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[6]"Ow@|Oa@ wO pOt l} 1u=t} Q j}kO |xOvvmpL R= xO=iDU= =@ 2006 p=U QOOQivt Gwt |wQq=@ u; ltmx@ w OWx=Q= jtasm ?; CqO=at |=Q@ OwOLt sHL2001 |=yp=U QO xOWx=Q= |Oa@ wO |=ypOt [7]"CiQo Q=Qk |UQQ@ OQwt xOvvmW\UwDt Q=W ẁv R= u=t} Q |@} QkD |xOvvmpL R= =yv; |xaUwD QO xm 2007 whrDNt |=y|oS} w |R=Ux}@W x@ QO=k |@ wNx@ 'xOW xO=iDU= 2�WAF� Q=OuRw

|t=vwU G=wt= |rL=U C=Q]N R= uDU=m QO Ovr@ G=wt= |wQq=@ |v}@V}B |}=v=wD?}yt G=wt= |U=U= |=yCrYN |R=Ux}@W Qw_vtx@ "Ovm|t =i}= |D=}L |WkvxO=iDU= OQwt |ivt w C@Ft OQivt G=wt= R= |@}mQD =} OQivt G=wt= ,qwtat '|t=vwUh}YwD QO |}xOQDUo OQ@ Q=m jtasm ?; |]NQ}e CqO=at uwvm =D "OvQ}o|t Q=Qk|=ypOt `k=w QO "Ov=xDW=O Q=O?}W pL=wU |wQ OQivt G=wt= |wQq=@ w pt=mD|}=yQDU@ |wQ xOvvmWQ}e G=wt= CmQL |R=Ux}@W x@ QO=k =yvD |wQq=@ |r}rLDl =D '�1958� uBUv} Qo w Q} Qm Q}_v |v}kkLt "OvDUy xO=U |=y|i=QowB wD =@Q}e CqO=at Q@ CW=ov p=ta= =@ �1986� T}m qwv}U w �1972� nv=wy wVwQx@ xm OvOQw; CUO x@ |]N CqO=at R= |y=oDUO 'jtasm ?; |]NCmQL |=y|oO}J}B R= |W=v |[=} Q CqmWt p}rOx@ [3�1]"Ow@ pL p@=k |r}rLDu}=Q@=v@ "CU= umttQ}e Gwt CUmW OQwtQO |Q_v ,qt=m |Di=}yQ |x=Q= 'p=}U"CU= xDiQo Q=Qk u}kkLt R= |Q=}U@ xHwD OQwt xQ=wty xrUt u}= |OOa pLp@k |D=@U=Lt |=yxQo R= |NQ@ 'Gwt |wQq=@ |OOa |R=UpOt QO xm =Hv; R='Qo}O u=}@x@ w pL |x}L=v pw] u}}aD 'OvDUy lWN =yv; x@ Gwt uO}UQ R=?wULt xOta |rmWt |v=tR s=o Qy QO Q+D � lWN lQLDt RQt |@=}u=mt

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%OwW|t xDWwv u}vJ xDUUoxt}v@U@t

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+ Si (3)x@ xm R}v Fi�(1=2) w Fi+(1=2) w CU= |D=@U=Lt pwrU |xR=Ov= �x u; QO xmw x = xi+(1=2) \=kv QO |OOa Q=W x@ ?}DQDx@ Ov=swUwt |rwrU u}@ |=yQ=Wu}tND |=Q@ xO=iDU= OQwt VwQ 'xt=O= QO "�2 pmW� Ovvm|t xQ=W= x = xi�(1=2)

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xR=Ov= w �h� jta |=yxO=O \ki '|v=tR *s=o Qy |=OD@= QO OwOLt sHL VwQ QO|r=L QO u}= "CU= OwHwt |D=@U=Lt pwrU Qy RmQt QO �hu� u=} QH CmQLRQt QO u=} QH |=yQ}eDt u}}aD sRrDUt pL OQ@W}B '3 |x]@=Q j@=]t xm CU=Q=DU}=B u}v=wk |=Q@ CUOq=@ |=wvm} |wor= R= xar=]t u}= QO "CU=ypwrU u}@

uwO@ pmW� OQivt Gwt l} |wQq=@ w Q=WDv= x@ \w@ Qt |=yQDt=Q=B h} QaD "1 pmW"�CU= T=}kt

\=kv "�x |xR=Ov= =@ |}=ypwrU ltmx@ pL xvt=O |v=mt |R=UxDUUo "2 pmW"OvyO|t u=Wv =Q =ypwrU RmQt QBwD \=kv w |rwrU u}@ |=yRQt |r=NwD

l}miD VwQ R= pOt wO u}= "OvDUy |wQq=@ w Q=WDv= Ov}=Qi |] Ovr@ G=wt=|Oa@l} |xrO=at wO x@ jtasm ?; |Oa@ wO CqO=at p}O@D Qw_vtx@ '3OvtCyH

[9w8]"Ov=xOQ@ xQy@CU= x=Qty Gwt s=k |xy@H |Q}opmW =@ xm Gwt CUmW |xO}OB O=ONQ\ki |D=@U=Lt pOt l} "CU= x=Qty |DqmWt =@ |OOa |R=Ux}@W Q_vx]kv R=lwW |R=Ux}@W x@ QO=k|@ wNx@ xmCU=?U=vt xOvvmW G=wt= p}rLD |=Q@|t=ovy|=Q=O *|OOa |=ywor= QDW}@ "OW=@ |OOa u=Uwv O=H}= uwO@ Gwt CUmW R= |W=vQ=DU}=B Cr=L QO jtasm ?; CqO=at pL |=Q@ uwvm =D xm '4lwW Q}NUD *C}Y=NR= |O=} R |=yxvwtv "Ov=xOW |L=Q] CqO=at uoty Cr=L |=Q@ 'Ov=xOW O=yvW}BOQ@ Q=m xm CW=O xHwD O}=@ [10]"CU= xOW VQ=Ro �2001� wQwD ?=Dm QO =ywor= u}=Gwt 'lwW Gwt Q=WDv= Q}_v |OQ=wt x@ =yvD uoty Cr=L QO jtasm ?; CqO=atQO w s}kDUt VwQl} "CU= OwOLt |ki= QDU@ |wQ Q@ OUCUmW Gwt w 5lQDW)=CqO=at R= |y=oDUO x@ =yv; l}miD uotyQ}e CqO=at pL |=Q@ xO=U p=L u}au}= [10]"CU= |rwtat p}Uv=Qi}O |xrO=at OvJ =} l} w uoty |xrO=at l} pt=W"OyO|t CUO x@ |i}a[ ,=D@Uv G}=Dv |t=Ox@W =} |t=O |=yu=} QH OQwtQO VwQl} |R=Ux}@W QO l}miD VwQ u}= R= xO=iDU= =@ xm O=O u=Wv u=oOvQ=ov |x@ QHDx@ O=R; K]U QO |m} R}iQ}e C=W=WDe= |QU l} 'Q=wty=v QDU@ =@ um =U ?; uRNt|}xY}kv u}vJ "OvwW|t QWDvt ?; K]U |wQ Q@ u=tR CWPo =@ xm O};|t OwHw'C}Y=N u}= "OwW|t xO=O C@Uv C C}Y=N x@ swUwt |oS} w |=[Q= sOa x@"Ovm|t u}t[D s=O u=} QH p=Ut QO =Q |OOa Q=W u=}O=Qo w xtWJ CQ=@a uR=wDpOt \UwD C C}Y=N |=[Q= C}ty= uwt=Q}B EL@ =@ u}kkLt '2008 p=U QO|xaUwD QO QwmPtC}Y=N x@ xHwD sOa xm OvDW=O u=}@ Ovr@ Gwt |wQq=@ |D=@U=LtQ}e G=wt= R= |}xawtHt O=H}= x@ OQ=wt R= |NQ@ QO CU= umtt |OOa pOt l}*|rY= Gwt `=iDQ= |=oQR@ x@ |a=iDQ= |=Q=O xm OwW QHvt |OOa ,=iQY w |m} R}iVwQ R= xm '2000 p=U QO xOWx=Q= |wQq=@ pOt 'xvwtv u=wvax@ [11]"Ov=xar=]t OQwt

[12]"CU}v 6uwRwt 'CU= xOQm xO=iDU= xtWJ CQ=@a |=Q@ l}miD|OOa |wor= QO xtWJ CQ=@a uOv=Hvo |=Q@ QFwt |O}ytD 1994 p=U QO|=Q@ u; R= 'VwQ u}= |R=Uxv}y@ =@ 1999 p=U QO u; R= TB [13]"OW `=O@= 7wQu}= VwQ p=L u}= =@ [14]"OW xO=iDU= |t=O |=yu=} QH pt=W hrDNt p=Ut pL

"CU= xO}J}B Q=}U@ u}kkLt|]NQ}e CqO=at |=Q@ OwOLt sHL |OOa |wor= l} |xaUwD x@ xar=]t u}=QO 8�SGM� K]U u=}O=Qo VwQ =@ xtWJ CQ=@a u; QO xm OR=OQB|t jtasm ?;|Q}op=QoDv= |=Q@ =Dwm�nv=Q xU x@DQt |xv}y@ |wor= R= w xOW xO}v=Hvo C=@U=LtxvwoI}y x@ R=}v uwO@ xOWx=Q= |wor= "CU= xOW xO=iDU= xDUUoxt}v CqO=at |v=tRx=Qty =]N w |aU =@ ,qwtat u; u}}aD xm |awvYt CHRr Q}_v �� |i=[= CQ=@a=Q C C}Y=N xm O=O u=Wv u=wD|t w CU= lwW Q}NUD C}r@=k |=Q=O �� CU=x@ R=}v xvwoQy uwO@ pOt u}= u}vJty "CU= uwRwt 'Qo}O u=}@ x@ w Ovm|t

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%OQm h} QaD u}vJ u=wD|t =Q O=R; K]U R=QD "�hr=3 pmW� OwQ|t Q=m�(x; t) = h(x; t) +B(x) (9)�+i+(1=2) Q}O=kt =OD@= 'CmQL xR=Ov= QO xO=iDU= OQwt OvwQ x@=Wt |OvwQ R= |wQ}B =@jta Q}O=kt 10 |x]@=Q ltmx@ TBU w �?3 pmW� Ov};|t CUO x@ ��i+(1=2) w

"OwW|t x@U=Lt pwrU RQt QO u=} QHh�i+(1=2) = �

�i+(1=2) �Bi+(1=2)

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?@U xOWO=} VwQ "CU= x = xi+(1=2) RQt QO QDU@ R=QD Bi+(1=2) u; QO xmxHwD O}=@ "OwW|t pwrU RQt QO jta |@=}uwQ@ R= |W=v |=]N Q}otWJ Vy=mw SGM R= pY=L |=y?=wH '3 |xrO=at QO xtWJ CQ=@a ?=}e QO xm CW=O

"Ow@ Oy=wN u=Um} DGM

|OOa |wor= ltmx@ pwrU RQt QO |OOa Q=W u}tND "5(FORCE)

Q=W |x@U=Lt 'pwrU Qy RQt QO CmQL xR=Ov= w jta |=yQ}eDt u}}aD R= hOyxOW O=yvW}B |vwo =vwo |=yVwQ uwvm =D Qw_vt u}= |=Q@ "CU= |rwrU u}@ |OOapw= x@DQt |RmQt |OOa |wor= R= |OOa Q=W u}tND Qw_vtx@ xar=]t u}= QO "CU=

T=U=Q@ [10]"OwW|t xO=iDU= 'OW O=yvW}B Q}r uw\UwD Q=@u}rw= xm '10�MUSCL�=@ pwrU RQt QO =yQ}eDt Q}O=kt '11=yxO=O CN=UR=@ x@ swUwt |xrLQt QO VwQ u}=CQwY |}xvwox@ |@=}uwQ@ u}= p=ta= "Ov};|t CUO x@ pwrU RmQt Q}O=kt |@=}uwQ@O=H}= uwO@ u=} QH |=y|oDUw}B=v w OwW pQDvm ?=wH |v=Uwv CrYN xm OQ}PB|t|]NQ}e |xOvvmOwOLtl} R= =yxO=OCN=UR=@ |=Q@ "OwW Q}NUD |OOa lqyDU=QO xOW |R=UR=@ |=yxO=O x@ ?U=vt |OOa lqyDU= =D OwW|t xO=iDU= 12?}WO=H}= R= 'pL |}=wvm} ^iL ut[ 'VwQ u}= R= xO=iDU= =@ "OwW xOwRi= pwrU RQt"OwW|t |Q}owrH O=} R u=}O=Qo =@ |L=wv |m}ORv QO |m} R}i Q}e w O}OW |=yu=Uwv

%CW=O s}y=wN s=i pwrU QO CmQL xR=Ov= |=Q@ T=U= u}tyQ@(hu)�i+(1=2) = (hu)i +

12�i(hu);

(hu)+i+(1=2) = (hu)i+1 � 12�i+1(hu): (4)|R=UR=@ |=yQ}eDt Q}O=kt |xOvyOu=Wv ?}DQDx@ - w + |=yT} wvq=@ TB u}= R=u=}O=Qo 4 |x]@=Q QO "Ow@ Ovy=wN Q_v OQwt RQt AJ CtU w CU=Q CtU QO xOW

"OwW|t xO=O 5 |x]@=Q j@] CmQL |xR=Ov=�i(hu) = i �

��i+(1=2)(hu) + �i�(1=2)(hu)

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(5)

pwrU wO QO CmQL xR=Ov= QO VQB �i�(1=2)h(u) w �i+(1=2)(hu) u; QO xm"OvwW|t h} QaD 6 |x]@=Q ?r=k QO w xOw@ Qw=Ht

�i+(1=2)(hu) = (hu)i+1 � (hu)i;�i�(1=2)(hu) = (hu)i � (hu)i�1: (6)

[10]"CU= Superbee ?}W |]NQ}e |xOvvmOwOLt '5 |x]@=Q QO

i =

8>>>>>>>>>>>>>:0 ri � 0;2ri 0 � ri � 12 ;1 12 � ri � 1;min(ri; 21+ri ; 2) otherwise:

(7)

Q@=Q@ 8 |x]@=Q j@] w CU= |r=wDt pwrU wO QO CmQL xR=Ov= C@Uv ri u; QO xm%=@ CU=

ri =�i�(1=2)(hu)�i+(1=2)(hu)

(8)

(SGM) K]U u=}O=Qo VwQ "4Q=m x@ CmQL xR=Ov= |=Q@ p@k CtUk QO xJv; x@=Wt |WwQ =@ u=} QH jta CN=UR=@pmW |=Q@ \ki VwQ u}= "OwW|t xO}t=v 13�DGM� jta u=}O=Qo VwQ 'CiQ|wQq=@ |R=UpOt QO p=L Qyx@ "CU=?U=vt �S(U) = 0 |va}� CqO=at uotyRQt QO ?; jta =Q} R 'OwW ^=Lr CqO=at QO QDU@ ?}W C=QF= O}=@ OQivt G=wt=O=R; K]U |v=tR C=Q}}eD u}vJty w QDU@ |i=QowB wD Q}F-=D CLD |D=@U=Lt pwrU Qy|=Q@ CkO |q=@ |x@DQt =@ |WwQ R= Qo = |DL '|Dr=L u}vJ QO "OQ=O Q=Qk u=} QHRQt QO u=} QH jta |ak=w C=Q}}eD 'OwW xO=iDU= pwrU RQt QO ?; jta CN=UR=@

109

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pOtQ=}at p=ta= =@ C=@U=Lt |Q=O}=B 'CU= K} QY |} wor= =Dwm � nv=Q |wor= xm =Hv;%OwW|t u}t-=D |v=tR s=o Q=Okt |wQ Q@ �CFL� |wr � V} QOQi � Cv=Qwm

�t = Cn mini

�xjuij+pghi ; 0 < Cn � 1 (16)

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OQivt Gwt |wQq=@ |R=Ux}@W |=Q@ sRq |RQt \}=QW "7|RQt \}=QW OQ=O CQwQ[ 'jtasm ?; |]NQ}e CqO=at K}LY pL |=Q@Ov=ptDWt =yRQt 'pL=U |wQ Q@ Gwt |wQq=@ Ov}=Qi|UQQ@ QO "OvwW p=ta= |DUQOx@QO =yv; |R=UxO=}B |ovwoJ x@ CtUk u}= QO xm '|@=DR=@ Q}e RQt w lQLDt RQt Q@

"OwW|t xDN=OQB xOW x=Q= |OOa pOt

lQLDt RQt "1"7?; |]NQ}e CqO=at R= xO=iDU= =@ Ovr@ G=wt= |wQq=@ Ov}=Qi |OOa p}rLD QOpL=U \N =Q} R &CU= pL=U \N C}akwt u}}aD R= |W=v CqmWt QDW}@ 'jtasmQo}O u=}@x@ "OwW|t =Hx@=H pL=U |wQ Q@ Gwt |wQu}}=B w |wQq=@ pqN QO"OHvo|t lQLDt RQt =@ p=Ut wQtrk QO pL=U |wQ Q@ Gwt |wQq=@ |xrUt\N h} QaD x@ C@Uv '|OOa pOt QO X=N |Di=}yQ P=ND= =@ xm CU= sRq u}=Q@=v@2000 |=yp=U QO xOWx=Q= |=yVwQ x@=Wt |WwQ R= |wQ}B =@ "OwW s=Ok= pL=Upw] QO [19w11]"OwW|t h} QaD ?; |xv}tm jta l} |OOa |wor= QO '2006 wu; OwW xv}tm Q=Okt R= QDsm |D=@U=Lt pwrU Qy QO ?; jta xJv=vJ C=@U=LtQiY Q=Okt pwrU u; QO u=} QH CmQL xR=Ov= x@ w 'OwW|t O=Otrk lWN pwrUQO "OwW|t |krD Q+D pwrU u=wvax@ pwrU CQwY u}= Q}e QO "OwW|t xO=O C@Uvu=wvax@ Ovm|t =OH lWN |=ypwrU R= =Q Q+D |=ypwrU xm |RQt |v=tR s=o Qyjta |xv}tm Q=Okt u}= xm CU= QmP x@ sRq u}vJty "OwW|t h} QaD pL=U \N|R=QD QO xm |}=yxQo � lWN pL=U Q@ `k=w |=yxQo |t=tD x@ x}rw= jta u=wvax@

"O@=}|t X=YDN= �OvQ=O Q=Qk ?; |x}rw= O=R; K]U R= QDq=@

|@=DR=@ Q}e RQt "2"7'O};|t Q=tWx@ |@=DR=@ Q}e RQt |awv xm �1 pmW 'x = L� =} QO CtU RQt QO*T=mav= uwO@ GwQN xm |RQt \QW u}= "OwW|t p=ta= xOvyOp=kDv= |RQt \QW

"|@=DR=@ Q}e |RQt \QW |R=UpOt |=Q@ |twywt |=yxQo "4 pmW

u}= [16]"OwW|t xO=iDU= 'OW x=Q= 1996 p=U QO Q=@u}rw= |=Q@ xm 14�FORCE�|wk Q=}U@ R}v OQ@ Q=m QO '|}xv=}=Q |xt=vQ@ QO |R=UxO=}B |oO=U Q@ xwqa 'Q=W |wor='�MUSCL� Q=DU}=B u}v=wk |=Q@ CUOq=@ |=wvm} |wor= VwQ =@ ?}mQD QO "CU=w u=tR QO wO |x@DQt R= |DkO |=Q=O �FORCE� pwrU RQt QO |OOa Q=W u}tND11 |x]@=Q CQwYx@ xi+(1=2) RQt QO |OOa Q=W wor= u}= T=U= Q@ "CU= u=mt

%OwW|t xDWwvFFORCEi+(1=2) =

12 (F

LFi+(1=2) + FLWi+(1=2)) (11)

CU= CQ=@a w CU= V} QOQi�Tmr |OOa Q=W |xOvyOu=Wv FLFi+(1=2) u; QO xm%R=

FLFi+(1=2) =12hF (U�i+(1=2)) + F (U

+i+(1=2))

i�12

�x�t

hU+i+(1=2) � U�i+(1=2)

i(12)

|=yQ}eDt |xOWx@U=Lt Q=OQ@ hQat ?}DQDx@ U�i+(1=2) w U+i+(1=2) xrO=at u}= QO'11 |x]@=Q QO u}vJty "OvDUy xi+(1=2) RQt AJ w CU=Q xHw QO Q=DU}=B%CU= hwQOvw � Tmr |}xrLQt wO |OOa Q=W =} Q}=tJ} Q |OOa Q=W FLWi+(1=2)FLWi+(1=2) = F (ULWi+(1=2)) (13)

"OwW|t xO=O 14 |x]@=Q =@ 'ULWi+(1=2) Q=DU}=B Q}eDt |v=}t Cr=L 'u; QO xmULWi+(1=2) =

12hU�i+(1=2) + U

+i+(1=2)

i�12

�t�x

hF (U+i+(1=2))� F (U�i+(1=2))

i(14)

pw= x@DQt |RmQt |OOa |wor= 'OQm C=@F= 2006 p=U QO wQwD xm Qw]u=ty|]N ?}mQD R= xm =yQ=W R= |}xawtHt u=}t QO w xOw@ =wvm} Q=W l} �FORCE�|w[a 'Ov};|t OwHw x@ hwQOvw �Tmr |}xrLQt wO Q=W w V} QOQi- Tmr Q=Wx@DQt |RmQt |OOa |wor= ?}mQD xm CU= QmP x@ sRq [17]"OwW|t ?wULt xv}y@

"CU= xOWv VQ=Ro u}W}B uwDt QO K]U u=}O=Qo VwQ w �FORCE� pw=

|Q=O}=B C=_Lqt w |v=tR |Q}op=QoDv= "6R= xO=iDU= =@ "CiQo Q=Qk xHwD OQwt sm =L CqO=at |v=mt |R=UxDUUo \ki =Hv}= =Dnv=Q |v=tR |R=UxDUUo VwQ ltmx@ 3 |xrO=at '?U=vt |x}rw= w |RQt \}=QW|=Q@ w OQ=O x=Qtyx@ =Q C=@U=Lt O=} R Q=}U@ |Q=O}=B VwQ u}= "OwW|t pL =Dwm �jtasm ?; |]N Q}e CqO=at Q}_v �� |wrwrPy p}Uv=Qi}O CqO=at sDU}U pLCqO=at j@] =Dwm � nv=Q xU |x@DQt |xv}y@ VwQ [18]"CU= ?U=vt Q=}U@ ��

%R= CU= CQ=@a 15U (1)i = U

ni + �t L(U

ni );

U (2)i =34U

ni +

14U

(1)i +

14�t L(U

(1)i );

Un+1i =13U

ni +

23U

(2)i +

23�t L(U

(2)i ): (15)

xm CU= 3 |x]@=Q CU=Q hQ] (W = Uni ; U (1)i ; U (2)i )L(W ) u; QO xmR= "OvDUy Q=DU}=B |v=}t |=yQ}eDt R}v U (2)i w U (1)i "CU= xOW x@U=Lt W QO

110

|DNO}@?

r=]QY=

vw|wO

yt|ra |=ypOt |HvUCLY |=Q@ ,qwtat jkLt u}= pOt "O=O x=Q= xOvvmWQ}e G=wt= O=R;

OwN |=yV}=tR; T=U=Q@ jkLt u}= u}vJty "OQ}o|t Q=Qk xO=iDU= OQwt |OOaxDiQo Q=Qk xO=iDU= OQwt =yv; QO QDj}kO |}=yxw}W 'u}W}B |=yQ=m =@ xU}=kt QO xmR= [3]"O=O x=Q= xOvvmW w xOvvmWQ}e OQivt G=wt= |wQq=@ xv}W}@ |=Q@ |]@=wQ 'Ow@OwN |OOa w |r}rLD |=ypOt CkO u=R}t u}}aD |=Q@ |Q=}U@ u}kkLt xm =Hv;j}kLD u}= QO xOWx=Q= pOt G}=Dv 'Ov=xOQm xO=iDU= QwmPt |y=oW}=tR; G}=Dv R=O=R; K]U |OOa |R=Ux}@W G}=Dv 5 pmW QO "OwW|t xU}=kt =yxO=O u}= =@ R}v|=Q@ "CU= xOW xU}=kt T}m qwv}U |r}rLD ?=wH w |y=oW}=tR; |=yxO=O =@w 'H=h0 = 0 0185 Gwt Oa@ uwO@ `=iDQ= |y=oW}=tR; |=yxO=O =@ |ovy=ty?wULt xOvvmWQ}e Gwt u}= |]}=QW u}vJ CLD "CU= 1%19 85 QDU@ ?}W|y=oW}=tR; w |r}rLD |OOa |=yxO=O u=}t xvqwkat |k@=]D |vWwQx@ "OvwW|t|QDy@ ,=D@Uv j@=]D R= |OOa G}=Dv '|r}rLD ?=wH =@ xU}=kt QO xJQo = "OQ=O OwHw|W=v O=R; K]U |v=tR C=Q}}eD '6 pmW QO "OvQ=OQwNQ@ |y=oW}=tR; |=yxO=O =@hr=6 pmW "CU= xOW xO}Wm Q} wYD x@ O=R; K]U R= x]kv wO QO xOWO=} Gwt R=|xOvyOu=Wv ?6 pmW w 'CU= x=h0 = 0 25 QO `k=w |x]kv x@ \w@ QtQ=O?}W K]U |v=}t |x]kv |m}ORv QO O=R; K]U |}=Hx@=H |v=tR C=Q}}eD|W=v pw= |xrk "OwW|t xOy=Wt =RHt |xrk wO u; QO xm CU= �x=h0 = 9 95�"CU= Q=O?}W K]U R= xDi=}?=DR=@ Gwt R= |W=v swO |xrk w xOvwQV}B Gwt R=,qta w xOW s=eO= Qo}Om} QO xrk wO u}= hr=6 pmW QO xm CU= |r=L QO u}=R= |m =L |r}rLD w |OOa G}=Dv |xvqwkat j@=]D "OwW|t xOy=Wt Gw= |x]kv l}Q=O?}W pL=U Q@ OQivt Gwt |wQ u}}=B w q=@ |=yOv}=Qi |R=Ux}@W QO pOt |}=v=wD

"CU=?}W R= xm H=h0 = 0 04 =@ |Hwt |=Q@ O=R; K]U |=yMQs}v 7 pmW QO"CU= xOW xU}=kt T}m qwv}U |y=oW}=tR; |=yxO=O =@ OwQ|t q=@ 1%19 85x=oW}=tR; QO 'ODi}@ j=iD= Gwt u}= CUmW O}=@ Gwt CUmW Q=}at |=v@t Q@ xJQo =

"Ovm|t O}}-=D =Q Ck}kL u}= R}v |OOa G}=Dv [3]"OyO|tv MQ ,qta Gwt CUmWGwt |=Q@ T}m qwv}U |y=oW}=tR; |=yxO=O =@ |OOa pOt G}=Dv 8 pmW QO'h0 C@=F jtax@ |@; |wQ Q@ Q=WDv= R= TB xm H=h0 = 0 3 `=iDQ= =@ |OQivtx@ Gwt xm |t=ovy "CU= xOW xU}=kt 'OwQ|t q=@ 1%19 85 ?}W =@ |L]U R=?}W u; CWB x@ C@Uv Gwt |xy@H 'Ovm|t CmQL Q=O?}W K]U |q=@ CtUQO xm |}=H =D OyO|t CUO R= =Q OwN uQ=kDt Cr=L Gwt pmW w O@=}|t |QDW}@Cr=L x@,qt=m Gwt |xy@H 'CU= hwQat GwtCUmW |x]kv x@ xm 'X=N|D}akwt=Q |OvwQ u}vJ t� = 20 w t� = 15 |=yu=tR QO |OOa G}=Dv "O};|tQO |OwtaQ=_Dv= |OOa |wor= lwW Q}NUD CrYN R= xm xvwou=ty w OvyO|t u=Wv |@ wNx@u}= =@ "CU= xO=Ov MQ Gwt s=k |xy@H |m}ORv QO |OOa u=Uwv xvwoI}y 'CiQ|tGwt `=iDQ= 'jtasm ?; |]NQ}e CqO=at 16|WNB Q}e C}Y=N p}rOx@ OwHwR}v Gwt CaQU w 'xOW xOR u}tND |y=oW}=tR; Q=Okt R= QDsm CUmW |x_Lr QO|q=@ x@ Q=WDv= |xt=O= QO Gwt "CU= xOt; CUO x@ |ak=w Q=Okt R= QDW}@ |QOkCOWx@ u; `=iDQ= w OR} Q|t wQi pL=U \N |x}rw= C}akwt |m}ORv QO 'K]UR= xm CU= p=}U R= |}xv=@ R p}mWD 'Ov}=Qi u}= pY=L "�t� = 25� O@=}|t Vy=m'O}UQ OwN OwaY |x]kv u} QDq=@ x@ xv=@ R xm u}= R= TB "OwQ|t q=@ Q=O?}W K]U|v}Wv?ka Gwt |xv=@ R 'Ov}=Qi u}= |=OD@= QO "OwW|t R=e; Gwt |wQu}}=B Ov}=QiK]U CtU x@ OwN Q=WDv= x@ Rwvy Gwt |xr=@vO xm CU= |r=L QO u}= w Ovm|tOvm|t OQwNQ@ Gwt |xr=@vO =@ |v}Wv?ka p=L QO *u=} QH "OyO|t xt=O= Q=O?}W"O};|t OwHw x@ pL=U \N |x}rw= C}akwt |m}ORv QO |m}rwQO}y VQB l} w

x@ 15|twywt |=yxQo uOwRi= =@ 'OR=U|t Q}PBu=mt= =Q |D=@U=Lt |xvt=O R= Gwt|D=@U=Lt xQo u} QN; T}Ov= xJv=vJ "�4 pmW� OwW|t p=ta= |D=@U=Lt |xvt=O"Ow@ Oy=wN M + 2 w M + 1 |twywt |=yxQo T}Ov= 'OwW xO=O u=Wv M =@

%CWwv u=wD|t CQwY u}= QO(hu)M+1 = (hu)M ; hM+1 = hM

(hu)M+2 = (hu)M ; hM+2 = hM (17)Q@ Owta |=DU=Q QO O}=@v u=} QH CmQL |xR=Ov= w jta xm Ovm|t u=}@ 17 |x]@=Q

"Ovvm Q}}eD |@=DR=@ Q}e RQt

OQivt Gwt |R=Ux}@W |x}rw= \}=QW "8Gwt u}= Q=WDv= |R=UpOt "OQ=Ov smW w CU= xrk l} |=Q=O \ki OQivt Gwt|x_Lr QO Gwt MQs}v p=ta= =@ w jtasm ?; |]NQ}e CqO=at pL =@ ,qwtatCQwY |OOa pOt QO '�(x; 0) O=R; K]U MQt}v |x}rw= Q}O=kt CQwYx@ t = 0QO xm OQivt Gwt l} MQs}v 'OQivt G=wt= pw= |x@DQt |x} Q_v T=U=Q@ "OQ}PB|t

[3]%OwW|t h} QaD 18 |x]@=Q j@=]t 'O@=}|t Q=WDv= x QwLt C@Ft CyH

�(x; t) = H sech2"s 3H

4h30 (x�X1 � Ct)#

(18)

C}akwt X1 '?; |x}rw= C@=F jta h0 'OQivt Gwt |x}rw= `=iDQ= H u; QO xmpmW j@=]t "CU= Gwt Q=WDv= CaQU C = pg(H + h0) w Gwt |xrk |x}rw=Q=O?}W pL=U ẁQW |x]kv w OQivt Gwt |xrk |x}rw= C}akwt u=}t |xrY=i '1

[3]%OwW|t xDiQo Q_v QO u}vJ �X0�

X1 �X0 =r4h303H arccosh

s1

0 05!

(19)

|x}rw= Q}O=kt "CU= OQivt Gwt |xYNWt pw] hYv `k=w QO xOWO=} |xrY=i"O};|t CUO x@ 20 |x]@=Q R= u=} QH CaQU

u(x; 0) = ��(x; 0)r

gh0

(20)

\N |x}rw= u=mt CtU x@ Gwt xm OyO|t u=Wv 20 |x]@=Q QO |ivt CtqaOQwt TB u}= R= xm |OQ=wt QO "Ovm|t CmQL �1 pmW QO C=YDNt -=O@t� pL=U�21 |x]@=Q� Oa@ uwO@ |=yQ}eDt R= G}=Dv u=}@ |=Q@ 'CiQo Ovy=wN Q=Qk |UQQ@

%OwW|t xO=iDU=�� = �

h0; x� = x

h0; t� = t

rgh0: (21)

EL@ w G}=Dv "9|]NQ}e pOt l} |x=Q= x@ OwN |xOQDUo C=k}kLD QO �1986� T}m qwv}Ul} R= xO=iDU= =@ jkLt u}= "CN=OQB xOvvmW Q}e G=wt= |wQq=@ |=Q@ |r}rLD wK]U MQs}v |=Q@ |r}rLD |@=wH w OQm pL =Q jtasm ?; |]NQ}e CqO=atCW=ov

111

"""|wQq

=@|R=Ux

}@W|=Q@

jtasm?

;|OOa

pOt

1%19 85 ?}W |wQ Q@ H=h0 = 0 0185 x}rw= `=iDQ= =@ |OQivt Gwt |wQq=@ |=Q@ �x� = x=h0� R= |a@=D CQwYx@ �� = �=h0� O=R; K]U MQs}v "5 pmWT}m qwv}U |r}rLD pL w |y=oW}=tR; |=yxO=O ?}DQDx@ =yu}J\N w \=kv 'Q[=L |xar=]t G}=Dv %ODtt \w]N "�t� = tpg=h0� Oa@ uwO@ hrDNt |=yu=tR QO

"OvyO|t u=Wv =Q �1986�

=@ u; |xU}=kt w x� = 9 95 %�?� w x� = 0 25 %�hr=� C}akwt QO �t� = tpg=h0� Oa@ uwO@ u=tR R= |a@=D u=wvax@ �� = �=h0� O=R; K]U |}=Hx@=H "6 pmW"�1%19 85 %QDU@ ?}W 'H=h0 = 0 0185 %Gwt x}rw= `=iDQ=� �1986� T}m qwv}U |r}rLD pL

T}m qwv}U w hwD}D w �1991�CrR |OOa |=yVwQ =@ xU}=kt QO |O=yvW}B |OOaQ=OQwNQ@ �1986� T}m qwv}U |y=oW}=tR; |=yxO=O =@ |QDy@ j@=]D R= �1995�pw] QO p=}U sQH |=k@ C}Y=N ^iL QO pOt |}=v=wD R= |W=v Qt= u}= "CU=|x}rw= sQH x@ C=@U=Lt u=}=B QO xDiQCUO R= sQHC@Uv xJv=vJ "CU= C=@U=Lt'OwW h} QaD 17�MCE� sQH |=k@ |=]N u=wvax@ |D=@U=Lt |xvt=O QO OwHwt

xm u}= "CU= xOy=Wt p@=k 50 < t� < 60|v=tR |xR=@ QO |@ wNx@ ẁ[wt u}=xOW |R=Ux}@W p=}U R= |mR=v sr}i CQwYx@ Gwt |v}Wv?ka p=L QO |xv=@ R?; |]NQ}e CqO=at uOQw; CUO x@ QO GRrQ}e p=}U ZQi p=ta= R= |W=v 'CU=Gwt |R} QwQi |x_Lr QO O=R; K]U |xOWx@U=Lt MQt}v '9 pmW QO "CU= jtasm|wor= |vWwQx@ "CU= xOW xU}=kt OwHwt |=yVwQ Q}=U =@ �8 pmW 't� = 25�

112

|DNO}@?

r=]QY=

vw|wO

yt|ra

|=yxO=O \=kv w Q[=L xar=]t G}=Dv ODtt \w]N "Oa@ uwO@ hrDNt |=yu=tR QO 1%19 85 ?}W |wQ Q@ H=h0 = 0 04 x}rw= `=iDQ= =@ OQivt Gwt |wQq=@ "7 pmW"OvyO|t u=Wv =Q �1986) T}m qwv}U |y=oW}=tR;

|=yxO=O \=kv w Q[=L xar=]t G}=Dv ODtt \w]N "Oa@ uwO@ hrDNt |=yu=tR QO 1%19 85 ?}W |wQ Q@ H=h0 = 0 3 x}rw= `=iDQ= =@ OQivt Gwt |wQq=@ "8 pmW"OvyO|t u=Wv =Q �1986� T}m qwv}U |y=oW}=tR;

113

"""|wQq

=@|R=Ux

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jtasm?

;|OOa

pOt?; u=} QH |R=UpOt QO u;|}=v=wD xm FORCE |OOa |wor= u; QO xm OW xDN=OQB|R=Ux}@W Qw_vtx@ Q=@u}rw= |=Q@ 'Ow@ xO}UQ C=@F= x@ u}W}B |tra uwDt QO jtasmu}vJty "CiQo Q=Qk xO=iDU= OQwt C}kiwt =@ lWN QDU@ |wQ Q@ Gwt Q=WDv=pL w |y=oW}=tR; |=yxO=O =@ |OOa |R=UpOt R= pY=L G}=Dv u=}t |}=yxU}=ktu}= QO xDiQoCQwY |=y|R=Ux}@W |xOwOLt QO p=L Qyx@ "CiQ}PB s=Hv= |r}rLD

%R= CU= CQ=@a xrY=L G}=Dv xar=]tQO xOW |Q}o\UwDt CaQU ZQi seQ|ra jtasm ?; |]NQ}e CqO=at �"OvDUy Ovr@ G=wt= |wQq=@ w Q=WDv= |xO}OB |UQQ@ |=Q@ Ovtu=wD |Q=R@= 'jta

|OOa lqyDU= Q]=Nx@ xOWx=Q= pOt G}=Dv 'OwHwt |r}rLD pL =@ xU}=ktQO �"OyO|t u=Wv |y=oW}=tR; |=yxO=O =@ |QDy@ j@=]D

|xOvvmW Gwt |wQq=@ xv}W}@ �1 pmW QO � |x} w=R� K]U ?}W V}=Ri= =@ �QDs}qt =@ xOvvmWQ}e Gwt |wQq=@ xm CU= |r=L QO u}= &O@=}|t V}=Ri= OQivt

"OwW|t QDW}@ ?}W uOW?@U �1 pmW 'X1� |rL=U \N x@ C@Uv Gwt |xrk |x}rw= |xrY=i V}=Ri= �

"OwW|t pL=U |wQ Q@ Gwt |wQq=@xv}W}@ Vy=m'sm =L CqO=at sQH |=k@ C}Y=N ^iL QO R}J=v |=]N p}rOx@ xOWx=Q= pOt �Gwt l} |R} QwQi |x_Lr QO u}W}B |=ypOt R= |NQ@ x@ C@Uv |QDy@ G}=Dv

"OyO|t CUO x@ xOvvmW|ovy=ty=v?Hwt 'jtasm ?;|]NQ}eCqO=at QO|WNB Q}eC}Y=Nu=Oki �"OW Gwt CUmW |x]kv |r=wL QO |y=oW}=tR; |=yxO=O w |OOa G}=Dv u=}tuOQmxi=[= w G=wt= |wQq=@ xv}W}@ |wQ Q@ QDU@ l =m]Y= R= |W=v QF= |xar=]t �KQ]t Qo}O |k}kLD ẁ[wt u=wvax@ Ov=wD|t sm =L CqO=at QO |WNB CQ=@a

"OwW

|wQq=@ |=Q@ OwHwt |=yVwQ Q}=U w xOWx=Q= pOt |OOa G}=Dv |xU}=kt "9 pmW"t� = 25 x_Lr QO 1%19 85 ?}W R= H=h0 = 0 3 =@ |OQivt Gwt

MCE��0 005 |xR=Ov=x@ |}=]N =@ QwmPt |xOvvmW Gwt |wQq=@ |R=Ux}@W|R=Ux}@W QO �1995� T}m qwv}U w hwD}D xm CU= |r=L QO u}= w CU= x=Qty"Ov=xOQm VQ=Ro MCE��0 7 |xR=Ov=x@ |}=]N x@=Wt |]}=QW CLD Gwt u}tyu=}t QwYLt K]U |Q}op=QoDv= R= |v=tR s=o Qy QO p=}U sQH xm CU= QmP x@ sRqR= |Qw@a sQH Q=W u; |x@U=Lt QO w xOt; CUO x@ u=} QH QDU@ w Gwt MQs}vlwW |R=Ux}@W QO R}J=v |=]N u}= OwHw "CU= xOW ^=Lr R}v =} QO CtU RQtsm =L CqO=at |Q=DU}=B C}Y=N ^iL QO pOt |}=v=wD Q@ 'Gwt CUmW R= pY=L

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CWwv=B1. exact Riemann solver2. weighted average ux3. directional splitting4. shock-capturing5. bore6. well-balanced7. Roe method8. surface gradient method9. conservative form10. monotonic upstream scheme for conservation laws11. data reconstruction12. nonlinear slope limiter13. depth gradient method14. �rst order centerd scheme15. ghost grid points16. non dispersive17. mass conservation error

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