Multiscale Element-Free Galerkin Method with Penalty for 2D Burgers Equation

Transcription

1 Jrl kolog ll ppr lscl lm-r Glrk o Pl for Brgrs o S Cg L S o Yk prm of mcl Sccs cl of Scc rs kolog ls 80 Joor Br Joor rl zm ls Corrspog or scg87@oo.com Arcl sor c 8 rc 0 c rs form 6 Aprl 0 Accp 7 0 Grpcl bsrc Absrc s ppr mrcl mo c s bs o coplg b mlscl mo mslss mo pl s lop for Brgrs o. g of mslss mo or f lm mo s rmsg procss s o rr. s s bcs mslss mo ppromo s cosrc rl rms of s of os. Sc mog ls srs LS sp fco os o ssf Krockr l propr so pl mo s op o forc ssl bor coos s ppr. orr o ob f scl ppromo locl rcm bss s ppl. locl rcm bss m op poloml bss fcos or or lcl bss fcos. r poloml bss fcos r cos s locl rcm bss. s mlscl mslss mo pl ll pro mor ccr rsl spcll crcl rgo c rrs gr ccrc. s bl s propos mo s rc pproc for solg mor grl problms c ol lrg formo. Kor Brgrs o; mlscl mo; mslss mo; pl mo; crcl r Absrk lm krs kr s k br brgk g brsrk gg r k mlsclr k mslss g pl l bgk bg prsm Brgrs'. Klb k mslss lb k sr rgg l b pross rmsg k prlk. l kr k pgmpr mslss b sp r sg s o. Sk r krg brgrk LS rgkp bk k mm l Krockr k pl pk k mgksk sr smp pg lm krs. lm s k mpk pgmpr skl ls ss pg mp gk. Ass pg mp bol mrm pk fgs ss poloml m-m fgs lss ss g l. s fgs ss poloml g pl sbg ss pg mp. lsclr k mslss g pl k mmbrk sl g lb p rm r gg g mmrlk kp g lb gg. Al prc b k g cgk l s pk g mrk k mlsk msl g lb mm g mlbk b bk g bsr. K kc Prsm Brgrs'; k mlsclr; k mslss; k pl; ks krkl 0 Prb Prss. All rgs rsr..0 ROCO Brgrs o s frs roc b Bm [] o s for r problm 95. Lr Brgrs r s o s mmcl mol for fr rblc sock []. Afr s orks of Brgrs s l o b cll s Brgrs o. s Brgrs o s olr prl ffrl o of sco orr. s occrs ros rs of ppl mmcs sc s mollg of gs mcs rffc rofol flo or cosc rsmsso rblc problms bor lr bor. Brgrs o s olr o c cos coco rm scos rm mp rm. f scos rm s cl Brgrs' o s prbolc ls s prbolc. s clrl so Brgrs o bs m propr. o compl gomr complc l bor coos of Brgrs o cs c solos for prccl pplcos r r rsrc. As rsl m rsrcrs propos ros k of mrcl mo o ob solo of Brgrs o. s solo moologs r f ffrc mo [ ] f lm mo [5 6] bor lm mo B [7 8]. B r r som problms rs sc s m cosmg of ms gro complc sglr grls of B. srcrs fc cllgg sk lopm of robs mrcl mo for skg ccr solos of Brgrs o. cl k of mrcl mo cll mslss or msfr mo rc rsrcrs os. o pprom solo s mo ol s s of os scr problm om o lm s ol. rfor ms gro m ll b s. s s r co for g prcls sr rgo o rf solo. o msfr mo r mor pproc o l cllgg problms sc s lrg formo olr or g gr. os r s m msfr mo sc s lm-fr Glrk mo G [9 0 ] rprocg krl prcl mo RKP [0] mslss locl Pro rlkolog.m.m SS 80 7 SS

2 50 S Cg L & S o Yk / Jrl kolog Sccs & grg Glrk LPG mo []. r r som pprs c pplg msfr mo o sol Brgrs o sc s Og l. [] comb crcrsc Glrk CG mo G o sol Brgrs o Yog l. [] s lr Lgrg mo of fml solos o sol o-msol s Brgrs os X l. [] s rprocg krl prcl mo RKP o sol o-msol Brgrs o Zg l. [5] mplo mlscl lmfr Glrk mo for solg Brgrs o. m-90s gs rs orgs of sblzo scms from rol mlscl pproc prs rol mlscl mo [6 7]. s mo ffr sblzo cs ppr s spcl css of rlg sb-gr scl mollg cocp. formlo rm s gs rol mlscl mo s propos. srg po of s mo s composo of sclr fl o cors f scls. s coorkrs lop mlscl/sblz formlos for lrz comprssbl r Soks os [8] rc flo o [9] co-ffso o [0]. Bss r r som rsrcs lso comb rol mlscl msfr mo. or mpl Zg coorkrs mplo rol mlscl lm fr Glrk mo for r problms [] Brgrs o [5] Soks problm []. Jog Sg [] mplo rol mlscl msfr ppromo for ffc lss of lsoplsc formo. A ol of ppr s s follos Sco prss fml prcpl of lm fr Glrk mo G. mpss ppr s scrpo of mlscl lm fr Glrk mo Sco. Sco prss rsls cssos coclsos r r Sco 5..0 AAL PRCPAL O G Accorg o mog ls sr LS rpol [9 0] locl ppromo o fco s g b r m s mbr of bss fcos p r mooml bss fcos r r ko coffcs. A commol s lr bss s pro s p p rc bss m p p p p ko coffc s ob po b mmzg follog g cr rror orm r r r r r r r r 0 r r r m s sz of om of flc m of o. mmm of J rspc o l follog lr os A B f A s rbl coffc c b prss s A B r A B r mrcs f b B p p p A... [ ppromo c b prss s p Sbs o ; A B r sp fcos of G s f b ] p A B mog ls srs LS ppromo os o pss rog s o f cr. rfor LS sp fcos lck of Krockr l propr. s l o mposo of ssl bor coos c b kr. l s p p o flfllm of s propr rsrcrs cor problms rg rcl bor coo. ros srgs b propos o sol s problm. grl cs r pl mo [0] Lgrg mlplr pprocs [0] mof rol prcpls [] prrb Lgrg [5] rc colloco mo [6]. s ppr pl mo s b slc o forc ssl bor coos. J [ p ] r s g fco of compc sppor s mbr of os gboroo of for c g fco 0. prs ork follog cbc spl s cos s g fco

3 5 S Cg L & S o Yk / Jrl kolog Sccs & grg ARAOAL LSCAL L R GALRK O. Brgrs o Cosr o-msol copl olr scos Brgrs os s f or om ] [0 0] [ g b follog os r r loc log -s -s s ol mbr. l coos r bor coos for loc r l bor coos for loc r. Sr k orm r crzo b lopg k formlo. L 0 C o spc of rl solos gg fcos for ko rbls. lplg bo ss of. b mssbl gg fco grg or om l sg Gr s orm g 5 So fcol s ob s 6 o forc ssl bor coos pl mo s s. rfor fcol bcoms r s loc o ssl bor. kg ro of. 7 rcs o Sc ll g follog k form for loc loc 8 9 r s gg fco for s L r proc spc

4 5 S Cg L & S o Yk / Jrl kolog Sccs & grg lscl composo of Clsscl k ormlo Assm sclr fl c b compos o cors scl f scl 0 r s cors scl û s f scl. l s lso ssm r s lr b û. rl fco spcs of c scl r f s g 0 r fco g rprss plcm bor coo prscrb o smoo bor ls o zro. Bss s fco c lso b compos o cors f scl compos c s ŵ g b ŵ s fco spcs of c scl c b so s follos 0 0. lscl rol Problm mplog bckr lr s m crzo for o 8 lrzo rl solos 0 gg fcos r sbs o sr rol form o ob r + rprs o c m pos msp. Assmg cors scl f scl r lrl p rfor o c b spl o cors f scl prs o sb-problms bcoms cors scl sb-problm f scl sb-problm Afr rrrg o rspcl g

5 5 S Cg L & S o Yk / Jrl kolog Sccs & grg Solo of Scl Problm or bo pro of P s s [7-0] orr o ob f scl ppromo. proc of P fcos locl ppromo fcos s spc of fcos s s for ppromo. ppromo of plcm fl po s g b. Locl rcm bss m mplo poloml bss fcos or or lcl bss fcos. Som poloml bss fcos r so s follog [7] rs orr p= flc fco s ol o rm Sco orr p= r orr p= Sco orr poloml bss fcos r s s ork. So ob 7 cors scl rgo ppromo sg clsscl G mo s follos l o rms r cosr s f scl ppromo c s 9 rfr sbs os 8 9 o os 5 6. mr form for r ob rspcl. L ob 0 K K K K

6 5 S Cg L & S o Yk / Jrl kolog Sccs & grg rprss mbr of cors scl os = = rprss mbr of f scl os = =.6 mrcl ro =.. 6 =.. 5 ro procrs r s follog 0 L ; plc cors rbl o rm rg- s K s from o sol cors scl problm o rm K. mplo for cors os mplo 0 gr Cors os f os Brgrs o gr sos gomr for cors os f os Brgrs problm. cs mbr of cors os l rprss mbr of f os.0 RSLS A SCSSOS. Brgrs o compg rsls of mlscl G pl for Brgrs o r prs s blo for f os o g rl rl 5 clcl rror = m rl rl ; 6 f rror <0-5 l go o ; ls go o. bl rprso of globl mbrg. Appromo mbr gr A mrcl llsro of ppromo solos loc b mlscl mo =0. o 5 5 os = = 6 = 7 = 6 = 7 5 Cors os os os os bl sos rprso of globl mbrg. or ppromo of clsscl G mo mbr from o 6 rprs cors os l mbr from 7 o c f os. or ppromo of poloml bss fco mbr c sr from o 6 for mbr from 7 o 5 for r f s f os. gr A mrcl llsro of ppromo solos loc b mlscl mo =0. o 5 5 os =00

7 55 S Cg L & S o Yk / Jrl kolog Sccs & grg bl comprso of mmm rror of loc loc b mlscl mo for m =0. =0.8 ffr o sz for =00 bl sos comprso of mmm rror of loc loc for os. o ffr ms =0. =0.8 b s s bl. rom bl c obsr mmm rror rc crm of os for mlscl mo pl for Brgrs o. s sos ccrc crs crm of os. bl comprso rrors of loc for mlscl r os b o mlscl mo mlscl mo =0. =0.8 o 55 os for =00 rom bl bo o 9 o 50 o 6 o 65 rprs os r clos o mlscl r. c b obsr rrors of s loc os b mlscl mo r smllr o mlscl mo m =0. =0.8. rrors bl c mlscl mo proc gr ccrc comprg o o mlscl mo. 5.0 COCLSO =0. o Sz s ppr mrcl mo cll mlscl lm-fr Glrk mo pl s lop for Brgrs o. s propos mo loc fl s compos o cors scls f scls. s ll bl f scl formo c b cpr. Accorg o mrcl rsls ob ccrc of mlscl G pl s mpro crm of os. Bss rsls of s propos mlscl mo s br o mlscl mo bs o rrors of loc os c r clos o mlscl r. Aor gos of s mo s lss compol m o b s comprg cool mlscl mslss mo. s s bcs mlscl mo ll ol b ppl crcl r c rrs gr ccrc. s brgs coc for g or lg os sr rgos s ms gro s o s mo. So =0.8 o Sz loc loc Crcl os o mlscl =0. lscl o mlscl =0.8 lscl o o o o s propos mo s sbl o l g gr olr lrg formo problms. Ackolgm. fcl sppor from Br5 P Kmr Pg gg ls src rs Gr GP Q.J J0 r fll ckolg. frcs [] Bm. 95. Som c srcs o oo of l. or [] Brgrs J A mcl ol llsrg or of rblc. Ac Appl ccs [] Br A. R. 00. A ll mplc -ffrc Scm for o msol Brgrs o. Appl mcs Compo [] Kl S. A. R. Br A. Ozs mrcl Solo of O msol Brgrs o plc c-plc ffrc o. Jorl of Compol Appl [5] roco A.. P. Lbb A. Gro R. Cmrro rpolo-fr Spc-m msg for Brgrs o. Commcos mrcl o grg [6] og A. 00. A Glrk lm Approc o Brgrs o. Appl. Comp [7] Co.. osk l cproc Bor lm Alss of m-p Brgrs o. grg Alss Bor lms [8] Kk K.. osk Grlz Bor lm Approc o Brgrs o. J mr o g [9] Blscko. Y. Y. L L. G. 99. lm-fr Glrk o.. J. mr. o g [0] Blscko. Y. Krogz. Org. lmg slss o A Or c lopms. Comp o Appl c g [] L G. R. Y.. G A roco o sfr o r Progrmmg. Brl York Sprgr. [] Og J. X.. Zg L. Zg lm-fr Crcrsc Glrk o for Brgrs o. grg Alss Bor lms [] Yog. L. C.. S. P. S.. Alr lr Lgrg o of ml Solos for o- msol s Brgrs os. grg Alss Bor lms. 95. [] X S. S. S. o S. Km G. oo S. Y mrcl Solo of O-msol Brgrs o sg procg Krl co. Jorl of Compol Appl mcs. 7. [5] Zg L. J. Og X. X. g.. Zg. 00. rol lscl lm-fr Glrk o for Brgrs o. Jorl of Compol Pscs [6] gs. J. R lscl Pom Gr s co rcl-o-m ormlo Sbgr Scl ols Bbbls Orgs of Sblz o. Comp. o Appl. c. g [7] gs. J. R G. R. oo. Lc Q. J-Bps rol lscl o-a Prgm for Compol ccs. Comp. o Appl. c. g. 66. [8] s A. 00. O sblz f lm formlo for comprssbl r-soks os. Procgs of or S Jp Cofrc o Compol l mcs [9] s A.. J. R. gs. 00. A sblz m f lm mo for rc flo. Comp. o Appl. c. g [0] s A. R. A. Krrm. 00. A lscl/sblz lm o for Aco-ffso o. Comp. o Appl. c. g [] Zg L. J. Og. Jg C. R. 0. rol lscl lm r Glrk o for r Problms. Jorl of Compol Pscs [] Zg L. J. Og X.. Zg. B. Zg O l-scl lm-fr Glrk o for Soks Problm. Appl mcs Compo [] Jog. Y. K. Y. Sg rol lscl Alss of lsoplsc formo sg sfr Appromo. rol Jorl of Sol Srcrs [] L Y. Y.. Blscko L. G. 99. A mplmo of lm-fr Glrk o. Comp.. Appl. c. grg. 97.

8 56 S Cg L & S o Yk / Jrl kolog Sccs & grg [5] C Y. A. B. or A Compol ol for clo of Sol-Sol Ps rsformos oll. Sml. r. Sc. grg [6] kr Y. X. S. kr O Bor Coos lm r Glrk o. Comp. c [7] S. C. X. L C. K. L. 00. rc Pro of lm o for Srss s cors Crck ps. Comprs & Srcrs [8] zr L. P. Bollr Srcrl mcs of scolsc Sc Pls b Pro of lm o. Comp. o Appl. ccs g [9] L S. C. S. C. L Y.. Cg rc slss fol o for o-msol Crck ollg. or. Appl. rc. c. 8. [0] lk J... Bbšk Pro of lm o Bsc or A Applcos. Comp. o Appl. g