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1、Runge-Kutta2009430Diagonally Implicit Runge-Kutta Methods for StiffProblems with Oscillatory SolutionsCandidateTingZhuSupervisorCollegeProgramSpecialityDegreeUniversityDateProfessor Aiguo XiaoMathematics and Computational ScienceComputational MathematicsNumerical Methods for StiffDifferential Equati
2、onsMaster of ScienceXiangtan UniversityApril 30th, 2009Runge Kutta23Runge Kutta33Runge Kutta34Runge KuttaRunge KuttaARunge Kutta44AP IAbstractStiffoscillatory problems are involved in various fields of modern science andtechnology. The research on their numerical methods has wide prospect in appli-c
3、ations. Due to its two-sided characteristics, namely, stiffness and oscillation, it israther difficult and challenging to obtain highly-efficient numerical methods, whichare attracting many scholars attention for a long time.This paper mainly consider diagonally implicit Runge-Kutta methods for solv
4、-ing stiffproblems with oscillating solutions based on other scholars research.Wemake our methods more effective by improving the algebraic order, stability condi-tions of the methods, and controlling the phase error and amplification error. Thispaper contains five parts.In chapter 1, we introduce t
5、he background of research and main work of thispaper.In chapter 2, we introduce the considered problem and the RK methods forsolving it, such as two-stage diagonally-implicit Runge-Kutta methods of orderthree, three-stage diagonally-implicit Runge-Kutta methods with an explicit firststage of order t
6、hree, three-stage symmetric diagonally-implicit Runge-Kutta meth-ods of order four and four-stage symmetric diagonally-implicit Runge-Kutta meth-ods with an explicit first stage of order four.In chapter 3, we analyze the scope which the coefficients of the methods shouldsatisfy when the methods are
7、A-stable.In chapter 4, we study the phase error and amplification error of the methodsto make the order of them as high as possible.In chapter 5, we take our methods to numerical experiments. It is shown thatthe constructed methods are efficient.Key Words: Stiffoscillatory problems; Diagonally impli
8、cit Runge-Kutta meth-ods; A-stability; Phase error; Amplification error; P-stabilityII. 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32.12.22.3Runge-KuttaRunge-KuttaRunge-Kutta.4.5.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9、 . . . . . . . . . . . . . . . . . . . . . . .12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24. . . . . . . . . . . . . . . . . . . . . . . . . . .
10、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11、. . . . . . . . . . . . . . . . . . . . . 33III2418Runge Kutta(RK) 1sRunge KuttasmRunge KuttaRKsm31RKRunge KuttaRunge KuttaGearRK20902 8Runge Kutta11BrusaNigro19801987HouwenSommeijer10Runge Kutta Nystrom1989HouwenSommeijer113Runge KuttaSommeijer12Runge Kutta NystromHouwen1997F ranco1311Runge Kutta13
12、Runge KuttaRunge KuttaRunge KuttaRunge Kutta 2213RungeKutta2133134Runge KuttaRunge KuttaRunge KuttaRunge KuttaRunge Kutta24Runge KuttaA y(t) = f(t, y(t), t a, b, Y iRunge Kutta. y(a) = , Rm,(2.1)Rmfmf yf : D = a, b Rm Rm i jRei 0,|Im(i)/Rej| 1,(2.1) = yn + h yn+1 = yn + h(2.1) ss aijf(tn + cjh, Yj),
13、 j=1 s bjf(tn + cjh, Yj), j=1(2.2)s h 0aij, bjcjci = cj, i = jaij = ci, j=1 i = 1, 2, . . . , s, yn(2.1)y(t)tn = a + nhYi y(tn+cih), i = 1, 2, . . . , ss(2.2)Butcherc1a11a12.a1scAbT=c2.csa21.as1a22.as2.a2s. ,ass(2.3)b1b2.bsc = (c1, c2, . . . , cs)T , b = (b1, b2, . . . , bs)T , A = (aij) Rss RKRunge
14、 Kuttaaij = 0(j i), i, j = 1, 2, . . . , s,DIRKDIRKRunge Kuttaaij = 0(j i), i, j = 1, 2, . . . , s,a11 = a22 = = ass = 0,SDIRKRunge Kuttaa11 = 0, a22 = a33 = = ass = 0,aij = 0(j i), i, j = 1, 2, . . . , s,ESDIRK3 Y1 = yn, y n+1 = yn + h2c21a1j = 0, c1 = 0(2.3)sRKs Yi = yn + h aijf(tn + cjh, Yj), i =
15、 2, . . . , s, j=1 s bjf(tn + cjh, Yj), j=1s 1RK2.1Runge-KuttaRunge KuttacAbTc1= c2a11a21b10a22 .b2(2.4)(2.3)B() : kbT ck1 = 1, k = 1, 2, . . . , ;C() : kAck1 = ck, k = 1, 2, . . . , ,ck = (ck1, ck2, . . . , cks)TqRunge Kutta (2.4)(2.3)22B(q), C(q)B(2) C(2)b1 + b2 = 1, b1c1 + b2c2 = 12,a11 = c1,a21 + a22 = c2,2a11c1 = c21, 2a21c1 + 2a22c2 = c22,a11 = c1 = 0, a21 = a22 =c2 2,b2 =12c2, b1 =2c2 1 2c2, c2 = 0.Runge Kutta00
