Perfectly matched layers for the convected Helmholtz equation
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Perfectly matched layers for the convected Helmholtz equation Eliane Bécache — Anne-Sophie Bonnet-Ben Dhia — Guillaume Legendre. N° 4690 Janvier 2003. ISSN 0249-6399. ISRN INRIA/RR--4700--FR. THÈME 4. apport de recherche.
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]oku|]hkuyyskla6klnqp:rt8rs]`c¨t;v`cy z cbcbnut;xBc z klv z cbxcbv z cbvrsnl{¤t; rskladc z t;aHp:klv p:xoxonlku|Dp:rskut;v`y«klv bÝox @b e dº8|t;v`ykuysry½klv p:xoxonl{3klvo pLrs&p:v`ysnqp:rskut;v ¥cZ¤Mt;cªrs]`c»yskla6klnqp:sklr { rt¿rs]`c pJ}3kqp:n ^pcbvmoa¥Bcby°±\^]okuy cbadt°cyrs]`cxocycbv`|ct:¤mov`ysr&p:¥onucadt z cyD± r klnlno¥c^cZ¤cbc z rt+klv6rs]`c¨¤Mt;nlnut° klvo p;xy ~ÄwÅ yD± \^]`cwt;¥Osc|br³t:¤²rs]okuydxp:xcb@kuydrs]`c«p:vp:nl{Oyskuy@t:¤frs]`c«|t;v<cbscbv`|cwt:¤f¥t;rs] ~ÄwÅ adt z cbnuy°p;ydrs]`c rs]oku|&É<v`cyyft:¤¡rs]`c yxBt;voc nqp{cb+rcbv z y+rt@klvO´`voklrº{± i<kla6klnqp:²|t;vcbscbv`|cHp:vp:nl{Oycy#]pc6p:nlcDp z {w¥Bccbv |Dp:sskuc z t;mor^¤t;^rs]`cfµ8cbnla6]`t;nlrs£²c¶<mp:rskut;v O3kqp ¥Bt;mov z p:s{@klvrcb&p:n$c¶<mp:rskut;v½rc|]ovoku¶<m`cyklw v bÝ @!d$t;^rs]`c xt;nucH|t;v z klrskut;vLklv bA@PCdº±½i<mosxoskuysklvonl{ ¡cHxotDcHrs]p:rD$¤t;²rs]`c³|t;vc|brc z µ8cbnla6]`t;nlrs£©c¶mp:rskut;v rs]`c r ¡t adt z cbnuy^p:nl ¨pD{3y^|t;v<cbsc± vHt;rs]`cb¡ ¨t; z yDO|t;vrs&p:s{³rt²rskladc z t;aHp:klv©p:xoxonlku|Dp:rskut;v`yD3rs]`c8xocycbv`|c t:¤mov`ysr&p:¥onucfadt z cy z t3cyv`t;r¢ep uàc|brrs]`c²cZ¹H|bkucbv`|b{½t:¤|bnqp;y&ysku|Dp:n ~ÄwÅ yD± v klvp:nlnl{Onucbr^m`ycba6xo]p;yskl£Dc#rs]p:rDOklv©adtysr^xp:xBcby^|t;v`|cbsvoklvoH~ÄwÅ y¨¤t;¡rskladcZ¦Ø]p:sadt;voku|p:xoxonlku|Dp:rskut;v`yD |t3cZ¹H|bkucbvryp:c z cysklv`c z klv t; z cb rt8yép:rskuy'¤0{+c¶<moklcbadcbvry cysr&p:¥onlkuy]`c z ¤t; rskladc z t;aHp:klv6p:xoxonlku|Dp:rskut;v`yD± Æc«ys]`tD ¯rs]p:rHrs]okuy@|]`t;ku|c«kuydrt3t»cysrssku|brskl c «¤Mt;Hklv`ysr&p:v`|crs]`c½xp:srsku|bmonqp: z cbxcbv z p:v`|cUt:¤frs]`cyc |t3cZ¹H|bkucbvry8cbp: z klvoHrs]`c#¤Mc¶m`cbv`|b{©]p;yv`thadt;cæm`ysrsk¬´|Dp:rskut;vU¤Mt;rs]`c²xocycbvr¢|Dp;y&c± \^]`cPt;morsnlklv`cPt:¤rs]okuyxp:xBcb¢kuyp;y^¤Mt;nlnut° yD± k ¶mp:rskut;v`y#t:¤ rs]`cPy|Dp:rsrcbsklvoHxot;¥onucbap:cfxocycbvrc z klvyc|brskut;vªO±· ¤Mt;samonqp:rskut;vUklv«ph¥t;mov z c z z t;aHp:klvÈkuy@klcbv klv<t;nl<klvo #rÊ |t;v z klrskut;v`y½t;v¸rs]`c´|brsklrskut;m`y©¥t;mov z p:skucyD ]oku|&] p:cwcZ}Oxonlku|bklrc z rs]ot;mo]«adt z p:n$cZ}Oxp:v`yskut;v`yD± v klvp:nlnl{ors]`c² ¡cbnln¬¦Øxtyc z v`cyykuyxotDcbvUm`yklvo voc z ]`t;nla rs]`ct;s{± Q¡nqp;y&ysku|Dp:n8p:v z v`cb ~Ä«Å rc|&]ovoku¶m`cy°¡ klrs] |t;v`ysr&p:vr@|t<cZ¹H|bkucbv<ryDp:c z cy|bskl¥c z klv yc|brskut;v rs]p:voÉOyfrtp@adt z p:np:vp:nl{3yskuy°rs]`cychnqp{cby p:c6ysÉcbr|]`c z rt©¥ c ºxBcb'¤c|brsnl{waHp:r|]`c z ±dÀ^cysk z cyD rs]`cb{ p:cPp:¥`y&t;s¥oklvo``cZ}O|cbxor¢rs]`c²|bnqp;yysku|Dp:n ~ÄwÅ y¢klvxocycbv`|c²t:¤rs]`cfyt:¦º|Dp:nlnuc z klv<cbyc²mox`ysrscDp:a adt z cyD± i3c|brskut;v e»kuy z cbt;rc z rt»rs]`cp:vp:nl{Oyskuydt:¤+rs]`ccbst; z m`c©rtrs]`c©rssmov`|Dp:rskut;v t:¤+rs]`c½nqp{cbyD± ·¢v c¶<mokl;p:nucbv<r¤Mt;samonqp:rskut;vdt:¤rs]`c¨xot;¥onucba klrs]6~Ä«Å$y kuy sklrsrcbv6klvhrs]`c¡xo]{Oysku|Dp:n z t;aHp:klv rs]`c¨rs]oku|É3v`cyy t:¤rs]`c^nqp{cby¨p:xoxBcDp:sklvo²klv6rs]`ccZ}Oxocyyskut;vdt:¤rs]`c #klsku|]onucbr'¦Ørt:¦ºÊ#cbmoaHp:vov©aHp:x`yD± \^]okuy¡p:nlnutD yrt+xot°c rs]p:r ¥Bt;rs]L~ÄwÅ adt z cbnuy|t;v<cbsc@rtrs]`c³xo]{Oysku|Dp:n¡yt;nlmorskut;v p;yPrs]`cHnucbvors]¿t:¤rs]`cHnqpD{cbyPrcbv z y rt klvO´`voklr {±HÄwt;chxoc|bkuycbnl{$rs]`c6cbst;Pklvrs]`chxo]{Oysku|Dp:n z t;aHp:klv z t3cyfv`t;r z cbxcbv z t;v»rs]`cd~Ä«Å adt z cbn mov z cb|t;v`ysk z cb&p:rskut;v6p:v zPz c|bcDp;ycy cZ}3xt;v`cbvrskqp:nlnl{²¤p;ysr ¤Mt; ¥t;rs]Padt z cbnuyD± Ê8t;rc]`tD ¡cbcbrs]p:r |bnqp;yysku|Dp:n ~ÄwÅ ynucDp z rtdp:vcZ}3xt;v`cbvrskqp:nlnl{©nqp:scfyt;nlmorskut;vklv½rs]`c+nqpD{cbyDo ]`cbcDp;yrs]`cfy&t;nlmorskut;vU|t;a6xomorc z klrs] v`cb ~Ä«Å y¢kuycbp:v`cy|cbvrklvUrs]`c+nqp{cbyD± k }3rcbv`yskut;v@rtfrs]`c8|Dp;yct:¤ànqp{cby¡ klrs]³ysxp:rskqp:nlnl{6p:s{3klvo|t3cZ¹H|bkucbvry¡kuy z kuy|bm`yy&c z klv³yc|brskut; 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(26)
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(34) & !"#. $&%(')*$&%,+-%$.')*/012'. Å cbr½m`y½|t;v`ysk z cb½rs]`cªklvO´`voklrcªsklk z z m`|brD+|Dp:ss{3klvo p adcDp:vÈ®`mok z ®t° ²+ys]`tD v klv¸´`mocl@± \^]`c xot;¥onucbakuy6pUr ¡t:¦ z kladcbv`yskut;vp:n#t;v`cycbrhklv¿rs]`c435xonqp:v`c ]`cbc©rs]`c3 _0cyx ±65-a pJ}Okuy kuyxp:&p:nlnucbn _0cysx ±v`t;saHp:n art³rs]`cP ^p:nlnuy#t:¤rs]`c z m`|brD±¢Äªp:rs]`cbaHp:rsku|Dp:nlnl{rs]`c z m`|br#kuy z cZ´`v`c z ¥<{rs]`cPmov<¥Bt;mov z c z z t;aHp:klv M87:9(;=< >2?A@CBØ ]`cbDc @ z cbv`t;rcy¢rs]`c z kuysr&p:v`|cf¥cbrº ¨ccbvUrs]`c²yt;mov z ]p: z ¨p:nlnuyD± \$t z cy|bskl¥Bc#rs]`c+xot;xp:p:rskut;vwt:¤p;|t;m`ysrsku|+ ^pcyklv½rs]`c z m`|brDO ¨cPp;yysmoadc#rs]`c#¤t;nlnutD klvo³p:xoxotD}3k¬¦ aHp:rskut;v`yrt6¥c+p:nlk z E \^]`c+®`mok z kuy]`t;adt;cbv`ct;m`yD`v`t;vU<kuy|t;m`y8p:v z v`t;v]`cDp:r¢|t;v z m`|brsklc± E \^]`cfrs]`cbsadt z {3vp:a6ku|²xot3|cyycy¢p:c²p z kqp:¥p:rsku|± { F)G+kuyymo¥`yt;voku|²p:v z movok¬¤t;sa± E \^]`cfadcDp:vUcbnut<|bklr 4 E \^]`cfxcbsrsmos¥p:rskut;v`y+p:cfysaHp:nln p:v z c¶<mp:rskut;v`y¢p:cfnlklv`cDp:klvUrs]`c²p;|t;m`ysrsku|²¶<mp:vrsklrskucyD± c HJICKMLONQP"R,SUTZV RXW8> ¥cbklvodrs]`c+xomonuy&p:rskut;v kuyp;yysmoadc z _p:nlrs]`t;mo] E · ]p:sadt;voku|²rskladc z cbxcbv z cbv`| rs]okuy¤p;|brt;kuyymoxoxocyyc z rs]ot;mo]`t;mofr aZ± \^]`c#p;|t;m`yrsku|¢xocyymoc´cbn Dz Y LZ3M?O5[Trs]`cbv©y&p:rskuy'´cy¡rs]`c#|t;vc|brc z µ8cbnla6]`t;nlrs£fc¶<mp:rskut;v³klv@rs]`cklvO´`voklrc z m`|br Y aY aY _ @8a L]\^N`_bacTd d PhgC_ d gCa Y 7ji«klv Mk? 3 a e 5 a eXf 3 e d. d. d. ]`cbc!imlon a L M^Tkuyp|t;a6xp;|brsnl{PymoxoxBt;src z ¤0mov`|brskut;v6p:v z _p7:F)GrqtsJGp:v z gu7vRwqtsAG¨p:ccysxc|brsklcbnl{ rs]`c²Äªp;|]v<moaP¥cb_]N \yxz_ x{\eap:v z rs]`cf ^pcbvmoa¥BcbD|sJG+¥cbklvodrs]`c²yt;mov z cbnut3|bklrº{©klvrs]`cf®`mok z ± vp zoz klrskut;v»rt _ @8aZrs]`c xocyysmoc cbsk¬´cy+rs]`chÊ8cbmoaHp:vov»]`t;adt;cbv`ct;m`yP¥Bt;mov z p:s{w|t;v z klrskut;vt;vªrs]`c rº ¨tdsklk z ^p:nlnuy¢t:¤rs]`c z m`|b^r }m7 d M Y _i a v }~ d 7{>6t;o 5 d \$t6t;¥or&p:klvUp ¡cbnln¬¦Øxtyc z xot;¥onucba`p º&p z kqp:rskut;vU|t;v z klrskut;v O ]oku|&]Uycbnuc|bry^rs]`c t;morst;klvo ^pcyD`]p;y ysrsklnlnrtP¥Bc z cZ´`v`c z p:r¨klvO´`voklrº{±\^]okuy¡|t;v z klrskut;v@kuyv`t;vonut3|Dp:nàp:v z kuy¨cZ}3xocyy&c z rs]p:voÉOyrtPrs]`c #klsku|]onucbr'¦ rt:¦ºÊ8cbmoaHp:vov _ #rÊ a^t;xcb&p:rt;D±\^]okuy¢c¶moklcy¢rs]`cfklvrst z m`|brskut;v«t:¤rs]`c²yt:¦º|Dp:nlnuc z adt z cy¢t:¤rs]`c z m`|brD â$â¯ Ô yz+{ !éö.
(35) . ,i!
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(37) pTV!p . ]oku|&] p:c³rs]`c©yt;nlmorskut;v`yHt:¤_ @8aº¦f_iaklv¿p:¥`y&cbv`|c½t:¤8yt;mo|c \^]`cycp:cfklcbv¥{ Y. . _.ij7 >iap:v. z klrs] ycbxp:&p:rc z p:skqp:¥onucyD±.
(38) LZ5[TA?. LZ3 ?O5VT7. ]`cbc 5 _i a G)LZ5VT7 @ \ ? LZ5VT7 @f @ ? l > ? p:v z pJ}Okqp:n ^pcbvmoa¥Bcby! p:cfrs]`cPyt;nlmorskut;v`yt:¤. ". N L]\ N`_ a T. a N f. #. gC_. e. g a 7. l Q~ a a ? @ a. Å cbr¢m`yklvrst z m`|c $ gV@ _0e a $ $ G 7 \^N`_ a ? p:v z nucbr('DG 7 < G B z cbv`t;rcfrs]`c+®t3t;t:¤ G± ¤ & % ' G+ kuycDp:n p:v z c¶<mp:nàrt _0Bia. . -,/.. N gC_ 7. *). 14032 0 0 L]\^N`_. g a N \ N`_ a. aT ~. vdrs]p:r|Dp;yc Y kuy|Dp:nlnuc z p+xot;xp:p:rsklc¢adt z c±\^]`cv<moaP¥cb¨t:¤xot;xp:p:rsklcadt z cykuy¡p:vdklv`|bcDp;ysklvo ¤Mmov`|brskut;vt:¤$rs]`c#Äwp;|&]@v<moaP¥cb _ ]oku|&]½kuy^p;y&ysmoadc z rtP¥c#xBtysklrsklc8klv©rs]`c¢cbaHp:klvoklvodt:¤ rs]`c8xp:xcbD± i<kla6xonucd|Dp:nu|bmonqp:rskut;v`y ys]`t° rs]p:r²rs]`cht;moxcbnut3|bklrº{655 7 kuyfxtysklrsklc ¤t; Y98 adt z cyPp:v z v`cbp:rsklc6¤Mt; Y : t;v`cyD±· ¨cbnln¬¦ØÉ<v`t° v«!c uàc|br¢t:¤ rs]`c²xocycbv`|c²t:¤ ®tD kuyrs]`cPcZ}3kuysrcbv`|cO ]`cbv . ;. . \^N`_ a @. x8g x. @ . ?. t:¤adt z cy Y 8 ]oku|]¿]pc©p½v`cbp:rsklc@xo]p;ycHcbnut<|bklr { 7 p:v z pxBtysklrsklc³t;moxLcbnut<|bklr {±U\^]`cb{ p:c |Dp:nlnuc z klv<cbyc²mox`yrscDp:a adt z cy°± \^]`cPpJ}Okqp:n ^pcbvmoa¥Bcb! kuy|t;a6xonucZ}½k¬¤ W<'DG . = ,=P . 40 20 0 L]\^N _ a T,N`g a ~ 7 _ia \ N`_ a v½rs]okuy¢|Dp;yc Y kuy¢cZ}3xt;v`cbvrskqp:nlnl{ z c|bcDp;ysklvo³ ]`cbvo3?>@,BA p:v z |Dp:nlnuc z p:vcbp:v`cy|cbvr8adt z c± C /w012' $" %{') ED8 Q M0w%/ / / M#E|D $% Æc v`tD ^p:vrfrt@ycbnuc|br+rs]`ct;morst;klvoyt;nlmorskut;v _ @8aº¦f_iaZ ]oku|&]ª|t;scyxBt;v z y+rt@ysmoxcbsxBtysklrskut;vªt:¤ Y98 _0cysx ± Y : aadt z cy8 ]`cbv 3F> A _0cysx ±-3F> N(A aZ`kرÃc±cbklrs]`cb#rtdrs]`cPxot;xp:p:rsklc adt z cy¢ klrs]ªp xtysklrsklc _0cysx ±v`cbp:rsklc8a¨t;moe xcbnut3|bklrº{½t;rthrs]`c²cbp:v`cy|cbvr8adt z cy°± \ t z cbsklc+rs]`c+p:xoxot;xoskqp:rc 8 klsku|&]onucbr'¦Ørt:¦ºÊ8cbmoaHp:vov«¥Bt;mov z p:s{©|t;v z klrskut;v O ¡c+klv<rst z m`|c²rs]`c8¥t;mov z c z z t;aHp:klv M3Ob nut<|Dp:rc z klvª¥cbrº ¡ccbv»r ¡t@¥t;mov z p:skucyBG cysxc|brsklcbnl{nut3|Dp:rc z p:r 367(3 : p:v z 367b3 8 _Myccf´`mocPiaZoysm`|]Urs]p:rrs]`c²ysmoxoxt;srt:¤ rs]`cPyt;mo|cyi«kuyklv`|bnlm z c z klv>M O M3O7I H LZ3M?O5[T-l>Mk?Q3 : 3 3 8KJ ~ ) ) N gC_. ÓSÚ â$ÓMí.
(39)
(40)
(41) !#"$ %
(42) '&()+*-,
(43) /.0 . . h. . . . . klmocP \^]`cf¥t;mov z c zz t;aHp:klv ± Æc²ycbr M rs]`c²|t;a6xonucbadcbv<r&p:s{ z t;aHp:klv`y M : 7IH LZ3M?O5[T-l>M ?Q3mx 3 : J p:v z M 8 7IH LZ3 ?O5VT-l \^]`c²yt;nlmorskut;v Y t:¤3_ @8a¡rs]`cbvUyép:rskuy'´cyrs]`cf]`t;adt;cbv`ct;m`y#c¶mp:rskut;v. . v. Mk?Q3 W 3. 8KJ ~. klv M ? :p v z rs]`cbcZ¤t;c²|Dp:v¥c z c|t;a6xBtyc z t;vrs]`c+adt z cy°±Q¨t;v`yc¶<m`cbvrsnl{`klv>M : OkرÃc± ¤Mt;Q3mx 3 : O ¡c²]pc aY L]\^N`_ a T d PhgC_ 3 a eXf d. _ ha. p:v z klv>M 8 OkرÃc±¤t;Q3. W 3. 8. d.
(44) Y LZ3M?O5[T7 L Y LZ3 G. 8. . 8. ? ~ TA?. Y. 3 e. d. d. aY. 5 a e. g a Y 7j>. T 0
(45) :
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(69) LZ5[TA?
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(81) @D. ]`cbc ¤Mmov`|brskut;v`y 2 B z kuysxcbyskut;vcbnqp:rskut;v . l. ,i!
(82) q p
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