High-order fundamental and general solutions of convection-diffusion equation and their applications with boundary particle method

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1 Hgh-order fndaental and general soltons of convecton-dffson eqaton and ther alcatons wth bondary artcle ethod W. Chen * Sla Research Laboratory, P. O. Bo. 34, 325 Lysaer, Norway E-al: wenc@sla.no (sbtted Oct. 2, revsed 8, Feb. 22) Abstract In ths stdy, we resented the hgh-order fndaental soltons and general soltons of convecton-dffson eqaton. To deonstrate ther effcacy, we aled the hghorder general soltons to the bondary artcle ethod (BPM) for the solton of soe nhoogeneos convecton-dffson robles, where the BPM s a new trly bondaryonly eshfree collocaton ethod based on ltle recrocty rncle. For the sae of coleteness, the BPM s also brefly descrbed here. Keyword: convecton-dffson eqaton, hgh-order fndaental solton; hgh-order general solton; bondary artcle ethod; radal bass fncton; eshfree; ltle recrocty ethod.. Introdcton The hgh-order fndaental soltons lay a votal role n the ltle recrocty BEM (MR-BEM) []. On the other hand, the bondary artcle ethod (BPM), recently ntrodced by the resent athor [2], also reqres the se of the hgh-order general soltons. In addton, an nderlyng relatonsh between the radal bass fncton (RBF) and Green second dentty hghlghts the ortance of these oerator-deendent soltons [3]. A ernel RBF-creatng strategy [3,4], whch eloys these hgh-order fndaental or general soltons as the RBF, s resented for the dal recrocty BEM * Ths research s sorted by Norwegan Research Concl.

2 [5], bondary not ethod [6], and Kansa s ethod [7]. To the athor s best nowledge, the elct hgh-order fndaental soltons are now avalable only for the Lalace [], Helholtz and odfed Helholtz oerators [8]. Based on the nderlyng relatonsh between the convecton-dffson eqaton and the odfed Helholtz eqaton [9], we derve the hgh-order fndaental and general soltons of the convecton-dffson eqaton []. In ths stdy, we wll gve a systeatc descrton of ths recent advance. In addton, to deonstrate the effcacy of these hgh-order soltons, we aly the to the bondary artcle ethod for the solton of soe nhoogeneos convecton-dffson robles. The BPM s develoed based on the ltle recrocty rncle [2,], whch ses ether hgh-order nonsnglar general soltons or snglar fndaental soltons to evalate hgh-order hoogeneos soltons, and then ther s aroates the artclar soltons. Le the MR-BEM, the BPM does not reqre any nner nodes for nhoogeneos robles and therefore s a trly bondary-only technqe. Unle the MR-BEM, the BPM does not need to generate ore than one nterolaton atr snce we develoed the effcent recrsve ltle recrocty forlas, whch treendosly redces cotng effort and eory reqreents. In addton, the BPM s essentally eshfree, sectral convergence, easy-to-se, and syetrc technqe. The rest of ths aer s organzed nto 4 sectons. We dscss the dfference between the snglar fndaental solton and nonsnglar general solton and then ntrodce the hgh-order fndaental and general soltons of the convecton-dffson eqaton n secton 2. The bondary artcle ethod s brefly descrbed n secton 3, and followed by the leentaton sses: overflow and ll condtonng n secton 4. Fnally, Secton 5 resented nercal testng the convecton-dffson hgh-order general soltons wth the BPM to soe 2D nhoogeneos convecton-dffson robles nder colcated geoetres. 2

3 2. Hgh-order fndaental and general soltons of Convecton-dffson oerator The general solton satsfes { } R, () where R s a dfferental oerator. The general solton at orgn s eqal to a lt vale rather than zero and nfnty. In contrast, the fndaental solton * has to satsfy * { } + R, (2) where reresents the Drac delta fncton whch goes to nfnty at the orgn ont and s eqal to zero elsewhere. Accordng to the above defntons, the essental dstnctness between the general solton and fndaental solton of a dfferental oerator s that the forer s nonsnglar whle the latter s snglar. 2.. Hgh-order general soltons Consder a hoogeneos sotroc regon Ω. The steady convecton-dffson eqaton s gven by 2 v D ( ) ( ) κ( ), n Ω (3) where v v denotes a velocty vector, D s the dffsvty coeffcent, κ reresents the reacton coeffcent. The varable can be nterreted as teeratre for heat transfer robles, concentraton for dserson robles [5]. Assng that convectve velocty v v eonental varable transforaton [9] and dffsvty D are constant, we have an 3

4 v v ν r 2D e w, (4) where w() s an nteredate feld varable, and r v s the dstance vector between the sorce and feld onts. Convecton-dffson eqaton (3) can be accordngly rewrtten as a odfed Helholtz eqaton where 2 2 w τ w n Ω, (5) 2 [( v 2D) κ D] 2 τ v +. (6) Accordng to Itaga s three-densonal hgh-order fndaental soltons of the odfed Helholtz eqaton [8], t s very easy to get the corresondng hgher-order general soltons,.e. w n 2+ + () r A ( r) I ( τr) τ n 2+, n 2, (7) where A A - /(2τ 2 ), A ; n s the denson of the roble; denotes the order of general solton; r s the Ecldean dstance nor; I reresents the odfed Bessel fncton of the frst nd, whch can be transfored to the odfed shercal Bessel fncton when n3. Usng coter software Male, we verfed that those hgherorder fndaental and general soltons are also establshed for ore than 3 densons. Throgh an nverse transforaton rocess, we derve the hgh-order general soltons of the convecton-dffson eqaton v v r τ n 2+ 2D n 2+ + () r A e ( r) I ( τr), n 2, (8) where τ s defned by forla (6). 4

5 The above hgh-order general soltons of the convecton-dffson eqaton are also valdated by atheatcal dedcton aroach wth the hel of coter algebrac acage Male. Naely, frst we rove the zero-order general solton, and then, t s fond that R } conssts only of (-)th or lower order general soltons of oerator R. { Therefore, the -order general solton s roved. It s noted that the general soltons (7) and (8) are nonsnglar at the orgn and actally nfntely contnos bt grow eonentally as r. Ths, t s often regarded as roer soltons []. In contrast, the fndaental solton decays eonentally and s consdered as roer solton. However, f or roble s to handle the data or PDEs on the bonded doans,.e. nteror robles, the growth at nfnty s no longer an sse, snce the eanson ters always rean fnte []. We have soe ersasve nercal deonstratons of the tlty of these general soltons wth the bondary not ethod [7,] n solvng nteror robles. In later secton, we wll show these general soltons n conncton wth the BPM cold be very sccessflly aled to evalaton of soe tycal convecton-dffson robles on bonded doans. In general, the fndaental soltons can be aled to both eternal and nteror robles, whle the general soltons are bt the solton of nteror robles. Bt the general soltons ay nevertheless be alcable even to eteror robles f we se the trncated doan aroach, e.g., for eteror wave robles wth a roer nonreflectng bondary condton at trncated bondary Hgh-order fndaental soltons The fndaental solton of steady convecton-dffson eqaton s deterned by 2 v D ( ) ( ) κ( ), n Ω (9) 5

6 where reresents the Drac delta fncton at a sorce ont. Followng the recedng rocedre for hgh-order general soltons, we can easly derve the -order fndaental solton * v v r τ n 2+ 2D n 2+ + () r B e ( r) K ( τr), n 2, () where K reresents the odfed Bessel fncton of the second nd. As n the odfed Helholtz case, the hgh-order fndaental solton of convecton-dffson eqaton has the snglarty order of (r 2-n ) ecet the 2D case where the only snglarty occrs n the zero-order fndaental solton. For nstance, the snglarty for 3D case s always r - rresectve of the order of fndaental solton. In other words, the snglarty order has nothng to do wth the order of a fndaental solton Evalatng fndaental and general soltons It s worth ontng ot that we can greatly slfy the above-gven standard fors of hgh-order soltons nvolvng soe Bessel fnctons. Ths, the cotng effort for the cold be trval. For eale, the zero-, frst-, and second-order general soltons of 3D convecton-dffson eqaton are resectvely slfed va coter algebra acage Male as () r 2 2 v v ( τr) v r A snh D e, () π τr () r A ( τr) snh( τr) r 2 v v r cosh D e 2, (2) π τr 2 () r ( τr) + 3snh( τr) 3r cosh( τr) r A 2 v v 2 2 r snh 2D e, (3) π τr 6

7 where snh and cosh resectvely denote the sne-hyerbolc and cosne-hyerbolc fnctons. 3. Bondary artcle ethods In secton 4 the above-derved hgh-order general soltons of convecton-dffson eqaton are aled wth the bondary artcle ethod to soe 2D nhoogeneos robles. For the coleteness and very recent orgn of the BPM, we gve a bref descrton of the ethod n ths secton. Consder a steady convecton-dffson syste 2 v D ( ) ( ) κ( ) f, Ω, (4) R, S, (5) n N, S T, (6) where n s the nt otward noral. The syste solton can be slt as h +, (7) where h and are the zero-order hoogeneos and artclar soltons, resectvely. The ltle recrocty rncle asses that the sffcently sooth artclar solton can be aroated by hgher-order hoogeneos soltons,.e. + h + h h, (8) 7

8 where serscrt s the order nde of hoogeneos solton. Throgh an ncreental dfferentaton oeraton va the convecton-dffson oerator, we have sccessvely hgher order bondary dfferental eqatons: R R R R h h n ( ) R( ) ( ) ( ) N ( ) n ( ) { h( )} f ( ) R { ( )} { ( )} ( f ( ) R { ( )}) h n n 2 { h ( )} R { f ( )} R { ( )} 2 { ( )} ( R { f ( )} R { ( )}) h n n, (9a), (9b), 2,3,, (9c) where R {} denotes the -th order of convecton-dffson eqaton, say R {}RR {}, and R {}R{}, and are nots ndces resectvely on Drchlet and Neann bondary. The -order hoogeneos solton can be aroated n ters of the - order general solton bass fncton h L ( r ), (2) where L s the nber of bondary nodes, and denotes the nde of sorce onts on bondary. are the desred coeffcents. Note that we can se the fndaental solton * nstead of general solton n Eq. (2). However, the se of fndaental solton reqres a fcttos bondary otsde hyscal doan as encontered n the MFS [2]. Collocatng Eqs. (9a,b,c) at all bondary nots n ters of reresentaton (2), we have the BPM bondary dscretzaton eqatons : 8

9 b n N n r R r L L, (2a) { } { } { } { } b n f n r f r L L R R R R, (2b) { } { } { } { } { } { } L L b n f n r f r R R R R R R 2 2, 2,3, (2c) Whenever the ed bondary condtons are resent, the resent BPM nterolaton atr s nsyetrc even for self-adont oerator. By analogy wth Fasshaer s Herte RBF nterolaton [3] the resent athor also roosed a syetrc BPM forlaton for self-adont oerator [2]. Snce the convecton-dffson eqaton s not self-adont, ths stdy sly ses the resent nsyetrc BPM schee. As n the MR-BEM [], the sccessve rocess s ternated at soe order M, naely, { } R M M. (22) The ractcal solton rocedre s a reversal recrsve rocess: L M M. (23) It s noted that de to 9

10 { () r } () r R, (24) the coeffcent atrces of all sccessve eqatons (2a,b,c) are the sae,.e., let Q b, M,M-,,,. (25) The LU decooston algorth s stable for ths tas. Then we can eloy the obtaned eanson coeffcents to calclate the BPM solton at any not,.e. M L ( ) ( ). (26) It s noted that the BPM solton (26) s also a RBF wavelet eanson seres, where the dfferental eqaton order s sed as the dlaton araeter and s the translaton araeter. It s also worth ontng ot that the above ltle recrocty solton rocedre s n fact eqvalent to that wthn the MR-BEM. The dstncton s that the resent one has a atheatcal rocedre cotatonally ore effcent snce t s a recrsve teraton solton whch does not need to evalate costly nterolaton atrces of hgh order artal dfferental eqatons. We wll gve a detaled dscsson on ths sse and resent the MR-BEM forlatons based on the general solton and the fndaental solton n a sbseqent aer. 4. Ileentaton sses: overflow and ll-condtonng As the araeter τ n forla (6) grows, the condton nber of the BPM dscretzaton atr Q n (25) ncreases qcly and the a and n entres resectvely assocated wth the general solton and the fndaental solton becoe rone to overflow. Conseqently, the solton blows at ed vale of τ. Wth the noral

11 nercal algebrac rocedre, the overflow and ll-condtonng are two aor bottlenecs n alyng the BPM to convecton robles wth large τ vale. To avod the overflow, ths stdy odfes the BPM nterolaton atr Q n (25) by ltlyng t wth an eonental araeter to scale down the a entry wth the general solton,.e. τ L Lc { e ( r )}{ e } c τ b, M,M-,,,, (27) where L c denotes a characterstc length. The entres of nterolaton atr n (27) s actally calclated n the followng anner: e τl c v v r [ n 2 τlc + + τr 2D n 2+ τr ( r ) A e ( τr ) e I ( τr ) ], n 2. (28) Here τr e I τ ) s seen as a new fncton, whch s never overflow and ch ore n 2( r sooth than the odfed Bessel fncton I ( τ ) n 2 r. Many software acages nclde the sbrotne or secal fncton for calclatng τr n 2( r e I τ ). It s worth notng that corresondng to (27), the forla (26) s revsed as M L τlc τlc ( ) ( e )[ e ( )], (29) where e τl c v v [ n 2+ v r τlc + + τr 2D n 2+ + τr ( r ) A e ( τr) e I ( τr )], n 2. (3) If the BPM ses the fndaental solton, we need to scale the n entry n coarson to (28),.e.

12 e τl c v v v r [ r ], n 2. (3) τlc + τr 2D n 2+ τr () r B e ( τr) e K ( τ ) * n 2 The reander of the rocedre s slar to that of the BPM sng the general solton. On the other hand, we stll lac a good devse to deal wth the ll-condtonng sse. Here we resent a concetal aroach called Hadaard rodct recondtonng. Defnton 4. Let atrces P[ ] and Q[q ] C N M, the Hadaard rodct of atrces s defned as Po Q [ q ] C N M, where C N M denotes the set of N M real atrces. Observng the BPM nterolaton atr Q n (25), we note that Q can be decoosed va the Hadaard atr rodct,.e. Q v v v r τr 2D n 2 r + τ { e } A e ( τr ) [ e I ( τr )] H Qˆ n 2 o. (32) Ths, the forlaton (25) s restated as [ H Qˆ ] b o, M,M-,,,. (33) Note that the atr Q ˆ wll never be overflow and s well condtoned. At sse s how to anlate and recondton the agented H, whch s a oorly-condtoned eonental fncton atr. Albet etreely sle n ts for, the Hadaard rodct s qte hard to analyze and anever [4-6]. For the te beng, the ll-condtonng s stll an oen sse n the BPM solton of the convecton-dffson roble. 2

13 5. Nercal valdatons The tested case s a 2D steady convecton-dffson roble descrbed by D 2 v η( y κ e + ) ( 2 4η + 2σ ) (34) wth the bondary condtons of the Drchlet tye (5) and Neann tye (6), where η σ + σ + κ. The eact solton s D, v v y -σ, κ3σ 2 2 /2, ( 2 ) 2 η ( + y 2 e ). (35) The Peclet nber Pe s defned as v ν Lc Pe. (36) D The Peclet nber ndcates the relatve ortance of convecton verss dffson. Fg.. llstrates the 2D rreglar geoetry. Ecet secfed Neann bondary condtons at and y edges, the otherwse bondary are all of the Drchlet tye. Eqally saced nots were aled on the bondary. The L 2 nors of relatve errors are calclated based on the nercal soltons at 46 nner and bondary nodes. The absolte error s taen as the relatve error f the absolte vale of the solton s less than.. Table dslays the eerental reslts, where L reresents the nber of the sed bondary nots. It s seen that the BPM schee based on the hgh-order general soltons rodces stable soltons wth ncreasng nber of bondary nodes for the resent nhoogeneos convecton-dffson robles. In artclar, the BPM soltons are fond very accrate even wth a sall nber of bondary nodes. It s stressed that we dd not se any nner nodes here. For hgher Peclect nber robles, the BPM soltons becoe nstable de to very oor condton of the BPM nterolaton atr. As n the 3

14 BEM, soe recondtonng technqes cold be sed to handle ths sse, whch s beyond the resent stdy. Now one can conclde that the resented hgh-order general soltons of convecton-dffson eqaton do wor well n ths nercal eerent. Table. L 2 nor of relatve errors for 2D nhoogeneos convecton-dffson roble. L Pe24 9.e-3.4e-3 3.6e-4 3.6e-4 L Pe72.2e-2 4.5e-2 2.2e-3 4.5e-4 4

15 References. Nowa AJ, Neves AC (ed.). The Mltle Recrocty Bondary Eleent Method. Sothaton: Cot. Mech. Pbl., Chen W. New RBF collocaton schees and ther alcatons. Internatonal Worsho Meshfree Methods for Partal Dfferental Eqatons, Bonn, Gerany, Set Chen W, Tanaa M. Relatonsh between bondary ntegral eqaton and radal bass fncton. The 52th Syos of Jaan Socety for Cotatonal Methods n Engneerng on BEM. Toyo, Chen W, He J. A stdy on radal bass fncton and qas-monte Carlo ethods. Int. J. Nonlnear Sc. & Ner. Slaton, (4); :2. 5. Partrdge PW, Brebba CA, Wrobel LW. The Dal Recrocty Bondary Eleent Method. Sothaton: Cot. Mech. Pbl., Chen W, Tanaa M. A eshless, eonental convergence, ntegraton-free, and bondary-only RBF technqe. Cot. Math. Al. (n ress), Kansa EJ. Mltqadrcs: A scattered data aroaton schee wth alcatons to cotatonal fld-dynacs. Cot. Math. Al. 99;9: Itaga M. Hgher order three-densonal fndaental soltons to the Helholtz and the odfed Helholtz eqatons. Engng Anal. Bondary eleents. 995;5: L BQ, Evans JW. Bondary eleent solton of heat convecton-dffson robles. J. Co. Phy. 99;93: Chen W. RBF-based eshless bondary not ethod and bondary artcle ethod. Proc. of Chna Congress on Cotatonal Mechancs 2, Gangzho, Chna, Dec. 2.. Dffy DG. Transfor Methods for Solvng Partal Dfferental Eqatons, CRC Press, Florda, Golberg MA, Chen CS. The ethod of fndaental soltons for otental, Helholtz and dffson robles. In Bondary Integral Methods - Nercal and Matheatcal Asects, (Ed. by M.A. Golberg),. 3-76, Cot. Mech. Pbl., 5

16 Fasshaer GE. Solvng artal dfferental eqatons by collocaton wth radal bass fnctons. Proceedngs of Chaon, 996, (Mehate, A. and Rabt, C. and Schaer, L. ed.), Horn RA. The Hadaard rodct, n Matr Theory and Alcatons, Seres: Proc. Of Syosa n aled atheatcs, V. 4, 87-69, Johnson C.R. Ed., Aercan Matheatcal Socety, Provdence, Rhode Island, 99. Acadec Press, Chen W. Relatonsh theore between nonlnear olynoal eqaton and the corresondng Jacoban atr, Int. J. of Nonlnear Scences and Nercal Slaton,, 5-4, Chen W, Sh C, He W, Zhong TX. The DQ solton of geoetrcally nonlnear bendng of orthotroc rectanglar lates by sng Hadaard and SJT rodct, Coters & Strctres, 74(), 65-74, 2. 6

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