Second Derivative Generalized Backward Differentiation Formulae for Solving Stiff Problems

Transcription

1 IAENG International Journal of Applied Mathematics, 48:, IJAM_48 Second Derivative Generalized Bacward Differentiation Formulae for Solving Stiff Problems G C Nwachuwu,TOor Abstract Second derivative generalized bacward differentiation formulae (SDGBDF) are developed herein and applied as boundary value methods (BVMs) to solve stiff initial value problems (IVPs) in ordinary differential equations (ODEs) The order, error constant, zero stability and the region of absolute stability for the SDGBDF are discussed The methods are A v, v -stable and v, v -stable with (v,v)-boundary conditions for values of the steplength, v<with order p = + Keywords: Linear Multistep Formulae, Boundary Value Methods, A v, v -stable AMS subject classification: 65L4, 65L5 Introduction The mathematical modeling in science and engineering problems often leads to systems of ordinary differential equations (ODEs) and many of these problems appear to be stiff A potentially good numerical method for the solutionofstiffsystemsofodesmusthavegoodaccuracy and some reasonably wide region of absolute stability, see [3] It is on this ground A-stable (stiffly-stable) methods are required Consider the initial value problem (IVP) y = f(x, y), x [t,t], y(x )=y () Definition (cf: Lambert [37]) The linear system where y = Ay + φ(x), y(a) =η, a x b () y =(y,y,y s ) and η =(η,η,,η s ) is said to be stiff if (i) Re(λ i ) <, i =,,,s (ii)max Re(λ i ) >> Min Re(λ i ) Corresponding Author: Advanced Research Laboratory (AR- LAB), Department of Mathematics, University of Benin, Benin City, Nigeria, gracenwachuwu@unibenedu ARLAB, Department of Mathematics, University of Benin, Benin City, Nigeria, freetega@gmailcom where λ i are the eigenvalues of s s matrix A, and the stiff ratio is Max Re(λi) Min Re(λ i) Definition ([3]) A numerical integrator is said to be A-stable if its region of absolute stability R incorporates the entire left half of the complex plane denoted C, ie, R = {z C Re(z) } (3) Bacward differentiation formulae (BDF) α j y n+j = hf n+ (4) are among the first most popular numerical methods to be proposed for stiff initial value problems (IVPs), see [, 9] These methods are found to be A-stable up to order p = with order p = and A(α) stable for = 3()6 In [9] is introduced a class of extended bacward differentiation formulae (EBDF) α j y n+j = hβ f n+ + hβ + f n++ (5) which has some advantage over the usual BDF It is found to be A-stable for = ()3 and A(α) stable for = 4()8 with order p = + The modified extended bacward differentiation formulae (MEBDF) by [] is { yn+ h(β v )f n+ = α (6) jy n+j + hv f n+ + hβ + f n++ where the choice of v canbedefinedtomaximizeinsome way the region of absolute stability of (6) The MEBDF is A-stable for = ()3 and A(α) stable for =4()8 with order p = + The second derivative extended bacward differentiation formulae (SDEBDF) by [] are of two classes in predictor-corrector pair They are A- stable for = ()3 and A(α) stable for = 4()8 with order p = + and p + 3 for class and p = +3 for class Other authors such as [34, 6, 4, 5] have also presented some modifications of the BDF That of [8] and [9] considered the BDF (4) as boundary value methods (Advance online publication: February 8)

2 IAENG International Journal of Applied Mathematics, 48:, IJAM_48 (BVMs) and obtained the generalized bacward differentiation formulae (GBDF) with better stability properties than the BDF This class of methods is α j y n+j = hf n+v (7) where { + v = for even + for odd It is v, v -stable and A v, v -stable for all with (v, v)-boundary conditions and order p = Asurvey of some BVMs can be found in the literatures [, 3, 4, 5, 6, 7, 33,,,, 3, 4, 3, 4] In this paper second derivative generalized bacward differentiation formulae (SDGBDF) shall be derived This class of methods is an extentionofgbdfproposedby[8]and[9] Themethods developed which are also applied as BVMs in the sense of [8] and [9] have improved order properties compared to the GBDF with respect to the steplength and are suited for the solution of stiff IVPs in ODEs () The paper is organized as follows In section we recall the main facts about BVMs The stability of BVMs is discussed in section 3 Section 4 deals with the second derivative BVMs, the derivation and stability Section 5 is devoted to the computational aspect for the implementation of the proposed class of methods to demonstrate how the class of methods are applied as BVMs Numerical experiments are carried out in section 6 Finally, in section7theconclusionofthepaperisgiven Boundary Value Methods (BVMs) To obtain the numerical solution of () it is usual to use a -step linear multistep formula (LMF), α j y n+j = h β j f n+j (8) where y n denotes the discrete approximation of the solution y(x n ) at x = x n and h = (T t )/N and f n = f(x n,y n ) If and are two integers such that + = then one may impose the conditions for the LMF (8) by fixing the first ( ) values of the discrete solution y,y,,y and the last = values y N +,,y N yielding the discrete problem α i+ y n+i = h β i+ f n+i,n=,,n, i= i= y,y,,y, y N +,,y N fixed (9) In this case the given continuous initial value problem () is approximated by means of a discrete boundary value problem The resulting methods are BVMs with (, )-boundary conditions Observe that for = and therefore =, one has the initial value methods (IVMs) So the class of IVMs is a subclass of BVMs for ODEs based on LMF [7] The continuous problem () provides only the initial value y In the sense of [7], to implement (9) as a BVM, the additional values y,,y, y N +,,y N are obtained by introducing a set of additional equations which are derived by a set of additional initial methods α (j) i y i = h β (j) i f i, j =,, () and final methods α (j) i y N i = h β (j) i f N i, () j = N +,,N The equations (9), () and () form a composite scheme assumed to be of the same order where () and () are the most suitable set of additional methods The discrete problem generated by a -step BVM with (, )- boundary conditions can be put in matrix form as A N y hb N f = (α iy i hβ i f i ) α y hβ f α y N hβ f N i= (α +iy N +i hβ +if N +i ) () where A N and B N are (N +) (N +)matricesgiven in (39) and (4) respectively, y =(y,,y N ) T is the discrete solution, f =(f,,f N ) T and h is the step size The matrix A N qb N,whereq = hλ, has a bloc quasi- Toeplitz structure which is as a result of the additional methods () and () in A N and B N as given in () 3 Stability of BVMs In order to characterize the stability of the family of methods to be considered the definitions of zero-stability and absolute stability for LMM (8) are generalized to BVM by introducing the following two inds of polynomials [6, 8]: Definition 3 Consider a polynomial p(z) such that p is a function of a complex variable z, calculatedbythe formula: p(z) = α j z j = α z +α z ++α (α ) (3) (Advance online publication: February 8)

3 IAENG International Journal of Applied Mathematics, 48:, IJAM_48 The zeros of the polynomial p(z) are denoted by z i,i =,,Ifthezerosz i are simple for all values of i their multiplicities are equal to one The polynomial p(z) is called the Schur polynomial if for all values of i =,, the condition z i < is satisfied The polynomial p(z) is called the Von Neumann polynomial if for all values of i =,, the condition z i is satisfied ([37]) Definition 4 Apolynomialp(z) of degree = + is a S -polynomial if its roots are such that z z z < < z + z and it is a N - polynomial if z z z < z + z being simple the roots of unit modulus Observe that for = and = a N - polynomial reduces to a Von Neumann polynomial and a S -polynomial reduces to a Schur polynomial Let ρ(z) = α jz j and σ(z) = β jz j denote the two characteristic polynomials associated with the LMM (8) Thus (z,q) =ρ(z) qσ(z),q = hλ, is the stability polynomial when (8) is applied on y = λy, Re(λ) < Then we have the following definitions (see [6, 7]): Definition 5 ABVMwith(, )-boundary conditions is O -stable if ρ(z) is a N - polynomial Observe that O -stability reduces to the usual zerostability from Definition 5 for LMM when = and = Definition 6 (a) For a giving q C, abvmwith (, )-boundary conditions is (, )-absolutely stable if (z,q) is a S -polynomial Again, (, )-absolute stability reduces to the usual notion of absolute stability when = and =for LMM (b) Similarly, one defines the region of (, )-absolute stability of the method as D = {q C : (z,q) is a S -pololynomial} Here (z,q) is a polynomial of type (,, ) (c) A BVM with (, )-boundary conditions is said to be A -stable if C D 4 Second Derivative BVMs, Derivation and Stability The second derivative bacward differentiation formulae (SDBDF) are based on the second derivative linear multistep formula (SDLMF) and can be defined generally as: α j y n+j = hβ f n+ + h γ f n+ (4) The conventional SDBDF provides -stable methods up to = 8 and are -unstable for 9withanorder p = + Following the idea of Brugnano and Trigiante [6,7,8,9],werewrite(4)as: α j y n+j = hβ i f n+i + h f n+i (5) where i =() and γ hasbeennormalizedto The choice i = is widely used to derive the conventional SDBDF as IVMs When (5) is used as BVMs with i, we gain the freedom of choosing the values of i which provide methods having the best stability properties for all values of In fact this is the case if i = v such that { + v = for even + (6) for odd Consequently (5) becomes α j y n+j = hβ v f n+v + h f n+v (7) where the + parameters allow the construction of methods of maximal order p = + The class of methods (7) called second derivative generalized bacward differentiation formulae (SDGBDF) must be used as BVMs with (v, v) boundary conditions These methods are found to be v, v -stable and A v, v -stable for all where v is the number of roots inside the unit circle and v are the number of roots outside the unit circle We rewrite the formula (7) as: α j y(x + jh)=hβ v y (x + vh)+h y (x + vh) (8) Expanding (8) in Taylor series and applying the method of undetermined coefficients yields a system of linear equations for the coefficients α j and β v (see [38] and [8]) ThesecoefficientsareinTables7and8for = () According to [37] and [7] the local truncation error associated with (7) is the linear difference operator L[y(x); h] = α jy(x + jh) hβ v y (x + vh) h y (x + vh) (9) Assuming that y(x) is sufficiently differentiable, we can find the Taylor series expansion of the terms in (9) about the point x to obtain the expression where L[y(x); h] = C y(x)+c hy (x)+ + C q h q y (q) (x)+ C = α j, C = jα j + β v,, j= () (Advance online publication: February 8)

4 IAENG International Journal of Applied Mathematics, 48:, IJAM_48 C q = j= j q α j q! We say that (7) has order p if + β vv q (q )! vq (q )! C j =, j =()p and C p+, () see [3] The C p+ is the error constant and C p+ h p+ y p+ (x) is the principle local truncation error at the point x The order equations () is equivalent to α 3 4 α 3 4 v α v = 6v α q 3 q 4 q q qv q β v q(q )v q () The order and the error constant of the SDGBDF (7) are shown in tables 7 and 8 for = () For the numerical solution of () the second derivative BVMs with (, )-boundary conditions, the main method α i+ y n+i = h β i+ f n+i +h λ i+ f n+i, i= i= i= n =,,N y,,y, y N,,y N+ fixed together with additional initial methods α (j) i y i = h β (j) i f i + h j =,, (3) λ (j) i f i, (4) and final methods { α(j) i y N i = h β(j) i f N i +h λ(j) i f N i, j = N,,N + can be expressed as A N y hb N f h C N f = (α iy n+i hβ i f n+i h λ i f n+i ) α y n+ hβ f n+ h λ f n+ α y n+n hβ f n+n h λ f n+n i= (α +iy n+n +i hβ +i f n+n +i h λ +if n+n +i ) (5) (6) where A N and B N are defined similarly as in (), while C N is given in (4), y = (y,,y N ) T is the discrete solution, f and f are the first and second derivatives respectively, h is the step size and A N,B N and C N are (N +) (N + ) matrices with the same structure as those in () To analyze the stability of a specific method (6), (see, [3]) we apply (3) on the test problem y = λy, y = λ y (7) to determine its boundary locus The class of methods (7) yields the characteristics equation: α j z j (qβ v + q )z v =, q = λh, q C (8) where v is defined as in (6) Letting z = e iθ we obtain two roots (since (8) is quadratic in q) for corresponding values of and v to give the stability regions defined by q given in Figures and for odd and even values of respectively Compared with the generalization of the BDF by Brugnano and Trigiante [8] discussed in section the proposed class of methods (7) are found to have higher order p = + and a smaller error constant for corresponding values of the steplength, although with need to compute the second derivative for which it is not expensive for some autonomous stiff systems 5 Implementation Procedure In this section the implementation procedure for the SDGBDF(7) of order 4 and 5 as BVMs in the sense of Brugnano and Trigiante [7, 8] is presented The proposed class of methods (7) is used with the following additional initial methods: αj y j = hβ i f i + h f i, i =,,,v (9) and final methods: αj y j = hβ i f i + h f i, i = v +,,N (3) The SDGBDF (7) of order 4 which is A, -stable and, -stable with (, )-boundary conditions requires two initial methods (y is already provided by the initial value defining the ODE ()) and one final method The fourth order SDGBDF (7) is given as: 6 y n+y n+ 5 y n++ 3 y n+3 = hf n+ +h f n+ (3) Themainmethod(3)canbewrittenintheform 6 y n +y n 5 y n + 3 y n+ = hf n + h f n (3) n =,,N (Advance online publication: February 8)

5 IAENG International Journal of Applied Mathematics, 48:, IJAM_48 and used with the following initial method and final method 3 y 5 y +y 6 y 3 = hf + h f (33) 9 y N 3 3 y N +6y N 85 8 y N = 3 hf N +h f N (34) Similarly the SDGBDF (7) of order 5 { 8 y n y n+ +3y n y n+3 + y n+4 = 5 3 hf n+3 + h f n+3 n =3,,N with the initial methods and (35) y 55 8 y +3y y y 4 = 5 3 hf + h f (36) y y 5 y y 3 y 4 =h f (37) and the final method { 8 y N y N 3 3y N +8y N 45 7 y N = 5 6 hf N + h f N will be taen together as a BVM (38) The methods (3 and 35) are implemented as BVMs efficiently by composing the main methods and the additional methods as simultaneous numerical integrators for the IVP() In particular for linear problems, we can solve () directly from the start with Gaussian elimination partially using pivoting and for nonlinear problems we can use a modified Newton-Raphson method In each case, the main methods and the additional methods are combined as BVMs to give a single matrix of finite difference equations which simultaneously provides the values of the solution and the first derivatives generated by the sequences {y n }, {y n},n =,,N,wherethesingle bloc matrix equation is solved while adjusting for boundary conditions ([35]) 6 Numerical Experiments with SDGBDF We consider the following stiff problems (linear and nonlinear) to illustrate the implementation and examine the accuracy of the SDGBDF (7) of orders p=4 (3) and 5 (35) implemented as bloc methods The fourth order method (7) and the fifth order method (7) are denoted by SDGBDF4 and SDGBDF5 respectively Problem : A linear stiff problem, see [4,, 5, 43] y = y +9y y 3, y () =, y =9y y +y 3, y () =, y 3 =4y 4y +4y 3, y 3 () = The SDGBDF4 is applied to this problem and the maximum absolute errors ( y(x) y n ) in the interval < x < are compared with the Adams type bloc method of Ainfenwa [4] (ATBM7), the generalized bacward differentiation formula of Brugnano and Trigiante [](GBDF8) and the L(α)-stable bloc multistep method of Ehigie and Ounuga [5] denoted by EH- OK5 The rate of convergence, Rate h = log ( errh err h ) where err h is the maximum absolute error at steplength h = n,n=,,, 3 and 4, is used to verify the order ofthemethods AlsoincomparisonwiththeCBDF 5 of degree s = 5 in Ramos and Garcia-Rubio [43], the AbsErr(t f ) is obtained by the SDGBDF4 in the interval x It is observed that the new method even though it is of order 4 performs better than the EH-OK5, theatbm7andthegbdf8oforders5,7and8respectively The details of the numerical results are displayed in Table In Table, it is noticed that the SDGBDF4 is comparable with the EH-OK5 in [5] and the CBDF 5 in [43] Table : Maximum absolute error, Max <i<n y i (x) y i,n for problem, h = n n SDGBDF4 EH-OK5 GBDF8 ATBM7 (Rate) (Rate) (Rate) (Rate) (387) (499) (64) (78) (393) (499) (7) (76) (44) (5) (664) (836) (64) (5) (684) (795) Table : Numerical results in comparison with CBDF 5 and EH-OK5 for problem the interval x Steps SDGBDF4 EH-OK5 CBDF 5 (Rate) (Rate) (Rate) (4) (496) (495) (397) (543) (495) (493) (453) (75) Problem : Non-linear stiff system solved by Wu and Xia [44] y = y + y, y () =, y = y y ( + y ), y () =, (Advance online publication: February 8)

6 IAENG International Journal of Applied Mathematics, 48:, IJAM_48 The exact solution of the system is given by y (x) =e x, y (x) =e x Problem is solved using SDGBDF5 with a steplength h =8 in the range x The maximum error (Max y i y(x i ) ) of the method is given in Table 3 From Table 3, it is obvious that the new method of order 5 is better than the methods of Ehigie et al [4] and Wu- Xia [44] and very comparable with the method of Jator and Sahi [36] which are of orders 5, 6 and 8 respectively Problem 3: Singularly Perturbed Problem Table 3: Maximum error, Max y i y(x i ), for problem Method N h y y (Max y i y(x i ) ) (Max y i y(x i ) ) SDGBDF Ehigie et al (BVM3) Jator-Sahi Wu-Xia y = ( + 4 )y + 4 y, y = y y y, y () =, y () = The exact solution is y = e t, y = e t, see [3] The SDGBDF4 and the SDGBDF5 are applied to Problem 3 and the results are compared with the theoretical solution The SDGBDF5 performs better than the SDGBDF4 as expected, see Table 4 Furthermore, tae note that the graphs of the exact and the numerical solutions in Figures 3 and 4 coincide Table 4: Absolute error in problem 3, h =, Error y i = y i y(x i ), i =, x y i Error in SDGBDF4 Error in SDGBDF5 y y y y y y y y y y y y y y y y y y y y Problem 4: Van der Pol equations, see [3] (nonlinear problem) y = y, y = y +y ( y ), y () =, y () = For problem 4, it is clearly seen from Table 5 and Figures 5 and 6 that the proposed class of methods compares favorably with the solution from the Ode5s in MATLAB Problem 5: Robertson s equation, see [3] (nonlinear Table 5: Errors in problem 4 using the modulus of the solution of SDGBDF minus the solution of Ode5s, h = Error y i = y isdgbdf y iode5s, i =, x y i Error in SDGBDF4 Error in SDGBDF5 y y y y y y y y y y problem) y = 4y + 4 y y 3, y =4y 4 y y y, y 3 =3 7 y, y () =, y () =, y 3 () = Problem 5 is solved by the SDGBDF4 and the SDGBDF5 and the results are compared with the solution from the Ode5s in MATLAB It is observed from Table 6 and Figures 7 and 8 that the new methods are very comparable with the Ode5s in MATLAB Table 6: Errors in problem 5 using the modulus of the solution of SDGBDF minus the solution of Ode5s, h = Error y i = y isdgbdf y iode5s, i =,, 3 x y i Error in SDGBDF4 Error in SDGBDF5 y y y y y y y y y y y y y y y (Advance online publication: February 8)

7 IAENG International Journal of Applied Mathematics, 48:, IJAM_48 Table 7: The Coefficients, Error Constant (EC) and Order p of SDGBDF(8) for =() v α α α α 3 α 4 α 5 α Table 8: Table 7 continued v α 7 α 8 α 9 α β v γ v EC p A N = α () α () α () α () 3 α () α ( ) α ( ) α ( ) α ( ) 3 α ( ) α α α α 3 α α α α α α α α (N α +) α (N +) α (N) α (N) (39) (Advance online publication: February 8)

8 IAENG International Journal of Applied Mathematics, 48:, IJAM_48 B N = β () β () β () β () 3 β () β ( ) β ( ) β ( ) β ( ) 3 β ( ) β β β β 3 β β β β β β β β β (N +) β (N +) β (N) β (N) (4) C N = λ () λ () λ () λ () 3 λ () λ ( ) λ ( ) λ ( ) λ ( ) 3 λ ( ) λ λ λ λ 3 λ λ λ λ λ λ λ β λ (N) λ (N) λ (N+ ) λ (N+ ) (4) (Advance online publication: February 8)

9 IAENG International Journal of Applied Mathematics, 48:, IJAM_48 Figure : Stability region (exterior of closed curves) of (7), = () 3 (Advance online publication: February 8)

10 IAENG International Journal of Applied Mathematics, 48:, IJAM_48 Figure : Stability region (exterior of closed curves) of (7), = () 3 (Advance online publication: February 8)

11 IAENG International Journal of Applied Mathematics, 48:, IJAM_48 Figure 3: Numerical Results for Problem 3 with SDGBDF4, h= Figure 4: Numerical Results for Problem 3 with SDGBDF5, h= (Advance online publication: February 8)

12 IAENG International Journal of Applied Mathematics, 48:, IJAM_48 Figure 5: Numerical Results for Problem 4 with SDGBDF4, h= Figure 6: Numerical Results for Problem 4 with SDGBDF5, h= (Advance online publication: February 8)

13 IAENG International Journal of Applied Mathematics, 48:, IJAM_48 Figure 7: Numerical Results for Problem 5 with SDGBDF4, h= Figure 8: Numerical Results for Problem 5 with SDGBDF5, h= (Advance online publication: February 8)

14 IAENG International Journal of Applied Mathematics, 48:, IJAM_48 7 Conclusion The second derivative generalized bacward differentiation formulae (SDGBDF) have been introduced in section (4) This class of methods is A v, v -stable and v, v - stable with (v, -v)-boundary conditions for values of withorderp = + The new class of methods is found to be suitable for the solution of stiff IVPs in ODEs for reason of their stability The class of methods (7) also finds application in the solution of boundary value problems, see [35, 39] Acnowledgement The authors are grateful to Prof MNO Ihile (Director of ARLAB) and the anonymous reviewers for comments that have improved the quality of this wor References [] L Aceto, P Ghelardoni and C Magherini, Boundary Value Methods as an extension of Numerovs method for Sturm-Liouville eigenvalue estimates, Appl Numer Math, 59, (9), [] L Aceto and C Magherini, On the relations between BVMs and Runge-utta collocation methods, Journal of Computational and Applied Mathematics, 3, (9), -3 [3] L Aceto, P Ghelardoni and C Magherini, PGSCM: A family of P-stable Boundary Value Methods for second-order initial value problems, Journal of Computational and Applied Mathematics, 36, (), [4] O A Ainfenwa, Seven step Adams Type Bloc Method with Continuous Coefficient for Periodic Ordinary Differential Equation, Word Academy of Science, Engineering and Technology, 74, (), [5] B I Ainnuawe, O A Ainfenwa and S A Ounuga, L-Stable Bloc Bacward Differentiation Formula for Parabolic Partial Differential Equations, Ain Shams Engineering Journal, 7, (6), [6] L Brugnano and D Trigiante, Convergence and stability of boundary value methods for ordinary differential equations, Journal of Computational and Applied Mathematics, 66, (996), 97-9 [7] L Brugnano, Boundary Value Methods for the Numerical Approximation of Ordinary Differential Equations, Lecture Notes in Comput Sci, 96, (997), [8] L Brugnano and D Trigiante, Solving Differential Problems by Multistep Initial and Boundary Value Methods, Gordon and Breach Science Publishers, Amsterdam, 998a [9] L Brugnano and D Trigiante, Boundary Value Methods: the third way between linear multistep and Runge-Kutta methods, Comput Math Appl, 36, (-), (998b), [] L Brugnano and D Trigiante, Bloc Implicit methods for ODEs in: D Trigiante (Ed), Recent trends in Numerical Analysis, New Yor: Nova Science Publ inc, [] L Brugnano and C Magherini, Blended Implicit Methods for solving ODE and DAE problems and their extension for second order problems, Jour Comput Appl Mathematics, 5, (7), [] L Brugnano, F Iavernaro and T Susca, Hamiltonian BVMs (HBVMs): implementation details and applications, AIP Conf Proc, 68, (9), [3] L Brugnano, F Iavernaro and D Trigiante, Hamiltonian BVMs (HBVMs): a family of drift-free methods for integrating polynomial Hamiltonian systems, AIP Conf Proc, 68, (9), [4] L Brugnano, F Iavernaro and D Trigiante, Hamiltonian boundary value methods (Energy preserving discrete line integral methods), Journal of Numerical Analysis, Industrial and Applied Mathematics, 5, -, (), 7-37 [5] L Brugnano, F Iavernaro and D Trigiante, A note on the efficient implementation of Hamiltonian BVMs Journal of computational and Applied Mathematics, 36, (), [6] L Brugnano, G Frasca Caccia and F Iavernaro, Efficient implementation of Gauss collocation and Hamiltonian Boundary Value Methods Numer Algor, 65, (4), [7] L Brugnano, F Iavernaro and D Trigiante, Analisys of Hamiltonian Boundary Value Methods (HB- VMs): a class of energy preserving Runge-Kutta methods for the numerical solution of polynomial Hamiltonian systems, Communications in Nonlinear Science and Numerical Simulation,, 3, (5), [8] R L Burden and J D Faires, Numerical Analysis (9th Edition), Broos/Cole, Cengage Learning, Boston, [9] J R Cash, On integration of stiff systems of ordinary differential equations using extended bacward differentiation formula, Numer Math, 3, (98), [] J R Cash, Second Derivative Extended Bacward Differentiation Formulas for the Numerical Integration of Stiff Systems, SIAM Journal of Numerical Analysis, 8(), (98), -36 (Advance online publication: February 8)

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