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In Copyright - Non-Commercial Use Permitted Rights ......Zm< ` ' . '& & '( H

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Research Collection Working Paper Integration of stiff mechanical systems by Runge-Kutta methods Author(s): Lubich, Christian Publication Date: 1992 Permanent Link: https://doi.org/10.3929/ethz-a-004283507 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection . For more information please consult the Terms of use . ETH Library
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Page 1: In Copyright - Non-Commercial Use Permitted Rights ......Zm< ` ' . '& & '( H

Research Collection

Working Paper

Integration of stiff mechanical systems by Runge-Kutta methods

Author(s): Lubich, Christian

Publication Date: 1992

Permanent Link: https://doi.org/10.3929/ethz-a-004283507

Rights / License: In Copyright - Non-Commercial Use Permitted

This page was generated automatically upon download from the ETH Zurich Research Collection. For moreinformation please consult the Terms of use.

ETH Library

Page 2: In Copyright - Non-Commercial Use Permitted Rights ......Zm< ` ' . '& & '( H

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Page 14: In Copyright - Non-Commercial Use Permitted Rights ......Zm< ` ' . '& & '( H

� 2 � 4�6���)�7�o CM:�B7HI\8ghZ]J>Z]HI\ ��� ��� ��� � Y*� D � � Y(?=V�<>:G:GY�@7?AJ>CM:G@�@D:=<>J>@DZmB7J>ZPOI:�?AJ ; @7<>JE<>Z]eIChJGLL J <>CMH bMr g�CMHAf :AOI:G@ w :(\MHIJ>:=g�J>C8?AJ4J>CM:(F8HIJ>:G\hJ>Zm? r :G\M:G@DeIV�Z]\�J>CM:(Y[:=BDC8?A\MZmBG? r <>Vh<>J>:GY �SPhL]W � Zm<��� � ��� � � Y*�DW � Lo CM:O@�4 � � +.-0/iZ]\L�:RhL t � Zm<OZ]Y[F r Z]:=g w V[B7HI\8ghZ]J>Z]HI\8<��:RhL T � ?A\8gF�:RhL t � L�L \EB7HI\hJ>@7?I<>JAKn?A\8? r VnJ>ZmBG? r <>H r]b QJ>Z]HI\8<OH ^ �SPhL]W � � )�;!+.2 254 � 2 ? w H b J)& � LlK;HIJ>:WKMCMHAf :AOI:G@CKAJ>C8?AJ Z]J Zm<O\MHIJ ?AJ;? r]r H w OnZ]H b <�4 � 6,+ � 6,+8J>C8?AJJ>CM:4gM?AY[FMZ]\Me B7HI\8ghZ]J>Z]HI\8<��:RhL T � K �:RhL t � ^ HI@;J>CM: r Z]\M:=?A@;J>:=<>Jd:CI b ?AJ>Z]HI\�`�'��� T �,�E?A@D:i< b H B7Z]:G\hJ ^ HI@gM?AY[FMZ]\Me4Z]\EJ>CM:i\MHI\ r Z]\M:=?A@OFM@DH wMr :GY �SPhL]W � Lo CM:GHI@D:GY(<�RhL]WE?A\8g�RhL P�f;Z r]r;w :EFM@DH;OI:=g�Z]\/J>CM:�B7H b @7<>:EH ^ J>CM: ^ H r]r HAf;Z]\Me�<>:=B7J>Z]HI\8<GL/o�H�FM@DH;OI:o CM:GHI@D:GY RhL]WWKOf :Ef;Z r]r JD?AjI:�?�@DH b J>:�<>Z]Y[Z r ?A@4J>HA7 s�9>��L \�J>CM: J �GQ :ANnF8?A\8<>Z]HI\�H ^ J>CM:E\ b Y[:G@DZmBG? r<>H r]b J>Z]HI\hK�:RhL N � � �

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Page 15: In Copyright - Non-Commercial Use Permitted Rights ......Zm< ` ' . '& & '( H

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Page 19: In Copyright - Non-Commercial Use Permitted Rights ......Zm< ` ' . '& & '( H

/�U� �D � � � �:L \�� b :G\8B7:EH ^ F8:G@DJ b @ w ?AJ>Z]HI\8< � �,-A4H@H@�+ � + � - �'� ��� 2 ; � --@�+ � + � -3) �>= 0�2 � � 4 �H7�$D 2 ),1�� � � ) 2 ��� 4 � 4�2 ) � ��� � �� ��� � Y*� D � ��C � � � � �� ��� � �� � � Y*� D ��C 4�--@ = 136 ��� 2 6 � � � � Y*� D ��C� �� � �Y*� D � 7O8 � 2 - ��� 2 6 2O2�: +5) � +.-+5 2 6 � + 3!2 2 � 4 � 6,+ ; 2 ) $ � $�� �� ��� 4�--@�� . ���%� �� � � � . ��C ),1+; ����� 4 ���� 2 � 6 4�-3) = � 6 � 2 @&5�4�6,+ 4H3!2 2 )� � `� .� `� . A� $ � ` .�� � � �

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M�6 ���>= 7;o CM:d@D: r ?AJ>Z]HI\8< ^ HI@ � �;?A\8gFD � � � ^ H r]r HAf�ghZ]@D:=B7J r V ^ @DHIY �:thL s � K �:thL v � ?A\8g[J>CM: w H b \8gM<lH ^u�:GY[Y(?�thL P�K b <>:=g(Z]\:�:thL]W=? � Lo CM:4<>:=B7HI\8g(:CI b ?AJ>Z]HI\�H ^ �:thL]WIW � K8f;Z]J>C ) � `� ) ?A\8g ) � `� ) HI\EJ>CM:4@DZ]eIChJ>Q C8?A\8g[<>Zmgh: w H b \8gh:=g w Vu�:GY[Y(?�thL P�Kh@D:=?IgM<�:thL]W=P � � `� . � W

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K;:ANnJqf :4B7HI\8<>Zmgh:G@;:G@D@DHI@;FM@DHIF8?Aen?AJ>Z]HI\EH;OI:G@;Y(?A\hVE<>J>:GF8<;H ^ J>CM: 9 b \MeI:GQ:M b J>JD?4Y[:GJ>CMHhg�?AFMF r Z]:=gJ>H=�SPhL]W � ?A\8gEZ]JD<dF8:G@DJ b @ w :=g�OA?A@DZm?A\hJGL���: gh:G\MHIJ>: w V=��� " � �� " � ?A\8g:� �� " � � �� " � J>CM: B7HI@D@D:=<>F8HI\8ghZ]\Me\ b Y[:G@DZmBG? r <>H r]b J>Z]HI\8<GL���:4?I<D< b Y[:�:thL]WCs � ��� � �� " � �"Y*� D � � � � � � � �� " � � �� " �"Y*� D � ^ HI@�R � ! D � $ �

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M�6 ���>= 7@�S? � u�:GJ b <d< b FMF8Hn<>: ^ HI@dJ>CM:4Y[HIY[:G\hJ;J>C8?AJq?[<>H r]b J>Z]HI\ ��� " � �� " � :ANnZm<>JD< ^ HI@qR � ! D � $�Kf;CMZmBDC(<D?AJ>Zm< ; :=<%�:thL]W;v � ^ HI@ ? r]r !�Llo CMZm<OBG?A\ w :1OI:G@DZI; :=g w V[?A\[Z]\8g b B7J>Z]HI\[?A@De b Y[:G\hJAKn?I<lZ]JOf;Z r]rMw :<>:G:G\4J>C8?AJ�J>CM: B7HI\8ghZ]J>Z]HI\8<�H ^ u�:GY[Y(?5thL]WO@D:GY(?AZ]\i<D?AJ>Zm< ; :=g b \MZ ^ HI@DY r VdH;OI:G@�J>CM:Of;CMH r :lZ]\hJ>:GeI@7?AJ>Z]HI\Z]\hJ>:G@ OA? r L7wMHI@ w @D:AOnZ]JSV[f :igh:G\MHIJ>:

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Page 27: In Copyright - Non-Commercial Use Permitted Rights ......Zm< ` ' . '& & '( H

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U �:UlB;U�rhEWUl>7 W!949iLdc w @7?AC8?AY�K �ML k;LdU�?A@7<Dgh:G\hK � � 13--@�4 � + � -3) �>=�� 2 ; � 4�- + ;,)�7 /O:G\ J ?AY[Z]\�� s b Y[Y[Z]\Me/x bMwMr Ls HIY[F�LPKM9;:=?IghZ]\MeFKhU�?I<D<GLPK8W=NlvAthL7 P�9lM L k;L /O@D:G\8?A\hK X L ulL s ?AY[F w : r]r KCulL 9iL=x�:GJ>vGH r g"K ��1 � 2 6,+ ;�4�2�( � 2 1 � + � - �>= �,- + � + 4�2 ��� 4�2 1 2 M�6 � 3!2 2 � )+.-��*+ G 2 6 2 - � + 4�2 � 2 / 2 3!6 4�+ ;)�!1 4 � + � -3)�7�K;HI@DJ>CMQSp;H r]r ?A\8g"KhK;:Gf��lHI@DjZK�W=NItINhL7 R�9 �ML s L�/ b JDBDCM:G@CK� %- ��� 2 + � � 2 2 � 2 - � 4 � + � - �>= + � � 2 + ;!+ � ��13-0/ 2���� 1 ��� 4 � 2 ����� @�)�7 / LSo�WCs#�DW=Nlv;s � KPWRlv=QSPATnRhL7 T�9�U�L s @DH b vG:GZPNZK �MLh9d?AFMF8?AvWK� %- ��1 � 2 6,+ ;�4�2 � � 6 � : + � 4 � + � - +.-���+ = 136A;�4 � + � -98 � 2 � 6[EH78U�?I<D<>HI\hKxl?A@DZm<AKh?A\8g X FM@DZ]\MeI:G@DQ��l:G@ r ?AeFK /O:G@ r Z]\hK�W=NINIRhL7 t�9Ok;L�klZmBDChK�U�L�pd?A\MjI:WK � 2 /�13254�6,+ � 4 � + � - � 2 ����� @�) = � 6�; � -3) � 6 4�+.- 2 @ � 2 ; � 4�- + ;�4�2 � 132 � + 3 � @ EL)[E�) �� 2 � )�78xl@D:GFM@DZ]\hJAKnp b Y w H r ghJ>Q��;\MZPOI:G@7<>Z]J=`?AJOv b /O:G@ r Z]\hK8W=NINhWIL7 s�9qk;L pd?AZ]@D:G@CK s C�L u bMw ZmBDChKOU�LO9;HhBDCM:WK��6,6 � 6 �>= ��13-0/ 2���� 1 ��� 4 � 2 ����� @�) = � 6L) � + G � 6 � 3!2 2 � )) � 1+@�+ 2 @�5�+ 4&@�+ G 2 6 2 - � + 4�2�4�2 / 2 3!6 4�+ ; 2 �!1 4 � + � -3)�7 / LSo*PIt'�DW=NItIt � KZslvAtAQ vARIRhL7 v�9qk;L;pd?AZ]@D:G@CK s C�L u bMw ZmBDChKOU�L 9;HhBDCM:WKO8 � 2 ��1 � 2 6,+ ;�4�2 ( � 2 1 � + � - �>= �*+ G 2 6 2 - � + 4�2 � 2 / 2 3!6 4�+ ;(+E�) � 2 � ) 3�E ��13-0/ 2���� 1 ��� 4 � 2 ����� @�)�7 X FM@DZ]\MeI:G@5u�KdU)WGTnRIN�K�W=NItINhL7 t�9Ok;L�pd?AZ]@D:G@CK �4L ��?A\M\M:G@CK ( � 2F5�+.-0/� %6A@�+.-�4�6[E��*+ G 2 6 2 - � + 4�2�)�!1 4 � + � -3) � ��7 ( � + G 4�--@��*+ G 2 6 2 - �� + 4�2 � 2 / 2 3!6 4�+ ;(M�6 � 3!2 2 � )�7 X FM@DZ]\MeI:G@DQ��l:G@ r ?AeFK�/O:G@ r Z]\hK�W=NINhWIL

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7 N�91ulL �n?=VWK J � 2 2 � ;�4 � + � - � 2 ����� @�) = � 6<@�+ G 2 6 2 - � + 4�2 � 4�2 / 2 3!6 4�+ ; 2 �!1 4 � + � -3) �>= +.--@ 2�: �H7 xl@D:GFM@DZ]\hJAKX :=B7J>Z]HI\�gh:iU�?AJ>C�LPK �;\MZPO8L8gh: �d:G\��:AOI:WK�W=NINIPhL7 W=R�9lulL;Mi? r ?IBDCM:AOZKA9iL k;L;]@ U�? r]r :GVWK �I@=LPK � 2 /�13254�6,+ � 4 � + � - �>= - � - 2 +.- 2 4�6�@�+ G 2 6 2 - � + 4�2 � 4�2 / 2 3!6 4�+ ; 2 �!1 4 �� + � -3)�7�9;:GF8HI@DJ;NIPAQ TFKF_d:GFMJGLMc;FMF r LMU�?AJ>C�LPK �;\MZPO8LMH ^ ��?I<>CMZ]\MeIJ>HI\hK8W=NINIPhL7 WIW!9hoqL;Mi?AJ>HFK�M 2 6 � 136�3�4 � + � -<8 � 2 � 6[E = � 6 0 +.- 2 4�6 � 2 6 4 �'� 6�)�7hPA\8g4kOg�LPK �d@ b \8g r :GCM@D:G\qY(?AJ>C�L�� Zm<D<GLWCRIP�K X FM@DZ]\MeI:G@DQ��l:G@ r ?AeFK�/O:G@ r Z]\hK�W=NItIRhL7 W=P�9��iL�MqZ]@7BDCMeI@7? w :G@CKWw L�u�?I<D?AeI\MZSKlM LhK;Z]FMFhKl_4L X J>H G :G@CK� %- ��� 2 4!� � 2 + ;�4 � + � - �>= +.-+5�4�6,+ 4�- ��� 4�- �+ = � 2 @ ��� 2 � 6[E C +.- �04�6 � + ;!13254�6 �'� - 1 � 2 6,+ ;�4�2 4�-�4�2FE�),+5)�7ZL \hJ>:G@D\�L X :G@=LIK b Y�LnU�?AJ>C�LPK �lH r LnNlv�DW=NINhW � K�W=tINAQDW=NlvnL7 WCR�9�U�LFMq\MHI@D@D:G\8<DBDCMZ r g"K�� 2 /�13254�6,+5),+ 2 6,13-0/&5 � -1@�+ G 2 6 2 - � + 2 2 2 � 4�2 / 2 3!6 4�+5)�; � 2 - (+E�) � 2 � 2 - � 8 � 2 � 6 2��� +5)�; � 2 13--@'- 1 � 2 6,+5)�; � 2 )�� 2 � � 2 7Z_dHhB7J>HI@7? r ghZm<D<>:G@DJD?AJ>Z]HI\hKh9 ��o;p�cd?IBDCM:G\hK�W=NItIthL7 WGT�9Ox�L"uO`HIJD<>J>:=ghJAK %-M4�� 2 -�4�2 � E = 13--; � + � - � 2 ����� @ = � 6 ��� 2 ),+ � 13254 � + � - �>= � 2 ; � 4�- + ;�4�2%)[E�) � 2 � )),1+3�� 2 ; ���'� ; � -3) � 6 4�+.- � )�7M9;:GF8HI@DJlo;91LSo�c;QSKdc;Q vANhW=N�KI9;HAVn? r L \8<>JGLAH ^ o�:=BDCM\MH r HIeIVWK X J>HhBDjnQCMH r Y�K�W=NlvANhL7 WCt�9�_4LIU�HI@DeI:G\8<>J>:G@D\hKALDL X v=? w��HFK � � 6,2 2 ),13-0/ 2 -��1+3 2 6 ��� 2 � 6 2 � +5)�; � 2 � 2 ; � 4�- + �H7 �d@ b \8g r :GCM@D:G\qY(?AJ>C�L� Zm<D<GL�WIW=P�K X FM@DZ]\MeI:G@DQ��l:G@ r ?AeFK�/O:G@ r Z]\hK8W=NWshWIL7 WCs�97M L8K;Z]FMFhK"_4L X J>H G :G@CK ��� 6 4H; � +65 2 +.-+5�4�6,+ 4�- � � 4�- + = � 2 @�) = � 6 � 4!�3) : +5) � 2 --; 2 C ) ��������� - 2 )�) C4�--@&; � - � +.- 1 � 10)O@ 2 � 2 --@ 2 --; 2 � - ��� 2 � 4!�-789;:GF8HI@DJ;NIPAQDWIWWK X cdU�KMkOo;p�ul`b @DZmBDChK8W=NINIPhL7 W;v�9n9iL k;L;]@ U�? r]r :GVWK �I@=LPK ( +.-0/�13254�6 M 2 6 � 136�3�4 � + � - � 2 ����� @�) = � 6 %6A@�+.-�4�6[E �*+ G 2 6 2 - � + 4�2 )�!1 4 � + � -3)�7X FM@DZ]\MeI:G@DQ��l:G@ r ?AeFKnK;:Gf��lHI@DjZK�W=NINhWIL

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