Phys., 7(), [3] Kewe Lang, Png Ln, Mng Tze Ong and Roger C.E. Tan, A splng movng mesh mehod for reacon-dffuson equaons of quenchng ype, J. Compu. Phys., o appear. [4] T. Lszka and J. Orksz, The fne dfference mehod a arbrary rregular grds and s applcaon n appled mechancs, Compuers & Srucures, (98), [5] D. Phllps, Esence of soluons of quenchng problems, Appl. Anal., 4(987), [6] Q. Sheng and A. Q. M. Khalq, A compound adapve approach o degenerae nonlnear quenchng problems, Numer. Mehods Paral Dfferenal Equaons, 5(999), [7] P. Sh, M. Shllor and X. L. Zou, Numercal soluons o one-dmensonal problems of hermoelasc conac, Compu. Mah. Appl., (99), [8] A. A. Tseng and S. X. Gu, A fne dfference scheme wh arbrary mesh sysems for solvng hgh-order paral dfferenal equaons, Compu. & Srucures, 3(989), [9] Dens Veynane and Luc Vervsch, Turbulen Combuson Modelng, Lecure of he Combuson School, 4. (hp:// langue=en) 7