仿真3数值积分法

1、第三章第三章 连续系统数值积分法连续系统数值积分法：就是利用数值积分方：就是利用数值积分方法对常微分方程建立离散化形式的数学模型（差法对常微分方程建立离散化形式的数学模型（差分方程）并求出数值解。分方程）并求出数值解。 最常用的数值解法有：最常用的数值解法有： 欧拉法、梯形法、欧拉法、梯形法、adams、rungekutta法。法。00)()(,()(xtxtuxtftx,210,nttttt dttuxtftxtxnttn 10)(,()()(01dttuxtftxntt0)(,()(01)(,(nnttdttuxtf)(ntxdttuxtftxtxtt0)(,()()(0连续解为：)(1n

2、txdttuxtfnntt1)(,( 1)(,(1nnttnnnndttuxtfqqxxnttt21dttuxtfnntt1)(,(nx)(ntxnq,21nxxxnnntth1dttuxtftxtxnnttnn 1)(,()()(1（精精确确值值）),(,),(),(21ntxtxtxdttuxtfnntt1)(,(,21,nttt,21,nxxx 21! 2)()()()()(htxhtxtxhtxtxnnnnn00)()(,()(xtxtuxtftx),(1nnnnnuxthfxx htxtxtxnnn)()()(1 huxtftxnnnn),()( 00)()(,()(xtxtuxtf

3、tx1,nntt1)(,()()(1nnttnndttuxtftxtx)(,(tuxtf1,nntt)(,(tuxtfnnnnnnhfxx1huxtftxtxnnnnn),()()(1f(t)tf fn n+ +1 1左矩形近似左矩形近似误差误差t tn nt tn n+ +1 1f fn n1)(,()()(1nnttnndttuxtftxtx)(,(tuxtf1,nntt)(,(111tuxtfnnn11nnnhfxxhuxtftxtxnnnnn),()()(1111nt)(,(tuxtfttdtdxnnnnnnnhfxx1ntnt)(,(nnnnnttuxtfxx1nt)(1ntx),(

4、)()(1nnnnnuxthftxtxnnxt ,1,nntt)(,(1111tuxtfttdtdxnnnn11nnnhfxx1nt1ntnnxt ,)(,(111nnnnnttuxtfxx1nt)(1ntx),()()(1111nnnnnuxthftxtx1)0(, 0)()(2ytyty10 t)(),(),()(22tyytftyty 即即),(111nnnnythfyy)(1.0121nnyy)1.01(11 nnyy1, 000yt9.01.01)1.01(, 1.00011yyyt819.091.09.0)1.01(, 2.01122yyyt4682.0)1.01(, 199101

5、0yyyt),(1nnnnythfyy211.0nnyy525101nnyy),(111nnnnythfyy前向欧拉),(1nnnnythfyy后向欧拉f(t)tf fn n+ +1 1左矩形近似左矩形近似误差误差t tn nt tn n+ +1 1f fn n),(111nnnnythfyy前向欧拉),(1nnnnythfyy后向欧拉),(111nnnnythfyy1)(,(nnttdttuxtf111,21)(,(1nnnnnnttuxtfuxtfhdttuxtfnn112nnnnffhxx梯形近似及其误差梯形近似及其误差f(t)tf fn n+ +1 1左矩形近似左矩形近似误差误差t t

6、n nt tn n+ +1 1f fn n)(,nntxt)(,11nntxt),(),()(2111121211nnnnnnnnuxtfkuxtfkkkhxxr，k，kk21riiirrwkkkkk12211 111222nnnnnnnfhfhxffhxx)(1ntx),(1nnnnpxthfxx1npx1nf1npf),(),(2111npnnnnncxtfxtfhxx1)0(, 0)()(2ytyty10 t)(),(),()(22tyytftyty 即即),(),(2111nnnnnnytfytfhyy)(05.02121nnnyyy1010020121nnnyyy1)0(, 0)()

7、(2ytyty10 t)(),(),()(22tyytftyty 即即2112kkhyynn),(1nnnnnpuythfyy),(1111nnnuytfk),(2nnpnuytfk)(,tuxtf )(,)2(txxxtftxtftxxtf ,00)(),()(xtxxtftxxtf ,),()(xtftx),(,xtfxxtftxtfmmxtx)( pmpmmmmmhtxphtxhtxtxhtxtx)(!1)(! 21)()()()()(21pmmpmmmmmhxtfphxtfhxtftx),(!1),(! 21),()() 1(2)(,),(),()(mpmmtxtxtx ) 1() 1

8、(),(!1),(! 21),(),(,(pmmpmmmmhxtfphxtfxtfhtxthhtxttxxtxmmm),(,()()(11hhtxtxxmm),(,(1)(11mmtxx与11)(mmxtx)2()2() 1() 1()()!2(1)()!1(1pmppmphtxphtxp 21! 2)()()()()(htxhtxtxhtxtxnnnnnnnnhfxx 1htxtxtxnnn)()()(1 huxtftxnnnn),()( 11)(nnxtx 32! 3)(! 2)(htxhtxnn 321! 3)(! 2)()()()()(htxhtxhtxtxhtxtxnnnnnn! 2

9、)()()()(21htxhtxtxtxnnnn htxtxtxhtxnnnn)()()(2)( 11)(nnxtx112nnnnffhxx hhtxtxtxtxhtxnnnnn)()()()(2)(1)()(2)(1nnntxtxhtxxtf ,hhtxtxxmm),(,(1) 1() 1(),(!1),(! 21),(),(,(pmmpmmmmhxtfphxtfxtfhtxtmmxt,),(,(htxtxtf ,),(,(htxthtt, duxftxhtxhtt)(,()()(htt,htttttm 21 mitxtfkiii，321)(, htxt),(, miiikchtxt1),(

10、, mitxtfkiii,1)(,， )(itx )(,),(11111txtftkt )(,)()()(111212txtftttxtx 1121)()(ktttx 222,txtfk 11212)()(,ktttxtf 333,txtfk 22323)()(,ktttxtf 22311213)()()(,kttktttxtf )(,)()()(222323txtftttxtx 2kmmmexhtxx )(1)(,(mmtxt),(amemxaht amx )(,(mmtxtammmexhtxx )(110 a),(1 memxht)(,(htxhtmm ),(1 memxht12111 m

11、ememxbxbxtx(t)tmtm+1x xe em+1m+1x(tx(tm+1m+1) )tm+ahx xe em+am+aamx 1 mexx(tx(tm m) )mmmexhtxx )(1),(amemxaht amx )(,(mmtxtammmexhtxx )(112111 mememxbxbx)(,(1mmtxtfk )(,(12ahktxahtfkmm )()()(2211211kbkbhtxbbxmm 12111 mememxbxbx ammmmxhtxbxhtxb )()(21ammxkxk 21, 22111)()(hktxbhktxbxmmm )()()(221121kb

12、kbhtxbbm )(,(1mmtxtfk )(,(12ahktxahtfkmm )()()(2211211kbkbhtxbbxmm ),(mmxt 12),(ahkxfahtfxtfkmm 2122122222221ahkxfkahxtfahtf 1),(ahkxfahtfxtfmmrrahkxfahtfxtfkmm 12),( )(,)2(txxxtftxtftx ),()(xtftx ),(,xtfxxtftxtfrtahxtxmm )()()2()()()(2211211kbkbhtxbbxmm )()()()()()2(2121rtahxtxbtxbhtxbbmmmm hrbtxah

13、btxhbbtxbbmmm2)2(222121)()()()()( 121 bb212 ab pmpmmmmmhtxphtxhtxtxhtxtx)(!1)(! 21)()()()()(21pmmpmmmmmhxtfphxtfhxtftx),(!1),(! 21),()() 1(2121 bb212 ab )(3ho hrbhtxxtxmmm23)3(11)(! 31)( ab2111 ab212 211 b212 b)(,(1mmtxtfk )(,(12hktxhtfkmm )(2)(211kkhtxxmm 01 b12 b)(,(1mmtxtfk )2)(,2(12khtxhtfkmm 21

14、)(hktxxmm 21 a)(,(1mmtxtfk ),(11 ijjijmimikahxhctfk piiimmkbhxx11 pi,3,2, 1 )(,(1mmtxtfk )2,2(12khxhtfkmm 321146kkkhxxmm )2,2(213hkhkxhtfkmm )(,(1mmtxtfk )3,3(12khxhtfkmm 31134kkhxxmm )32,32(23khxhtfkmm )(,(1mmtxtfk )2,2(12khxhtfkmm 43211226kkkkhxxmm ),(34hkxhtfkmm )2,2(23khxhtfkmm ),()(2111nxxxtftx

15、 ),()(2122nxxxtftx ),()(21nnnxxxtftx ),(mmmxtff mmmhfxx 1 ),(111 mmmxtff11 mmmhfxx ),(mmmxtff 112 mmmmffhxx ),(11mmmmhfxtff ),(111 mmmxtff 112 mmmmffhxx),(mmmxtff )(,(1111mmntxtfkkk )2,2(12122khxhtfkkkmmn 43211226kkkkhxxmm ),(34144hkxhtfkkkmmn )2,2(23133khxhtfkkkmmn )()()(tbutaxtx )(mmtbuax221htbukh

16、xamm222htbukhxammhtbuhkxamm3)(,(1mmtxtfk )2,2(12khxhtfkmm),(34hkxhtfkmm)2,2(23khxhtfkmm1)0(, 0)0(, 0)(2)(5.0)( yytytyty 21xyxy ，令 21215.0210 xxxx得： 01)0()0()0()0()0(21yyxxx初始条件：0 u; 05.0210 ，ba系数阵：）计算所有变量的11k 20015.0210021111axkkk：）计算所有变量的22k21112010102212221 . 02kkxxakhxakkk 95.11.02021.0015.0210：）

17、计算所有变量的33k22122010202313321 . 02kkxxakhxakkk 9612.10975.095.11.021.0015.0210：）计算所有变量的44k2313201030241441 . 0kkxxahkxakkk9214. 11961. 09612. 10975. 01 . 0015 . 0210：）计算第一步的近似值5 432101226kkkkhxx 9214.11961.09612.10975.0295.11.022061.001 1957.00099.1仿真时间结束。的近似值，直到所需及）计算下一步的）转到xkkkk432116xy)0(y1 . 1)(*1

18、 . 0000001 yxyhfyy22. 1)1 . 11 . 0(*1 . 01 . 1112 hfyy362. 1)22. 12 . 0(*1 . 022. 1223 hfyy前前向向欧欧拉拉法法习习题题 23 nyyy,10nfff,10 00)()(,()(xtxtuxtftx 1)(,()()(1mmttmmdttuxtftxtx)()()(2211 mmmmmmftftft，、，、，0122)(atatatp )(,(tuxtf)(tp)(,(tuxtf 11)(,()(mmmmttttdttuxtfdttp0122)(atatatp 212102221212111mmmmmmm

19、mmfffaaatttttt0122)(atatatp 121122121)()()()()( mmmmmmmmmmmmmmfttttttttftttttttttp21221)()( mmmmmmmftttttttthttm httmm 1httmm22 httttmm 1httttmm22 h) 1( h)2( 121122121)()()()()( mmmmmmmmmmmmmmfttttttttftttttttttp21221)()( mmmmmmmftttttttt1 mtt2 mtth) 1( h)2( 221222) 1()2(2)2() 1()( mmmfhhhfhhhfhhhp

20、21) 1(21)2()2)(1(21 mmmfff 22122)(21)2()23(21 mmmfff 11)()(,(mmmmttttdttpdttuxtfhttm 有 httm 10)( dph hddt 有22122)(21)2()23(21)( mmmfffp 10)( dp01)23(21)3()2233(2122312323 mmmfff 21)2131(21) 131()22331(21 mmmfff21125341223 mmmfff 10)( dp21125341223 mmmfff 1)(,()()(1mmttmmdttuxtftxtx )()(1mmtxtx则： 11)

21、()(,(mmmmttttdttpdttuxtf 10)( dph 10)( dph )()(1mmtxtx 21125341223mmmfffh mmxx1 215162312 mmmfffh)(1101kmkmmmmfbfbfbhxx 0113/2-1/2223/12-16/125/12325/24-59/2437/24-9/2441901/720 -2774/720 2616/720 -1274/720 -19/720)(11110111 kmkmmmmmfbfbfbfbhxx0111/21/225/128/12-1/1239/2419/24-5/241/244251/720 646/7

22、20 -264/720 106/720-19/720321,fff)9375955(243211 mmmmmmffffhxx0f)9375955(24012334ffffhxx 321,xxx1)0(, 0)()(2ytyty10 t)(),(),()(22tyytftyty即 32115162312 nnnnnfffhyy)51623(121.03222121 nnnnyyyy1010020121 nnnyyy7683.0)51623(121.00212222 yyyy9087.01010020020 yy1, 000yt 11, 1.0yt8328.01010020121 yy 22, 2

23、.0yt 33, 3.0yt7132.0)51623(121.01222323 yyyy 44, 4.0yt4991.0)51623(121.07282929 yyyy 1010, 1 yt1 mxmt1 k 112110 mkmkkmkmxxxx )(12110 mkkmkmfffh kjkjmjkjkjmjfhx0101 kjkjmjkjkjmjmfhxx011011 1 mx kjkjmjkjkjmjmfhxx011011 mkmkmxxx、21 mkmkmfff、21 )(121101mkkmkmmxxxx )(12110 mkkmkmfffh 0 k 0 k )(121101mkkm

24、kmmxxxx )(12110 mkkmkmfffh 01101321 kkkk，)(1101kmkmmmmfbfbfbhxx )(110111kmkmmmmmfbfbfbfbhxx kjkjmjkjkjmjmfhxx011011 kjjj:0 ，、 1)( mmxhtxr )(,()()(101101kjmkjkjmjkjkjmjmtxtfhtxhtx )()(101 kjmjkjkjmjtxhtx )(1 kjmtx1* kmtt在点)(1 kjmtx1)( mmxhtxr )()()(*)(*) 1(1*0txhcthxctxcrpppkc 2100)(210211kkkc )2(!12

25、1kpppkpc )2()!1(11211kppkp , 3 , 2 p )()()(*)(*) 1(1*0thxcthxctxcrppkjjj:0 ，、 0210 pcccc01 pc)()(2*) 1(11 pppphotxhcr010pccc122，令k)(121101101mmmmmmfffhxxxkc2100)2(!121kpppkpc)2()!1(11211kppkp, 3 , 2p )(210211kkkc)(121101mkkmkmmxxxx)(12110mkkmkmfffh01100c0)(221011c0)2()4(212112c0)4(218612113c为独立参数令0)

26、51()1(8)5(12)1(1111mmmmmmfffhxxx11 )51(1210)1(321)5(1212 )1(! 414c)1317(! 5315c时当104c时当10054cc，)4(31111mmmmmfffhxx)51()1(8)5(12)1(1111mmmmmmfffhxxx时当004c此时，)85(12111mmmmmfffhxx0)5(12124245101，)24(54111mmmmmffhxxx01100c0)(21011c0)4(21112c 0令)1(1)3(211)1(210)1()3(2)1(111mmmmmffhxxx0若取)3(211mmmmffhxxkj

27、kjmjkjkjmjfhx010101112101kkk，）取02110k）取101mkkjkjmjfhx02100kc02211kkkc0)!1()2(!1121kkkkkkkkkkc 1-111101，即通常取k11!10mmmfhxx0100c0111c 11mmmhfxx11-1122/31/3-4/3136/11-2/119/11-18/111412/253/25-16/2536/25-48/2515161k0123451377561371213760147601372001373001373001471014772147225147400147450147360kx kjkjmjk

28、jkjmjmfhxx011011 121kxxx、121kfff、)(12110mkkmkmxxx)(12110 mkkmkmfffh 0 x1)0()(10)(xtxtx，ttexetx1010)0()(mmmhfxx1)10(mmxhxmxh)101 ( 12)101 (mxh01)101 (xhm，发散，则）当11012 . 01hh，等幅振荡；）当2 . 02h，收敛。）当2 . 03h为复常数为复常数，型：型：时，都针对同一试验模时，都针对同一试验模讨论方法的数值稳定性讨论方法的数值稳定性为摆脱这种依赖性，在为摆脱这种依赖性，在，估计其大小很困难。，估计其大小很困难。计算误差太依赖于计算误差太依赖于单步法单步法 xxtxfhtxhxxnnnn ),(),( 1xfxtf ，),(mmmhfxx 1 1、前前向向欧欧拉拉法法mmmmxhxhxxeuler)1( 1