A High-Order Discontinuous Galerkin Method for Modeling Micro-Pulsed Plasma Thrusters*+
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1 Hgh-Od Dcnn Gln Mhd Mdlng Mc-Pld Pl h* G. Ln nd G.. Knd Dn ld Mhc n n Pdnc RI gln@c.bn.d g@c.bn.d IP--54 n n ncl hd dl c-ld l h ngl -ld/ l l. cl/h ln l dcn ld bh cd nd ncd h n - nd h-dnn. h hd bd n dcnn Gln n h dcn nd dn cnn nd bl n h L n. Pln ln MHD l n dn g nd nd -nn dnd h ll hgh d ccc h -hng. Indcn h ld l h PP h bn dd cll h []. h h' nl lc nd bn lld n cl nl nl dng h 96' nd 97'. Rnd n ll ll n h d h ngn PP bh n d ch d nc nd b ngh clng n [-5]. Mc-PP h cnnnl PP.. ln bld nd nd dng ld dchg. H h ggng chn b dn nd n c n ll [6]. c-pp h chcc dnn h d hch n d gnd l hn n h ll cnnnl PP dgn. h cl ng dchg h d nd h bld l l b µg. l cl c-pp bcd dd RL h nd d nd bl ln. µg/l [6]. Mdlng h n c-dn n ch c c n nd h b n l lc-dnc nd g dnc cl. cbnd c -cnn ch d dd cd h nn-lb c cll h ll c-pp dgn. In cl n ddn h l-c n h l n PP nd h clng h h nl cc nd h blng ld c n c-pp l nd dd: nn nnl nd d g. Mll nd lngh cl. l nd hh ncn. Lcl nn-lb c. Dn lcc cc. N l-ggng chn. In h d cnn-bd hdlg h h c c b h n ncd l bnd cndn. hgh-d bnd cndn lc l nd j h bn dld n [7] nd ll b lnd n h cn h cnn ch. l lll PI N. Gn cncn cd n c dlng c-pp. * Pnd P IP--54 h 7 h Innnl lcc Pln nnc Pdn 5-9 Ocb. gh b n n. Pblhd b h lcc Rc Pln c h n.
2 h cbl gn-hd-dnc MHD n dcbng l l n PP dndn. In ddn h n ngl cld nd hb d hblc/blc chc dndng n h ng h lg ng l nd l cl nld. M h h bn dl ddd n h blhd [8-]. H n h ln h cn ncl hd h h l-d cc. Hgh-d ccc n cng h nhn nn bh PP. M hgh-d hd bl lng ll cl n h c c-ld l h. Nglcng h n cnbn h lcn nd cndng h n nd lcn h dn h l l cn b dld ngl-ld/- bl. h n h dgnc- cndn h gnc ld l n l h hblc h dl MHD n. h dln bl Rnn l b Pll [] cn b l ndd l-dnn nd l hghd dcn. In h llng l h dcnn Gln hd h dcn nd dn n. hn n h cc lgh h MHD n. bnl n ncl ln MHD l n c-pp n - nd h dnn. cncld h b. Dcnn Gln ln n h dcnn Gln DG ln gnc dcndn n h µ h nd cnd ncd nd c l cnbn cl. cc lnn h MHD ll b dcd l n h n cn. lng h dcn-dn n h ll n h ncd nd c cnbn hch n gnl hb dn hcl. Dcnn Gln dcn ln h ln cnd h ln -dnnl n dcn cnd n n gn h g h l c hch dn h n. h h nl n h ndd Gln ln b llng b ncn nd ngng b d n d d h n h n nl nd h l. In h dcnn Gln ln ch ln d l cndng nl n ngng b nc nd H ncl l h nc bn ln : :n h chn bd n nd cndn. n bnd ng n Rnn l [4]. Dcnn Gln Dn h n d n h dcnn Gln ln dn l h n n d hd [5].. h n l bl. H cnd dl bl blc n h bl ccn µ dn h n h c cnbn: µ n L 4
3 . n g hn ndc h l bl µ h L nd - h blc n n n / µ n g h ln h bl hn ll. nd L L ch h L n b / L n b µ n g ngng b gn bn n ln ln hch ln. h n nl bl L n b / L n b µ n g H h bc dn cnbn ld h n d h bnd. h h bnd dnd b b ld bd n gng h cndng l h d h ln. h b cnl ld lcl b ln ch lc l nd cndn. In c cl/h dcn ld h nl cb hgnl lnl h n ch ln [4]. Gnng n h ngl-ld/- l n cn b d n cn n cc nn τ I n cn σ σ n cn I l l P P σ σ τ φ τ P h j j j δ τ l ε ε nd l h Gnld Oh L n P cn η lnl n l h h lcl d l h d MHD
4 P P P P P P MHD φ τ h l κ c µ P µ L η L h nn h nc h cnn l h h n d n h cl hblc chc. I h bn hn n [6] h n ll dgnc n h gnc ld cn dcll chng h chc l ncl ln. H ll h ln Pll []. ngl h d -l h cbn ncld n "ghh-".. h dgn d h cnd lc. In h h dgnc cd h h dgnc- cndn dd; h h gnl h cbn n h. h dcn cl cnd ddng h MHD n c nl h gh-hnd-d h ln n. h cn ln h dn h n nd lcn nd h l nb gnl nn. Ilnn h Incd l l h ncd l nd h d n h n h ln nd dd ccn ll
5 h dcnn n h l bn n djcn ln dcd n bcn.. In d l h ncd l n ln nc n-dnnl Rnn l l ncl l h. dn bnd h cd cndn nd h bnd h bnd gh ln. h cn h Rnn l ll ln bnd. ln h n-dnnl l n n h nl dcn hd ln-bnd ng h g h c h d h ln bnd. h nc n nnln ncn h c h g n n h cbn h l c c. h cbn h l c h ln n d n bl l hn h cnd. h h lnn h nd n h cnd. h cbn 99 nd h h h cld l- nd gh-gnc h cbn d Pll [] n λ l ln λ l λ l d [] h n nnh hch cnd h lcn ng: l h: * 4 4 * * 4 4 * * gn
6 cn n bn h nd cnd bl h h llng n c h ω υ h cnd bl nd ω υ h bl. h g: ω υ ω υ nd h. n n n l h ncl l h ln bnd. h llng ln h ndd l: 9 5 I I λ 6 I l H h l nd h dd l nd gh gnc h cbn. h l h h gh gnc clcl h cnd l. h λ h d cd h h gnc. Ilnn h h c ld n. bn h l d h bl ng h dcnn Gln ch. hn h c ch h c l ng h d. Dchl bnd cndn h n nd ng chcc bl cn b d l lcl h l h bn ld nd hn jc h l ng h hgnl cb lnl b. Ncl ln Mc-PP dld h c MHD ld nl cnd -dnnl dl bl h l h c-pp d n [6]. ccll h dld -dnnl MHD cd b bl dl h -dndn bnd cndn ll bl. h bnd cndn nln Inl : n. π..
7 7.4 lcn l h cndng nl cndn nln : l lcn g - lcn nnl ng n c MHD nd ld nl. g - ncd h PP dl h c MHD nd ld nl. g 4 - In nnl ng n c MHD nd ld nl. In g h h ncd h h b gn. h lnl d d n ch ln 4. In g nd 4 h cl l h l l. - nd h-dnnl MHD l n L/6 PP H cnd h L/6 PP g; nc [7] dnn. -dnnl MHD l n Mc-PP L/6 g - Mn n dcn. h h ld n h -dnnl ln hn n g 5. h bnd cndn l : Inl:.6 g/
8 n l. 5 µ j h: 7 H 4 µ 4π σ. 6 j σ h. h / R l µ 6 K R R l µ Ol:.6 5 g / g 5 - cd h -dnnl PP L/6 dl. l nl. 6K > R l µ ld h llng nl cndn:.6 g/ g/.45 <.6 g 6 - PP L/6 dl nl ccln. µ j.45.6 j.45µ.45 < <.6 h: 7 H 4 µ 4π σ. 6 j σ h. h g 7 - lc n PPL/6 ng n l hd. 6 K K.6 < <.6
9 lng cl dn d bn h l. h-dnnl MHD l n Mc-PP L/6 R L ll c c g 8 lc n PPL/6 ng cnd-d cl/h ln hd. I.c.4c Inl 6K 6R ll Ol.5 6K 6R >. g ch h h-dnnl PP L/6 nd bnd cndn. g 9 - lc n PPL/6 ng h-d cl/h ln hd. N l h h-dnnl L/6 PP hn n g. h cndng cd h hn n g. h bnd cndn : Inl:.6 g/ g - lc n PPL/6 ng h-d cl/h ln hd. In g 6 h h nl ccln bnd n h ln L/6. g 7 h cc -nn.. h d h cb lnl ncd h chng h h g 5. h h ln d gncnl nd bnd h-d h ll cngnc. N h h n l gn h c h dl h l h hc nd gdn hch cnnnl cl hd cnn hndl. H h dcnn Gln hd bl n h L n nd n lc Ol: h:.5µ j 7 H 4 µ 4π σ. 6 j σ h. h 6 K R l R µ
10 5.6 g/. l nl 6 K > l R µ h nl cndn l : g - cd h h-dnnl PP L/6 dl..6 g/ g/.45 <.6 µ j.45.6 j.45µ.45 < <.6 h: 7 H 4 µ 4π σ. 6 j σ h. h 6 K K.6 < <.6 g - lc n hdnnl PP L/6 dl. / R l µ In g nd l nnn cn h lc nd h l ng h ln ln. g 4 - l ng n h-dnnl PP L/6 dl.
11 h nd hgh-d hd dl c MHD ld ngl ld b n nd lcn. In h h hn ll h h hd nd h -nn.. cl nn cn ld hgh ccc n n h nc gdn nd n hc. In ll cl h MHD l n bln dl nd h nl cc nd ll ln lc l nd j bnd cndn. h ddn nc n d d nngl n nn h d h dgn c-pp. cnldgn h d b h nl Mhc g OR. n d NPI nd n IM P. ld l hn P. N. Gn PI l ggn gdng h. Rnc []. lb nd R.. nd Pnc d ld l-ld lcc ch. cc [] H. Kh nd P.. ch Dgn n nd ngn n ndcl-dn ld l h I h I/M// n Pln nnc & hb l llnd OH. [] P.. ch I.G. Mld P.G. Mld nd H. Kh On ld l h cll ln I-99-5 h I/M// n P ln nnc & hb -4 n 999 L ngl. [4].. chlnn R. I.G. Mld P.. ch nd P.G. Mld Pld l h n d n cbl I h I/M// n Pln nnc & hb 6-9 l Hnll lb. [5] H. Kh P.. ch P.G. Mld nd I.G. Mld nl nd hcl ngn n n-nch cl ld l h I h I/M// n Pln nnc & hb 6-9 l Hnll lb [6].. Glcn M.. Dllgn.P. L nd G.G. nj Mcln ch RL I h n Pln nnc & hb 6-9 l Hnll lb. [7] G.. Knd nd. Mc l: ndnl nd ln ng. [8]. D nd P.R. dd l Rnn l nd hgh-d Gdn ch hblc cnn l.. Ph [9]. L. Zch. Mlgl nd P. lll hghd Gdn hd ldnnl dl gnhddnc IM. c [] R..Pn M.H. nd.r.nc n gnc l n n b cdn L cd.. Ph [] K.G.Pll n Rnn l Mgnhddnc chncl R I R 94-4 I N Lngl 994. [] O.. n. hl nd D.. bhd n lc ch nndl gnhddnc.. Ph. 997 [] P.lll M.D nd D.D. n n Dnc Mhd h Ncl ln Pl ld n chncl R RL-- 99 Lnc L Nnl Lb 998. [4].. bn nd G.. Knd Dcnn Gln Mhd h MHD n.. Ph [5]. Gl nd P.. R n ln Mhd N- n ng-lg N Y/ln 986. [6].. cbll nd D.. n h c nn dgnc n h ncl ln h gnhddnc n.. Ph [7] P.G.Mlld nd P.. ch Mdlng l bln n ln ld l h I 96-7 nd I/M// n Pln nnc & hb l n L.
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DEPARTMENT OF MATHEMATIC EDUCATION MATHEMATIC AND NATURAL SCIENCE FACULTY
HANDOUT ABSTRACT ALGEBRA MUSTHOFA DEPARTMENT OF MATHEMATIC EDUCATION MATHEMATIC AND NATURAL SCIENCE FACULTY 2012 BINARY OPERATION We are all familiar with addition and multiplication of two numbers. Both