ALGEBRAIC SCHUR COMPLEMENT APPROACH FOR A NON LINEAR 2D ADVECTION DIFFUSION EQUATION

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1 st Annual Internatonal Interdsplnary Conferene AIIC Aprl Azores Portugal - Proeedngs- ALGEBRAIC SCHUR COMPLEMENT APPROACH FOR A NON LINEAR D ADVECTION DIFFUSION EQUATION Hassan Belhad Professor Samr Khallouq PhD Canddate Unversty of Abdelmalek Essaad FST Department of Mathemats Tanger Moroo Abstrat: Ths work deals wth a doman deomposton approah for non statonary non lnear adveton dffuson equaton. The doman of alulaton s deomposed nto q non-overlappng sub-domans. On eah sub-doman the lnear part of the equaton s desretzed usng mplt fnte volumes sheme and the non lnear adveton term s ntegrated expltly nto the sheme. As nonoverlappng doman deomposton we propose the Shur Complement (SC) Method. The proposed approah s appled for solvng the loal boundary sub-problems. The numeral experments appled to Burgers equaton show the nterest of the method ompared to the global alulaton. The proposed algorthm has both the propertes of stablty and effeny. It an be appled to more general non lnear PDEs and an be adapted to dfferent FV shemes. Key Words: Non lnear adveton-dffuson problems Strutured mesh Burgers equaton Fnte volumes method (FVM) Shur Complement (SC) The system of equatons Let us onsder the followng ntal boundary value problem: Fnd : Ω x (0 T) R suh that fs () ν + = g n Ω (0 T ) t x s (.) xt () = D () xt on Ω (0 T) ( x0) = 0 ( x) n Ω Where Ω R s a bounded polygonal doman and (0; T) T > 0 tme nterval. By Ω and Ω we denote the losure and boundary of Ω respetvely. We assume that the data have the followng propertes [6 7 8]: a) f s C (R) f s (0) = 0 f s C f s = b) ν > 0 ) g C( [ 0 T] ; L ( Ω )) * d) D C C( 0 T ; H ( Ω)) L ( Ω (0 T )) on Ω (0 T ) e) 0 L ( Ω ). In vrtue of assumpton a) the funtons f s satsfy the Lpshtz ondton wth onstant C f the funtons f s are fluxes of the quantty n the dreton x s ts represent onvetve terms the onstant ν > 0 s the dffuson oeffent. We use the standard notaton for funton spaes (see e.g. [9]): L p (Ω) L p (Ω (0 T)) denote the Lebesgue spaes W kp (Ω) H k (Ω) = W k (Ω) are the Sobolev spaes L p (0 T; X) s the Bohner spae of funtons p-ntegrable over the nterval (0 T) wth values n a Banah spae X C s the trae of some [ ] 7

2 st Annual Internatonal Interdsplnary Conferene AIIC Aprl Azores Portugal - Proeedngs- C([0 T]; X) (C ([0 T]; X)) s the spae of ontnuous (ontnuously dfferentable) mappngs of the nterval [0T] nto X. We shall assume that problem (.) has a weak soluton (f. [67]) satsfyng the regularty ondtons: p+ (.) L (0 T; H ( Ω)) an nteger p wll denote a gven degree of polynomal approxmatons. Suh a soluton satsfes problem (.) pontwse. Under (.) p+ C( [ 0 T ]; H ( Ω)) and C( [ 0 T ]; L ( Ω)) Fnte volume approah The fnte volumes approah onssts n dvdng the doman of alulaton Ω nto a fnte N M number of ontrol volumes (CVs) V (=.. N M) wth Ω= V. For a general CV we use the notaton of the dstngushed ponts (md-pont mdponts of faes) and the unt normal vetors aordng to the notaton as ndated n Fgure (rght). The mdponts of neghborng CVs we denote wth aptal letters W S et. (see Fgure left) these notatons are gven n [3]. = Fgure. FV strutured mesh of doman Ω By ntegratng the equaton (.) over an arbtrary CV V P and applyng the Green formula we obtan: (.) () t dv P+ f s( ()) t n as ds ap ν () t n a ds ap = g () t dv P VP S ap S ap VP a a S ap (a = e n w s) are the four faes of volume V P (see Fgure ) na = ( na na ) are the unt normal vetors to the fae S ap and µ(v P ) s the volume of ell V P. Approxmatng the lnear operator ν by the mplt Euler method and the non-lnear term by an explt approxmaton we get: n+ n P P n n+ n (.) µ ( VP) + fs( ) nas dsap ν nadsap = µ ( VP) gp t S ap SaP a a n n g = P g( x t ) dvp µ ( V ) VP P and 0 0 p = 0( x) dvp or P 0( xp). µ ( V ) = VP P 8

3 st Annual Internatonal Interdsplnary Conferene AIIC Aprl Azores Portugal - Proeedngs- - For the dsretzaton of dffuson term we have onsdered a entred dfferene sheme. - For the onvetve terms we use the numeral flux for the CV V P and S ap (a = e n w s): n fs( P) nas f K > 0 n (.3) fs( ) nas = n fs( I) nas ( I = E W S N) f K 0 n n n n K = fs( ) nas = ( P+ I). - For the approxmaton of the volume and surfae ntegrals we have employed the mdpont rule. Let us denote that n I s the onentraton on the volume V P (I=P E W N or S) at tme t n. The onentraton varables n+ I and n I (I=P E W N or S) n equaton (.) an be arranged as follows: n+ n+ n+ n+ n+ (.4) a = b P P E E W W N N S S P b P s a onstant dependng on the soure term g n P n P the dsretzed onveton flux the boundary and the ntal ondtons. Fnally the numeral sheme s expressed as the lnear system: AC n+ P = b A s a (N M N M) type matrx of oeffents a I (I=P E W N or S) C n+ P and b are the vetors of n+ P and b P respetvely. Shur omplement method Doman deomposton The doman Ω s deomposed nto mult-doman nonoverlappng strp deomposton q Ω Ω q Ω = = Ω and Ω Ω = when (fgure ). Let Γ denote the nterfae between Ω and Ω and Γ = Γ and by n the normal dreton (orented outward) on Γ for = q- and =+. For smplty of notaton we also set n = n. Ω Ω Ω 3. Ω q Fgure. Non-overlappng strp deomposton Consderng a retangular mesh of Ω eah subdoman Ω s parttoned nto n (= q) ells n X dreton and m ells n Y dreton (fgure 3). Fgure 3. Doman deomposton and strutured onformng mesh of doman Ω 9

4 st Annual Internatonal Interdsplnary Conferene AIIC Aprl Azores Portugal - Proeedngs- The problem (.) an then be expressed as : fs( ) ν + = g n Ω (0 T ) =... q t x s () xt = D () xt on ( Ω ( Γ+ Γ ) (0 T) (3.) ( x0) = 0 ( x) n Ω = on Γ =... q = on Γ n n The last two nterfae ondtons are known as transmsson ondtons on Γ. The deomposed problem (3.) s dsretzed on eah sub-doman Ω = q usng the mplt fnte volume sheme desrbed n Seton. For the nterfae ondtons we have used the entred dfferenes sheme. We obtan the followng system for = q- and =+: n+ n+ n+ n+ ap ( ) P W W N N S S = b P n Ω a n+ n+ n+ n+ ap ( ) P E E N N S S = b P n Ω b (3.) n+ n+ e = () w on Γ n+ n+ n+ n+ e + 0 ( ) e p p = on Γ d σ = e and σ = e f V P Γ ( = q ) σ = E and σ = W else b P s a onstant dependng on the soure term g n P n P the dsrtzed onveton flux the boundary and the ntal ondtons n Ω =...q. Shur omplement The methods based on Shur Complement exsts n two versons. The frst one uses the Steklov Ponaré operator and the seond one s an algebra verson. For exemple n [ 4] and n [5] one fnds presentatons of these methods (for lnear adveton dffuson equaton) used n the ontext of a fnte elements method and fnte volumes method respetvely. In ths work we have used an algebra verson of Shur Complement tehnque. Let C n+ and C n+ Γ denote the vetor of the unknowns of Ω (= q) and Г at tme t n+ (respetvely) and b denote the vetor of b P. The deomposed problem (3.) an be wrtten n the followng matrx form: n+ A A Γ C b n 0 A... 0 A + b Γ C (3.3) =. n Aq AqΓ C b q q n+ AΓ AΓ... AΓ q A ΓΓ C 0 Γ wth A A Γ desrbe respetvely (a) and (b) of system (3.) and A Γ A ΓΓ (= q) desrbe respetvely () and (d) of system (3.). The matrx A present the ouplng of the unknowns n Ω A ΓΓ t s related to the unknowns on the nterfae A Γ and A Γ representng the ouplng of the unknowns of eah sub-doman Ω wth those of the nterfae Γ + for (= q-). The system (3.3) an be sought formally by blok Gaussan elmnaton. 0

5 st Annual Internatonal Interdsplnary Conferene AIIC Aprl Azores Portugal - Proeedngs- Elmnatng C n+ (= q) n the system (3.3) yelds the followng redued lnear system for C Γ n+ : (3.4) SC Γ n+ = χ Γ and χ Γ = A Γ A b = q S = A ΓΓ A Γ A A Γ = q S s the Shur Complement matrx. After alulatng C n+ Γ C n+ an be obtaned mmedately and ndependently (n parallel) by solvng (3.5) A C n+ n+ = b A Γ C Γ (= q) Numeral Smulatons In ths seton we shall verfy the proposed approah by numeral experments. Let us apply FV mono-doman (FV-MonoD) and the ombned FV method Shur Complement (FV-SC) to the D vsous Burgers equaton [6 7 8]: (4.) νδ + x + x = g The spatal doman s the square Ω = ( ) the tme nterval T = (0) ν = 0.0 the ntal data 0 = 0 and the Drhlet ondtons D = 0. The rght-hand sde g s hosen so that t onforms to the exat soluton [8]: (x t) = ( e t )( x ) ( x ) As we want to examne the error of the spae dsretzaton we overkll the tme step so that the tme dsretzaton error s neglgble. Fgure 4 (abd) show respetvely the analytal the numeral mono-doman the multdoman (q=) and the mult-doman (q=9) solutons. Fgure 5 shows the onvergene of the proposed algorthm when varyng the mesh of alulaton. Fgure 4a. Analytal soluton Fgure 4b. Numeral monodoman soluton.

6 st Annual Internatonal Interdsplnary Conferene AIIC Aprl Azores Portugal - Proeedngs- Fgure 4. Numeral mult-doman (q=) soluton Fgure 4d. Numeral mult-doman (q=9) soluton Fgure 4. Numeral and analytal soluton Fgure 5. Convergene of numeral sheme Conluson A new approah ouplng mplt FV and Algebra Shur Complement methods appled to a sem lnear adveton-dffuson equaton on D strutured and onformng mesh s presented. The numeral experments show that the proposed algorthm appled to a non-overlappng mult-subdoman deomposton has both the propertes of stablty and auray. On the other hand ts redues the alulaton ost ompared to global FV alulaton. As perspetve of ths work we proet to develop a new algorthm ntegratng the non lnear adveton part mpltly. Ths algorthm wll nlude for example Newton method to ompute the adveton term after eah tme step of the numeral sheme. Referenes: [] A. Quarteron A.Vall Doman deomposton methods for partal dfferental equatons Clarendon Press Oxford 999. [] A. Quarteron A.Vall Theory and applaton of steklov-ponarré operators for boundary value problems Kluwer Aadem Publshers 99. [3] Mhael Shafer Computatonal Engneerng - Introduton to Numeral Methods. Sprnger- Verlag Berln Hedelberg 006.

7 st Annual Internatonal Interdsplnary Conferene AIIC Aprl Azores Portugal - Proeedngs- [4] P. Tarek A. Mathew Leture Notes n Computatonal Sene and Engneerng 6- Doman Deomposton Methods for the Numeral Soluton of Partal Dfferental Equatons. Sprnger- Verlag Berln Hedelberg 008. [5] S. Khallouq and H. Belhad Shur Complement Tehnque for Adveton-Dffuson Equaton Usng Mathng Strutured Fnte Volumes Advanes n Dynamal Systems and Applatons ISSN Volume 8 Number pp. 5-6 (03). [6] V. Doleší M. Festauer J. Hozman Analyss of sem-mplt DGFEM for nonlnear onvetondffuson problems on nononformng meshes Comput. Methods Appl. Meh. Engrg. 96 (007) [7] V. Doleší M. Festauer V. Sobotíkovà A dsontnuous Galerkn method for nonlnear onveton-dffuson problems Comput. Methods Appl. Meh. Engrg. 94 (005) [8] M.Beček M.Festauer T.Gallouët J.Háek and R. Herbn Combned trangular FV-trangular FE method for nonlnear onveton-dffuson problems ZAMM Z. Angew. Math. Meh. 87 No (007) / DOI 0.00/zamm [9] A. Kufner O. John S. Fučík Funton Spaes Aadema Prague