1、2009530A New Class of Ecient Numerical Methodsfor Nonlinear Sti Dierential EquationsCandidate Li LinhaiSupervisor Professor. Huang YunqingCollegeProgramSpecialityDegreeUniversityDateMathematics and Computational ScienceComputational MathematicsNumerical Methods for Dierential EquationsMaster of Scie
2、nceXiang Tan UniversityMay 30, 2009B- EBDFNew BDF NBDF k NBDF kB-kkEBDF k 1kB-EBDF k = 2, 3, ., 8NBDFEBDFNBDFEBDFNBDFB-B- EBDFINBDFAbstractBased on B-theory of numerical methods for nonlinear sti dierential equations,we modify the existing EBDF methods, and construct a new class of ecient numericalm
3、ethods which are known as New BDF methods with abbreviation NBDFs. It is provedthat the k step NBDF method is B-consistent of order k and convergent of order k in theclassical sense, and has the same perfect numerical stability properties as the k-th orderEBDF method, where k = 2, 3, , 8. However, t
4、he k-th order EBDF method has anessential drawback that its Bconsistency order is one less than the convergence order.Fortunately, this drawback is overcome by our New BDF methods constructed.Theoretical analysis and numerical experiments show that for solving nonlinear stiproblems the computational
5、 accuracy and eciency of an NBDF method is usually muchhigher than the corresponding EBDF method of the same order, the latter usually causesorder reduction, but the former usually has observed order closely near to its B-consistencyorder. Therefore, we can conclude that the NBDF methods constructed
6、 in the presentpaper are of importance in practice.Key Words:Nonlinear sti dieretial equations; B-theory; EBDF methods;NBDF methods; accuracy and stability analysisII. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7、. . . . . . . . . . (I)Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (II). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8、. . . . . . . . . . (1)NBDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(7). . . . . . . . . . . . . . . . . . . . . . . . . .
9、. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10、. . . . . . . . . . . . . . . . . . (17). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(18). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11、 . . . . . . . . . . . . . . . . . . . . . . (20)IIIy (t) = f(t, y(t),(Backward Dierentiation Formula, BDF)(kj=1jyn+kj1)= hf(tn+k, yn+k) (1.1)(y(a) = ,a t b, Rmf : a, b Rm Rm(1.2)y Lipschitz y(t) a t b)DIFSUB( 2) LSODE( 3)BDFBDF 3-6BDF6 BDFA()- BDFBDFEnrightGaussian Runge-Kutta ( 4)8)Radau IIA Runge
12、-Kutta(9-14)( 5-7)BDFEnright1980 Cash (Extended Backward DierentiationFormulae, EBDF( 15) 2-4 A- 5-9 A()-BDFBDFEBDFEBDFGaussianRunge-kutta1983 CashRadau IIA Runge-kutta EnrightEBDFEBDF Modied Extended Backward Dierentiation FormulaeMEBDF( 16)1j yn+j = hk fn+k + hk+1fn+k+1k = 1j (j = 0, , k 1) k k+1
13、1 28 4 17 279 18 728 4008 144 394 96000 26550 600 5756 46800 77940 1200 7545 1189475 1324470 14700- 1120080 14471072 35354480 28187040 235200 112 137 1089 34 16 147- 15981 18 200 1089 48 400 65856 300 1089 147 109760 1089 47040Cash BDFk k +1 EBDFkj=0(k) (k) (k) (1.3)k = 1, 2, ., 8 h 0 yn+j (1.2)y(t)
14、 tn+j y(tn+j) tn+j = tn + jh, tn = a + nh, fn+j = f(tn+j, yn+j)y(t) tn+j y(tn+j) (k)(k) (k)1.1,(k)k k +1 EBDF( 1) 1.1k (k)0 (k)1 (k)2 (k)3 (k)4 (k)5 (k)6 (k)7 (k)8 (k)k (k)k+11 -1 1 32 22 523 231 2223 23319799197 1971 150197 1974 1112501 250121242501 25011 16442501 2501514919292514919 14919187001491
15、9 149191 882014919 149196 69039981 399812137539981 399816845039981 399811 2178039981 39981762670970070626709292334626709723975626709 6267091393070626709 6267091 319620626709 6267098 10930512403947 12403947520184012403947 124039472688630012403947 124039473453128012403947 124039471 598836012403947 124
16、03947(1.3)kk kj=0jyn+j = hkf(tn+k, yn+k) (1.4)j(j = 0, 1, ., k) k k = 1j (j = 0, 1, , k 1) k1.2, k kBDFBDF(1.1) 1.2k 0 1 2 3 4 5 6 7 8 k1 -1 1 1234567813325121014760735159819112575137724901089672011136251372251471764274401598112530013714736751089159811137450147490010290015981136044101089159811294082
17、32015981115981 1236111225601376014742010895880159812yn, yn+1, ., yn+k1 k EBDF(1)h 0yn, yn+1, ., yn+k1 k kyn+k yn+k+1 ,(1.4)fn+k+1 =f(tn+k+1, yn+k+1)(2) fn+k+1 (1.3) fn+k+1 yn, yn+1, ., yn+k1hk(1.3)EBDFyn+k,k +1k +1 MEBDF EBFDkB-k +1 EBDFk,(k = 1, , 8)k +1B-B-( 1,16,21-23)k +1EBDFkk k = 2, 3, , 8EBDF
18、k + 1,New BDF NBDF k k NBDFk kk k NBDFk kNBDFB-Bk BNBDFkk = 2, 3, , 8A-k NBDFNBDFA()-kEBDFB-NBDFEBDF3EBDFNBDFNBDFEBDF EBDFk k h 0 yn+j(j = 0, 1, ., k)k1jyn+j + hkf(tn+k, yn+k),j=0k2n+k+1 = jyn+j+1 k1 n+k + hkf(tn+k+1, n+k+1),j=1k1jyn+j + hkf(tn+k, yn+k) + hk+1f(tn+k+1, yn+k+1),j=0yn+k(2.1a)(2.1b)(2.
19、1c)yn+k+1 Rm(1.2) y(t) tn+j tn+k tn+k+1 y(tn+j) y(tn+k)y(tn+k+1) tn+j = tn + jh tn = a + nhk 1j, j, j, k, k, k, k+1k+1-33(2.1)( )k k1 k1 k1 j = 1,j=0 j=0 j=0k1jjp + kpkp1 = kp, p = 1, 2, ., k,j=0k1j(j + 1)p + kp(k + 1)p1 = (k + 1)p, p = 1, 2, ., k,j=0k1j=0 jjp + kpkp1 = kp k+1p(k + 1)p1, p = 1, 2, ., k,(2.2) 0 (2.1) (2.2)(2.2)1 k+1 k k (2.1)(n) = yn+k, (n) = yn+k+1, (n) = yn+k,(n)T (n)T (n)T T (2.1)-(2.2) 3 k
