Top Banner
ISSN 0249-6399 ISRN INRIA/RR--4189--FR+ENG apport de recherche THÈME 4 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE On the behavior of upwind schemes in the low Mach number limit : II. Godunov type schemes. Hervé Guillard — Angelo Murrone N° 4189 Mai 2001
26

On the behavior of upwind schemes in the low Mach number ...

May 05, 2023

Download

Documents

Khang Minh
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: On the behavior of upwind schemes in the low Mach number ...

ISS

N 0

249-

6399

ISR

N IN

RIA

/RR

--41

89--

FR

+E

NG

ap por t de r ech er ch e

THÈME 4

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

On the behavior of upwind schemes in the low Machnumber limit : II. Godunov type schemes.

Hervé Guillard — Angelo Murrone

N° 4189

Mai 2001

Page 2: On the behavior of upwind schemes in the low Mach number ...
Page 3: On the behavior of upwind schemes in the low Mach number ...

Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles, B.P. 93, 06902 Sophia Antipolis Cedex (France)

Téléphone : 04 92 38 77 77 - International : +33 4 92 38 77 77 — Fax : 04 92 38 77 65 - International : +33 4 92 38 77 65

�������������� ������������������������! #"$����%&�' ����������)(*��� +, �"��!�-��%.��'��(���%/�0�213#3$465 �7�8�9���:�;�=<-�>� #"$����%/�' 4

?�@BA�CED6FHGJILKMKONPARQ$SHT�U9VJWE@BKMXZY[G�ARA�XEV�@�\

]_^a`�b-cedgf�hjikb�lamonqpritsvu9c�p'sxwypritb-ioz{n|p{iksxu} c�z�~yz�p{`�bgcRz_�*sxb-wamkc*�ycRz

�$�rs|��c*p:ha�H��hy���n|waw=sx�{p } c��rc���^ac��r��^ac�uE�Jd��R�x��f���nxi#�|�x����f��|��w�n|�vc�z

�������� q¡a¢x�x£ ]_^aioz�w�n|w=c���wa�{cRz�c�uBprz�n|u¤nxu�n|mk~yz�iozes|¥:¦§s } lauasq¨>z{��^ac�b-c7itu©pr^ac�mksqª«��nv��^6ujlab�¬Pc����rc��xikb-cx­�®6c�z�p{l } ~pr^acg��itc�b-nxuau�w��{sv¬amtc�b¯n|u } z{^asqª°p{^Pnqp§p{^acgituBprc��{¥Mnx�*c�wa�rc�zrz�la�rc���sxuBprnxitu�z�nx��sxl�z�p{io��ª_n±¨xcRz's|¥²sv� } c��§�Hnx��^Hc�¨xc�u�i³¥´pr^acikuaitp{ion|m } nqp�n�n|�rc�ªµc�mtm�wa�rc�w�nx�{c } nxu } �*svuBprn|iku�sxu�mt~9wa�rc�zrz�la�rce¶�l��*p{l�n|p{iksxu�z:s|¥$sx� } c�����nx��^Hzr·Bl�n|�rc } ­¸®6c�pr^ac�u�w��{svwPsBz�cprs�b-s } i³¥¹~�p{^ac�¶�ly�yc�z´��sxb-walyprc } ¬j~g¦§s } l�uasq¨�p�~jw=c:z{��^�c�b-c�z�¬j~-z�svmt¨jikua��newa�{cR�*svu } itp{iksxuac } ��itc�b-nxuau-wa�rsx¬amkc�bºitu�z�p{cRn }sx¥»p{^acesv�{ik�xiku�n|m=sxu�cx­$]_^aioz¸z�p{��nqprc��v~gioz_n|waw�mtikc } p{sg¼�½��'sxcez{sxmk¨xc���z´ª'^�c��rc�ª¸c§z{^asqªZpr^�nqp'itp_nxmtmksqª�z´prs��rc���sq¨xc��µng�*sx�r�rc��0pzr��nxmtikua�-s|¥�p{^ac�wa�rc�zrz{la�{c§¶�lP�0p{lPnqp{iksxuPz�­$¾�l�bgc��{io��nxm¿c*�yw=c��ritb-c�uBp�z'�*sxuaÀ��{b�p{^acRz�cepr^ac�sv�{c�p{io��n|m¿�rc�z{lamtprz�­Á�ÂBÃEÄ{Å�Æ  qÇJ�x£ È svb-wa�{cRz{z{ik¬amtc�¶�sqª�z{sxmk¨xc��rz�ɲÊ#sqª���nx��^Ëujlab�¬=c��Rɵ�'ikc�b7nxuauËw��{sv¬amtc�b8ɸ¦§s } lauasq¨Ìzr��^ac�bgcvɲ¼§½J��sxczr��^ac�b-cvÉa�$�{cR�*svu } itp{iksxuaikua��­

Í_Î�Ï�ÐMÑ*Ï0Ò ÓJÔ±Õ×Ö×ÖÙØ�ÐMÚ|ÛJÜLÝ0ޱ߱ÕÙØRÒ Õ×à±ÐMÕÙØRÒ ákпâ=ãPÐMÝ�äMÏ�å�æRçeè�æRÎé è�à±ê0Ï�Ö×Ý±Ò çeÔ±ÐMÐLÝ0à±Ï{ÛJÜLÝ0ޱ߱ÕÙØRÒ Õ×à±ÐMÕÙØRÒ ákвâ$ã�ÐLÝ�äMÏëå�æRçeè�æRÎÌÏ�å_ìJí#ãEî0æ�í#ÎJîrïyçeíyèËÚRÔ�ð¿ñPè�ð¿Ø�ÚqØ�ÐLØ�òóß±Ï0ô�õ�öRõ�÷0ø�æ�Ø�Õ×à�åMâ¹ãaØ�Ô±Ö âùïjÏ�úëâ¹ìJÔRÐOØ�à±ò�ÏðEÏ�ÚRÏ�û�ütýqÐOØ�à±ò�Ï{þ

Page 4: On the behavior of upwind schemes in the low Mach number ...

� ��%&�>���»�J��%&�����«�9�' � #"$����%& � ���'"¸���������' ��� ��� �(� &(���%;����� �9�' �q ���9(��' ����% 9�J�' ��� +, �"$� 1 3�3�4�� "$����%/ � ��� �P<7�>� 5 �7�8������� 4

��� ��� � £eÈ c�p_nx��prik��mtc:wa� �Rz�c�uBp{c�lau�c§nxu�n|mk~yz�c } l��*svbgw=sx�{p{c�b-c�uBp } l�z{��^���b7n } c�¦§s } lauasq¨ } n|u�zµmon�mkitb-itp{c } c�z²¥Mn|ik¬amtcRzuasvb�¬a�rc�z } c��Hnx��^»­$¾�svl�z���p{l } iksxuPz_mtc�wa�rsx¬�mt`�bgc } c��'ikc�b7nxuau�c�p�b-sxuBp{�rsxu�z_·Blac�mongwa�{cRz{z{iksxu��gm�� ikuBp{c���¥Mnv�*cec�uBp{�rc } c�la���c�mkmtlamkc�z��*svuvpritc�uBp } c�z�sxu } c�z�nv�*svl�z�prik·BlacRz } � sx� } �rc���nx��^©b���b-c9z�iµmtcRz } svuau���cRz�ikuaitp{ion|mkc�z�z{sxuBp�¬�itc�uÌwa����w�nx� ��c�zec*pguac��sxuBp{ikc�uau�c�uBp�·Blac } cRz'¶�lP�0p{lPnqp{iksxuPz } c�w��{cRz{z{itsvu } � sv� } �{c���nv��^Hn|l���nx�{���x­_¾�svl�z�w��{svwPsBz�svu�z�n|mksx��z } c�b-s } itÀ�c���mtcRz'¶�la���n|mo�*lam���zgw�nx��mkc�z-zr��^���b7nxz } c9p�~jwPc�¦§s } lauasq¨©c�uË� �Rz�svmt¨qnxuvpglau w��{sv¬amt`�b-c } c8�'ikc�b7n|u�u wa������sxu } itp{iksxu�u���nxl mkitc�l } lwa�rsx¬�mt`�bgc } c��'ikc�b7n|u�u¤sx�rit�vitu�nxmL­ È c*p{p{c�z�p{��nqp����xikc7c�z�p�nxwawamkik·Bl���c���lauÌz{sxmk¨xc�l�� } c7p�~jw=c�¼�½��'svc�c*p�u�sxl�z�b-svuvpr�{svu�z·Bl�� c�mkmtcµwPc��{b-c�p } cµ�{c�p{�rsxla¨vc��»lau�¬Psvu��*svbgw=sx�{p{c�b-c�uBp } cµmkn:wa�{cRz{z{iksxu»­���c�z#c*�yw����ritc�u��*cRz#ujlab����rio·vl�c�z»¨jikc�uau�c�uBp��*svuyÀ��rb-c����c�z'����z{lamtprnqp�zµp{^���sx�rik·BlacRz�­� Æ ��� Ä ¢�� � �¤£ hysxmk¨xc�l��rzew=sxla��mtcRz�����sxlamkc�b-c�uvp�z��*svb-wa�{cRz{z{ik¬amtcRz�ÉJ½�nxit¬amkc7uasvb�¬a�rc } c���nv��^»É��´�{sv¬amt`�b-c } c��'ikc�b7n|u�u»Éhy��^ ��b7n } c�¦§s } lauasq¨EÉ�hy��^���b7n�¼§½��'sxcvÉ��$�����*svu } itp{iksxuau�c�b-c�uBp�­

Page 5: On the behavior of upwind schemes in the low Mach number ...

�������������� �������������������� �"!$#&%'���&(�'#)� �*�����,+-���/.* 0%'�1�2��(3��'�4+5� (�� �7698�8;: �

<>= ���'��(*�'�@?���%&������

]_^aceª¸sx��A7s|¥$��ua�vc�mks-��la�{�rsxu�c§^Pnxz¸¬Pc�c�u�z{lawaw=sx�{p{c } ¬j~�ng�x��n|uBp_s|¥Jp{^acCB È sxu�z{c�ikm¿�'c��vitsvu�n|mED�sx¥��rc��xiksxu9�E� È ��n|u }pr^ac �§]'�GF|ha]'�4FqÊJ��]J� s|¥ È9H � È n } nx�rnv��^acx­J��ua�vc�mks��8la�r�rsxuac'ª'ioz�^�c�z$prs�nIAjuasqª'mkc } �xc¸p{^ac§z�l�wawPsv��pµs|¥��©­j¦§�rnxu } s|p{p{sJ È9H � È n } n|��nx��^ac�K0ÉJ��­�¦§la¬=c��ru�nqprikz J È9H � È n } nx�rnv��^ac�K�n|u } ��­Jhynxla�{c�m JMLON hy] L ��n|��z{c�ikmtmkc�K*­9�:c�ªµsvlam } nxmkz{s�mki-Avc-p{spr^�n|uPA�^aioz¸¥Mn|b-itmk~�¥¹sv�_p{^ac�it���*sxuPz�p�n|uBp_z{lawaw=sx�{p:nxz_ª¸c�mkm»nxz È ­y¼�itsRQ�n|p'¥¹sx�'^ac��'wa�{cR�*iksxl�z_n } ¨jik��cx­

STS àIUPV±õ�ø�W

Page 6: On the behavior of upwind schemes in the low Mach number ...

d � :������ + +- I� ! I� !�� :T.1����� ���"�

� ������������

���� ��  Æ Ç��¢|�� Æ � �

����  ¢ Æ � �� � Æ ����#  Æ � �  � �

�����  �  �©¡ ��� ��  Æ ���  �� � � �  � ÆjÅ � ¡�¢ ��� ��H�   ��� ��ù� ���

����� ÂH �²Â ¢|� Æ�! �#   ¢ Æ � Ç�ù�� Æ � ��" �$#

#�% Æ � ¢�� #�� Æ � �'&

� � ���  ( � *)

+�,TS-+ è

Page 7: On the behavior of upwind schemes in the low Mach number ...

�������������� �������������������� �"!$#&%'���&(�'#)� �*�����,+-���/.* 0%'�1�2��(3��'�4+5� (�� �7698�8;: �

� �* »������� �&?����J�'

� � � Æ�� ¡��  � Æ*! � �  �#   ���#  Â���Æ � ¡ �-��� � � Æ Ç Â �  � � !óÆ  �� ���������ë� Æ ��� � �� ����� ���� ��¹ÇJÇ �  � � ������� ��������� Æ ��� Æ ����¡ � Ç !OÆ   ��ÆP ��¢ �  �  � �  ! ��� ��� ����ÆP �R¢ �  �  � ��¹ÇJÇ �  � �! Æ Ç� � Æ�� ��¢ �  �  �ë  " � ����" ­§­�­e­�­e­�­�­e­�­e­�­§­�­e­�­�­e­�­e­�­e­�­�­§­�­e­�­e­�­�­e­�­e­�­§­�­e­ #

� � �  !óÆ #  � Æ ����¹��� ÂËÅ ¡ �= �J¡y���    � �� � � �  � Æ � »�� Æ � Æ�! � �  �  �©¡ � � ��  Æ ���  � £��¡�� � Ä q¡y   ! ¡a¢|�� Æ � � ¢ Æ � ��¡�¢|� � � Ä � � Æ ¢%$ � ���&� � Ä � � Æ ¢%$ � ¢ Æ � ��¡�¢|� � � Ä  q¡y   ! ¡a¢|�� Æ � � ��¢'� � Ä  ±¡a   ! ¡a¢x� Æ � �¢ Æ � �R¡a¢|� � � Ä  q¡y   ! ¡a¢x� Æ � � ��Ç�� � Ä � � Æ ¢%$ � ¢ Æ � �R¡a¢|� � � Ä � � Æ ¢%$�" ­�­e­�­e­�­�­§­�­e­�­e­�­�­e­�­e­�­§­�­e­ �R�

� � � �   q�  ¢|�� Æ � Æ�!  ±¡a   ! ¡a¢x� Æ � ¢�#  �P � !óÆ  )(+*-, .�/0(213, .546�7� �  ! ����¡ � ÇË� � Æ ¢8$Ì¢ »  �= � !óÆ  9(2*-, .&/(213, .5:6�;��  "�� �<��" ­e­�­e­�­e­�­�­§­�­e­�­e­�­�­e­�­e­�­§­�­e­�­�­e­�­e­�­e­�­�­§­�­e­�­e­�­�­e­�­e­�­§­�­e­ ���

d % Æ ����¡a  M� Æ � �  � Å�Âj � ! Æ Ç� � Æ�� � Æ � �P  7¡ � ÇZ¡a� à ����� Æ � M¢�¡ � ¡�� à ��¹� !óÆ  =(2*>4�(216� �  ! ����¡ � Ç!óÆ  ?(2*>:�(21@��  "�� �<��" ­�­e­�­�­§­�­e­�­e­�­�­e­�­e­�­§­�­e­�­�­e­�­e­�­e­�­�­§­�­e­�­e­�­�­e­�­e­�­§­�­e­ ��d

� A    �R� #  Â>B �¢|�#¡y�� Æ � � � � � B�ÆjÅ � ¡�¢ � � ��H�   �C !óÆ  �¢ Æ ���J¡y  ¹� Æ � � �  ¢�#  �=ÂÌÃED (&F ( ¹�Ç ¹��� �M¡ Ã= ÇG" ­e­�­�­e­�­e­�­e­�­�­§­�­e­�­e­�­�­e­�­e­�­§­�­e­�­�­e­�­e­�­e­�­�­§­�­e­�­e­�­�­e­�­e­�­§­�­e­ ��#

H � � Æ�� ¡��  � Æ*! � �  �#   ���#  Â���Æ � ¡ �-��� � � Æ Ç Â �  � � !óÆ  � ���������ë� Æ ��� � � ����� ���� ��¹ÇJÇ �  � � � �I��� �����J��� Æ ��� Æ ���<" ­e­�­�­e­�­e­�­§­�­e­�­�­e­�­e­�­e­�­�­§­�­e­�­e­�­�­e­�­e­�­§­�­e­ ���

STS àIUPV±õ�ø�W

Page 8: On the behavior of upwind schemes in the low Mach number ...

H � :������ + +- I� ! I� !�� :T.1����� ���"�

� 3 �����J�7�8��"²�J�*���

L u���sxb-wa�rc�zrz�ik¬amkc�¶�sqª�z�n|�rc�n�w�n|�{p{io�*lamon|�§��nvz�c�s|¥µ��sxb-wa�rc�zrz�ik¬amkc�sxuacRz�nxu } pr^ac��rc*¥¹sv�{c�itu>wa�{iku���itwamkcxÉ¿n��*sxb-wa�rc�zrz{it¬amkc�¶Psqªz{sxmk¨xc��¸z{^asxl�m } ¬Pc�nx¬amtc�p{s7��sxb-walyprc�pr^ac�z{c§¶�sqª�z�­ N uy¥¹sx�{p{lauPnqp{c�mt~vÉjp{^ac��{c�nx�{c§c*�yw=c��ritb-c�uBp�n|mEc�¨ji } c�u���c�z_z{^asqª'itu���p{^Pnqp'sxun9Àa�yc } b-cRz�^»É#p{^ac�z�svmtlypritsvu�zes|¥¸p{^�c��*sxb-wa�rc�zrz{it¬amkcg¶�sqª } ikzr�*�rc*pri-Q�c } c�·Bl�n|p{iksxu�z�nx�{c-uasxp�n|u¤nx���*la��nqprcgnxwawa�rs±�jikb7nqpritsvusx¥²p{^�c�z�svmtlypritsvu�z§s|¥µp{^ac-iku��*svbgw��{cRz{z{it¬�mtcgb-s } c�m J cv­ ��z{c�c�� � H�� K0­7]_^aioz�·vl�c�z�p{iksxu�^Pnxze�rc��*c�it¨vc } n���sxu�z{i } c���n|¬amkc�ikuBp{c��{cRz�piku�pr^ac-mknvz�pe~xcRn|��z:nxu } z{c�¨vc���n|m��{cR�*c�uBpeªµsv� Ayz�^�n±¨xc�p{�ritc } prs�c��jw�mknxitu�p{^ac-�rc�nxz{sxuPz:sx¥´p{^aioz } i����*lamtp�~Hn|u } p{s8�*svu�z�p{�rl��0pujlab-c��rio��n|mµz{��^ac�b-c�z�¨qn|mki } ¥¹sx��nxmtmµ�Hnx��^¤ujlab�¬=c���z�­9hjsvbgc�s|¥_p{^acRz�c7ª¸sx��Ayzec*�jp{c�u } p{sHp{^ac���sxb-wa�rc�zrz�ik¬amkc-�{c��xikbgcvÉ#pr^acujlab-c��rio��n|m�bgc�p{^as } z´l�z�c } ¥¹sx��p{^�c:�*svbgw�lyprn|p{iksxu-s|¥EituP�*sxb-wa�rc�zrz{it¬amkc¸¶�sqª�z�­ H �an|b-wamkc�z´s|¥=p{^acRz�c_p�~jw=c's|¥Ebgc�p{^as } zµn|�rc¸¥¹sx�iku�z�prn|uP�*c��k� � sx��� �'# � ­���uasxp{^ac��§nxwawa�rsvnx��^�c�z_�rc�mk~�svu�z�svb-c�b-s } i³ÀP��nqpritsvu�z�sx¥´^ait�v^�sx� } c��§z�^�sj� A8��nxwyp{l��{ikua�-p{cR��^auaio·Blac�z�­]_^acRz�cen|w�wa�{sBnx��^acRz´n|�rc�¥¹sv�µiku�z�prnxu��*c } c�zr�*�rit¬=c } itu� # � É�×� � ¥¹sx�_�'sxc } ioz{���{c�p{i Q�nqpritsvu»ÉBitu� � � ¥¹sv�²pr^ace�:Ê#Ê H zr��^ac�b-c§n|u } itu�×� � � ¥¹sx�'½Jmtly�8z{��^�c�b-c�z�­$]_^ac�ik�_wa�{iku���itw�nxm¿itu��x�rc } ikc�uBp'ikz¸pr^acel�z{c§sx¥�wa�rc���sxu } i³pritsvuaitu���prc���^auaio·Blac�z_sv�{ik�xiku�nxmtmk~ } c�¨vc�mksxwaw=c }¥¹sv��z�p{cRn } ~�z�prn|p{c'�*svb-walyprn|p{iksxu�z��×�|� � É��×�y� � É�� d � p{^Pnqp´n|�rcµ^�c��rc¸z{c�mkc��*p{ik¨xc�mk~�n|w�wamtikc } sxuamk~ep{s§pr^ac_lawjª'itu } n|�{p{itÀP��n|mB¨Biozr�*svz{itp�~x­

L u � # � É$ªµc�^�n±¨xc�c��an|b-itu�c } p{^�ikz�p{cR��^auaio·vl�c7¥¹sx�-�'sxc�p�~Bw=c�z{sxmk¨xc��rzeª'^�c��rc7p{^ac�ujlab-c��rik��n|m´¶Ply�¤¬Pc�p�ªµc�c�u¤p�ª¸sH�*c�mtmozp�nIAxcRz¸p{^ace¥¹svmtmksqª'ikua��¥¹sv�{b��

����� 1�� � *���� 1 *�� � �� ��� � �&1�* ��� 1�� � � �&1�* ��� *!�" $#�% � � �&1�*% �

# & 1�* � � J ��K

�:c��rc �&1�* ikzµpr^ac�l�uai³p_uasv�{b7n|m=nqp¸p{^ac§ikuvprc��{¥Mnx��c�n|u }�& 1�* � � � 1 / � * iozµp{^acµ��lab-w�¬=c*p�ª¸c�c�u7p{^ac§¨qn|mklac�z'� 1 n|u } � * sxucRnx��^8z�i } c�s|¥�p{^ac�ikuBp{c���¥Mnv�*cx­

(¸~�w=c��{¥¹sx�rbgikua�8n|u6nxz{~Bb-wyprs|prik�gn|u�nxmt~yz{ikz�s|¥²p{^ac } ioz{���{c�p{cgc�·Bl�n|p{iksxu�z�É¿ªµcgi } c�uBp{itÀ�c } pr^ac-mtikb-i³p�c�·Bl�n|p{iksxu�zezrnqprikz�À�c }¬j~��'svc } ikzr�*�rc*pri-QRnqp{iksxu>n|u } z�^asqª¸c } p{^�n|p�pr^ac�z{c-mtikb-i³p�cR·vlPnqp{iksxuPz�z{lawaw=sx�{p�wa�{cRz{z{la�rc�¶�lP�0p{lPnqp{iksxuPz�sx¥´p{^�cgsv� } c��§s|¥²pr^ac��nv��^7ujlab�¬=c��R­J]_^�c��rc*¥¹sx�rcxÉvitu-pr^ac�mksqª ��nx��^-ujlab�¬=c��µmtikb-i³pRÉxpr^ac�z{sxmklypritsvu�z$s|¥Epr^ac:�'svc:�*svbgw��{cRz{z{it¬�mtc�z{sxmk¨xc��´��n|u�uas|p²¬=cnxu9nv���*l��rn|p{cen|wawa�rs±�yitb7n|p{iksxu�sx¥#pr^ac�ituP�*sxb-wa�rc�zrz{it¬amkcez�~yz�p{c�b�ª'^ac��rc§p{^acewa�rc�zrz�l��{c�¶�l��*p{l�n|p{iksxu�z'zr��nxmtcRz_ª'i³pr^�pr^ac #*)'�2 I� �sx¥�pr^ac���nx��^9ujlab�¬=c��R­

L u�p{^aioz'w�nxwPc���Éyª¸cec*�an|b-ikuac§p{^ac�zrn|b-cewa�rsx¬amkc�b ¥¹sv��¦§s } lauasq¨7p�~jw=c�z{��^ac�b-c�z�­ L u8�*svuBp{��nxz�p¸ª'itp{^��'svc�p�~jw=c�z�svmt¨vc���z�Épr^ace¶�ly�9^ac��rceikz'sx¥#pr^ace¥¹sx�rb+�

������1 � ��* �,� 1�* ��� � �-� 1�*.�/�%1�* � J �0Kª'^ac��{c0��1�*©ioz_p{^�c�z�svmtlypritsvu�sx¥Jnxu�c��ynv�0p'sx��nxwawa�rs±�yitb7nqprc§��itc�b-nxuau9wa�{sv¬amkc�b } c�À�uac } ¬j~�p{^acep�ª¸sgz�prn|p{cRz1�%1�n|u } �%*µ­

]_^acRz�c§p�~jw=c�s|¥Jzr��^ac�bgcRz_c*�y^aik¬aitp'p{^ac�zrn|b-cewa�rsx¬amkc�b�sx¥�nv����la�rnv�*~gp{^Pn|u��'sxc�� z'nxu } s|p{^�c��'lawjª'iku } z{��^ac�b-c�z�­½Jik��­���z{^asqª�z�n|u c��an|b-wamkc�s|¥:p{^aiozg¬Pc�^�n±¨jitsv��­32�u¤p{^�ikz�À��vla�{c�ikz-z{^asqª'u n>z�cR·vl�c�u���c�s|¥§�*sxb-walaprnqpritsvu�z�s|¥�pr^ac�¶Psqªnx�{svlau } n�¾�� È �:�x���R�-nxit�{¥¹sxikm#nqp } c����{cRnxz{itua���Hnx��^�ujlab�¬=c���sxu8pr^ac�zrn|b-c�b-c�z{^»­ L u8pr^ac�z{c���sxb-walyp�nqp{iksxuPz�É�p{^ac�ituy¶Psqª¨vc�mksj��i³p�~7ioz�Axc�wyp��*svu�z�p�n|uBp_c�·Bl�n|m=prsgl�uai³p�~�ª'^aitmkc§p{^aceikuy¶�sqª w��{cRz{z{la�rc:ioz_itu����{cRnxz{c } ­$]_^ac�À��vla�{cRz_z�^asqª p{^aceuasv�{b7n|mki Q�c }wa�rc�zrz{la�{c�4 / 465�7 8:9;4�5�<>= / 4�5�7 8y­>��z�pr^ac9��nv��^¤ujlab�¬=c�� } c����{cRnxz{c�z�É#pr^ac��rc�z{lamtprz�¬PcR�*sxb-c�ªµsv�rz{c�n|u } p{^ac9z�svmtlypritsvu�z} s7u�s|p���sxuj¨xc��{�vc:prs�n-�{cRnxz{sxu�nx¬amtcen|w�wa�{s±�yikb7nqp{iksxu�s|¥�p{^�c�itu���sxb-wa�rc�zrz�ik¬amkc�z�svmtlap{iksxu»­²]_^�c�À��xl��{cRz } ikz{wamon±~7¥¹�rsxb�mkc*¥ùp:p{s�rit�v^Bp²p{^�c§�rc�z{lamtprzµsx¬yp�n|ikuac } ª'i³pr^��'svc§zr��^ac�bgc�ª'^ac��rc�pr^ac§¶�ly��ioz²sx¥»p{^�c�¥¹sv�{b J ��K0Éjprsx�vc*p{^�c��¸ª'i³pr^�p{^�c��rc�z{lamtprz¸sx¬yp�n|ikuac }¬j~�pr^ac�¼�½��'svc�z{��^�c�b-c J p{s7¬=c } cRz{���{ik¬Pc } ¬Pc�mtsqªCK¸n|u } ¦§s } l�uasq¨�zr��^ac�b-ceª'^ac��rcep{^�ce¶�ly�yc�z'nx�{ces|¥Jp{^�ce¥¹sx�rb J �RK0­ L p�iozz{c�c�u�pr^�nqp_pr^ac�z{nxb-c§�rc�z{lamtprz'nx�{cesx¬aprn|ikuac } ¥¹sv�_p{^acRz�cepr^a�{c�celawjª'itu } zr��^ac�b-cRz�­

N u } c���z�prn|u } itu���pr^ac_�{cRnxz{sxu�s|¥�pr^aioz$¬Pc�^�n±¨jitsv��ikz�w�nx��prik��lamon|�rmt~�ikuBp{c��rc�z�p{ikua�eiku�pr^ac���nvz�c_s|¥¿¦§s } lauasq¨�zr��^ac�bgcv­Jhjiku���c'itupr^aikz_zr��^ac�b-cvÉxpr^ac:¶Ply���*svbgw�lyp{c } nqp¸p{^ac§ikuvprc��{¥Mnx��c�ikz¸c��ynv�0pRÉxpr^ac�svuamk~gc��{�rsx�µz�svla�r��c��rc�z{lam³p�z´¥¹�rsxb«pr^ac�w��{sx��c��0pritsvu7s|¥»pr^acz{sxmklyp{iksxu�svu�p{^�c�¬�nvz�ioz�sx¥´waitcR�*c�ª'ikz{c��*svu�z�prn|uBp�¥¹l�u��0pritsvu�z�­��:z:¥¹sx����sxc�zr��^ac�b-cvÉPª¸c�ª'ikmtm�z�c�c�pr^�nqp�p{^acgn|u�nxmt~yz{ikz'�rc�¨vc�nxmkzpr^�nqp7pr^ac } ioz{���{c�p{i Q�c } z{sxmklyp{iksxuPz7n|�rc�w=sxmkmtlap{c } ¬j~Ìw��{cRz{z{la�rc�¶�lP�0p{lPnqp{iksxuPz7s|¥§sx� } c�����nv��^»­�]_^ac��rc*¥¹sv�{cvÉ´pr^ac�·Blac�z�p{iksxu¬=c���sxb-c�z�p{s�i } c�uBp{it¥¹~8p{^acgb-c���^�nxuaioz�b p{^Pnqp��*�rc�nqprc�z�wa�rc�zrz�la�rc�ª_n±¨xcRz�s|¥´sv� } c�����nx��^�n|p:pr^ac�ikuBp{c���¥Mnv�*c�¥¹�rsxb nxuHikuai³priknxmÀ�c�m } pr^�nqp:��sxuBprnxitu�z_svuamk~�wa�rc�zrz�la�rc�¶�l��0prl�nqpritsvu�z_s|¥�sx� } c��'�Hnx��^8z{·Bl�nx�{c } ­

+�,TS-+ è

Page 9: On the behavior of upwind schemes in the low Mach number ...

�������������� �������������������� �"!$#&%'���&(�'#)� �*�����,+-���/.* 0%'�1�2��(3��'�4+5� (�� �7698�8;: #

Roe Mach=0.1

PRESSURE : MIN=71.0920 MAX=73.5269

VFRoe Mach=0.1

PRESSURE : MIN=71.0920 MAX=73.5254

Godunov Mach=0.1

PRESSURE : MIN=71.0919 MAX=73.5208

Roe Mach=0.01

PRESSURE : MIN=7138.78 MAX=7159.35

VFRoe Mach=0.01

PRESSURE : MIN=7138.78 MAX=7159.35

Godunov Mach=0.01

PRESSURE : MIN=7138.78 MAX=7159.35

Roe Mach=0.001

PRESSURE : MIN=714244 MAX=714446

VFRoe Mach=0.001

PRESSURE : MIN=714244 MAX=714446

Godunov Mach=0.001

PRESSURE : MIN=714244 MAX=714446

½Jik�xla�rc7� � � � Æ�� ¡��  � Æ�! � �  ��   �R�»  Â��$Æ � ¡ �-�*� � � Æ Ç Â �  � � !OÆ  0� � ����� �ë� Æ ��� � � ���� ����� ��¹ÇJÇ��  � �� ��� � � ������� Æ ��� Æ ��� ¡ � Ç !OÆ   ��ÆP �R¢ �  �  � �  ! �<� �0� ����Æ= �R¢ �  �  � ��¹ÇJÇ��  � � ! Æ Ç� � Æ�� �R¢ �  � Â��  "�� �<��"

hjiku���c-p{^ac�ª¸sx��AHsx¥'hE­�hy��^asy��^ac�p �t�<# � J z{c�c�nxmkz{s � � � Kep{^ac7¬=c�^�n±¨jiksx��ª'^ac�u©p{^ac��Hnx��^©ujlab�¬=c����vsxcRz§p{s1Q�c��{s�sx¥µpr^acz{sxmklyp{iksxuPz#sx¥ap{^ac_�*svb-wa�{cRz{z{ik¬amtc H lamkc���c�·Bl�n|p{iksxu�z#¥¹sx�J�xc�uac���n|mvitu�i³priknxm } nqprn�ÉqikzJª¸c�mkmvl�u } c���z�p{sjs } nqpJmtcRnxz�p�¥¹sv�#pr^ac¸ikz{c�uBp{�rsxw�ik�cR·vlPnqp{iksxuPz���pr^acemtikb-i³p:z{sxmklypritsvu�z{c�w�nx�rn|p{c�zµikuvprs7n|u�nv�*svl�z�prik��ª_n±¨xc�p{^�n|p } c�w=c�u } z_svu9n�¥Mnxz�p_p{ikb-c§¨qnx�{ion|¬amkc � 9 ��� wamkl�z'nz{mtsqª w�n|�{p�pr^�nqp } svc�zJuasxp } c�w=c�u } sxu�p{^aiozJ¥Mnxz�p�p{ikb-c¸¨qnx�{ion|¬amkcx­��8sx�rc�sq¨vc��#p{^ac'z{mtsqªÌwPn|�{p } svc�z�zrnqp{ioz�¥¹~ep{^ac_iku��*svbgw��{cRz{z{it¬�mtcH l�mtc��$c�·Bl�nqpritsvu�z�­�½asv�´n���c��{prnxitug�*monxzrz�sx¥Pikuaitp{ion|m } n|prn J p{^�c¸ª¸c�mkmywa�rc�w�nx�{c } ��nxz{c�K»p{^ac'nx��sxl�z�p{io�¸�*sxb-w=sxuac�uBpJioz�uas|p$wa�rc�z{c�uBpn|p§mkc�n } itu���sx� } c���n|u } p{^ac-z{sxmklyp{iksxu�z�s|¥$pr^acgitu���sxb-wa�rc�zrz�ik¬amkc H l�mtc���cR·vlPnqp{iksxuPz�nx�{c�pr^ac-z�pr�{svua��mkitb-itprz§s|¥$pr^ac-z�svmtlypritsvu�zsx¥�pr^ac��*svbgw��{cRz{z{it¬�mtc§sxuacRz'nxz_ª_nxz�Ajuasqª'u�¥¹�{svb�z{sxb-cec�n|�rmkitc��¸�rc�z{lamtprz_sx¥.� � � n|u } � � � ­

½�sx�gp{^ac8uasvu��óikz{c�uBp{�rsxw�ik��bgs } c�mLɲp{^ac�z{i³prl�nqpritsvuËiozgmtcRz{z-��mtcRn|�R­ L uËp{^ac8ª¸c�mkm¸w��{c�w�n|�rc } ��nxz{cxɸnxz-z�^�sqª'u iku �k� H�� ɲitpioz)Ajuasqª'u>p{^�n|p�p{^ac�z{sxmklypritsvu�zes|¥¸p{^ac7iku���sxb-wa�rc�zrz�ik¬amkcgb-s } c�m²n|�rc7z�pr�{svua��mkikbgitprz�s|¥¸p{^ac�z{sxmklypritsvu>sx¥µpr^ac��*svbgw��{cRz{z{it¬�mtcb-s } c�mO­�]_^�c�z{nxb-c��{cRz�l�m³p§ikz�p{�rlac�nxu } uas|p§uac���c�zrz{nx�{~�¥¹sx�§ªµc�mtm�wa�{c�w�n|�rc }8} n|prn7iku�pr^ac�ª'^asvmtcgz�wPnx�*c �k�x� � ­���sqª¸c�¨xc���É�itupr^ac�wPc��{iks } io����nxz{cxÉyn-mks|p's|¥$·Blac�z�p{iksxuPz_�{c�b-nxitu�sxw=c�u J z{c�c �k�R� � ¥¹sx��n|u8law��Lprs�� } nqprc��rc�¨jikc�ª[sx¥�pr^aioz'wa�rsx¬amkc�b3K0­

STS àIUPV±õ�ø�W

Page 10: On the behavior of upwind schemes in the low Mach number ...

� � :������ + +- I� ! I� !�� :T.1����� ���"�

]_^ac�lau } c��rz�prnxu } ikua�Hsx¥¸pr^aioz���sxb-wamkc*�©¬Pc�^�n±¨jitsv��ioz�l�z{c*¥¹lamµp{s6��n|pr��^©p{^�c��rc�nxz{sxuPz�ª'^j~¤¦§s } l�uasq¨6zr��^ac�bgc�¥Mn|ikmkz�p{s��sxb-walyprc�pr^ac�iku���sxb-wa�rc�zrz�ik¬amkc�mkitb-itp�­��'�0prl�n|mkmt~vÉ»nxz�ªµc�ª'itmkm$z�c�c�iku>z{c��*p{iksxu���ÉPpr^acgituBp{c���¥Mnv�*c�¨±nxmtl�c�ze�*svbgw�lyp{c } ¬j~�¦§s��} l�uasq¨gzr��^ac�bgc��*sxuBp�n|iku�ne¥Mnxz�pµnx��sxl�z�p{io�'�*svb-wPsvuac�uBp²pr^�nqp¸�*sxb-cRz$¥¹�rsxbºp{^ac�wa�{sx��c��*p{iksxu-s|¥Ep{^�c:ikuai³priknxm } nqprn�sxu7waikc���c�ª'ioz�c��sxu�z�prnxuvp§À�c�m } z�­�]_^acgwa�rc�zrz�la�rc�w=c��{p{la�r¬�n|p{iksxu�zenxzrz{sj��ikn|p{c } ª'i³pr^�pr^aikze��sxb-wPsvuac�uBp�n|�rc�sx¥²sx� } c��e��nx��^6n|u } ^ai } cg��sxb �wamkc*prc�mk~�p{^�c�iku��*svb-wa�{cRz{z{ik¬amtc�w�n|�{p�­��º��sxu�z{c�·Blac�u��*c�s|¥²pr^aioz:¥Mnv�0peikz§pr^�nqp§p{^�c�ÀPc�m } �*svb-walyp{c } ¬j~9p{^�c } ikzr�*�rc*prcgzr��^ac�b-c�rc�z{lamtprz¸¥¹�rsxb�ng¬�nxmknxu��*c§s|¥$nx��sxl�z�p{io�§ª¸n±¨vc�zµituPz�prc�n } sx¥Jzrnqp{ioz�¥¹~Bikua�gp{^�c�c�·Bl�nqpritsvu�z_s|¥�p{^ac�iku���sxb-wa�rc�zrz�ik¬amkc§b-s } c�mL­

]_^ac-z{lab-b-nx�{~8sx¥´p{^aioz§ªµsv� A9ioz§nvz�¥¹svmtmksqª�z�­ L u6z�cR�0pritsvu>��ª¸c��rc���n|mkmJz{sxb-c ABu�sqª'uH�{cRz�l�m³p�z�svu�p{^�cg¬=c�^Pn±¨Biksx��s|¥´pr^acz{sxmklyp{iksxuPzµsx¥#pr^ac H lamkc��'cR·vlPnqp{iksxuPzµiku�p{^acemksqª ��nx��^�ujlab�¬=c��'mkitb-itp�­´]_^acewal��{w=svz{ces|¥»pr^aikz:z�cR�0p{iksxu�ikz¸prs7z�^asqª p{^�n|p¸pr^acz{sxmklyp{iksxuPz$s|¥Ep{^�c:iku��*svbgw��{cRz{z{it¬�mtc'b-s } c�m=n|�rc'sxuamk~�ª¸c�n0A�mkitb-itprzµs|¥=p{^ac§z�svmtlap{iksxu�z$sx¥Ep{^ac:��sxb-wa�rc�zrz�ik¬amkc H lamkc��µbgs } c�mL­$½�sx�pr^aikz�É�ª¸ceituBpr�{s } l���c�n7z�ikb-wamtc�mkikuac�nx�'bgs } c�mEp{^�n|p } ioz�wamon±~yz¸p{^ac�zrn|b-c§p�~jwPc�sx¥J¬=c�^Pn±¨Biksx�_pr^�n|u�p{^�c H lamkc��'cR·vlPnqp{iksxuPz'n|u }ª¸c�itu } ik��nqprc§^�sqª p{^�c��{cRz�lamtprz_svu�pr^aioz'b-s } c�m»c��Bprc�u } prs�pr^ac�uasvu�mkitu�c�n|� H lamtc���b-s } c�mO­

L u�z{c��0pritsvu��aÉaª¸c�z�prl } ~�p{^ac�z�svmtlypritsvu�z_s|¥Jp{^�c��'ikc�b7n|uau8wa�rsx¬�mtc�b�iku8p{^ac�mksqª°�Hnx��^8ujlab�¬=c���mkikbgitp�­¸®6c�z�^asqª p{^Pnqpc�¨xc�u�i³¥�p{^ac§itu�i³priknxm } n|prn�n|�rc:ª¸c�mkmPwa�rc�wPn|�rc } nxu } �*mksvz{c�prsgnxu�iku��*svbgw��{cRz{z{it¬�mtc'ÀPc�m } ª'itp{^�w��{cRz{z{la�rc�¶�l��*p{l�n|p{iksxu�zµs|¥�sx� } c����nv��^Ìz{·Bl�nx�{c } É»pr^ac�z{sxmklypritsvu6sx¥_p{^ac9�'itc�b7n|uau¤wa�rsx¬�mtc�b c*�y^aik¬aitprz�nx�*svl�z�p{io�g¶�l��0prl�nqpritsvu�z�pr^�nqp-z{��n|mkc7ª'i³pr^©p{^ac8��nx��^ujlab�¬Pc���­$]_^aioz´wa^�c�uasvbgc�u�n���nxu7¬Pc:�{c�mkn|p{c } prs�p{^ac�¥Mnx�0p²p{^�n|p²p{^�c�wa�rs|��c��*p{iksxu-s|¥Epr^ac�ikuai³priknxm } n|prn�svu-waitcR�*c�ª'ikz{c:�*svu�z�prn|uBp¥¹lauP�0p{iksxuPzµ���{cRnqp{cRz´n|�{p{itÀP��iknxm } ioz{��sxuBp{ikujlai³pritcRz$pr^�nqp_n|�rc'�{cRz�w=sxu�z{ik¬amtc_¥¹sv�´pr^ac:�vc�uac��rn|p{iksxugs|¥#nx��sxl�z�p{io�_ª_n±¨xc�z$nvz´iku-p{^ac§��nvz�csx¥Juasxp'ªµc�mtm»w��{c�w�n|�rc }�} nqp�na­

½�sx�'pr^ac��'svc�z{��^ac�b-cxÉ�itp:ioz7Ajuasqª'u9p{^�n|p�p{^�c�l�z{c�s|¥$wa�rc���sxu } itp{iksxu�itua�-ioz:n-w=sqªµc���¥¹l�m»�{c�b-c } ~�p{s���la�{cepr^ac�nv���*l��rnv�*~wa�rsx¬�mtc�b � # � É"�×� � Évp{^�c�z{nxbgc���sxu���mtlPz�iksxu�z�^Pn±¨xc¸¬Pc�c�u7�rc�nx��^�c } ¥¹sx�$s|pr^ac��´p�~Bw=c�z´s|¥Elawjª'itu } zr��^ac�bgcRz$nxzJpr^ac:�:Ê�Ê H zr��^ac�b-c� � � sv�:pr^ac-¼�½�½ È p�~Bw=c7z{��^�c�b-c�z �×� � � ­ L u©z�cR�0pritsvu�d�É#ªµc-z�prl } ~�pr^ac7n|waw�mtio��n|p{iksxu�sx¥²wa�rc���sxu } itp{iksxu�itua��p{s9¼�½��'svcgp�~Bw=czr��^ac�b-cRz �k��� � ­Ì®6c8z�^asqª�p{^�n|p-nvz�¥¹sv�gp{^ac9wa�rc�¨jitsvl�z{mt~¤�*itp{c } l�wBª'iku } z{��^ac�b-c�z�É´wa�rc��*svu } itp{iksxuaikua�>ikz-n|moz{s>n>wPsqª¸c��{¥¹lam�rc�b-c } ~7p{^�n|p:n|mkmtsqª�zµpr^ac�nx���*la��nqprce��sxb-walyp�nqpritsvu�z¸s|¥�uac�nx�'itu���sxb-wa�rc�zrz�ik¬amkc§z{sxmklypritsvu�z�­

� � ��� "¸�����J���g������ ��9����8(*��%

]�s¤^ac�mkwZp{^�c�lau } c��rz�prnxu } ikua�©s|¥ep{^ac�¬=c�^Pn±¨Biksx�7sx¥§l�wBª'iku } zr��^ac�b-c�z�itu p{^�cHmtsqª2��nx��^Zujlab�¬Pc��7mtikb-i³pRÉ'ªµc��rc���n|mkm'itupr^aikz�z�cR�0pritsvu©z{sxb-c$Ajuasqª'u©�rc�z{lam³p�z�sxu6pr^ac��*svuBp{ikuBl�sxl�z�w��{sv¬amtc�b8­�®6c�¬Pc��xiku©ª'i³pr^©p{^ac�ioz{c�uBp{�rsxwaio�7z{~jz�p{c�b ª'^ac��{c-pr^acb7nqpr^ac�b7n|p{io��n|mEpr^ac�sv�{~�ikz�nxmtb-sBz�p��*svb-wamtc�p{c�n|u } �rc�z�p{�rik�*p_p{^ac } iozr�*l�zrz�iksxu�p{sgpr^ac�wPc��{iks } io����nxz{cx­

J ¾�s|prc�pr^�nqp-ikuÌp{^�c9��nxz{c�s|¥�pr^ac9ª'^asxmkc�z{w�nx��c�nxu } c�¨vc�u¤¥¹sv���vc�uac��rnxmµikuaitp{ion|m } nqp�naÉ$pr^ac9�{cRz�lamtprzgn|�rc�z�p{�rsxua�vc�� } lacprs8p{^ac } cR��n±~�n|p�ituyÀPuai³p�~>s|¥¸p{^ac7c�uac��r�x~�sx¥µpr^ac�nx��sxl�z�p{io�-�*svbgw=sxu�c�uBp�­7�:z�z�^asqª'u6iku �×�x� � p{^ac7mkikbgitp�z{sxmklyp{iksxu�zesx¥µpr^ac��sxb-wa�rc�zrz�ik¬amkc�c�·Bl�n|p{iksxu�z�z{n|p{ioz�¥¹~9pr^ac�cR·Bl�nqpritsvu�z:sx¥$p{^acgiku��*svbgw��{cRz{z{it¬�mtc H lamtc��ebgs } c�m$n|mtp{^asvla�x^�lau�i³¥¹sv�{b ��sxuj¨xc��{�vc�u���c¬a�rc�n0Ayz_uac�nx� � � � ­ K

]_^aceioz�c�uBp{�rsxwaio� H lamkc���cR·vlPnqp{iksxuPzµª'�ritp{c �

%��% �

� } ik¨�� � ��� � �@�

% �% �

§} it¨"����� � � ����� 4 � �

J d K

L u>p{^acRz�c7cR·Bl�nqpritsvu�z�É�� � � (����6� ioz�n9¨xcR�0p{sv�eÀ�c�m } n|u } � n8z{��n|mon|�§À�c�m } É } c*À�u�c } sxu�� ����� � ª'^ac��{c�� � ioz�n�p�ªµs} ikb-c�u�z{iksxu�nxm#w=c��rits } io��¬=s±�E­ ioz:n�z{b7n|mkm#w�nx�rnxbgc�p{c���wa�rsxw=sx�{p{iksxu�nxmEp{s7pr^acg��nv��^�ujlab�¬=c��R­']_^ac�wa�rc�zrz{la�{c14�ikz:�{c�mkn|p{c }prs7p{^ac } c�uPz�itp�~�¬j~�p{^ac�ioz{c�uBp{�rsxwaio��z�prnqprc�mon±ª 4 ��È p{c ��� ­']�s�b7nIAxcepr^ac�nxu�n|mk~yz�ioz�z{itb-wamkc��RÉ�ª¸c�wPc���¥¹sv�{b�p{^�c�¥¹sxmkmtsqª'ikua���^�nxua�xcesx¥�¨qn|�riknx¬amkc �

�� � �� /��! 4#"�� � �&�

��� 4�" � � � J H K

nxu } z{c*p �� � �%$ &� ª'^�c��rc �&$ ioz_n���sxu�z�prnxuBp�­ L u�prc��rb s|¥»pr^aceuac�ª[¨±nx�{ion|¬�mtcRz�� � � � � Éjpr^ac�z�~yz�p{c�b J d K´prn0Axc�zµpr^ac§¥¹sx�rb �

% �% �

� � � � ��#�&$ } ik¨'� �I�

% �% �

§} ik¨ �(�)*� � ���� � � ��#� $ � � � �J �RK

+�,TS-+ è

Page 11: On the behavior of upwind schemes in the low Mach number ...

�������������� �������������������� �"!$#&%'���&(�'#)� �*�����,+-���/.* 0%'�1�2��(3��'�4+5� (�� �7698�8;: �

®6c�ª'�{itp{c§pr^aikz:z�~yz�p{c�b�itu9p{^�c�z�~jb�¬=sxmkio�:¥¹sv�{b �

% �% �

�� ��� � � � �� � $ �!� �I� J �RK

ª'itp{^ � � � � � ( � � ��� É��!� � � } ik¨'� � � � ��� ioz¸p{^ac�mkikuac�nx��Éa��sxu�z�prnxuvp���sBc*���*ikc�uBp�Éanv�*sxlPz�prik�§sxw=c���nqprsx�R­��zJn�À���z�p�z�p{c�w�iku�pr^ac'n|u�nxmt~yz{ikz�s|¥ J �RK*Éx�*svu�z{i } c��#pr^ac¸¥¹sxmkmtsqª'ikua�§�*sxuPz�p�n|uBp���sxc*�7��itc�uBpJmkitu�c�n|��b-s } c�myw��{sv¬amtc�b�sx¬yp�n|ikuac }

¥¹�rsxb J �RK:¬B~�n�mtikuacRn|�rikzrnqpritsvuHn|�rsxlau } p{^ac-z�p�nqprc � � � � � � � ������ ­�½�sx��p{^ac7zrnIAxc�s|¥µz{itb-wamkio�*itp�~xÉ¿ikuHp{^�c�¥¹svmtmksqª'itu���ª¸c�z{c*p�� �%$ � � ­

% �% �

�+��� � � ��(� � � �% �% �

�+��� � � � � � �J �v��K

®6c�ª'�{itp{c§pr^aikz�mtikuac�nx��z�~yz�p{c�b�lau } c��_p{^ace¥¹sv�{b��

% �% �

� � � �!� �@� J �±�0K

ª'^ac��{c � ���+��� ��ikz¸n���sxu�z�prnxuBp²¨xc�mtsy�*itp�~gmtikuac�nx�¸n } ¨xcR�0pritsvu7sxw=c���nqp{sv��­�®>�ritp{ikua� J �R�0K´itu�½asxl��{ikc��¸z�w�nv�*c'¥¹sv�²pr^ac�½�sxla�rikc����sxb-wPsvuac�uBp�z��� ��� � sx¥ �jÉyª¸ce�xc*p �

% �� ��� �% �

���� � ��� � � �� ��� ��� �� ��� �&� � ¥¹sx� ����� � J �R�RK

ª'^ac��{c§pr^ac�b7nqp{�rit� � ��� � @� 9 ��1��� � ioz'cR·vlPn|mEp{s �������

�+� � � . 9 � � 9 � . 9 �+� � �

� � 9 � �+� �

�! " J ��d K

]_^aioz'b7nqpr�{it��ioz } ion|�xsvu�n|mki Q�n|¬�mtcvÉBitprz�c�ik�xc�uj¨vc��0prsx��zµnx�{c �

� . ��� �&� � �� # � # �

�� �/ ��./ � �

�" � � � ��� �&� �# � #

�� �/ � ���.

�" � �$# ��� �&� � � # � # �

�� ��'.� ��" J � � K

ª'itp{^�nxzrz{sj��ikn|p{c } c�it�vc�uj¨qn|mklac�z&% . ���+� � / # � # � % � ���+� �©n|u } % # �'�2� � # � #

­

]_^ac��{c�¥¹sx�rcxÉBp{^�c�z�svmtlypritsvu�sx¥ J ��� K²ioz �

�� ��� � � �&�())))))))))* ))))))))))+

� �� # � # � �$�� ��� � � �2/ � . �( ��� � ���-/ � � �� ��� � ���>��,.-0/2143�5 67-98 6:8 ;=<?>2� � . ��� � �# � # � / � � �( ��� � ��� � . �� ��� � ���>��, -0/@3�5 6A� � � ��� �

� �� # � # � �$�� ��� � � � � . �( ��� � ��� � � �� ��� � � � ��, -B/C143�5 6EDF8 6:8 ;=<?>2� �$# ��� �

J � H K

½��{svb J � H K�i³p§ikz§z{c�c�uHp{^�n|p:p{^�c�z{sxmklyp{iksxuHz{c�w�nx�rn|p{cRz'ituBprs�n7¥Mnvz�p�svzr�*ikmtmonqprsx�r~��*svb-wPsvuac�uBp:pr^�nqp } c�w=c�u } z:svu�p{^�c�¥Mnxz�ppritb-c�¨qn|�riknx¬amkc � 9

��EG�HJI ��� � � � � 9 � � � � # � # �

())))* ))))+�$�� ��� � ���-/ � . �( ��� � � �2/ � � �� ��� � ���>�K,�-0/2123E5 67-98 6:8 ;=<?>C� � . ��� � �$�� ��� � ��� ��.��( ��� � � �" � � �� ��� � ���>�K, -0/2123E5 6EDF8 6:8 ;=<?>2� � # ��� �

J �<#IK

STS àIUPV±õ�ø�W

Page 12: On the behavior of upwind schemes in the low Mach number ...

�R� � :������ + +- I� ! I� !�� :T.1����� ���"�

wamkl�z�n-z{mtsqª°�*svbgw=sxu�c�uBp_p{^�n|p } sxcRz_uas|p } c�wPc�u } sxu�p{^aioz_¥Mnvz�p'¨qn|�riknx¬amkc �

��EH��@G � ��� � � �&� �# � # � / � � �( ��� � ��� � . �� ��� � � � ��, -B/ 3E5 6A� � � ��� � J �R�RK

L u ½asxl��{ikc��-z�wPnx�*cvÉ�p{^ac@Axc��{u�c�m¸s|¥'p{^�c�mkitu�c�n|�-nx��sxl�z�p{io��svwPc��rn|p{sv� ��1��� � ioz�p{^ac8z�wPnx�*c9z�wPn|uac } ¬j~6p{^ac9¨xcR�0p{sv� � � ��� �=�� ��� / � � � � .���� ­´]_^aioz'�*sv�{�rc�z{w=sxu } z¸iku8wa^j~jz{io��n|m¿z{w�nx��c§p{s-p{^�c�z�w�nv�*c } c*ÀPuac } ¬j~ �Avc��&� ��� � � � � ��� � � �0prc �r} it¨'� � ��� J �R�RK

]_^ac¸z�mksqª©�*sxb-w=sxuac�uBp�� H��@G � � 6 �� H�� G � ��� � � ��,A/4675 � ioz¿pr^BlPz¿p{^�c²wa�rs|��cR�0p{iksxuesx¥Bpr^acµz{sxmklypritsvu§svuep{^aioz#z{w�nv�*c²nxu } �{c�wa�rc�z{c�uBprzpr^ac � � % ��( ������# #����+ � ��sxb-wPsvuac�uBp's|¥�p{^ac�z{sxmklypritsvu»­ L p:z{n|p{ioz�ÀPc�zµp{^acCB�iku��*svbgw��{cRz{z{it¬�mtcez�~yz�p{c�b,D �

% � H��@G �% �

� � H��@G � � � J �x�RK

ª'itp{^8ikuai³priknxm } n|prn � H�� G � � � �I� � �� � � � �I��� ª'^�c��rc � ioz¸p{^ac�wa�rs|��cR�0pritsvu�svu Avc��&��­½��{svb�c*�ywa�rc�zrz{itsvu J �<#IK*É�itp�ikz�z�c�c�u9p{^�n|p � G�HJI �*sxuj¨vc��r�xc�zµiku8p{^ac�z{c�uPz�ces|¥ } ioz�p{�rit¬alap{iksxu�z_prs � ª'^ac�u � � ­²��sqª¸c�¨vc��

pr^aikzµioz´sxu�mt~-n�ª¸c�nIAg�*svuB¨vc��r�xc�u��*c'n|u } p{^ac��{c�¥¹sx�rc'c*�a�*c�wyp´¥¹sx�¸n�z{wPcR�*ion|m���mknvz{z²s|¥¿itu�i³priknxm } n|prnaÉvp{^�c:z�svmtlap{iksxu��*svuBprn|iku�z$¨vc��r~¥Mnvz�p�svzr�*ikmkmkn|p{sx�r~8�*sxb-w=sxuac�uBprz�p{^�n|p } s�u�s|p } ikzrn|w�wPcRn|�eª'^ac�u � � ��iku>�vc�uac��rnxmLÉEc�¨xc�u�¥¹sx�§¨xc��{~Hz{b7n|mkm ª¸cg^�n±¨xc�p{sª'�ri³prc �

� ��� � � �&� � H�� G � ��� � � � � G�HJI ��� � � � � 9 � J �a��K®6c'uasqª } c�zr�*�rit¬=c'n�z�w=c���iknxma�*monxzrz�s|¥Eitu�i³priknxm } nqp�n�z�lP��^�pr^�nqp´ª¸c_ª'itmkm�^Pn±¨xc_n�z�pr�{svua���*sxuj¨vc��r�xc�uP�*c´prsqª¸nx� } p{^ac:itu���sxb �

wa�rc�zrz{it¬amkc�mkitb-itp�­_� uac���c�zrz{nx�{~���sxu } i³pritsvu�¥¹sv� � ��� � � � p{s���sxuj¨xc��{�vc�u���c§z�p{�rsxu��xmk~7p{s-pr^ac�iku��*svbgw��{cRz{z{it¬�mtcemkitb-itp�� H�� G � ��� � � �ioz´pr^�nqp�� G�HJI ��� � � � � 9 � ��sxuj¨xc��{�vc�z$z�p{�rsxua�vmt~�p{s � ­J]_^�ikzµikz²uas|pµwPsBz{z{it¬�mtc ����% � � � it¥¿p{^ac:ituaitp{ion|m } n|prn�nx�{c���mtsBz�c'prs�p{^�cCAxc��{u�c�msx¥F�:­²]_^aioz'ioz_p{^ac���nxz{ce�{c�¥¹c��r�{c } prs�nxz_pr^acCB�ªµc�mtm»w��{c�w�n|�rc } ��nvz�c'D�­²½�sx�_pr^ac�ioz�c�uvpr�{svwaio� H lamtc���c�·Bl�n|p{iksxu�z�Éypr^aikz:z�itp{l�n|p{iksxu^�nvz_¬Pc�c�u8c*�anxbgikuac } iku � � � n|u } � � � ª'^aikmtce¥¹sv�_p{^ac�u�sxu��óikz{c�uBpr�{svwaik��b-s } c�mOÉaª¸c§�rc*¥¹c��_p{s �k� H�� ­

½�sx��svla�'b-s } c�m#w��{sv¬amtc�b J �x��Kµpr^ac B�ªµc�mtm#wa�{c�w�n|�rc } ��nvz�c'De��n|u8¬Pc } c�zr�*�rit¬=c } nvz¸¥¹sxmkmtsqª�z�­ L p�ikz���mtcRn|�'¥¹�rsxb J � H Kµp{^Pnqpit¥ (* + �$�� ��� � ���-/ � . �( ��� � � �-/ � � �� ��� � � �>���� � �

�$�� ��� � ��� � . �( ��� � � � � � �� ��� � � �>���� � �J �v�0K

pr^ac�u�p{^ac§svzr�*ikmkmkn|p{sx�r~-�*sxb-w=sxuac�uBp¸sx¥»p{^acez{sxmklyp{iksxu�� G�HJI �*svuB¨vc��r�xcRz´z�pr�{svua�xmk~gp{s � nxu } pr^BlPz¸p{^ac§ituP�*sxb-wa�rc�zrz{it¬amkc���sxb-w=s��uac�uBp � H��@G � ioz¸p{^�c�z�pr�{svua�-mtikb-i³p�sx¥ ��nxz � � ­

L u8wa^j~yz�io��nxm¿z�w�nv�*cvÉjp{^acep�ª¸s7�*sxu } i³pritsvu�z J �v�0Kµb-c�nxu�pr^�nqp'pr^ac�itu�i³priknxm } nqp�ng^�n±¨xc�p{^ace¥¹svmtmksqª'ikua�g¥¹sx�rb+�� � ��� � ���&� &� . ��� � � ������ � � ��� � $ ��� � � � � . ��� � � � J �x�RK

ª'^ac��{c � . ��� � � � ikzJn|u � � � � z{��n|mon|�¿À�c�m } É � $ ��� � � � n|u } � . ��� � ��� p�ª¸s � � � � ¨vc��0prsx�#À�c�m } z ��"! � $ ��� � � � zrnqp{ioz�À�c�z } ik¨�� $ � � ­Ê#c*p�l�z:uasqª �rc*prla�ru�prs7p{^ac�ikz{c�uBp{�rsxw�ik� H lamkc��:c�·Bl�nqpritsvu�z J �RK0­¸��mtp{^asvla�x^8pr^ac�nxu�n|mk~yz�ioz'n|u } prc���^auaio·Blac�z�s|¥�wa�{sjsx¥�nx�{c

b�l���^�b-sx�rc��*sxb-wamkc*��p{^�nxu8¥¹sx��svla��mkitu�c�n|�:bgs } c�m#wa�rsx¬�mtc�b9É�p{^ac��{cRz�lamtprz:n|�rc�c�zrz�c�uvpriknxmtmk~�pr^ac�zrn|b-cx­_�:z:wa�rsq¨xc } iku �t�<# �J z�c�c�n|moz�s�� � � K*Éaiku�pr^ac�mksqª°��nv��^�ujlab�¬=c��:mtikb-i³pRÉyp{^ac�z{sxmklypritsvu9��n|u9¬Pc�ª'�ri³p{p{c�u8nvz �

� ��� � � �&� � H��@G � ��� � � � � G=H?I ��� � � � � 9 �" �� � � J �|d Knxu } z{c�w�nx�rn|p{c�ikuBp{s�n�¥Mnxz�p¸nx��sxl�z�p{io����sxb-wPsvuac�uBp.� G=H?I ��� � � � � 9 � pr^�nqp } c�w=c�u } zµsvu-p{^ac�¥Mnxz�p´pritb-c§¨±nx�{ion|¬�mtc � 9 nxu } n�z{mksqªw�nx��p � H��@G � ��� � � � ­²�8sx�rc�sq¨vc��RÉxpr^ac�z{mtsqª w�n|�{p'ikz'z{sxmklyp{iksxu9s|¥Jp{^�c�itu���sxb-wa�rc�zrz�ik¬amkc�cR·Bl�nqpritsvu�z �

} it¨�� � � � � ��� � � �� $ � %% �

� .�� ���� � �>� ��� � � � ��� � � �

+�,TS-+ è

Page 13: On the behavior of upwind schemes in the low Mach number ...

�������������� �������������������� �"!$#&%'���&(�'#)� �*�����,+-���/.* 0%'�1�2��(3��'�4+5� (�� �7698�8;: �x�

L u7�vc�uac��rnxm6� H�� G � ��� � � � ioz$sxu�mt~-neªµcRnIA�mtikb-i³p²sx¥Ep{^ac:z{sxmklypritsvu�z$c*�a��c�wyp´¥¹sv�´ª¸c�mkm�wa�rc�w�nx�{c } ikuaitp{ion|m } nqp�n��*mksvz{c_p{s�pr^ac Axc��{u�c�msx¥ �:­���z_¥¹sx�_pr^ac�b-s } c�m»w��{sv¬amtc�b8Éjp{^aioz�z{wPcR�*ion|m»�*monxzrz¸s|¥�itu�i³priknxm } nqp�ngikz'��^�nx�rnv�0prc��ri-Q�c } ¬j~ �

� � ��� � � ��� &� . ��� � � ��!��� � � � � � $ ��� � � � � . ��� � ��� ª'i³pr^ } ik¨�� $ � � J � H K

¦§svitua�-¬�nv� A7p{sgpr^ac�sx�rit�vituPn|mE¨qn|�riknx¬amtcRz ��4 � � � Éaª¸c�z�c�c�pr^�nqp'pr^aioz'itb-wamkikc�z¸p{^Pnqp'p{^�c�ituaitp{ion|m } nqp�ngnx�{c�z{l���^�p{^�n|p �� 4���� � � � �[È prc � 4 � ��� � ����!��� � � � � � $ ��� � � � � . ��� � ��� ª'i³pr^ } ik¨�� $ � � J �'#IK

ª'^ac��{c14 � ��� � � �,� � $ ��� � � �,� �&.���� � � � n|�rc§z{b-sBsxp{^»Éy�rc��vlamon|�¸¥¹lau��*p{iksxu�z�­

� � �����2�*��%/ ������9����9(���% �����J���(��'� +, �"$� �-�9% ��'��(���%;���

®6cgnx�{c�ituBp{c��{cRz�prc } iku��*svb-walyp{ikua��¶�sqª�z:��mtsBz�c�prs�pr^ac�iku��*svb-wa�{cRz{z{ik¬amtc�mtikb-i³pR­��:z§z{c�c�u�itu�pr^acgwa�{c�¨jitsvl�z�z{c��0pritsvu»É=i³¥´pr^acikuaitp{ion|m } nqp�n�n|�rc�ªµc�mtm�wa�rc�w�nx�{c } nxu } s|¥�pr^ac�¥¹sv�{b J �'#IK_ª'itp{^Hw��{cRz{z{la�rce¶�l��0prl�nqpritsvu�z�sx¥�p{^ac�sx� } c���sx¥�p{^acgz{·Bl�nx�{c�sx¥Jpr^ac��nv��^�ujlab�¬=c��RÉ�ª¸c�^�n±¨xc�z�p{�rsxu����*sxuj¨vc��r�xc�uP�*c§s|¥�p{^ac�z�svmtlypritsvu�s|¥Jp{^acg�*svbgw��{cRz{z{it¬�mtcec�·Bl�n|p{iksxu�z'p{s7pr^ac�z{sxmklyp{iksxu8sx¥Jpr^aciku��*svb-wa�{cRz{z{ik¬amtc7cR·Bl�nqpritsvu�z�­���sqª¸c�¨xc��ec�¨xc�u©i³¥_p{^�c7ituaitp{ion|m } nqp�n } s�¨xc��{it¥¹~�pr^aikz�nxzrz�l�bgwap{iksxu»É#p{^ac�w��{sx��c��0pritsvu>sx¥µpr^ac�z{c} n|prn8svu>waikc���c�ª'ioz�c���sxu�z�prnxuBp§¥¹l�u��0pritsvu�z��*�rc�n|p{cRz } ikzr�*svuBp{ikuBl�i³pritcRz�nqp�p{^�c7ituBp{c���¥Mnv�*c-¬=c*p�ª¸c�c�u6pr^ac��*c�mtmoz�­�®�^ac�u¤z{sxmk¨Bikua�pr^ac��'ikc�b7n|u�u�wa�{sv¬amkc�b8Éyp{^�c�z{c } ikzr�*svuBp{ikuBl�i³pritcRz'�xc�uac���nqprc§w��{cRz{z{la�rc§ª_n±¨xcRz¸s|¥$sx� } c��:�Hnx��^»­²��z:n7��sxu�z{c�·Blac�u��*cvÉyitu�z�p{cRn }sx¥��*svbgw�lyp{ikua�©n|u nxwawa�rs±�jikb7nqpritsvu¤s|¥'p{^�c9z{sxmklyp{iksxuËs|¥'pr^ac�iku���sxb-wa�rc�zrz�ik¬amkc�c�·Bl�n|p{iksxu�z�É$lawjª'iku } zr��^ac�b-cRz���sxb-walyprc9n¬�nxmknxu��*cesx¥�nv�*sxlPz�prik��ª¸n±¨vc�z�­

È svu�z�i } c��µpr^ac��'ikc�b7n|u�u9w��{sv¬amtc�b } c�À�uac } ¬j~ �% �% �

%�� �/� �%�� �I�

� � � � � � � � � � � 1 i³¥ � : �� � � * i³¥ � 4 �J �x�RK

ª'itp{^

� ������ (� �� ,

��� � ��� �&�

���� (� ( � 4� (��� � ,' 4 � (

��� J �x�RK

L u�sv� } c��¿prs�b7nIAxc²c�nxz{ikc��¿pr^ac } cRz{���{ikwypritsvu§sx¥yp{^aioz»w��{sv¬amtc�b iku�pr^ac²mksqª©�Hnx��^�ujlab�¬=c���mtikb-i³pRÉqªµc²�{cR��n|mkm|¬=c�mksqª©^asqª6p{s:z{sxmk¨xcitp´itu�p{^�c¸�vc�uac��rnxmB��nxz{cx­J½asx� } c*prnxitmoz�Éqsxuac'��nxu��rc*¥¹c���¥¹sv�$ituPz�p�n|u���c²prs J �t�R� � É6� H�� K*­$]_^ac¸z{sxmklypritsvug��sxu�z{ikz�przJsx¥Pd } i��Ec��rc�uBp´�*svu��z�prnxuvp²z�prn|p{cRz��%1 � � �1 � � �* � �%*��*svuauacR�0p{c } ¬B~�p{^a�rc�c:ª¸n±¨vc�z�p{^�n|pµ��nxu-¬Pc'¥¹�rsxbºmtc�¥ùp´p{s��rik�x^Bp�ÉBn-� �ó�rnx�{c�¥Mnx�0pritsvu�sx�µn-� �ëz�^asy� AEÉxn��sxuBprnv�0p } ioz{��sxuBp{ikujlai³p�~�n|u } ne� �ó�rnx�{c�¥Mnx�0pritsvu�sx�´ne���óz{^asy� AE­#]_^ac_¥¹sxl��$wPsBz{z{it¬�mtc_ª_n±¨xc¸w�nqp{p{c��ru�z$nx�{c¸�{c�wa�rc�z{c�uBp{c } svug½Jik��­×�y­

]_^a�rsxl��x^�n�� �ó�rnx�{c�¥Mnx�0pritsvu»É|pr^ac§c�uBp{�rsxwj~xÉBpr^ac:p�n|ua�vc�uBp{ion|m=¨xc�mtsy��i³p�~7n|u } p{^�c§·Bl�nxuvpri³p�~ ( � � 9 � � /;� � ª'^ac��{c � ioz²pr^acz{sxlau } z{wPc�c } � 4"9 � É»n|�rc���sxu�z�prnxuBp�­�]_^jl�z�p{^�cgz�prn|p{c�z � � ( � �"� 4�p{^�n|pe��n|u>¬Pc7�*svuauacR�0p{c } prs�pr^ac7z�p�nqp{c � 1 � ( 1 ��� 1 � 4 1pr^a�{svla�x^8n9� �ó�rnx�{c�¥Mnx�*p{iksxu�^�n±¨vc§p{s7z{n|p{ioz�¥¹~ �

4� �� 4 1� � 1 � ( � �

� /�� �@( 1 � ��1� /�� J ����K

nxu } � � � 1 J �v�RKÈ svb�¬aikuaikua�-p{^acep�ª¸s7c�·Bl�n|p{iksxu�z J �a��K0Éaª¸cesx¬yp�n|iku�pr^ac�� �O��n|�rc*¥Mnv�0p{iksxu��*la�r¨xc§p{^Pnqp } c�À�uacRz'itu9pr^ac � ( � 4 � wamknxuacep{^�c�z�p�nqprc�zpr^�nqp:��n|u9¬Pc���sxuauacR�0prc } p{s � 1 �

( �I(+1 � ��1� /6� � � / �

44�1 ��� � ª'itp{^ � � � � /6� � 9 � � n|u } 4 : 4 1 J � � K

hjikb-itmon|�rmk~xÉBª'�{itp{ikua��pr^�nqp´pr^ac���itc�b-nxuau7ikuB¨qnx�{ion|uBprz²n|�rc:�*svu�z�p�n|uBp$p{^��{svla�x^�n����O��n|�rc*¥Mnv�0pritsvu»Éqªµc:sx¬yp�n|iku-p{^ac§� �ó�rnx�{c�¥Mnx�0pritsvu��la�{¨vc§p{^�n|p } c*À�u�c�z'n|mkm¿pr^acewPsBz{z{it¬�mtc�z�prn|p{c�z¸pr^�nqp:��n|u9¬Pc���sxuau�c��0prc } p{s ��*6p{^��{svla�x^8n-� �ó�rnx�{c�¥Mnx�0pritsvu �

(J� ( * / � � *� /�� � � / �

44 * � � � ª'i³pr^ 4 : 4�* J ��#IK

STS àIUPV±õ�ø�W

Page 14: On the behavior of upwind schemes in the low Mach number ...

�±� � :������ + +- I� ! I� !�� :T.1����� ���"�

Case (a)

tcontact

X

1−rarefaction 3− shock

q qL*

R*

qL

qR X

Case(b)

t

1−shock 3− rarefaction

q q

q q

L*

R*

L R

contact

Case(c)

t

X

1− rarefaction 3−rarefaction

q q

q q

L*

R*

L R

contactt

contact

X Case (d)

q q

qq

L*

R*

L R

1−shock 3−shock

½Jik�xla�rcÌ� � ���  !óÆ #  � Æ �R� M� �  Š¡ �= ��¡y���    � � � � �  � Æ � #� Æ � Æ�! � �  �  �©¡ � � ��  Æ ���  � £ ��¡3� � Ä q¡y   ! ¡a¢|�� Æ � � ¢ Æ � ��¡a¢x� � � Ä � � Æ ¢%$ � ����� � Ä � � Æ ¢%$ � ¢ Æ � ��¡a¢x� � � Ä  ±¡a   ! ¡a¢x� Æ � � ��¢�� � Ä  q¡y   ! ¡a¢|�� Æ � � ¢ Æ � ��¡a¢x� � � Ä q¡y   ! ¡a¢|�� Æ � � ��Ç&� � Ä � � Æ ¢%$ � ¢ Æ � ��¡a¢x� � � Ä � � Æ ¢8$�"

]_^acg�*svu�z�i } c��rn|p{iksxu�s|¥²p{^ac�p�ªµs��*la�r¨xcRz J � � K � J �'#0K'n|mkmtsqª�z:sxuac�p{s8�*svbgw�lyp{c�p{^ac-z�svmtlap{iksxuHiku>��nxz{c J ��K�sx¥µ½Jit�P­ �a­§½�sx�pr^aces|p{^�c��'����nxz{c�z�Éjsxuac§^�nxzµprsg��sxu�z{i } c��¸p{^acesy����la�rc�u���c:s|¥�z�^�sj� A-ª_n±¨xcRz�­J®>�ritp{ikua��pr^ac���n|u2ABikuac �ó�:la�xsvuaitsxpµ�rc�monqpritsvu�z �

())* ))+& � (0��� & �& � ( � 4 ��� & � (& � ( �=��� & � �& � � ,' 4 � ( ��� & � ,

J �v�RK

pr^a�{svla�x^�p{^�c�z�^asy� Ayz�Éyªµcesx¬aprn|iku�p{^ac7� �óz{^asy� A�n|u } � �ëz{^asy� A��*la�r¨xcRz��

(0�I(+1 / 4 / 4 1 4 �� � 4 1

� �5/�� � � 9 � 1 ª'itp{^ 4 4 4 1 J dP��K

(0�@(+*� 4 / 4 * 4 �� � 4 *

� �5/�� � � 9 � * ª'i³pr^ 4 4 4 * J dB�RK

ª'^ac��{c � � � � � /7� � 9 � � �� � ­J]_^�c�z{sxmklyp{iksxu�sx¥¿p{^ac§�'ikc�b7n|uau�wa�{sv¬amkc�b ikz¸�*svb-wamtc�p{c } ¬j~�ÀPu } ikua��pr^ac:¨qn|mklacRz � ( � � 4 ��� itupr^ac¸z�prn|p{cRz�� 1�� nxu } � * � ­�½a�rsxb n|ugn|mk�xc�¬a�rnxik��w=sxikuBpJsx¥a¨jitc�ª�ɱp{^aioz���sxu�z{ioz�p�z»prs�À�u } p{^ac¸ikuvprc���z�cR�0pritsvu�s|¥apr^ac:� �O��n|�rc*¥Mnv�0p{iksxu��� �óz{^asy� A��*la�r¨xc§sx�rik�xiku�nqpritua��¥¹�{svb � ( 1 � 4 1 � ª'itp{^�p{^�ce���O��n|�rc*¥Mnv�0pritsvu��j���óz{^asy� A��*l��{¨vc�w�nvz{z{ikua��pr^a�rsxla�v^ � ( * � 4 * � ­ H �a�*c�wypiku�pr^ac§��nxz{c:sx¥¿p�ªµs�c*�jp{�rc�b-c�mk~7z�pr�{svua��¥Mnv�*ikua����n|�rc*¥Mnv�0p{iksxuPz�Éxp{^acRz�c:p�ªµsg�*la�r¨xcRz²n } bgitprz¸n�l�uaik·Blac§ikuvprc���z�cR�0pritsvu�wPsvituBp_n|u }pr^aikz:�*sxb-wamkc*prc�mk~ } c�p{c��{b-ikuac�z¸pr^ac�z�svmtlap{iksxu9s|¥#pr^ac��'ikc�b7n|u�u9w��{sv¬amtc�b8­

®6c�n|�rceuasqª ikuBp{c��{cRz�prc } ¬B~�p{^ac�z�svmtlypritsvu�sx¥�pr^aioz�wa�rsx¬amkc�b�itu9p{^�c�mtsqª ��nx��^8ujlab�¬Pc��'mtikb-i³pR­_�8sv�{c�z{wPcR�*itÀP��nxmtmk~xÉ�ªµc��sxu�z{i } c��'ikuai³priknxm } n|prn���� 1�� � *�� ��mtsBz�c§p{s7n-�*svu�z�prn|uBp } c�u�z�itp�~�iku���sxb-wa�rc�zrz�ik¬amkc:¶�sqª itu9pr^ace¥¹sxmkmtsqª'ikua�-z�c�u�z{c �

� 1 � ���� � � $ �� �� � 1�, � @��� ���(21 � � ��� � � 6��� (+13, . I� ������ 10��� ��� � �� �� � � 13, .� I� �����4�1 � � ��� � ���� ��4 $ �� � � 4 13, � @����� �J dvd K

� * � ���� � � $ �� �� � *-, � I��� ���(2* � � ��� � � 6��� (+*-, .� I� ������ *7��� ��� � �� �� � � *-, .� I� �����4 * � ����� � ���� ��4 $ ���� � 4 *-, � @����� �J d � K

+�,TS-+ è

Page 15: On the behavior of upwind schemes in the low Mach number ...

�������������� �������������������� �"!$#&%'���&(�'#)� �*�����,+-���/.* 0%'�1�2��(3��'�4+5� (�� �7698�8;: ���

2�¬�z{c��r¨xc¸pr^�nqp´iku7nx����sx� } n|uP�*c¸ª'i³pr^-p{^ac'�rc�z{lamtprz�pr^�nqp´ª¸c'^�n±¨xc_�rc���n|mkmtc } iku7z�cR�0p{iksxu��aÉqp{^ac:ituaitp{ion|m } nqp�n�n|�rcµpr^ac'wa�rs|��cR�0p{iksxusvu>n�waitcR�*c�ª'ikz{cg��sxu�z�prnxuBp:À�c�m } s|¥ B�ª¸c�mkm�wa�rc�wPn|�rc } D�ikuaitp{ion|m } nqp�na­ L u6w�n|�{p{io�*l�mknx��ÉPp{^ac-wa�rc�zrz{la�{c�À�c�m }�} sxc�z§uasxpe��sxuBprnxitu¶�lP�0p{lPnqp{iksxuPz�sx¥�sx� } c�� ��� ­©]�sHÀPu } pr^ac9z{sxmklypritsvuÌs|¥'pr^ac9��itc�b-nxuau w��{sv¬amtc�bitu p{^ac9mtikb-i³p ��� � É$ª¸c�ª'�ri³prc�pr^acz{sxmklyp{iksxu9iku�pr^ace¥¹sx�rb+�

� � ���� � � $ �� � � .� 6� �� � � @��� ���( � ���� � � 6��� (+.� I� ������=� ���� � � 6��� � .� I� ��� �

4 � � ��� � ���� ��4 $ �� �>4�. 6� � � 4 � @����� �J d H K

L uBp{�rs } l���itua��p{^�c�z{c�c��jw��{cRz{z{itsvu�z�itu�p{^ac-��n|�rc*¥Mnx�*p{iksxu�nxu } z{^asy� A��*la�r¨xcRz J � � K � J �'#0K � J dP��K � J dB�RK*ÉPitp�ikz§c�nvz�~9prs9z{c�c�p{^�n|p�itupr^ac�mtikb-i³p � � � Éap{^acRz�c��*l��{¨vc�z_¬=c��*svb-c�z'z�p{��n|ik�x^�mkikuac�z'sx¥Jz{mtsvwPcRz�� � 9�� � 4 $ � $ �

� �ë��n|�rc*¥Mnv�0p{iksxu (2. �I(213, . / �� � 4 $ � $

4 . ¥¹sx� 4 . : �

� ��hj^asy� A (2. �I(213, . / �� � 4 $ � $

4 . ¥¹sx� 4 . 4 �

� �ë��n|�rc*¥Mnv�0p{iksxu (2. �I(2*-, .� �� � 4 $ � $

4 . ¥¹sx� 4 . : �

� ��hj^asy� A ( . �I( *-, . �� � 4 $ � $

4 . ¥¹sx� 4�. 4 �

J d #IK

��z�z{^asqª'u©itu̽Jik��­8��s|¥¸p{^ac�d�wPsBz{z{ik¬amtc7ª_n±¨xc-w�n|p�prc��ru�z } ioz{wamkn±~vc } itu̽Jik��­H�yÉJsxu�mt~�p{^acH� �O��n|�rc*¥Mnv�0pritsvu��j���O��n|�rc*¥Mnv�0p{iksxuJ ª'^ac�u (2*-, . / (213, .546� K¸nxu } � �ëz�^asy� A �j���óz{^asy� A J ª'^ac�u (2*-, . /7(+13, .5:�� K_z�svmtlypritsvu�z_n|�rcewPsBz{z{it¬�mtcv­

U

P

U

P < 0

U

U

1

γ P0

R,1

L,1

1 1

1

1

ρ0

3−rarefaction

3−shock

1− rarefaction

1− shock

−1

γ P0ρ 0

U

P

U

U

U

L,1

R,1

1

1

1> 0 1P

1− rarefaction

3− rarefaction

1−shock

3−shock

−1

γ P0ρ

0

γ P0ρ0

1

½Jik�xla�rc²��� � � �   q�  ¢|�� Æ � Æ*!  q¡y   ! ¡a¢|�� Æ � ¢�#  �P � !óÆ   (+*-, . / (213, .546� � �  ! �<�²¡ � Ç�� � Æ ¢8$8¢�#  �P � !óÆ  5(+*-, . / (213, .5:����  "�� �<��"

½�sx�_p{^�c�z{c�p�ª¸s7��nvz�cRz�Éjpr^ac��*svbgw�lyprn|p{iksxu9s|¥#pr^ac�ikuvprc���z�cR�0pritsvu�w=sxikuBp�s|¥#pr^ac��*l��{¨vc�z J d #IK¸�xik¨xcRz �(+. � � (213, .� ;(2*-, . � 9 �4 . � / � � 4 $ � $ & (2. 9 �

J dB�RK

ª'itp{^ & ( . � & *+1 ( . � ( *-, . /;( 13, .R­e]_^ac��rc*¥¹sv�{cvÉPitpeikz§z�c�c�u�pr^�nqpec�¨vc�uHit¥´pr^ac�ikuaitp{ion|m } nqp�n�nx�{c�sx¥´p{^�c�¥¹sx�rb J dvd K � J d � Kª'itp{^ 4�1�, . � 4�*-, . � � Éjpr^ac�ikuvprc��{¥Mnx��c§w��{cRz{z{la�rc§��sxuBprnxitu�z'n�¶�lP�0p{lPnqp{iksxu8sx¥�sx� } c�� � � �

4 � � ��� � ���� ��4 $ / ���� � � 4 $ � $ &9( . I� ��� J dB�RK

�'�*p{l�nxmtmk~xÉ_pr^aikz�wa�rc�zrz�la�rc8¶�l��*p{l�n|p{iksxu�ioz�wal��{c�mt~Ës|¥�nx��sxl�z�p{io�8sx�rit�vitu#­°]_^ac>z�^asy� A �*la�r¨xcRz7s|¥ J d #IK7nx�{c9pr^ac>��nxuPAjitu�c ��:la�xsvuaitsxp¸�rc�monqpritsvu�z²¥¹sx�_n|u�nv�*svl�z�prik��z�~yz�p{c�b�n|u } J dv� K´ioz¸itu�¥Mnx�*pµpr^ac�z�svmtlypritsvu�s|¥#p{^ac +5� �"�� I� 0% ����#�� ��% w��{sv¬amtc�b } c*ÀPuac }¬j~ �

%% �

�� ( .4 .�"

�� � � 9 � $� 4 $ �

�" %%��

�� ( .4 .�" �@� J � �RK

STS àIUPV±õ�ø�W

Page 16: On the behavior of upwind schemes in the low Mach number ...

��d � :������ + +- I� ! I� !�� :T.1����� ���"�

nxu } p{^ac��{c�¥¹sx�rcxÉvp{^ac�¶�ly���*svbgw�lyp{c } nqp´pr^ac:ikuBp{c��{¥Mnx��c�iku�z�prc�n } s|¥¿¬Pc�itu���n|u�nxwawa�rs±�jikb7nqpritsvugsx¥Ep{^ac�B�iku��*svb-wa�{cRz{z{ik¬amtc¸¶�la�RDiozep{s8mkc�n } itu��9sv� } c��§p{^acg¶�la�6�*svb-itua��¥¹�rsxb nxu R% ����#�����% wa�{sv¬amkc�b¯�xc�uac���nqprc } ¬j~�p{^ac7nx��pri³ÀP��iknxm } ioz{��sxuBp{ikujlyp{ikc�z�n|p§pr^acikuBp{c��{¥Mnx��c�z_¬=c*p�ª¸c�c�u�pr^ac��*c�mtmoz�­

]�s�ikmkmtl�z�p{��nqprcgp{^aioz�n|u�nxmt~yz{ikz�ÉEªµc-z{^asqª iku6½Jik��­gd�ÉEp{^�c-z{sxmklypritsvu�s|¥µp{^ac7�'ikc�b7nxuau>w��{sv¬amtc�b J �x�RK§�*svbgwPn|�rc } prs�pr^acnvz�~jb-wyp{sxp{io�§c*�ywa�rc�zrz�iksxu J dv� Kµ¥¹sx�'ikuaitp{ion|m } n|prngs|¥Jp{^ace¥¹sv�{b J dxd�K � J d � K } c*ÀPuac } ¬j~ �

())* ))+� 1 � � �(+1 � � �� 1 � ���4 10� �%����� ���

nxu }())* ))+� * � ���(2* � ����� � sv� � � � �� * � � � � �4 * � �8��� ��� � � �

J � ��K

]_^ace�rc*¥¹c��{c�u��*c���nx��^�ujlab�¬=c��_¥¹sv�µpr^aioz���nvz�c§ikz¸sx¥�sx� } c�� �%�:- � ­ L u9½Jik��­�d�Éjpr^ac�mkc*¥ùp'wamks|prz } ikz{wamon±~-p{^ac���nvz�c§s|¥$���rnx� �c�¥Mnx�0pritsvu�z{sxmklyp{iksxu J ª'^ac�u (2*>4�(+1 K²ª'^aikmtc�p{^ace�rit�v^Bpµwamks|p�z¸z{^asqª�z²p{^ace��nvz�c�s|¥�n�p�ª¸s�z{^asy� A-z{sxmklypritsvu J ª'^ac�u (2*>:�(21 K*­2�¬�z{c��r¨xc�p{^�nxu�itu�p{^ac�p�ªµs���nxz{c�z�Évp{^ac�wa�{cRz{z{la�rc$��lab-w�¬=c*p�ª¸c�c�u7p{^�c:p�ª¸s�ikuai³priknxm¿z�p�nqp{cRz´ioz²c��jp{�rc�b-c�mk~7z�b7n|mkm J s|¥¿pr^ac§sx� } c��sx¥ � � � ��� Kµ¬alyp'p{^�c�¨xc�mksy�*itp�~e��lab-wH�*�rc�n|p{c�z_nxu�c��Bpr�{c�b-c�mk~�mknx�{�vce¨±nx�{ionqpritsvu�sx¥#p{^�c�ituBp{c���¥Mnv�*cewa�rc�zrz�l��{cv­

½��{svb n|u�c��an|b-ituPnqp{iksxu s|¥epr^acH�'ikc�b7n|uau�w��{sv¬amtc�b.ikuZp{^ac�mtikb-i³p � � � ɸª¸cH��nxu p{^ac��{c�¥¹sx�rcH�*sxuP�*mkl } c8p{^Pnqp�c�¨xc�uit¥´p{^�c�ikuai³priknxm } nqprn�n|�rc��*mksvz{c�prs�n���sxu�z�prnxuBp } c�uPz�itp�~8itu���sxb-wa�rc�zrz�ik¬amkceÀ�c�m } É=p{^�c�À�c�m } �*sxb-walap{c } ¬B~�nxuHlawjª'itu } z�svmt¨vc����sxuBprnxitu�zµnq¥ùprc��²svuac'pritb-c�z�p{c�w»Éynx��sxl�z�p{io�_wa�rc�zrz�la�rc'ª_n±¨xc�z´s|¥¿sx� } c�� � �¸pr^�nqp_n|�rc'b�l���^�mknx�{�vc���p{^Pn|u7pr^ac:wa�rc�zrz{la�{c_¶Pl��0prl��n|p{iksxu�z } lac�prs7p{^ac�ituP�*sxb-wa�rc�zrz{it¬amkc��*svb-wPsvuac�uBp�­']_^acRz�c�nx�*svl�z�p{io���*sxb-w=sxuac�uBprz��rc�b7nxitu8wa�rc�z{c�uBp:iku8p{^ac�¥¹sxmkmtsqª'ikua�-pritb-cz�p{c�w�z�n|u } pr^ac�c�·Blaikmkit¬a�riklab�z�prnqprc�s|¥Jp{^ac } ikzr�*�rc*prc§cR·Bl�nqpritsvu�z J it¥Jitp:c��jioz�prz Kµ��n|u9¬Pc } c�zr�*�rit¬=c } nxz'n-¬Pn|mon|u���c�sx¥�nv�*sxlPz�prik�ª_n±¨xcRz:iku�z�p{c�n } sx¥²�{c�wa�rc�z{c�uBp{ikua��nxu>nxwawa�rs±�jikb7nqpritsvu�s|¥µnxu�iku��*svbgw��{cRz{z{it¬�mtc�À�c�m } ­ L u©z{c��0pritsvu H É¿ª¸cgwa�{cRz�c�uBp�pr^aikz§mkitb-itpz{~yz�prc�b8ÉEp{^acg�rc�n } c��:ª'ikmkmJu�s|p{io�*c�p{^�cg��mtsBz�c��{cRz�c�b�¬amon|u���c�s|¥²p{^aiozez�~yz�p{c�b2ª'itp{^�p{^acgmtikb-i³p�cR·vlPnqp{iksxuPz:sv¬yprnxituac } ¥¹�rsxb¯n|unvz�~jb-wyp{sxp{io�en|u�nxmt~yz{ikz_sx¥#p{^�c��'sxc�zr��^ac�b-c§p{^�n|p�ªµce^�n±¨vcewa�{cRz�c�uBp{c } itu � # � ­

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-250 -200 -150 -100 -50 0 50 100 150 200 250

(U-m

in(U

L,U

R))

/(m

ax(U

L,U

R)-

min

(UL,

UR

))

X/t

GodunovAsymptotic

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-250 -200 -150 -100 -50 0 50 100 150 200 250

(U-m

in(U

L,U

R))

/(m

ax(U

L,U

R)-

min

(UL,

UR

))

X/t

GodunovAsymptotic

-60

-40

-20

0

20

40

60

-250 -200 -150 -100 -50 0 50 100 150 200 250

(P-m

in(P

L,P

R))

/(m

ax(P

L,P

R)-

min

(PL,

PR

))

X/t

GodunovAsymptotic

-60

-40

-20

0

20

40

60

-250 -200 -150 -100 -50 0 50 100 150 200 250

(P-m

in(P

L,P

R))

/(m

ax(P

L,P

R)-

min

(PL,

PR

))

X/t

GodunovAsymptotic

½Jik�xla�rced6� % Æ ���J¡y  ¹� Æ � �  � Å�Âj � ! Æ Ç � Æ�� � Æ � �=  �¡ � Ç�¡a� à ���#� Æ ��¹¢8¡ � ¡ � à � M� !óÆ   ( * 4�( 1 � �  ! ���g¡ � Ç !óÆ  ( * :�( 1 �ë  " � ����"

+�,TS-+ è

Page 17: On the behavior of upwind schemes in the low Mach number ...

�������������� �������������������� �"!$#&%'���&(�'#)� �*�����,+-���/.* 0%'�1�2��(3��'�4+5� (�� �7698�8;: � �

� � ��� ���6�'"´�����Z�9���'"¸�����9�0�J�*������� ?

½�sx�e�'sxc�p�~jw=cgzr��^ac�bgcRz�É=ª¸c�^�n±¨xc�z�^asqª'u�itu3� # � p{^Pnqp:¥¹sv��¨vc��r~�z{la¬�z{sxuaio��¶Psqª�z:n���mtcRn|��itb-wa�rsq¨xc�bgc�uBp�s|¥´pr^ac�nv���*l��rnv�*~��n|u8¬Pcesv¬yprnxituac } ¬B~�bgs } it¥¹~jitu��-p{^ac�ujlab-c��rio��n|m=¶�la� J ��K¸itu�p{^�ce¥¹sxmkmtsqª'ikua�-ª¸n±~ �

������1 � ��* �,� 1�* � � �� ��� � � 1�*!����1 � � � � 1�*!����* � � - . # � % � � � 1�*% �

# & 1�*�� � J � �0K

ª'^ac��{c � ioz�n�wa�{cR�*svu } itp{iksxuaikua��b7n|p{�ri³�¿­ L u�p{^aiozez�cR�0pritsvu»ÉEªµc�wa�{svwPsBz�cgn�z{itb-ikmknx��z�p{��nqprc��x~9prs�sq¨xc��r��sxb-cep{^�cgnv���*l��rnv�*~wa�rsx¬�mtc�b�c�u���sxlauBprc��rc } ¬B~9¦§s } lauasq¨7p�~jw=c�z{��^ac�b-c�z�­

�8sv�{c�z�w=c���i³ÀP��n|mkmt~vÉ�ª¸c�ª'ikmtmJn|waw�mt~8wa�rc���sxu } itp{iksxu�itua�-prs�pr^ac�¼§½J��sxc���mknvz{z:s|¥²zr��^ac�b-c�z'p{^�n|p�ª¸c } cRz{���{ik¬Pc�uasqª�­�]_^ac¼§½J��sxc�zr��^ac�b-c�z �k��� � l�z�c�n�¶�la��s|¥�p{^ace¥¹sv�{b �

����� 1�� � *.�,�&1�* ��� � �-� 1�* �/� 1�*�� J � �RK

ª'^ac��{c � 1�* ioz�n8z�prnqprc��*svbgw�lyp{c } ¬j~�p{^ac�z{sxmklyp{iksxu©sx¥¸n��'ikc�b7n|u�u6w��{sv¬amtc�b ¬=c*p�ª¸c�c�u6pr^ac7p�ª¸s8z�prn|p{cRz0� 1�� � * ­ L ¥¸p{^aioz�'ikc�b7nxuau�wa�rsx¬amkc�b�ikz } c*À�u�c } ¬j~�pr^ac�uasvu�mtikuacRn|� H l�mtc���c�·Bl�n|p{iksxu�z�Éapr^ac�zr��^ac�b-c } c�À�uac } ¬j~ J � �RK¸ioz:z{itb-wamk~8¦§s } l�uasq¨zr��^ac�b-cv­���sqª¸c�¨vc��RÉ�¥¹�rsxb n��*sxb-walaprnqpritsvu�n|m�w=sxikuBp§sx¥²¨jitc�ª�É¿i³p�b7n±~8¬Pc-ikuvprc��rc�z�p{ikua��prs } c*À�u�c�pr^aikze�'ikc�b7nxuau>w��{sv¬amtc�b¬j~�ngmkitu�c�n|�ri Q�c } wa�rsx¬amkc�b�¬Pc�p�ªµc�c�u�p{^acep�ª¸s7z�p�nqp{cRz ��� 1�� � *�� �

% ��% � :�� 4 % ��

%�� �@��� � � � � � � � �� � �� 1 i³¥ � : ��� � �� * i³¥ � 4 �

J � d K

ª'^ac��{c :���4 ikzen9�*sxuPz�p�n|uBp§b7n|p{�ri³��nxu } �����n|u>¬Pcgpr^ac7�*sxuPz�c��{¨qnqprit¨vc�¨qn|�rion|¬amkc�z �7sx��nxuB~�sxp{^ac���z�c�p§sx¥µiku } c�w=c�u } n|uBp¨qn|�rion|¬amkc�z�­$��c��rcxÉj¥¹sv�'ituPz�p�n|u���cxÉyª¸c�ª'itmkm¿l�z{cep{^acCB�c�uBp{�rsxw�ik��D�¨qn|�riknx¬amtcRz �� � ��4 � ( � �"��� ��� nxu }8} c*ÀPuac :�� 4 ¬j~ �

:�� 4 ����:6(�4 � : 4 4 � �� 9 : � 4 :6(�4 � �� � :�( 4 �� � � :�( 4

�� J � � K

ª'^ac��{c :��&4 � �>� � � 1� � � � *.� 9 � } c�uas|prcgp{^ac7n|�ritp{^ab-c*prik�-n±¨vc���n|�xc�¬Pc�p�ªµc�c�u �� ��� 1 n|u } �� ��� * ­7]_^jl�z§itu J � �RK*É�� 1 * ikz§pr^ac��sxu�z{c��r¨qnqp{ik¨xc�¨qn|�riknx¬amtcRz���sx�r�{cRz�w=sxu } itu���p{s �� 1 * z�svmtlap{iksxu�sx¥²pr^ac-mtikuac�nx�{i Q�c } �'ikc�b7n|u�u>wa�rsx¬�mtc�b J � d�K0­g��u©nxz{~Bb-wyprs|prik�nxu�n|mk~yz�ioz�sx¥�p{^aioz�lawjª'itu } z{��^�c�b-c�z{itb-ikmknx��prs�pr^ac9nxu�n|mk~yz�ioz�w��{cRz�c�uvprc } ikuËz{c��*p{iksxu ��~Bikc�m } z�p{^ac8zrn|b-c���sxu���mtl�z{iksxu�z�nxz¦§s } lau�sq¨Hz{��^ac�b-cx­ L p�z{^asqª�z�p{^�n|p§pr^acgituBp{c���¥Mnv�*cgwa�rc�zrz�la�rc�ioze�xik¨xc�u�¬j~�c*�ywa�rc�zrz{itsvu J dB�RK�n|u } p{^jl�z§p{^Pnqpep{^ac-wa�rc�zrz{la�{cÀ�c�m } ��sxuBprnxitu�z¸¶Pl��0prl�nqpritsvu�z_s|¥�sx� } c�����nv��^»­

L u§p{^�c²z�w�it�ri³p»sx¥jp{^ac$wa�rc���sxu } i³pritsvuaitu��'s|¥y�'sxc´zr��^ac�b-c J � �0K*É�ª¸c$wa�rsxw=svz{cJp{s'b-s } i³¥¹~�p{^�c$ituBp{c���¥Mnv�*c�¶�la�0������1 � ��* �,� 1�* � �� �-� 1 *.���%1 * � ¬j~���sxb-walypritua� ��1�*©nxzµpr^ac�z�svmtlypritsvu�sx¥Jn ��� � % ���"!�� � �����"��! �'itc�b7n|uau9wa�rsx¬amkc�b8­´�8sx�rcez�w=c���i³ÀP��n|mkmt~vÉjitu J � �RKª¸ceprn0Axc �%1�*Ìnxz¸p{^�c�z�svmtlypritsvu�sx¥�pr^acewa�rc��*svu } itp{iksxuac } w��{sv¬amtc�b+�

% ��% �

� :� 4 % ��% � �I�

�� � � � � � � � �� � �� 1 i³¥ � : ��� � ���* i³¥ � 4 �J � H K

ª'^ac��{c � ikz¸pr^ac�wa�rc���sxu } itp{iksxu�itua�-b7nqpr�{it��wa�rsxw=svz{c } itu �k��� �

� ���� � � � �� � � �� � � �� � � �

�� J � #IK

ª'itp{^ n-w�n|��n|b-c�p{c��_sx¥#p{^�c�sx� } c��_s|¥Jp{^�c���nx��^9ujlab�¬=c��0� �� � .�ª'i³pr^ . �� � � � ­

STS àIUPV±õ�ø�W

Page 18: On the behavior of upwind schemes in the low Mach number ...

� H � :������ + +- I� ! I� !�� :T.1����� ���"�

®6c-wa�{sy��c�c } uasqª�prs9z{^asqª�pr^�nqp§p{^ac-ikuBp{c��{¥Mnx��c�w��{cRz{z{la�rc�z{sxmklyp{iksxu>s|¥²pr^ac ��� � % ���"!�� � �����"��! ��itc�b-nxuau6wa�{sv¬amkc�b J � H K} svc�z$u�s|p²�*svuBprn|iku�¶�l��0prl�nqpritsvu�z$sx¥=sv� } c�� ��� it¥Ppr^ac�ikuai³priknxmawa�rc�zrz�l��{c'�*svuBprn|iku�z�svuamt~�wa�{cRz{z{la�rc¸¶�l��0prl�nqpritsvu�z$sx¥Psx� } c�� ��� � ­]_^ac�mkc*¥ùp:nxu } �{ik�x^Bp'z�p�nqprc�z�nx�{c�p{^ac�u } c*À�u�c } ¬j~ �

� 1 � ����� � � 13, $ 6��� � 1�, .� �� �� � 13, � I� �����(+1 ������ � �� �� � (213, .� @����� �� 1 �'���� � � 6��� � 13, .� @����� �

4�1 � � ��� � ���� � 4 $ 6� � � 4�1�, � I��� ���J � �RK

� * � ����� � � *-, $ 6��� � *-, .� 6� �� � *-, � I� �����(+* �'���� � � �� � (2*-, .� @����� �� * �'���� � �� 6��� � *-, .� @����� �

4 * � ����� � ���� � 4 $ 6��� � 4 *-, � I��� ���J � �RK

¾:s|p{c§pr^�nqp�iku J � ��� � �RK0Éjª¸c§nxmtmksqª�pr^ac�wa�rc�z{c�u���c�sx¥ } c�u�z�itp�~ } iozr�*sxuBpritujlaitp{ikc�z_sx¥�sx� } c�� � � � � ¬alyp'p{^Pnqp_p{^ac�w��{cRz{z{la�rc´��lab-w¬=c*p�ª¸c�c�u�pr^ac�mkc*¥ùp:n|u } �rit�v^Bp'z�prn|p{cRz_ikz'sx¥�sv� } c�� � �� ­

]�s�sx¬aprn|ikuHn|u�c��ywamtio�*itp�c��ywa�{cRz{z{iksxu9¥¹sx�:p{^ac�z{sxmklypritsvu�s|¥ J � H K*É�ª¸c } ion|�xsvu�n|mki Q�cep{^�c�b7nqpr�{it� : � 4 ­ L p�z�c�ik�xc�uj¨±nxmtl�c�znx�{ce�xik¨xc�u�¬j~ � % .5� � � �. � ��:6(�4�/ � � � 9 �% � � % # �9:�(�4%�� � � � �. � ��:6(�4 � � � 9 � J H �RK

ª'itp{^�� � � � � / � � :�( 4 � � � � � : 4 4 9 : � 4 n|u } pr^ac�nxzrz�sy��ikn|p{c } c�ik�xc�uB¨vc��*p{sx��z ���gn|�rc �

� . ������

�% . / � :�( 4 �%� : 4 4�

����� � � � �

�������

�� � � # �

�������

�� � ��� �

�����

�% � / � :�( 4 � � : 4 4�

����� J H ��K

½Jiku�n|mkmt~vÉjmtc�pµl�zµituBpr�{s } l���c'p{^�c��*sjsv� } iku�n|p{c�z � � � & *+1 �� � sx¥¿p{^ac²��lab-w & *+1 �� � �� * / �� 1 itu7pr^ac§¬�nxz{ikz²s|¥Ep{^�c�c�it�vc�uj¨xcR�0p{sv�rz� � � � . � / ��% .�/�:6(�4 � & 4 � � : 4 4 &9(� �

� � � &�� # � & �

�� � ��% � /:6( 4 � & 4 � � : 4 4 &9(� �

J H �0K

pr^ac�u»Éap{^ac�z{sxmklyp{iksxu9s|¥ J � H Kµioz'�xik¨xc�u�¬j~�d7�*svu�z�prn|uBp�z�prnqprc�z ���1 � �� �1 � �� �* � ���* } c�À�uac } ¬j~ �

�� � � � � �&�())))))))* ))))))))+

�� 1 ¥¹sx� � 9 � : % .�� �1 � �� 1 � . � . ¥¹sx� % . : � 9 � : % ��� �* � �� * / ������� ¥¹sx� % � : � 9 � : %����%* ¥¹sx� % � 4 � 9 �

J H �RK

N z�ikua�©p{^�c�c*�ywa�rc�zrz�iksxu�zgsx¥�pr^ac�mtc�¥ùp�n|u } �rik�x^Bp7c�ik�xc�uj¨vc��0prsx��z J H �RK � J H ��K0ɵp{sv�xc*pr^ac��7ª'itp{^ pr^acHnxz{~jbgwap{s|prik�8c��yw�n|u�z{iksxu�zJ � �RK � J � � K0Éyª¸c§sv¬yprnxitu�p{^�ce¥¹sxmkmtsqª'ikua�7nxz{~jbgwap{s|prik�§c��jw��{cRz{z{itsvu�z²¥¹sx�_p{^�cep�ªµs7z�prnqprc�z �� �1 � �� �* �

+�,TS-+ è

Page 19: On the behavior of upwind schemes in the low Mach number ...

�������������� �������������������� �"!$#&%'���&(�'#)� �*�����,+-���/.* 0%'�1�2��(3��'�4+5� (�� �7698�8;: ��#

J ¾�s|prcep{^�n|p'p{^ac�¨qnxmtlacesx¥ ( nxu } 49nx�{cecR·vlPn|m¿iku�pr^acep�ªµs7z�prn|p{c�z �� �1 � �� �* K

( � �'����

� � 6� � � :�( . 4/ :6( . 4� � � $ &9( . /& 4 �: � $ 4 � � $ �

��� � � :�( � 4 /:�( . 4� � � $ &9( � � ����� � & (2.� : � . 4 & 4 �: � $ 4 � � $ � � ��� � & 4 � � ��� ���

4 � � � ��� � ���� ��4 $ �� � � � : 4 � 4/ . � � 4 $� � $ &9( . :6( . 4� � � $ & 4 � � � �����

J H d K

ª'^ac��{c � $ � � : (2. 4 � � � . � � 4 $ 9 : � $ 4 É : ��4 � � � ��� 1� � � � *!� 9 � } c�uas|prc�zepr^ac�nx�{itp{^ab-c�p{io�7n±¨xc���n|�vc�nxu } � ��� ���z�prnxu } z¸¥¹sx��z�svbgce�*svb-wamtc���prc��rb-z¸pr^�nqp:nx�{c§uasxp�c*�ywamkik��i³prc } ^ac��rcx­

]_^ac�¨qnxmtlacRz_s|¥Jp{^ac } c�u�z{i³pritcRz'n|u } p�n|ua�vc�uBp{ion|mE¨xc�mtsy��i³pritcRz'n|�rc§�vit¨vc�u�¬B~�

��1 � ���� � � 13, $ 6��� � 1�, .*

� � � � � 13, � / . � � 13, $� � $ &9( . � 13, $ :�(2. 4� � 4 $ � � $ & 4 � � 13, $� � 4 $ & 4 � �" ����� �

� �1 � � 1

��* � � ��� � � *-, $ �� � � *-, .

� � � � � *-, � / . � � *-, $� � $ &9( . � *-, $ :6( . 4� � 4 $ � � $ & 4 � / � *-, $� � 4 $ & 4 � � ����� �

� �* � � *

J H � K

½��{svb�c*�ywa�rc�zrz�iksxu J H d K0Éyitp�ioz�z�c�c�u�p{^�n|p'p{^ac�ikuBp{c���¥Mnv�*cewa�rc�zrz�la�rc§ikz¸¥¹�rc�c�sx¥�sx� } c��'�Hnx��^�¶�l��0prl�nqpritsvu�z�­

¾:sqª�Évmkc*p¸l�z´�rc*prla�{u-prs�pr^ac:ujlab-c��rio��n|mPc*�yw=c��ritb-c�uBp�z$p{^Pnqpµª¸c�^�n±¨vc�wa�rc�z{c�uBp{c } itu�p{^ac�ituBp{�rs } l��*p{iksxu»­�½Jit�P­ H wa�{cRz�c�uBprzpr^ac�w��{cRz{z{la�rc�À�c�m } z:¥¹sx��p{^ac-zrn|b-c�pr^a�{c�c } cR�*��nxz{itu�����nx��^Hujlab�¬Pc��:pr^�n|u>itu>½�ik��­g�x­§®6cgz{c�cvÉ�pr^�nqpeitu6�*svuvpr�rnvz�p:p{s�pr^ac�rc�z{lamtprz_sx¬aprn|ikuac } ª'itp{^9pr^ac�sx�rit�vituPn|m=¶�ly�yc�z�Éjp{^�c�z�svmtlypritsvu9��sxuj¨xc��{�vc�prs-nxu8lauaio·vl�c�z�svmtlypritsvu»­

½Jik��­ � wa�{cRz�c�uBprz�pr^ac�w��{cRz{z{la�rc7¶�l��*p{l�n|p{iksxu�z�ª'i³pr^ �rc�z{wPcR�0p�prsHp{^�c9��nv��^¤uBl�b�¬=c��R­��:z�itb-wamkikc } ¬j~6c*�ywa�rc�zrz�iksxu J H d K*Épr^acewa�{cRz{z{la�rc�¶�lP�0p{lPnqp{iksxuPz'z{��n|mkc:c*�anv�0p{c�mt~7ª'itp{^�p{^�cezr·Bl�n|�rc:sx¥»p{^�c��Hnx��^�ujlab�¬Pc��_itu8n|�v�{c�c�b-c�uBpµª'itp{^�p{^�c§¬=c�^Pn±¨Biksx�¸s|¥pr^ac��*svuvpritujlasvl�z_c�·Bl�n|p{iksxu�z¸iku9p{^ac���nxz{c§sx¥�ª¸c�mkm �ówa�rc�w�nx�{c } ikuaitp{ion|m } n|prn�­

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

0.001 0.01 0.1 1

(Pm

ax-P

min

)/P

max

MACH

VFRoe-Turkely=x*x

½Jik�xla�rc � � A    �R�»   B #¢x� �¡y� Æ � � � �� � BJÆjÅ � ¡a¢ � � ���    C !óÆ  [¢ Æ ����¡a  M� Æ � � �  ¢�#  �P ÃED�(&F ( ¹�Ç ¹��� �¹¡ ÃP ÇG"

STS àIUPV±õ�ø�W

Page 20: On the behavior of upwind schemes in the low Mach number ...

�R� � :������ + +- I� ! I� !�� :T.1����� ���"�

VFRoe-Turkel Mach=0.1

PRESSURE : MIN=71.2412 MAX=71.9025

VFRoe-Turkel Mach=0.01

PRESSURE : MIN=7142.67 MAX=7143.33

VFRoe-Turkel Mach=0.001

PRESSURE : MIN=714285.5 MAX=714286.2

½Jik�xla�rc H � � � Æ�� ¡��  � Æ�! � �  ��   �R�»  Â��$Æ � ¡ �-�*� � � Æ Ç Â �  � � !OÆ  0� � ����� �ë� Æ ��� � � ���� ����� ��¹ÇJÇ��  � ��� � ��� �����J��� Æ ��� Æ ���<"

+�,TS-+ è

Page 21: On the behavior of upwind schemes in the low Mach number ...

�������������� �������������������� �"!$#&%'���&(�'#)� �*�����,+-���/.* 0%'�1�2��(3��'�4+5� (�� �7698�8;: ���

� � ����"¸(��� J�*���

N z�ikua��nxu©nvz�~jb-wyprs|p{io�7n|uPn|mk~jz{iozes|¥:¦§s } lauasq¨6z{��^ac�b-cxÉ#ª¸c�^�n±¨vc-c*�y^aik¬ai³prc } pr^ac�b-c���^�nxuaioz�b p{^�n|p�wa�rc�¨xc�uBprz§p{^�ikz�p�~Bw=csx¥:n|wawa�rs±�yitb7n|p{iksxu>p{s���sxb-walyprc�uacRn|��iku��*svb-wa�{cRz{z{ik¬amtc�¶�sqª�z�­�]_^ac�p{�rsxla¬�mtc���sxb-c�z�¥¹�{svb pr^ac�¥Mnv�0p�pr^�nqp�p{^ac�ituBp{c���¥Mnv�*cwa�rc�zrz{la�{c���sxb-walyprc } ¬j~�pr^ac��'ikc�b7n|u�u8z{sxmk¨xc����*svuvp�n|iku�z'wa�rc�zrz{la�{c§¶�lP�0p{lPnqp{iksxuPz's|¥�sx� } c��:�Hnx��^8c�¨vc�u8i³¥�pr^ac�ikuai³priknxm } n|prnpr^�nqp } c*À�uac�p{^ac§�'itc�b7n|uau-wa�rsx¬amkc�b �*svuBprn|ikug¶�l��*p{l�n|p{iksxu�z�pr^�nqp¸z{��n|mkc'ª'i³pr^-p{^ac§z{·Bl�nx�{c_s|¥Epr^ac���nv��^gujlab�¬Pc���­�]_^aioz$¥Mnv�0pioz } c�c�wamt~7mkikuPAxc } p{s�p{^�c�¬=c�^�n±¨jiksx�¸s|¥»pr^acez�svmtlypritsvu�zµs|¥»pr^ac H lamkc��_c�·Bl�n|p{iksxu�z¸nqp_mtsqª[�Hnx��^�ujlab�¬Pc����$it¥»pr^ac�ikuaitp{ion|m } n|prnnx�{c�u�s|p�ª¸c�mkmJwa�{c�w�n|�rc } Éap{^�c�uHiku>n }a} itp{iksxu8p{s�p{^ac�ituP�*sxb-wa�rc�zrz{it¬amkc���sxb-wPsvuac�uBpRÉ�pr^acgz{sxmklypritsvu�z�sx¥$p{^ac H lamkc��§c�·Bl�nqpritsvu�z��sxuBprnxitu©n|moz�s8n|u6c*�jpr�{c�bgc�mt~H¥Mnxz�p�nv�*svl�z�prik��w�nx��pR­-]_^�ikz�nv�*sxlPz�prik��w�n|�{p�ikz���sxb-walyprc } ¬j~�p{^ac } ikzr�*�rc*prc-zr��^ac�bgc�p{^Pnqp��rc �nv�0prz'p{s7pr^ac�n|�{p{itÀP��iknxm } ikzr�*svuBp{ikuBl�i³pritcRz } lac�prs7p{^�c�wa�rs|��c��*p{iksxu9s|¥Jp{^ac�ituaitp{ion|m } nqp�n-sxu8waikc���c�ª'ioz�c���sxu�z�prnxuBprz_¬j~����{cRnqp{ikua�nv�*sxlPz�prik�§ª_n±¨xc�z_n|p¸pr^ac�ikuvprc��{¥Mnx��c§¬=c*p�ª¸c�c�u�pr^ac��*c�mtmoz�­

®6cgnxmkz{s�z{^asqª pr^�nqp�nxz�iku�pr^acg��nxz{c�sx¥µ�'sxc-zr��^ac�b-c�� # � ÉEp{^�ikzez{i³prl�nqpritsvu>��nxu�¬=cg��sxuBp{�rsxmkmkc } ¬j~�b-s } i³¥¹~jikua��p{^acgujl��b-c��rio��n|m=¶�la�jcRz¸ª'i³pr^�pr^ace^ac�mkw�sx¥#wa�rc���sxu } itp{iksxu�itua��p{cR��^auaio·Blac�z�­´�8sx�rc�z{wPcR�*itÀP��nxmtmk~xÉyª¸c§wa�{svwPsBz�c:p{s7��^�nxua�xc:p{^acesx�rik�xiku�n|m�'ikc�b7nxuauHwa�rsx¬amkc�b ¬j~�n ��� � % ���"!�� � �����"��! �'ikc�b7n|u�uHwa�rsx¬amkc�b8­�®6c�nxwawamk~�pr^aikz§z�p{��nqp{c��x~�p{s�n�¼�½��'sxcgz{��^�c�b-cx­ L uHp{^aiozmonqp{p{c�����nxz{cxÉyª¸c�z�^asqª ¬Psxp{^�p{^ac�sx�rc*prik��n|mkmt~�nxu } uBl�bgc��{io��nxmtmk~-p{^Pnqp'p{^�c�z�svmtlypritsvu�sx¥#pr^ac�wa�rc��*svu } itp{iksxuac } ��itc�b-nxuau9wa�{sv¬��mkc�b�wPsBz{z{c�zrz�cRz¸wa�{cRz{z{la�rc:¶Pl��0prl�nqpritsvu�z's|¥�p{^ac���sx�r�{cR�0p_b7n|�xu�i³prl } cx­

�$�rc���sxu } i³pritsvuac } ��sxb-wa�rc�zrz�ik¬amkc�z�svmt¨vc���z � # � É �×� � É�� � � n|�rc�n|¬amkc�prs } cRn|m#ª'itp{^Hp{^acgz�ikua�xl�mknx�{itp�~�sx¥�p{^ac H lamkc��§c�·Bl�nqpritsvu�zikuÌpr^ac�mksqª���nx��^ uBl�b�¬=c��g�{c��xikbgcv­ È svbgwPn|�rikz{sxu�z�ª'i³pr^ iku��*svbgw��{cRz{z{it¬�mtc�nxu } mksqª���nx��^¤ujlab�¬Pc��gz{sxmk¨xc��rz�z�^asqª pr^acc��y��c�mkmtc�uBp�¬=c�^�n±¨jiksx��sx¥$p{^acRz�c�bgc�p{^as } z J z{c�c�cx­ ��­ �×�qd � É �k��� � É �t��d � É �k� � � K0­§��sqª¸c�¨vc��RÉaitu�sxla��sxwaikuaiksxu»É=p{^�c��rc�nxz{sxu8sx¥$p{^aioz�vsBs } ¬=c�^�n±¨jiksx�´�rc�b7nxitu�z´lau��*mkc�nx��­��:zµb-c�uBp{iksxu�c } ¬j~ (§­B¨qn|u�Ê»c�c��²nxu } � �óʲ­��§n|�rb-s|¥Mn|mPitu �×�|� � pr^ac:¬=c�uac�ÀP�*ion|mPitua¶�lac�uP�*csx¥Jwa�rc���sxu } i³pritsvuaitu��7sxu�p{^ac�nv����la�rnv�*~7sx¥���sxb-wa�rc�zrz�ik¬amkcez{sxmk¨xc��rz_iku9p{^ac�mtsqª ��nx��^8ujlab�¬=c��'�rc��vitb-c�ª_nxz'nxu9l�uac*�yw=c��0prc }z{i } c§c �=cR�0pR­ L u�p{^�ikz_w�nxwPc���Éynvz¸n�À���z�p'z�p{c�w9iku�p{^�cenxu�n|mk~yz�iozµs|¥#pr^ac�z{c§bgc�p{^as } z�Éyª¸c�^�n±¨vc�z{^asqª'u�pr^�nqp_pr^ac�b-sBz�p'sx¬j¨jiksxl�z�rc�z{lamtp´s|¥¿wa�rc��*svu } itp{iksxuaikua���*svu�z{ikz�prz$sx¥=�rc�b-sq¨jikua�§p{^ac'¶Pl��0prl�nqpritsvu�z$s|¥¿sv� } c��²��nx��^-ikugp{^ac�ikuBp{c���¥Mnv�*c'wa�rc�zrz�l��{cRz�­J]_^BlPz�Évituz{sxb-c�z{c�uPz�cvÉxpr^ac:wa�rc���sxu } i³pritsvuac } z{sxmk¨xc��rz´z{lawaw��{cRz{z´p{^ac�laujª¸nxuvprc } nv�*sxlPz�prik�'ª_n±¨xcRz$�xc�uac���nqprc } ¬j~gp{^ac�ituBprc��{¥Mnx�*c } ikzr�*svu��pritujlaitp{ikc�z�­ L p'ª'itmkm»�rc�b7n|iku�p{s-wa�rsq¨xc�p{^�n|p¸pr^ac��rc�z{lam³pritu���¶Ply�yc�z_wa�rsq¨ji } cRz_n��vsjs } n|w�wa�{s±�yikb7nqp{iksxu�s|¥#pr^ac�iku��*svbgw��{cRz{z{it¬�mtcz{~yz�prc�b8­�®�^aikmtcgpr^ac��rcgnx�{cgc*�yw=c��ritb-c�uvp�n|mJc�¨ji } c�u���c�zez{^asqª'itu���p{^Pnqpep{^aioz§ikzeiku } c�c } p{^ac7��nvz�cvÉ=p{^acgp{^�c�sx�rc*prik��n|mJwa�rsBsx¥²s|¥pr^aikz'¥Mnx�0p��rc�b7nxitu�z¸prs-¬Pce¥¹svlau } ­

STS àIUPV±õ�ø�W

Page 22: On the behavior of upwind schemes in the low Mach number ...

�x� � :������ + +- I� ! I� !�� :T.1����� ���"�

� < ������� 3

�:c��rc>nxz�itu[z{c��*p{iksxu ��É'ªµc6�*sxuPz�i } c���p{^ac>mtikb-i³p8z�svmtlypritsvu�s|¥�pr^ac>�'ikc�b7nxuau�wa�rsx¬amkc�b J �x�RK-iku sx� } c��7p{sËi } c�uBp{it¥¹~Ëpr^acmkitb-itp�z�~yz�p{c�b2pr^�nqp§p{^�c } ikzr�*�rc*prc�lau2ABu�sqª'u�z�zrnqprikz�¥¹~x­ L u©z{c��0pritsvu��aÉ¿ª¸cg^�n±¨xcgz�^asqª'uHp{^�n|p§c�¨xc�u>i³¥µp{^acgikuai³priknxm } nqp�n�nx�{cB�ªµc�mtm�wa�rc�w�nx�{c } Denxu } s|¥�p{^�c�¥¹sx�rb J dxd �Od � K0Éap{^ac�z{sxmklypritsvu8n|¥ùp{c��:sxu�c�p{ikb-c�z�prc�wH��sxuBprnxitu�z�wa�{cRz{z{la�rc�ª¸n±¨vc�z_sx¥�sv� } c�� � �|­]_^ac��{c�¥¹sx�rcxÉ¿ª¸c�^Pn±¨xc�prs9��sxu�z{i } c��eb-sx�rc��vc�uac��rnxm�ikuaitp{ion|m } n|prn�p{^�nxu J dxd��ód � K0ÉEiO­ cgªµcg^�n±¨vc�prs9��sxu�z{i } c���p{^ac7��nvz�cgª'^ac��{cpr^ac�ikuaitp{ion|m } n|prn�nxm³pr^asxl��x^�z�p{ikmtm¿��^�nx�rnv�0p{c��{i Q�c } ¬B~7n�mtsqª[��nv��^�ujlab�¬=c��_n|�rc�uasxpµª¸c�mkmEwa�{c�w�n|�rc } ­ L u�n }a} itp{iksxu»Éyz{ituP�*c�pr^aciku��*svb-wa�{cRz{z{ik¬amtc H lamkc��'b-s } c�m»z�law�wPsv��p�z } c�u�z{itp�~ } ikzr�*svuBp{ikuBl�i³pritcRz�Éyª¸c�n|moz�sgn|mkmtsqª iku�p{^ac�ikuaitp{ion|m } n|prnaÉBpr^ac�wPsBz{z{ik¬aitmkitp�~-p{s^�n±¨vc � � � �²} c�u�z�itp�~ } iozr�*sxuBpritujlaitp{ikc�z�­´]_^jl�z¸p{^acep�ª¸s7z�p�nqprc�z1� 1 n|u } � * n|�rcenxzrz�lab-c } prs7^�n±¨xc�p{^ace¥¹svmtmksqª'ikua��¥¹sv�{b��

� 1 � ����� � � 13, $ �� � � 1�, .� ���� � � 13, � ��� ���(21 �'�

��� � �� 6���G(213, .� �� � � (213, � ��� ���� 1 �'�

��� � � �� � � 13, . �� � � � 13, � � �����

4 1 � � ��� � ���� ��4 $ 6� � 4 13, . 6� � � 4 13, � ��� ���J H H K

nxu }� * � � ��� � � *-, $ 6� � � *-, . 6� � � � *-, � ��� ���(+* �'�

��� � � ���� (+*-, .� 6��� � (+*-, � � ��� �� * �'�

��� � � 6��� � *-, .� 6��� � � *-, � ��� ���4 * � ����� � ���� ��4 $ 6��� 4 *-, .� 6��� � 4 *-, � � ��� �

J H #IK

��z7itu z{c��*p{iksxuZ�aÉ_ª¸c9ikuBp{�rs } l��*c8pr^ac�z{cHnxz{~jb-wyp{sxp{io�8c*�yw�n|uPz�iksxu itu pr^acHz�^�sj� A n|u } ��n|�rc*¥Mnx�*p{iksxuZ�*la�r¨xcRz�­ L p�ioz7cRnxz{itmk~z{c�c�u�pr^�nqp�n|�Bn|iku7p{^�c�z{ce��la�r¨xc�z } c��xc�u�c���nqp{c§ikuvprsgz�p{��n|ik�x^Bp¸mtikuac�z_sx¥Jz{mtsvwPcRz / � 9 � � 4 $ � 13, $ n|u } 9� 9 � � 4 $ � *-, $ ­��:sqªµc�¨xc���ɬ=c���n|l�z{c 4 1�, . n|u } 4 *-, . n|�rc�uasxpCQ�c��{sPÉapr^ac�¥¹sxl��:w=svzrz�ik¬amkc�ª¸n±¨vc�z'wPnqp�prc��ru�s|¥´½Jit�P­:�7��nxu�sy���*l���­��8sv�{c�sq¨xc��RÉ } lac�p{s7pr^ac� � � �²} c�u�z{itp�~ } ioz{��sxuBp{ikujlaitp{ikc�z�Éjp{^ac�z{wPc�c } z_sx¥�z{sxl�u } nx�{ceuasxp'c�·Bl�n|m¿svu9p{^acep�ª¸sgz{i } c�z_sx¥�pr^ac��*svuvp�nx�*p } ikzr�*svuvpritujlaitp�~x­�:sqªµc�¨xc��RÉjitu9pr^ace¥¹sxla�'w=svzrz{it¬amkc���nvz�cRz�Éjpr^aceituBp{c��rz{c��*p{iksxu9s|¥Jp{^ac�z{^asy� A�n|u } ��n|�rc*¥Mnv�0pritsvu��*l��{¨vc�z¸�xik¨xcRz��

( � �'� ��� � � �� � � � � 13, $ (+13, . � � *-, $ (+*-, .� � 13, $ � � *-, $

/ �� � 4 $

& 4 .� � 13, $ � � *-, $

� � � � � �����

4 � � ����� � ���� � 4 $ 6��� � � � *-, $ 4 1�, .� � � 13, $ 4 *-, .� � 13, $ � � *-, $

/ � � 4 $ � � 13, $ � *-, $� � 13, $ � � *-, $

&9(2. � � � � � �����

J H �RK

ª'^ac��{ce^ac��{c & � ���&� & *+1�� � ��� � ��� * / � � � 1J­$]_^ace¨±nxmtl�c�z_s|¥�p{^ac } c�u�z{i³p�~�nxu } p�n|ua�vc�uBp{ion|mE¨vc�mksj��i³pritcRz_sxu�p{^acep�ª¸sgz{i } c�z¸s|¥pr^ac��*svuvp�nx�*p } ikzr�*svuvpritujlaitp�~�nx�{ce�xik¨xc�u�¬j~ �

��1 � � ��� � � 13, $ 6� � � � 13, . / � 13, $

� � 13, $ � *-, $� � 13, $ � � *-, $

&9(+.� � 4 $ � 13, $ � � 13, $

� � 13, $ � � *-, $& 4 .� 4 $ �

��� � � ��� �� �1 � � 1

��* � � ��� � � *-, $ 6� � � � *-, . / � *-, $

� � 13, $ � *-, $� � 13, $ � � *-, $

&9(+.� � 4 $ � *-, $ � � *-, $

� � 1�, $ � � *-, $& 4 .� 4 $ �

��� � � ��� �� �* � � *

J H �RK

È svu�z�i } c��§u�sqª p{^�c-nxwawamkik��nqpritsvu�s|¥'¦§s } lauasq¨�zr��^ac�bgcgiku6n8½Jitu�i³prc7¼$sxmklab-c�¥¹��n|b-c�ªµsv� AE­�½�sx��z�ikb-wamkik��i³p�~vÉ#ª¸cg��sxu�z{i } c��pr^�nqp�ª¸c7l�z{c�n9�{c��xlamon|����n|�{p{c�z{ion|u��v�{i } s|¥_lauait¥¹sx�rb b-c�z{^¤z�i Q�c���iku©p�ªµs } ikb-c�u�z{itsvu�z�­�� � � �����6� iozep{^ac7iku } c��6sx¥µpr^acuas } c�ª'^asvz{c7�*sjsx� } iku�nqprc�z�n|�rc � � � �� � � n|u } ª¸c7l�z�c-pr^ac�uasxprn|p{iksxu� ��� � � � � ��/ �:���6� � � � � ���6�,� � ����=/ � �,� � �,�� � � �sv�� ��� �?� �� � � ��� ��� � ¥¹sx��mon|¬=c�mkmtikua�¤pr^acHuac�it�v^j¬Psv�rz-sx¥�pr^acH�x�ri } u�s } c��±­°]_^acH��c�mkm�nvz{z{sy�*ionqp{c } ª'itp{^�u�s } c���ioz� � � � � �2/�� 9 � � � � � �6 � 9 � � � � � � � � /6� 9 � � � � � �1 � 9 � � � � É��� ikz¸pr^acesxlyp�ª_n|� } l�uai³p'uasv�{b7nxm=¨vc��*p{sx�_svu % � � n|u } ªµc§uasxp{c�� ��� �

�������� ��

� ­

+�,TS-+ è

Page 23: On the behavior of upwind schemes in the low Mach number ...

�������������� �������������������� �"!$#&%'���&(�'#)� �*�����,+-���/.* 0%'�1�2��(3��'�4+5� (�� �7698�8;: �y�

®�^ac�uÌnHz�p{c�n } ~>z�prn|p{c�ioz��rc�nv��^ac } É#p{^ac9n|waw�mtio��n|p{iksxu©s|¥'n8À���z�p �Osv� } c��eÀ�uaitp{c�¨xsxmklab-c�zr��^ac�b-c�~jikc�m } zep{^�c7¥¹sxmkmtsqª'ikua�z{c�b-i � } iozr�*�rc*p{cecR·vlPnqp{iksxu � �

����� 1 � > �� ��� � � � � �� ��� �I� J #|�RK

ª'^ac��{c � ��� } c�uas|prc�p{^�cg¨qnxmtlac�s|¥´pr^ac7z�p�nqprc�sxu�p{^acgikuvprc��{¥Mnx��c�z�¬Pc�p�ªµc�c�uHpr^ac-�*c�mkmoz ��nxu } �0­�½�sxmkmtsqª'ikua�7pr^ac7n|u�nxmt~yz{ikzpr^�nqp�ª¸c�^Pn±¨xc§wPc���¥¹sv�{b-c } sxu�p{^ac�nvz�~jb-wyprs|p{io�§z�svmtlypritsvu�sx¥#pr^ac��'itc�b7n|uau9wa�rsx¬amkc�b8Éyª¸c�ª'�ri³prc§p{^acRz�ceikuvprc��{¥Mnx��c�¨qn|mklacRz_itupr^ace¥¹sxmkmtsqª'ikua�g¥¹sx�rb �

� ��� � � ��� � � $��� � � � .��� � �� � �� � @�8� �*�( � � � �

��� � � � � ( .��� � �� ( ���� @�8� �;�

� ��� � ���� � � � � � .��� � �� � ���� I� �8�;�

4 � � � ����� � ���� � 4 $� � � � 4 .��� � �� 4 �� � @�8� � �

J #y��K

pr^ac�u�p{^ac§¶�la��¥¹lau��*p{iksxu9c�¨qn|mkl�nqprc } sxu�pr^aioz¸z�prn|p{ce^�nvz´pr^ac§¥¹sxmkmtsqª'ikua�gc*�yw�n|uPz�iksxu J ^ac��rc ( � � � � ��� } c�u�s|p{c§�rc�z{wPcR�0p{ik¨xc�mt~-pr^acuasv�{b7nxm»n|u } prnxua�xc�uvpriknxm¿¨xc�mtsy�*itp{ikc�z_svu�pr^ac�ikuvprc��{¥Mnx��c�K �

� �� �/� ��� � � �� � � � . � ����� � ��� � ��� � $� � ( .� � � � � �� �1�

� �� �/� ��� � � �� � � � � � ����� � ���� � 4 $��� 6��� 4 .��� 6� �� � � $��� � ( .��� � � 4 ���� � � � � #� �1�� �� �/� ��� � � �� � � � # � ����� � ���� � � �� � $��� ( .��� � .��� � � � #� �1�� �� �/� ��� � � �� � � � � � ����� � #��� � ��� �

� /6� 4 $��� (.� � � � � �� �1�

½��{svbºpr^ac§c*�ywa�rc�zrz�iksxu J H �RK´s|¥¿pr^ac�ikuBp{c���¥Mnv�*c:w��{cRz{z{la�rcxÉxitp_ikz¸z�c�c�u�p{^�n|p 4 $��� � 4 $�� ���´n|u } p{^ac�u�pr^ac�sv� } c�� � b-sxb-c�uBp{l�bcR·vlPnqp{iksxuPz } c��vc�uac��rn|p{c�z¸ikuBp{s �4 $

������ 1 � > �� ��� �I�

¬alap�p{^acesv� } c�� � ��b-sxb-c�uBprlab�cR·vlPnqp{iksxuPz¸nx�{c:� ������ 1 � > 4

.��� �� ��� �@� J #x�0K

ª'^ac��{c§pr^ac�¨qn|mklac�s|¥4 .��� ioz'�xik¨xc�u�iku J H � K0­´]_^aioz_�xik¨xcRz²¥¹sv�'p{^�c§^�sx�ri-Q�sxuBprnxm=b-svbgc�uBp{lab�c�·Bl�nqpritsvu�

/� � $� 4 . � $ 4 .�

� � $� � $ � $ � 4 .� � � $� 4 .�� � $� � $ �

� � 4 $ � � � $� � $ �� � $� � $ � & � � (. / � � $� � $ � � $� � $

& � ( . �J #|�RK

nxu } ngz{ikbgikmon|�'c*�ywa�rc�zrz{itsvu�¥¹sx��p{^ac�¨vc��{p{io��n|mEb-sxb-c�uvprlab�cR·Bl�nqpritsvu»­´½a�rsxb J #q� K0Éaitp�ikz'��mtcRn|�_pr^�nqp!4 . ikz'u�s|p��7nxz'z{sjsxu8nxzpr^ac�¨vc�mksj��i³p�~�À�c�m } ioz�u�s|p�l�uai³¥¹sv�{b8­:¾:s|prc�n|moz{s-p{^�n|p�c�·Bl�n|p{iksxu J #q�RK'ikz§n|mkb-svz�p:i } c�uBp{io��nxm»p{s�p{^ac�c�·Bl�nqpritsvu�sx¬yp�n|ikuac } ¥¹sx��'svc } ikzr�*�rc*prikzrnqpritsvu�iku � # � ­

STS àIUPV±õ�ø�W

Page 24: On the behavior of upwind schemes in the low Mach number ...

�v� � :������ + +- I� ! I� !�� :T.1����� ���"�

� �'�q�'������"¸�'

�t� � ��­ (¸i ��mBnxu } �8®>cRz{z{c�mkikua��­|� lauaitÀ�c } b-c*pr^as } ¥¹sx���*sxb-walap{ikua��iku��*svbgw��{cRz{z{it¬�mtc´nxu } �*svbgw��{cRz{z{it¬�mtc�¶�sqª�z»itu�¬=sxlau } n|�r~À�p�p{c } �*sjsx� } iku�nqprcx­�� :����I(C� ��� :��9����# É#��d�� �k� � � �E��#q��Éy���v�x�a­

� � � h¿­ È mtc��r�x­$¾�lab-c��rio��n|m»z{ikb�lamonqpritsvu�s|¥Jp{^�c�^asxb-sx�vc�uac�sxl�zµc�·Blaikmkit¬a�riklab�¥¹sv�_p�ªµs �Owa^Pnxz{c�¶�sqª�z�­�� �I����� I+ �O��� ��( ����� � �����" �+��9����#���%�# ÉJ� H � � � � d �j��# � Éx�x�x�v�a­

� � � ��­���c��r¨jitc�ly�¿É ��­¿¦§lac&Q�c�ua�Bn|�RÉ=�:c��r¨ ��¦§l�itmkmknx� } É=nxu }>È ­=¼:iks0Q�n|p�­§��u�nxmt~yz{ikz:s|¥´mksqª ��nx��^Hz{itb�lamonqp{iksxuPz�ª'itp{^>��sxb �w��{cRz{z{it¬�mtcelawjª'iku } �*s } c�z�­ L u � ���9�9. ������� ­��|sv^au9®�itmkc�~9n|u } hjsvu�z�É¿�R�x�v�a­

� d � ��­B��n } c�z{waikc�m H ­j]#la��Axc�mPnxu } ¾�­��e�rsxmkmL­»��z{z{c�zrz�b-c�uvpµs|¥»wa�rc���sxu } itp{iksxu�itua��bgc�p{^as } z²¥¹sx�µb�lamtp{i } ikb-c�u�z{itsvu�n|mEn|c��{s } ~ �uPn|b-ik��z�­ � ��( ����� �&� # I� !�� +5����!�# ÉP� H � H ��� � H �|d�É����x��#j­

� � � H ­E¦§�rc�uaikc��R­ 2§zr�*ikmtmonqprsx�r~�w=c��{p{la�r¬�nqpritsvu�z�sx¥$p{^acg¾:n±¨jikc�� �ëhBprs0Avc�z_c�·Bl�n|p{iksxu�z�­�� : .* �����:��9��� �'#&:����0��+ : É2# H � d�# # �BdB�x��É�R�x��#j­

� H � ��­#¦§laikmtmon|� } nxu } ��­»�:¬a�x��n|mkmO­ .���!��&+5� #& �� ����� !0�'#��9����!R��#�2��(��'��� )&�2�'#% ��( ��� �'# #����+-�'# É#¨vsxmklab-c � s|¥ � ��� 8 � �*8! �"�#�#$ 8 ��% . �'& �"� . �'& 8� ­µ¦enxlyp{^�itc�� �ó¼:ikmtmon|��z �ó¾:sx�{p{^��:sxmkmknxu } Éa��n|�rikz�É��|�v�a�v­

� # � ��­a¦§laitmkmon|� } n|u }9È ­j¼:iks0QRnqp�­ 2�u�p{^�c�¬=c�^�n±¨jiksx�¸s|¥#l�wBª'iku } z{��^ac�b-c�zµitu�p{^acemksqª[��nx��^�ujlab�¬Pc��_mtikb-i³pR­ � ��( ����� �&� # I� !(�G����!�# ÉP�|�6� H � �j� H ÉP���x�v�a­

� � � h¿­)�emknxitu�c��rb-nxu nxu } ��­¸�Hnq� } n�­ È svb-wa�{cRz{z{ik¬amtc�nxu } itu���sxb-wa�rc�zrz�ik¬amkc9¶�lai } z�­ � �I( (����2��% �� �����2#*���*����� �* I� !���0� +5��� !$.* ������&($ �� ��%�# ÉP� � � H �x� � H � ��É����v�v�a­

� � � ��­ �>2�­+�e�rc�ioz{z�­��$�rsx¬�mtc�b-z�ª'i³pr^ } i��Ec��rc�uBp�pritb-c�zr��nxmtcRz§¥¹sx��w�nx��priknxm } i �=c��{c�uvpriknxm´c�·Bl�n|p{iksxu�z�­ � ��(�(����2��% �� �����2# �������� �, I� ! �9�0� +5��� !$.* I�����'($ �� ��%�# É=�x��� �x�v� �jdxdv��Éa���v�x��­

�k��� � ��­×�©­B��nvz�c�mtmonaÉ L ­v½�nxitmkmkcxÉjn|u } ]e­B¦enxmtsvl-,*pR­ 2�u�nxu7n|wawa�rs±�yitb7n|p{c:¦§s } lauasq¨gz{��^�c�b-cx­ 8���� : � : ���'� ��( �":.� +5����! % ���": É�±� �k���v� �=��dv��Éa���v�x�a­

�k�x� � ¦g­#��c*p{ik¨jitc���n|u } hE­�ha��^asy��^ac*pR­g]_^ac-iku��*svb-wa�{cRz{z{ik¬amtcgmkitb-itp�s|¥²pr^ac7uasvuaikz{c�uBpr�{svwaik� H lamkc���cR·Bl�nqpritsvu�z�­ � � %'��: � I� :.*� %'��:-� �" �+-: ÉP�x�x�v�a­�p{s-¬Pc�w�la¬amkikz{^ac } ­

�k�R� � ¦g­J�8c*prit¨jikc���nxu } hE­Jhy��^�sj��^�c*p�­�Ê»ikbgitp{c�itu���sxb-wa�rc�zrz�ik¬amkc } c�z��R·Bl�nqpritsvu�z } � H lamkc���u�sxu��óikz{c�uBpr�{svwaik·BlacRz�­�h ��b-ituPn|ik�{c�R·Bl�nqpritsvu�zHn|la� } ���{ik¨ ��c�z�wPn|�{p{ikc�mkmtcRz } cÌm�� ���*svmtcË��svmt~Bprc���^auaio·BlacxÉ�p{s ¬=cÌ¥¹svlau } nqp � ^Bp{p{w �5F0Fqª'ª'ª�­ b7nqpr^�z�­ lau�it¨ ��rc�u�uac�z��x­ ¥¹��F�b-c*prit¨jikc��'F±wa�rc�wa�rikuvp�z�­ ^Bp{b-mOÉy�x�x���x­

�k��� � ��­»��n|ikmtmkc��rcxÉ»hE­ È mkc����|É È ­E¼:iks0QRnqp�É L ­¿]�sxlab-iOÉ»n|u } � �ë�J­»��n|�vu�n|l } ­�¾�lab-c��rio��n|mJb-c*pr^as } z§¥¹sx�§mtsqª«��nx��^Hujlab�¬Pc��pr^ac��rb7n|m �O^j~ } ��n|lamkio��¶�sqª�z�­ L u/� �����9. ������� ­.�|sv^au9®�itmkc�~9n|u } hjsxuPz�É¿�R�x�x��­

�k��d � ��­»��n|ikmtmk`��rcxÉ È ­¿¼:iks0QRnqpRÉE��­��elab�¬Pn|�rs�É¿n|u } L ­¿]#svlab-iL­ È svb-w�n|�rikz{sxu�s|¥²mksqª ��nx��^>ujlab�¬=c��eb-s } c�moz�¥¹sx�eu�nqprla��n|m��sxuj¨xcR�0pritsvu�wa�rsx¬amkc�b7z�­ L u/� ��� �������&��(0� �'(�� � I�1 ,�3254 ­µ¦§c�uasvn�É L prnxmt~vÉ H �ë�-hyc�wyprc�b�¬=c����R�x�v�a­

�k� � � H ­=hy��^�nxmtmOÉ È ­�¼�itsRQ�n|p�É�(§­���sjsx¬al�z�Éyn|u } ��­ ��c��r¨Bikc�la�E­'2�u�pr^ac��*svb-walyprn|p{iksxu8s|¥�lau�z�p{cRn } ~�n|u�z�z�p{cRn } ~�mtsqª���nx��^¶Psqª�z_ª'i³pr^9ikb-wamkik��i³p�lawjª'iku } b-c*p{^�s } z�­ L u � �����9. �6�87�9�9�: É=�x�x�a�v­

�k� H � h¿­jhy��^�sj��^�c*p�­»��z�~jb-wyprs|p{io��z´¥¹sx�¸z�~jb-b-c*p{�rio��^j~jwPc��{¬=sxmkik�:z�~yz�p{c�b7z²ª'i³pr^�n�mon|�r�xc�w�nx�rnxb-c*p{c���­;� �������" �+ �O��!�� <C�&� �&�2� �� �+��)&�2 �� �����2# É # � �k� �a� #yÉP�R�x�x��­

�k��# � h¿­Phy��^asy��^ac�p�­$½�nvz�p:z{itu��xlamon|�_mkitb-itprz'sx¥�^j~jwPc��{¬=sxmkik�§w } c�z�­=� �������" �+T���,!�� <C�&� �&�2� �� �+T��)&�2 �� �����2# É#�x��d � d # H � � �±�yÉa�R�x�xd�­�k��� � H ­ ½¸­�]�sx�rs�­#� ���'($ ����>�"�I+5���'� #, ��"!3����(�&����% �+ ($�&�����;!�#��'�I�)�9����!3!���" �(���%�# ­_hjwa�ritua�vc��_¼$c��rmon|��É�(¸c��{mkitu#É¿���x��#j­�k��� � H ­E]�la��Axc�mL­e�$�rc��*svu } itp{iksxuac } b-c*pr^as } z�¥¹sv�§z{sxmk¨jitu���p{^�c�iku��*svb-wa�{cRz{z{ik¬amtc�n|u } mksqª z{wPc�c } ��sxb-wa�rc�zrz�ik¬amkc�c�·Bl�n|p{iksxu�z�­

� :�� ��( ����� :��G�?��# É+#|��� �'#�# �a�|�v�aÉa�R�x�'#y­�×�|� � H ­x]#l�� Avc�mO­��'c�¨jikc�ª s|¥Ewa�rc��*svu } itp{iksxuaikua�§b-c*p{^�s } z�¥¹sx�$¶Plai }-} ~ju�nxb-ik��z�­ ���0� +5��� !� ��(�&����% �+2.� I�����'($ �� ��%'# É=�R��� � � # �

�x�|dPÉ=�R�x�v�a­�×�y� � H ­=]�la��Axc�mOÉ���­P½Ji³prc��rb7n|u»ÉEn|u } (§­�¨qnxuHÊ»c�c���­ �9� � % ���"!�� � ������� �?@1 ��"! �����,+E� ( � �C���)�����%��I(C� � �'#�#���'+-�� �3����� � �"%��I(A

� � �'#�#���'+-�6�7��� �,)&�2 �� �����2#C�'����B �2� � � !�� <C�&� �&� % � #&%����'($��# É´wPn|�xcRz��y� � �a�|�|dP­C�xsx^au¤®�ikmkc�~¤n|u } hysxu�z�É È ^aik��^�c�z�p{c��RÉ�R�x�xd�­

+�,TS-+ è

Page 25: On the behavior of upwind schemes in the low Mach number ...

�������������� �������������������� �"!$#&%'���&(�'#)� �*�����,+-���/.* 0%'�1�2��(3��'�4+5� (�� �7698�8;: �|�

�×�x� � h¿­ N A±nxiL­»]_^�c�ituP�*sxb-wa�rc�zrz{it¬amkc�mkitb-itp¸nxu } p{^ac�itu�i³priknxmPmon±~xc��$s|¥¿p{^�c��*svb-wa�{cRz{z{ik¬amtc H lamtc��µcR·Bl�nqpritsvu�z�­;� :".* �����:��A�0��� �� �2� ��: ÉP� H � �B�|� �j�x���xÉa�R�x� H ­

�×�|� � (§­»¨qnxuÌÊ»c�c���n|u } � �óʲ­ �§n|�rb-s|¥Mn|mO­�hBprc�n } ~ H l�mtc��ghjsvmtlypritsvu�z�iku¤� J u K 2�wPc��rn|p{iksxu�z�­ L u ��­»¼:ikc��rc�u } c�c�moz H ­��:io� AEÉ�-­P�'itc�b7z�mon|�v^»Éyc } itp{sv��É . ��+E��� @0����!3($�&�����;!�#�� 8,6�� � ��%�� � !�� ��@�# ���)����� ��� �R��� � ��� ����� I�*(���+5��� @0����!@%��I�;�'�&� �&� % ��� ���&+-!� � � �'������ �&+ @I� ��(�(� � � � �&(3 �&��7�� 4�9 ��:������ É$w�n|�vc�z��|d �j�v�a­¿hjwa�ritu��xc��RÉJ�x�x�v�a­7Ê»cR�0p{l��{c7uasxp{cRz�itu¤��sxb-walyp�nqp{iksxuPn|mzr�*ikc�uP�*c�n|u } c�u��xikuac�c��{ikua� � ��dP­

�×�qd � È ­¿¼�itsRQ�n|p�­ � I+-%'��+9!�� � % ����+-�&(�&�2��#�# � �� ������� I� � ��#3�&�C� �P#�� �� ������� �� ����#����P�'� � �C�"��(3�� � !0�3.� R%�� �'�4�'� ($ �� + + @ ��#�&� � � ��# ­´�$^ ��p{^�c�z{ikz�É N uaik¨xc��rz{itp�~�s|¥�¾�io�*c ��hjsvwa^aion �ë��uBp{ikwPsvmtioz�É¿���x�v�a­

�×� � � È ­�¼:iks0Q�n|p�­ È nxmk��lam } � ����sxlamkc�b-c�uBp�z } ikwa^�nxz{io·vl�c�z } nxu�z'lauaceprla~x`��{c � L ua¶�lac�uP�*c } c�mon-�{c�uasx�rb7n|mkikzrnqpritsvu } l�zr��^���b7n} ce¶�ly�¿­´]#cR��^auaio��nxmE��c�w=sx�{p�h��eh È 2)F|Ê�¦eÊJh2Fq��]CFq�v� �ë�a��d�É È9H ��É=�x�x�x��­

�×� H � ¦g­�¼$sxmkwPcv­´��c���¥¹sv�{b7nxu��*c§s|¥$�*svbgw��{cRz{z{it¬�mtc�¶�sqª ��s } c�z'n|p_mksqª°�Hnx��^9uBl�b�¬=c��R­ � 8 � � � ������� I+ É���� � dv� � � H É����v�x��­�×� # � 2�­ È ­��=ikc�u2ABikc�ª'io�'QvÉ���­Ph�Q�b-c�m³prc��RÉ�n|u } ��­��Jc��r��n|ik�{cv­ È sxb-wa�rc�zrz�ik¬amkcenxu } ituP�*sxb-wa�rc�zrz{it¬amkc§¶�sqª�z �µ�:uHn|mk�xsv�{itp{^ab�¥¹sx�

nxmtm»z{c�nvz�svu�z�­ � ��( ����� : .*�&�����;!�# �9�0� +-:T.*� %���: � ��@R: É�#|� J ��K �k��� � �E�R�y�vÉa���v�x��­

STS àIUPV±õ�ø�W

Page 26: On the behavior of upwind schemes in the low Mach number ...

Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles - B.P. 93 - 06902 Sophia Antipolis Cedex (France)

Unité de recherche INRIA Lorraine : Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - B.P. 101 - 54602 Villers lès Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38330 Montbonnot St Martin (France)

Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - B.P. 105 - 78153 Le Chesnay Cedex (France)

ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, B.P. 105 - 78153 Le Chesnay Cedex (France)��������� ���� ���������� ��� ���

ISSN 0249-6399