On second derivative 3-stage Hermite Birkhoff Obrechkoff methods for stiff ODEs: A-stable up to order 10 with variable stepsize

Transcription

1 Computational Methods for Differential Equations Vol 5, No 4, 2017, pp On second derivative -stage Hermite Birkhoff Obrechkoff methods for stiff ODEs: A-stable up to order 10 with variable stepsize Truong Nguyen-Ba Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5 Thierry Giordano Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5 Rémi Vaillancourt Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5 Abstract Variable-step (VS) second derivative k-step -stage Hermite Birkhoff Obrechkoff (HBO) methods of order p = (k + ), denoted by HBO(p) are constructed as a combination of linear k-step methods of order (p 2) and a second derivative two-step diagonally implicit -stage Hermite Birkhoff method of order 5 (DIHB5) for solving stiff ordinary differential equations The main reason for considering this class of formulae is to obtain a set of k-step methods which are L-stable and are suitable for the integration of stiff differential systems whose Jacobians have some large eigenvalues lying close to the imaginary axis with negative real part The approach, described in the present paper, allows us to develop L-stable k-step methods of order up to 10 Selected HBO(p) of order p, p = 9, 10, compare favorably with existing Cash L-stable second derivative extended backward differentiation formulae, SDEBDF(p), p = 7, 8 in solving problems often used to test stiff ODE solvers Keywords Hermite Birkhoff methods, generalized DIRK methods, A-stable, oscillatory stiff DETEST problems, confluent Vandermonde-type systems 2010 Mathematics Subject Classification 65L06, 65D05, 65D0 1 Introduction For solving stiff ordinary differential equations (ODE), y = f(t, y), y(t 0 ) = y 0, where = d and y R n, (11) dt a linear k-step method of order p 2 and a second derivative two-step diagonally implicit -stage Hermite Birkhoff method of order 5 (DIHB5) are cast into a k-step -stage Hermite Birkhoff Obrechkoff (HBO) methods of order p = (k + ), denoted Received: 26 March 2017 ; Accepted: 24 September 2017 Corresponding author 24

2 CMDE Vol 5, No 4, 2017, pp by HBO(p) The method s name was chosen because it uses Hermite Birkhoff interpolation polynomials, first and second order derivatives of y like Obrechkoff methods [20] Here, the DIHB5 is defined in Section 2 with p = 5 and step number k = 2, and has three degree of freedom since the coefficients in (24) are free parameters And, following the approach of Cash [], the abscissae c i are allowed to be 0 c i 2, i = 2,, 4 There is a variety of variable step (VS) methods designed to solve nonstiff and stiff systems of first-order differential equations (ODEs) Gear advocated a quasi-constant step size implementation in DIFSUB [9] This software works with a constant step size until a change of step size is necessary or clearly advantageous Then a continuous extension is used to get approximations to the solution at previous points in an equally spaced mesh This was largely because constant mesh spacing is very helpful when solving stiff problems Another possibility is fixed leading coefficient, which is seen in Petzold s popular code DASSL [21] Finally, the actual mesh can be chosen by the code as done in MATLAB s ode11 This is the equivalent of a PECE Adams formula in contrast with the Adams Moulton formula of DIFSUB and DASSL In this paper, a fully variable step size implementation is used with actual mesh A more basic point about the implementation of a method is the choice of the form The present method uses a generalized Lagrange form and much of the paper is devoted to computing the coefficients efficiently Remark 61 in Subsection 62 connects the computation of coefficients for three well known forms: generalized Lagrange form, generalized Newton divided differences form (similar to Krogh s modified divided differences [16]) and Nordsieck form [19] A brief survey of methods for the numerical integration of (11) reveals that there are many advances in the class of generalized linear multistep methods for stiff ODEs, methods like second derivative multistep methods (SDMM) proposed by Enright [5], second derivative extended backward differentiation formulas (SDEBDF) by Cash [], second derivative BDF methods (SDBDF) by Hairer et al [10], special classes of SDMM introduced by Ismail et al [14], Hojjati et al [12] and Khalsaraei et al [15] The first modification, introduced by Cash [], was the SDEBDF in which one super-future point has been applied The current investigation and the results are offered as potentially useful additions to the contemporary repertory of variable step (VS) L-stable second derivative multistep solvers for stiff differential equations (ODEs) This paper explores an alternative way to improve the order of L-stable second derivative multistep methods Forcing a Taylor expansion of the numerical solution of HBO(p) methods to agree with an expansion of the true solution leads to multistep and Runge Kutta type order conditions which are reorganized into linear Vandermonde-type systems The solutions of these systems are obtained as generalized Lagrange basis functions by new fast algorithms This approach allows us to develop L-stable methods of order up to 10, while we know well the difficulty of deriving multistep A-stable formulae of order greater than 2 A-stable linear multistep methods are limited to having maximum order 2 while high order, A-stable Runge-Kutta formulae can be very expensive to implement

3 26 T NGUYEN-BA, T GIORDANO, AND R VAILLANCOURT The extra stability is particularly important when integrating stiff differential systems whose Jacobians have some large eigenvalues lying close to the imaginary axis Stiff oscillatory problems happen often in practice In particular, they frequently develop when the method of lines technique is applied to a system of partial differential equations (PDE) that have some hyperbolic type of behaviour Typical examples of such problems are the integro-differential equations describing the stiff beam problem [10], and advection-dominated PDE problems, as described, for example, in [11, 2] A good description of the difficulties involved in integrating these hyperbolic type equations can be found in [10, pp 12] The selected HBO(p), p = 9, 10 compare favorably with SDEBDF(p), p = 7, 8, [] in solving problems often used to test highly stable stiff ODE solvers on the basis of number of steps (NS), CPU time (CPU) and the error at the endpoint (EPE) of the interval of integration The paper is organized as follows: in Section 2, we introduce new general VS HBO(p) methods of order p Order conditions of general VS HBO(p) are listed in Section In Section 4, particular variable step HBO(p) are defined by fixing a set of parameters and are represented in terms of Vandermonde-type systems In Section 5, symbolic elementary matrices are constructed as functions of the parameters of the methods in view of factoring the coefficient matrices of Vandermonde-type systems Fast solution of Vandermonde-type systems for variable step HBO(p) is constructed in Section 6 Section 7 considers the regions of absolute stability of constant step L-stable HBO(p), p = 5, 6,, 10 Section 8 deals with the step control In Section 9, we compare the numerical performance of L-stable methods considered in this paper Appendix A lists the algorithms Appendix B lists the coefficients of constant step L-stable HBO(p) methods of order p = 9, 10 2 General variable step HBO(p) of order p Variable step -stage HBO methods are constructed by the following formulae to perform integration from t n to t n+1 Let h n+1 denote the step size The abscissa vector [c 1, c 2, c, c 4 ] T defines the offstep points t n + c j h n+1 with c 1 = 0 and c 4 = 1 Following the approach of Cash [], c i are allowed to be 0 c i 2, i = 2, Let F 1 = f n and F j := f(t n + c j h n+1, Y j ), j = 2,, 4, denote the jth stage derivative With the initial stage value, Y 1 = y n, HB polynomials are used as implicit predictors P i to obtain the stage values Y i to order p 2, Y i = h n+1 a ii f(t n + c i h n+1, Y i ) + h 2 n+1γ ii f (t n + c i h n+1, Y i ) [ p 4 i 1 ] i 1 + y n + h n+1 β ij f n j + a ij F j + h 2 n+1 γ ij F j, i = 2, (21) j=0 j=2 j=2

4 CMDE Vol 5, No 4, 2017, pp An HB polynomial is used as implicit integration formula IF to obtain y n+1 to order p, y n+1 = h n+1 b 4 f(t n + h n+1, y n+1 ) + h 2 n+1g 4 f (t n + h n+1, y n+1 ) [ p 4 ] + y n + h n+1 β j f n j + b j F j + h 2 n+1g F (22) j=0 An HB polynomial is used as implicit predictor P 4 to control the step size, h n+2, and obtain ỹ n+1 to order p 2, ỹ n+1 = h n+1 a 44 f(t n + h n+1, y n+1 ) + h 2 n+1γ 44 f (t n + h n+1, y n+1 ) [ p 4 ] + y n + h n+1 β 4j f n j + a 4j F j + h 2 n+1γ 4 F (2) j=0 Here, the forms (21) and (22) are used by the implicit algebraic equations system defining Y i, i = 2, and y n+1 to handle implicitness in the context of stiffness The distinct implicit algebraic equations systems (21) and (22) defining Y i, i = 2, and y n+1 are solved iteratively by the modified Newton Raphson method similar to the usual resolution of system of implicit algebraic equations of BDF method [17, p 11 1] The following terminology will be useful An HBO(p) method is said to be a general variable-step HBO method if its backstep and the coefficients c 2, c, a 22 = a = b 4, (24) in (21) and (22) are variable parameters Hence, the general variable-step HBO method has three degrees of freedom (c 2, c, a 22 = a = b 4 ) If the coefficients in (24) are fixed, the method is said to be a particular variable-step method If the step size is constant, and hence the backsteps and the coefficients in (24) are fixed parameters, the method is said to be a constant-step method l=1 j=2 j=2 Order conditions of general HBO(p) To derive the order conditions of -stage (p )-step HBO(p), we shall use the following expressions coming from the backsteps of the methods: { p 4 η j 1 l+1 B i (j) = β il (j 1)!, i = 2,, (1) j = 1, 2,, p, and η j = 1 h n+1 (t n t n+1 j ) = 1 h n+1 j 2 h n i, j = 2,,, p (2) In the sequel, η j will be frequently used without explicit reference to (2) Forcing an expansion of the numerical solution produced by formulae (21) and (22) to agree with the Taylor expansion of the true solution, we obtain multistep- i=0

5 28 T NGUYEN-BA, T GIORDANO, AND R VAILLANCOURT and several Runge Kutta(RK)-type order conditions that must be satisfied by -stage HBO(p) methods To reduce a large number of RK-type order conditions (see [18]), we impose the following simplifying assumptions: i j=2 γ ij c k 1 i j (k 1)! + j=2 c k j a ij k! + B i(k + 1) = ck+1 i (k + 1)!, { i = 2,, k = 0, 1,, p () Thus, there remain only two sets of equations to be solved: 4 i= 4 i= g i g i c k 1 4 i (k 1)! + i=2 c p 2 i (p 2)! + i=2 c k i b i k! + B(k + 1) = 1, (k + 1)! k = 0, 1,, p 1, (4) ] b i [ i j=2 γ ij c p i j (p )! + where the backstep parts, B(j), are defined by p 4 B(j) = l=1 β l j=2 a ij c p 2 j (p 2)! + B i(p 1) c p b 4 (p 1)! + B(p) = 1 p!, (5) η j 1 l+1, j = 1, 2,, p + 1 (6) (j 1)! These order conditions are simply RK order conditions with backstep parts B i ( ) and B( ) 4 Vandermonde-type formulation of particular variable step HBO(p) The general HBO(p) methods obtained in Section contain free coefficients in (24), and depend on h n+1 and the previous nodes, t n, t n 1,, t n (p 4), which determine η 2, η,, η p in (2) To obtain A-stability of particular HBO(p) methods, p = 9, 10, the coefficients listed in Table 1 were chosen In Table 1, since a 22 = a = b 4, only values of a 22 are listed It is to be noted that, to obtain the coefficients in Table 1, the well-known exhaustive search method is used with possible candidates (c 2, c, a 22 ), for c 2 = 01, 015, 02,, 20, c = 01, 015, 02,, 20 and a 22 with the same increment (or decrement) size 005 The value of a 22 which yields the largest α of A(α)-stability is used as a starting value (the exhaustive search method can be repeated with new starting values of a 22 ) The remaining of this paper is concerned with particular VS HBO(p) p = 9, 10, with coefficients c i, i = 2,, and a 22 given in Table 1

6 CMDE Vol 5, No 4, 2017, pp Table 1 Coefficients c i, i = 2, and a 22 (= a = b 4 ) of particular VS HBO(p), p = 9, 10 k 6 7 coeffs\p 9 10 c e e+00 c e e+00 a e e Predictor P 2 The (p 2)-vector of reordered coefficients of the predictor P 2 in (21) with i = 2, u 2 = [β 20, β 21,, β 2,p 4, γ 22 ] T, is the solution of the Vandermonde-type system of order conditions where M 2 u 2 = r 2, (41) η 2 η η p 1 η η 2 η p 2 2! 2! c 2! 2 M 2 = η 2 0 η η p c!! 2 2! 2! η 0 p 2 η p η p p (p )! (p )! (p )! and r 2 = r 2 (1 : p 2) has components c i 1 2 c p 4 2 (p 4)! r 2 (i) = ci 2 i! a 22, i = 1, 2,, p 2 (i 1)!, (42) A truncated Taylor expansion of the right-hand side of (21) with i = 2 about t n gives p+1 S 2 (j)h j n+1 y(j) n with coefficients c j 1 2 j=0 S 2 (j) = a 22 (j 1)! + M 2 (j, 1 : p 2)u 2 c j 1 2 = a 22 (j 1)! + r 2(j) = cj 2, j = 1, 2,, p 2, j! p 4 S 2 (j) = a 22 S 2 (j 1) + i=1 β 2i η j 1 i+1, j = p 1, p, p + 1 (j 1)! We note that P 2 is of order p 2 since it satisfies the order conditions S 2 (j) = c j 2 /j!, j = 1, 2,, p 2,

7 0 T NGUYEN-BA, T GIORDANO, AND R VAILLANCOURT and its leading error term is [ S 2 (p 1) ] cp 1 2 h p 1 n+1 (p 1)! y(p 1) n 42 Integration formula IF The p-vector of reordered coefficients of the integration formula IF in (22), u 1 = [β 0, b, b 2, β 1, β 2,, β p 4, g ] T, is the solution of the Vandermonde-type system of order conditions where M 1 u 1 = r 1, (4) c c 2 η 2 η η p 1 c M 1 = 0 2 c 2 2 η2 2 η 2 η p 2 2! 2! 2! 2! c 2! c 0 p 1 c p 1 2 η p 1 2 η p 1 η p 1 p (p 1)! (p 1)! (p 1)! (p 1)! (p 1)! and r 1 = r 1 (1 : p) has components r 1 (1) = 1 b 4, c i 1 4 c i 2 4 r 1 (i) = 1 i! b 4 (i 1)! g 4, i = 2,,, p, (i 2)! where b 4 = a 22 and g 4 = γ 22 The leading error term of IF is c [g p (p 1)! + b 4 c p p 4 4 p! + j=1 β j η p j+1 p! + j=2 c p 2 (p 2)!, (44) c p j b j p! + g c p 1 ] (p 1)! 1 h p+1 n+1 (p + 1)! yp+1 n 4 Predictor P We consider the (p 1)-vector of reordered coefficients of the predictor P in (21) with i =, u = [β 0, a 2, β 1,, β,p 4, γ 2 ] T, is the solution of the Vandermonde-type system of order conditions where M u = r, (45) c 2 η 2 η η p 1 M c = η 2 2 η 2 η 2! 2! 2! 2 p 2! c 2 c 0 p 2 2 η p 2 2 η p 2 η (p 2)! (p 2)! (p 2)! p 2 p (p 2)! c p 2 (p )! (46)

8 CMDE Vol 5, No 4, 2017, pp The first (p 2) components of r = r (1 : p 1) are r (1) = c a, c i 1 c i 2 r (i) = ci i! a (i 1)! γ, i = 2,,, p 2, (i 2)! the (p 1)th component is where c p 2 c p r (p 1) = S (p 1) a (p 2)! γ (p )!, (47) c p 1 4 S (p 1) = 1 [ 1 b p! b 2S 2 (p 1) b 4 (p 1)! c p 2 g (p 2)! g c p 2 ] 4 4 (p 2)! B(p), a = a 22 and γ = γ 22 The equation for r (p 1) in (47) corresponds to order condition (5) A truncated Taylor expansion of the right-hand side of (21), with i =, about t n gives with coefficients c j 1 p+1 S (j)h j n+1 y(j) n j=0 c j 2 S (j) = a (j 1)! + γ (j 2)! + M (j + 1, 1 : p 1)u c j 1 = a (j 1)! + γ c j 2 (j 2)! + r (j) = cj, j = 1, 2,, p 2, j! S (j) = a S (j 1) + γ S (j 2) + a 2 S 2 (j 1) + γ 2 S 2 (j 2) p 4 + β i i=1 η j 1 i+1, j = p 1, p, p + 1 (j 1)! 44 Step control predictor P 4 The (p 2)-vector of reordered coefficients of predictor P 4 in (2), u 4 = [β 40, a 42, β 41,, β 4,p 4 ] T, is the solution of the system of order conditions M 4 u 4 = r 4, (48)

9 2 T NGUYEN-BA, T GIORDANO, AND R VAILLANCOURT where c 2 η 2 η η p c M 4 = η2 2 η 2 η p 2 2! 2! 2! 2! c 0 p 2 η p 2 η p (p )! (p )! (p )! and r 4 = r 4 (1 : p 2) has components r 4 (1) = 1 (b 4 + ω 4 ) (b + ω ), c i 2 4 η p p (p )! r 4 (i) = 1 i! (g 4 + ω 4) (i 2)! (b 4 + ω 4 ) (i 1)! c i 2 c i 1 c i 1 4, (49) (g + ω ) (i 2)! (b + ω ), i = 2,,, p 2 (i 1)! For arbitrary nonzero ω and ω, P 4 yields ỹ n+1 to order (p 2) A good experimental choice is ω = 0025, ω = 0025, ω 4 = 0025, ω 4 = 0025 The solutions u l, l = 1, 2,, 4, form generalized Lagrange basis functions for representing the HB interpolation polynomials 5 Symbolic construction of elementary matrix functions Consider the matrices M l R m l m l, l = 2, 1,, 4, (51) of the Vandermonde-type systems (41), (4), (45) and (48), where m 2 = p 2, m 1 = p, m = p 1, m 4 = p 2, (52) and p is the order of the method The purpose of this section is to construct elementary lower and upper triangular matrices as symbolic functions of the parameters of HBO(p) These matrices are most easily constructed by means of a symbolic software These functions will be used in Section 6 to factor M l into a diagonal+last-1-column matrix, W1, l l = 2, 1,, which will be further diagonalized by a Gaussian elimination, M 4 into the identity matrix I 4 This decomposition will lead to a fast solution of the systems M l u l = r l, l = 2, 1,, 4 in O(p 2 ) operations Since the Vandermonde-type matrices M l can be decomposed into the product of a diagonal matrix containing reciprocals of factorials and a confluent Vandermonde matrix, the factorizations used in this paper hold following the approach of Björck and Pereyra [2], Krogh [16], Galimberti and Pereyra [8] and Björck and Elfving [1] Pivoting is not needed in this decomposition because of the special structure of Vandermonde-type matrices

10 CMDE Vol 5, No 4, 2017, pp Symbolic construction of lower bidiagonal matrices for M l, l = 1, 2,, 4 We first describe the zeroing process of a general vector x = [x 1, x 2,, x m ] T with no zero elements The lower bidiagonal matrix L k = I k τ k τ m 1, (5) defined by the multipliers τ i = x i x i 1 = L k (i, i 1), i = k + 1, k + 2,, m, zeros the last (m k) components, x k+1, x k+2,, x m, of x This zeroing process will be applied recursively on M l, l = 1, 2,, 4, as follows For k = 2,,, left multiplying Tk l = Ll k 1 Ll k 2 Ll L l 2M l by L l k zeros the last (m l k) components of the kth column of Tk l Thus we obtain the upper triangular matrix L l M l = L l m l 1 L l L l 2M l, l = 1, 2,, 4, (54) in (m l 2) steps, for l = 1, 2,, 4 We note that L l does not change the first two rows of M l Process 1 At the kth step, starting with k = 2, M l(k 1) = L l k 1 Ll k 2 Ll 2M l is an upper triangular matrix in columns 1 to k 1, The multipliers in L l k are obtained from M l(k 1) (k+1 : m l, k) since M l (i, k) 0 for i = k + 1, k + 2,, m l Algorithm 1 in Appendix A describes this process The input is M = M l ; m = m l The output is L k = L l k, k = 2,,, m l 1, l = 1, 2,, 4 52 Symbolic construction of upper bidiagonal matrices for M l, l = 1, 2,, 4 For matrix L l M l, l = 1, 2,, 4, we construct recursively upper bidiagonal matrices U1, l U2 l, U l such that right multiplying L l M l by the upper triangular matrix kend l U l = U1U l 2 l U l transforms L l M l into a matrix W l kend l C l = L l M l U l with nonzero diagonal elements, WC l l (i, i) 0, i = 1, 2,, m l, the last C l nonzero columns WC l l (1 : m l, j) 0, j = m l C l + 1, m l C l + 2,, m l, and zero elsewhere We call such a matrix a diagonal+last-c l -column matrix Here C 1 = 1, C 2 = 1, C = 1, C 4 = 0, (55) k 1 end = m 1 2, k 2 end = m 2 2, k end = m 2, k 4 end = m 4 1, (56) for M l, l = 1, 2,, 4, respectively We describe the zeroing process of the upper bidiagonal matrix Uk l on the two-row matrix (L l M l )(k : k + 1, 1 : m l ):

11 4 T NGUYEN-BA, T GIORDANO, AND R VAILLANCOURT (L l M l )(k : k + 1, 1 : m l )U1U l 2 l Uk 1 l [ yk1 y = k,k y k+1,1 y k+1,k 1 y k+1,k y k+1,ml C l y k,ml C l +1 y k,ml C l +2 y k,ml y k+1,ml C l +1 y k+1,ml C l +2 y k+1,ml ] (57) The divisors σ i = 1 y k+1,i y k+1,i 1 = U l k (i, i), i = k + 1, k + 2,, m l C l, define the upper bidiagonal matrix U l k = I k σ k σ k+1 σ k σ ml C l 1 σ ml C l σ ml C l (58) Right multiplying (57) by U l k zeros the 1 s in position k + 1, k + 2,, m l C l in the first row and puts 1 s in position k + 1, k + 2,, m l C l in the second row: (L l M l )(k : k + 1, 1 : m l )U1U l 2 l Uk 1U l k l [ yk1 y = k,k y k+1,1 y k+1,k 1 y k+1,k 1 1 y k,ml C l +1 y k,ml C l +2 y k,ml y k+1,ml C l +1 y k+1,ml C l +2 y k+1,ml ] (59) Thus, U l = U1U l 2 l U l transforms the upper triangular matrix L l M l into the kend l diagonal+last-c l -column matrix W l C l = L l M l U l 1U l 2 U l k l end, (510) in kend l steps Here kl end is defined in (56) Process 2 At the kth step, starting with k = 1, M l(k) = L l M l U1U l 2 l Uk l is a diagonal+last-c l-column matrix in rows 1 to k, The divisors in Uk l are obtained from M l(k 1) (k + 1, k : m l C l ) since M l(k 1) (k + 1, j) M l(k 1) (k + 1, j 1) 0, j = k + 1, k + 2,, m l C l Algorithm 2 in Appendix A describes this process for M l, l = 1, 2,, 4 The input is M = M l ; m = m l The output is U k = Uk l, k = 1, 2,, kl end where kl end is defined in (56)

12 CMDE Vol 5, No 4, 2017, pp Fast solution of Vandermonde-type systems for particular variable step HBO(p) Symbolic elementary matrix functions L l k and U k l, l = 1, 2,, 4, are constructed once as functions of η j, for j = 2,,, p, by Algorithms 1 and 2 in Appendix A to factor for l = 1, 2,, M l into a diagonal+last-1-column matrix, W1 l which will be further diagonalized by a Gaussian elimination, M 4 into the identity matrix I 4 These elementary matrix functions are used, first, to find, successively, the solution u l, l = 2, 1,, 4 in elementary matrix functions form and, then, to construct (a) fast Algorithm in Appendix A, to solve systems (41), (4), (45) and (b) fast Algorithm 4 to solve system (48), at each integration step 61 Solution of M l u l = r l, l = 1, 2, We let m 1 = p, m 2 = p 2 and m = p 1 as defined in (52) (1) The elimination procedure of Subsection 51 is applied to M l to construct m l m l lower bidiagonal matrices L l k, k = 2,,, m l 1, with multipliers τ i = M l (2,k) i 1 = L l k (i, i 1), i = k + 1, k + 2,, m l Left multiplying the coefficient matrix M l by the lower bidiagonal matrix L l = L l m l 1 L l L l 2 transforms M l into the upper triangular matrix L l M l in column 1 to m l 1 of the form (54) (2) The elimination procedure of Subsection 52 is used to construct m l m l upper bidiagonal matrices Uk l, k = 1, 2,, m l 2, with multipliers σ i = k M l (2, i) M l (2, i k) = U l k (i, i), i = k + 1, k + 2,, m l 1 (61) Right multiplying L l M l by the upper triangular matrix U l = U l 1U l 2 U l m l 2 transforms L l M l into a diagonal+last-1-column matrix W l 1 of the form (510) () Factored Gaussian eliminations, L l m l +1L l m l, will eliminate column m l of W l 1 and transform W l 1 into the identity matrix I l = L l m l +1L l m l W l 1 where L l k, k = m l, m l + 1 have nonzero entries listed in Table 2 and zeros elsewhere Table 2 Nonzero entries of Gaussian elimination matrices L l k, k = m l, m l + 1, l = 1, 2, Gaussian elimination matrices L l m l L l m l +1 L l m l (i, i) = 1, i = 1, 2,, m l 1 L l m l +1(i, i) = 1, i = 1, 2,, m l L l m l (m l, m l ) = 1/W l (m l, m l ) L l m l +1(1 : m l 1, m l ) = W l (1 : m l 1, m l ) This procedure transforms M l into the identity matrix I l = L l m l +1L l m l L l 2M l U l 1U l 2 U l m l 2

13 6 T NGUYEN-BA, T GIORDANO, AND R VAILLANCOURT Thus we have the following factorization of M l into the product of elementary matrices: M l = ( L l m l +1L l m l L l 2) 1 ( U l 1 U l 2 U l m l 2) 1, and the solution is u l = U l 1U l 2 U l m l 2L l m l +1L l m l L l 2 r l, (62) where fast computation goes from right to left Procedure (62) is implemented in Algorithm in Appendix A in O(m 2 l ) operations, l = 1, 2, The input is M = M l ; m = m l ; r = r l ; L k = L l k, k = 2,,, m l + 1; U k = U l k, k = 1, 2,, m l 2 The output is u = u l, l = 1, 2, 62 Solution of M 4 u 4 = r 4 We let m 4 = p 2 as defined in (52) Similar to steps (1) and (2) of Subsection 61, the matrix L 4 = L 4 m 4 1 L 4 L 4 2 transforms the coefficient matrix M 4 into the upper triangular matrix L 4 M 4 in column 1 to m 4 1 of the form (54) Next, the right-product of the U 4 k, k = 1, 2,, m 4 1, will transform L 4 M 4 into the identity matrix I 4 of the form (510) Thus we have the following factorization of M 4 into the product of elementary matrices: M 4 = ( L 4 m 4 1L 4 m 4 2 L 4 2) 1 ( U 4 1 U 4 2 U 4 m 4 1) 1, and the solution is u 4 = U 4 1 U 4 2 U 4 m 4 1L 4 m 4 1 L 4 2 r 4, (6) where fast computation goes from right to left Procedure (6) is implemented in Algorithm 4 in Appendix A in O(m 2 4) operations The input is M = M 4 ; m = m 4 ; r = r 4 ; L k = L 4 k, k = 2,,, m 4 1; U k = U 4 k, k = 1, 2,, m 4 1 The output is u = u 4 Remark 61 Formulae (21) to (2) can be put in matrix form For instance, (22) can be written as y n+1 y n = F 1 u 1 + G 1 v 1 where and F 1 = h n+1 [f n, F, F 2, f n 1, f n 2,, f n (p 4), h n+1 F [ G 1 = h n+1 f(t n + h n+1, y n+1 ), h n+1 f (t n + h n+1 ], ], u 1 = [β 0, b, b 2, β 1, β 2,, β p 4, g ] T, v 1 = [b 4, g 4 ] T It is interesting to note the three decomposition forms of the system F u: F (U Lr) (generalized Lagrange interpolation), (F U)Lr (generalized divided differences), (F U L)r (Nordsieck s formulation)

14 CMDE Vol 5, No 4, 2017, pp The first form is used in this paper, the form similar to the second form for Vandermonde systems is found in [16], and the third form is found in [19] 7 Regions of absolute stability The regions R of constant step HBO(p), p = 5, 6,, 10, listed in Appendix B, are obtained by applying formulae (21) and (22) of the predictors P i, i = 2, and the integration formula IF with constant h to the linear test equation y = λy, y 0 = 1 This gives the following difference equation and corresponding characteristic equation k k η j (z) y n+j = 0, η j (z) r j = 0, (71) j=0 j=0 respectively, where k = p is the number of steps of the method and z = λh A complex number z is in R if the k roots of the characteristic equation in (71) satisfy the root condition (see [17, pp 70]) The scanning method used to find R is similar to the one used for Runge Kutta methods (see [17]) The stability functions η j (z), j = 0, 1,, k in (71) are rational functions of the form 5 l=0 η k (z) = 1, η j (z) = n jlz l 6 l=0 d, j = 0, 1,, k 1 l jlz Hence, in the difference equation of (71), y n+k 0 as z This implies that HBO(p), p = 5, 6,, 10 are L-stable since these methods are A-stable For each given step number k, Table lists the α angles of A(α)-stability for HBO(5 10), SDEBDF(5 10) (class 1) of Cash [], SDMM(5 9) of Enright [5], SDMM(5 10) of Ismail et al [14], SDBDF(5 10) of Hairer et al [10, p 270], SDMM(5 10) of Hojjati et al [12] and SDMM(5 10) of Khalsarei et al [15], respectively It is seen that α of HBO methods compare favorably with α of the considered methods of comparable order p 8 Controlling step size The estimate y n ỹ n and the current step h n are used to calculate the next step size h n+1 by means of formula [1] { [ ] } 1/κ tolerance h n+1 = min h max, β h n, 4 h n, (81) y n ỹ n with κ = p SCP + 1 and safety factor β = 081 Here p SCP is the order of the step control predictor (SCP) (it is to be noted that the value ỹ n is obtained by the formula (2)) The procedure to advance integration from t n to t n+1 is as follows (a) The step size, h n+1, is obtained by formula (81) (b) The numbers η 2, η,, η p, defined in (2), are calculated

15 8 T NGUYEN-BA, T GIORDANO, AND R VAILLANCOURT Table For each given step number k, the table lists the order p, the α angles of A(α)-stability for the listed methods HBO(p) SDEBDF(p) SDMM(p) SDMM(p) class 1 of Cash [] of Enright [5] of Ismail et al [14] k p α k p α k p α k p α SDBDF(p) SDMM(p) SDMM(p) of Hairer et al [10, p 270] of Hojjati et al [12] of Khalsarei et al [15] k p α k p α k p α (c) The coefficients of predictors P 2, integration formula IF, P and step control predictor P 4 are obtained successively as solutions of systems (41), (4), (45) and (48) (d) The values Y 2, Y, y n+1, and ỹ n+1 are obtained by formulae (21) to (2) (e) The step is accepted if y n+1 ỹ n+1 is smaller than the chosen tolerance and the program goes to (a) with n replaced by n+1 Otherwise the program returns to (a) and a new smaller step size h n+1 is computed 9 Numerical results of comparing L-stable methods The error at the endpoint of the integration interval (EPE, endpoint error) is taken in the uniform norm, EPE = { y end z end }, where y end is the numerical value obtained by the numerical method at the endpoint t end of the integration interval and z end is the exact solution obtained by MATLAB s ode15s with stringent tolerance The necessary starting values at t 1, t 2,, t k 1 for HBO(p) were obtained by MAT- LAB s ode15s with stringent tolerance Computations were performed on a PC with the following characteristics: Memory: 58 GB, Processor 0,1,,7: Intel(R) Core(TM) i7 CPU 267GHz, Operating system: Ubuntu Release 1104, Kernel Linux generic, GNOME 221 We compare L-stable methods on different types of problems The following test problems are considered Problems 1 and 2 are often used to test stiff ODE solvers Problem is representative of some stiff oscillatory problems which arise frequently in practice In particular, they often arise when the method of lines technique is applied to a system of partial differential equations that have some hyperbolic type of

16 CMDE Vol 5, No 4, 2017, pp behaviour We have chosen Problems and 4 where the eigenvalues of the Jacobian matrix lie close to the imaginary axis, since it is problems of this type that cause major difficulties to many existing codes (1) The Oregonator equation describing Belusov-Zhabotinskii reaction [7] Problem 1 y 1 = 7727(y 2 + y y1 2 y 1 y 2 ), y 1 (0) = 1, y 2 = (y (1 + y 1 )y 2 )/7727, y 2 (0) = 2, y = 0161(y 1 y ), y (0) =, (91) with t end = 60 (2) The van der Pol s equation [10, pp 4 6], [12] Problem 2 y 1 = y 2, y 1 (0) = 2, y 2 = µ 2 [(1 y 2 1)y 2 y 1 ], y 2 (0) = 0, (92) where µ = 500 and with t end = 08 () The stiff DETEST problem B5 [6] Problem y 1 = 10y 1 + αy 2, y 1 (0) = 1, y 2 = αy 1 10y 2 y 2 (0) = 1, y = 4y y (0) = 1, y 4 = y 4 y 4 (0) = 1, y 5 = 05y 5 y 5 (0) = 1, y 6 = 01y 6 y 6 (0) = 1, (9) with α = 1000 and t end = 20 (4) As above with α = 1500 (5) A problem with large eigenvalues lying close to the imaginary axis [] Problem 4 y 1 = αy 1 βy 2 + (α + β 1)e t y 1 (0) = 1, y 2 = βy 1 αy 2 + (α β 1)e t y 2 (0) = 1, y = 1 y (0) = 0, (94) with α = 1, β = 0, fixed step h = 009 and t end = 20 The exact solution is y 1 (t) = y 2 (t) = e t, y (t) = t (6) As above with β = 42 and fixed step h = 100

17 40 T NGUYEN-BA, T GIORDANO, AND R VAILLANCOURT Figure 1 log 10 (global error) versus log 10 h at t n = 20 for the listed HBO(p) applied to problem (92) with µ = 1 over t [0, 20] with constant stepsizes h HBO(9) HBO(10) Numerical verification of the order p of HBO(p) To show the relevance of the theoretical order of HBO(p), we have applied these methods with various constant stepsizes h on Problem (92) (van der Pol s equation) with µ = 1 over t [0, 20] In Figure 1, the global error of y 1 and y 2 at t n = 20, max{ y 1,n y 1 (t n ), y 2,n y 2 (t n ) } = O(h p ), is plotted in a log-log scale for the listed HBO(p) methods applied to problem (92) over t [0, 20] with different constant stepsizes h so that the curves appear as straight lines with slope p whenever the leading term of the global error is of order p For HBO(p), the slopes of the straight lines which approximate the data in the leastsquares sense are very close to p, which confirms the orders of the methods 92 Comparing number of steps of selected L-stable HBO(p) and highly efficient existing L-stable methods In our first tests, we numerically compare our most efficient methods HBO(p), p = 9, 10 with the highly efficient L-stable methods which are the second derivative extended backward differentiation formulae, of class 1, of order 7 and 8 [], denoted by SDEBDF(p), p = 7, 8, on the basis of the EPE, endpoint error as a function of number of steps (NS) These classical SDEBDF(p) methods have been widely used to compare stiff ODE solvers Figure 2 depicts the graph of log 10 (EPE) (vertical axis) as a function of number of steps (NS) (horizontal axis) for the considered test problems Here, similar to Cash [4], we use also Problem It is seen that, in general, HBO(p), p = 9, 10, compare favorably with SDEBDF(p), p = 7, 8 The number of steps percentage efficiency gain (NS PEG) of method 1 over method 2 is defined by the formula (cf Sharp [22]), [ j NS PEG = 100 NS ] 2,j j NS 1, (95) 1,j

18 CMDE Vol 5, No 4, 2017, pp log 10 (EPE) Figure 2 log 10 (EPE) (vertical axis) as a function of number of steps (NS) (horizontal axis) for Oregonator equation (top left), van der Pol s equation (top right), problem B5 with α = 1000 (bottom left) and problem B5 with α = 1500 (bottom right) HBO(9) HBO(10) SDEBDF(7) SDEBDF(8) log 10 (EPE) HBO(9) HBO(10) SDEBDF(7) SDEBDF(8) number of steps number of steps log 10 (EPE) HBO(9) HBO(10) SDEBDF(7) SDEBDF(8) log 10 (EPE) HBO(9) HBO(10) SDEBDF(7) SDEBDF(8) number of steps number of steps where NS 1,j and NS 2,j are the estimates of NS of methods 1 and 2, respectively, and j = log 10 (EPE estimate) To compute NS 2,j and NS 1,j appearing in (95), we approximate the data (log 10 (EPE), log 10 (NS)) in a least-squares sense by MAT- LAB s polyfit Then, for chosen integer values of the summation index j, we take log 10 (EPE estimate) = j and obtain log 10 (NS estimate) from the approximating curve, and finally the estimate of NS Table 4 lists the NS PEG of HBO(p), p = 9, 10, over SDEBDF(p), p = 7, 8, for the listed problems It is seen that HBO(p), p = 9, 10, win 9 Comparing CPU time of selected L-stable HBO(p) and highly efficient existing L-stable methods In our second tests, programmed in C++ code, our most efficient methods HBO(p), p = 9, 10 and the highly efficient L-stable methods SDEBDF(p), p = 7, 8, are compared on the problems shown in Table 5 on the basis of EPE as a function of CPU time in seconds (CPU)

19 42 T NGUYEN-BA, T GIORDANO, AND R VAILLANCOURT Table 4 NS PEG of HBO(p), p = 9, 10, over SDEBDF(p), p = 7, 8 for the listed problems NS PEG of HBO(9) over: NS PEG of HBO(10) over: Problem SDEBDF(7) SDEBDF(8) SDEBDF(7) SDEBDF(8) Oregonator equation 9% % 2% 26% van der Pol s equation 7% 7% 48% 17% Prob B5 with α = % 0% 7% 8% Prob B5 with α = % 56% 6% 16% Table 5 CPU PEG of HBO(p), p = 9, 10, over SDEBDF(p), p = 7, 8 for the listed problems CPU PEG of HBO(9) over: CPU PEG of HBO(10) over: Problem SDEBDF(7) SDEBDF(8) SDEBDF(7) SDEBDF(8) Oregonator equation 17% % 5% -% van der Pol s equation 7% 7% 26% -11% Prob B5 with α = % 0% 28% -15% Prob B5 with α = % 56% % -9% The CPU percentage efficiency gain (CPU PEG) of method 1 over method 2 is defined by the formula (cf Sharp [22]), [ j (CPU PEG) i = 100 CPU ] 2,ij j CPU 1, (96) 1,ij where CPU 1,ij and CPU 2,ij are the estimates of CPU time of methods 1 and 2, respectively, associated with problem i, and estimate of EPE = 10 j The computation of CPU 2,j and CPU 1,j appearing in (96) is similar to the computation of NS 2,j and NS 1,j appearing in (95) Table 5 lists the CPU PEG of HBO(p), p = 9, 10, over SDEBDF(p), p = 7, 8, for the listed problems It is seen that HBO(9) wins For HBO(10), the NS PEGs for HBO(10) in Table 4 are positive and tend to be larger than the CPU PEGs in Table 5 since the equations of the problems listed in Table 4 or 5 are relatively not expensive to evaluate This would suggest that, when these equations become more expensive to evaluate, HBO(10) will be more efficient in CPU time Table 6 compares the efficiency of HBO(9) and HBO(10) on 4 considered problems (91), (92), (9), (94) under listed LT = log 10 (TOL) The comparison is based on CPU time in seconds, number of steps (NS), number of rejected steps (NRS) and end point error (EPE) It is seen that, to solve these 4 particular problems, in general, HBO(9) is more efficient 94 Comparing errors of methods on problems with large eigenvalues lying close to the imaginary axis Our final result is a comparison of the errors of HBO(p), p = 9, 10, and second derivative extended backward differentiation formulae of class 1 of order 9 and 10 [], denoted by SDEBDF(p), p = 9, 10, on problems whose

20 CMDE Vol 5, No 4, 2017, pp Table 6 For 4 considered problems (91), (92), (9), (94) and LT = log 10 (TOL), the table lists CPU time in seconds, number of steps (NS), number of rejected steps (NRS) and end point error (EPE) in corresponding left and right columns for HBO(9) and HBO(10), respectively HBO(9) and HBO(10) Problem LT CPU time NS NRS EPE Oregonator -05 5e-02 6e e e e-02 42e e e e e e e-08 van der Pol e-0 282e e e e-0 58e e e e-0 88e e e-10 B5-0 10e e e e-08 α = e-02 48e e e e e e-11 12e-10 B e-0 148e e-07 28e-07 α = e-02 58e e e e e e-10 11e-09 Table 7 Error results obtained for the solution of Problem 4 (with α = 1, β = 0, fixed step h = 009) as a function of step number k and t k = 6 k = 7 HBO methods SDEBDF methods Error in y 1 Error in y 2 Error in y 1 Error in y 2 t = t = t = t = t = t = Jacobians have some large eigenvalues lying close to the imaginary axis For this comparison, similar to Cash [], we use Problem 4 Table 7 presents error results obtained for the solution of Problem 4 (with α = 1, β = 0, fixed step h = 009 suggested by Cash []) as a function of step number k and t It is seen that HBO(p), p = 9, 10, and SDEBDF(p), p = 9, 10 remain stable for the integration of this problem Next, we present a numerical example which demonstrates the superior stability of the class of high order HBO(p), p = 9, 10 The problem integrated was Problem 4 with large eigenvalues lying close to the imaginary axis, α = 1, β increased to 42 and fixed step h = 100 Table 8 shows HBO(p), p = 9, 10, remain stable while there is instability for SDEBDF(p), p = 9, 10

21 44 T NGUYEN-BA, T GIORDANO, AND R VAILLANCOURT Table 8 Error results obtained for the solution of Problem 4 (with α = 1, β = 42, fixed step h = 100) as a function of step number k and t k = 6 k = 7 HBO methods SDEBDF methods Error in y 1 Error in y 2 Error in y 1 Error in y 2 t = t = t = t = t = t = Conclusion Second derivative multistep -stage Hermite Birkhoff Obrechkoff (HBO) methods of orders p were considered It is seen that HBO(p) are L-stable up to order 10 Selected HBO(p) of order p, p = 9, 10, compare positively with existing Cash modified extended backward differentiation formulae, SDEBDF(p), p = 7, 8 in solving differential equations problems often used to test highly stable stiff ODE solvers HBO(p) of order p, p = 5, 6,, 10, are members of general variable-step variableorder (VSVO) highly stable -stage k-step of order p = k + which appear to be promising highly stable stiff ODE solvers in the light of the numerical results obtained in this paper 11 Acknowledgment Thanks are due to the reviewer whose deep and extended comments contributed to substantially improve the manuscript Thanks are also due to Martín Lara for supplying the authors with his programs and sharing his experience This work was supported in part by the Natural Sciences and Engineering Research Council of Canada Appendix A Algorithms Algorithm 1 This algorithm constructs L k (i, i 1) entries of lower bidiagonal matrices L k (applied to IF, P l, l = 2,, 4) as functions of η j, j = 2,, For k = 2 : k end, do the following iteration: For i = m : 1 : k + 1, do the following two steps: Step (1) L k (i, i 1) = M(i, k)/m(i 1, k) Step (2) For j = k : m, compute: M(i, j) = M(i, j) + M(i 1, j)l k (i, i 1), where k end = m l 1, l = 1, 2,, 4 for IF, P l, l = 2,, 4 Algorithm 2 This algorithm constructs diagonal entries U k (j, j) of upper bidiagonal matrices U k (applied to IF, P 2, P and P 4) as functions of η j, j = 2,, For k = 1 : k end, do the following iteration: For j = j 0 : 1 : k + 1, do the following two steps: Step (1) U k (j, j) = 1/[M(k + 1, j) M(k + 1, j 1)]

22 CMDE Vol 5, No 4, 2017, pp Step (2) for i = k : j, compute M(i, j) = (M(i, j) M(i, j 1))U k (j, j) where k end values are defined in (56), j 0 = m l 1, l = 1, 2, for IF, P 2, P respectively and j 0 = m 4 for P 4 Algorithm This algorithm solves the systems for P l, l = 1, 2, in O(m 2 ) operations Given [η 2, η,, η p ] and r = r(1 : m), the following algorithm overwrites r with the solution u = u(1 : m) of the system Mu = r Step (1) The following iteration overwrites r = r(1 : m) with L m 1 L m 2, L 2 r: for k = 2,,, m 1, compute r(i) = r(i) + r(i 1)L k (i, i 1), i = m, m 1,, k + 1 Step (2) This step forms the two matrices L m and L m+1 which transform W l 1 into the identity matrix I l = L m+1 L m W l 1 : this step computes the coefficients G m(i), i = 1, 2,, m used to form the two matrices L m and L m+1, whose nonzero entries are listed in Table 2, as follows First set G m (1 : m), G m (1 : m) = M(1 : m, m) The following computation overwrites G m(1 : m) with L m 1L m 2 L 2G m(1 : m): for k = 2,,, m 1, compute G m (i) = G m (i) + G m (i 1)L k (i, i 1), i = m, m 1,, k + 1 Step () The following computation overwrites the newly obtained r with L m+1 L m r: r(m) = r(m)/g m (m), next, for k = m 1, m 2,, 1, compute r(k) = r(k) G m (k)r(m) Step (4) The following iteration overwrites r = r(1 : m) with U 1 U 2 U m 2 r: For k = m 2, m,, 1, compute r(i) = r(i)u k (i, i), i = k + 1, k + 2,, m 1, r(i) = r(i) r(i + 1), i = k, k + 1,, m 2 Algorithm 4 This algorithm solves the systems for P 4 in O(m 2 ) operations Given [η 2, η,, η p ] and r = r(1 : m), the following algorithm overwrites r with the solution u = u(1 : m) of the system Mu = r Step (1) The following iteration overwrites r = r(1 : m) with L m 1 L m 2 L 2 r: for k = 2,,, m 1, compute r(i) = r(i) + r(i 1)L k (i, i 1), i = m, m 1,, k + 1 Step (2) The following iteration overwrites r = r(1 : m) with U 1U 2 U m 1r: For k = m 1, m 2,, 1, compute r(i) = r(i)u k (i, i), i = k + 1, k + 2,, m, r(i) = r(i) r(i + 1), i = k, k + 1,, m 1 Appendix B Coefficients of HBO(p), p = 9, 10 The appendix lists the coefficients of HBO(p), of order p = 9, 10, considered in this paper It is to be noted that, in Table 9, since a 22 = a = b 4 and γ 22 = γ = g 4, only values of a 22 and γ 22 are listed

23 46 T NGUYEN-BA, T GIORDANO, AND R VAILLANCOURT Table 9 Coefficients of the implicit predictors P i, i = 2, and of the integration formulae of HBO(p), p = 9, 10 k 6 7 coeffs\p 9 10 c e γ e e-01 a e e-01 β e e+00 β e e-01 β e e+00 β e e+00 β e e+00 β e e-01 β e-02 c e γ e e-02 a e e-01 β e e-01 β e e-01 β e e-01 β e e-01 β e e-01 β e e-01 β e-02 g e e-01 b e e-01 b e e-02 β e e-01 β e e-02 β e e-02 β e e-02 β e e-02 β e e-0 β e-04 References [1] A Björck and T Elfving, Algorithms for confluent Vandermonde systems, Numer Math, 21 (197), [2] A Björck and V Pereyra, Solution of Vandermonde systems of equations, Math Comp, 24 (1970), [] J R Cash, Second derivative extended backward differentiation formulas for the numerical integration of stiff systems, SIAM J Numer Anal, 18 (1981), 21 6 [4] J R Cash, A comparison of some codes for the stiff oscillatory problem, Comput Math Appl, 6 (1998), [5] W H Enright, Second derivative multistep methods for stiff ordinary differential equations, SIAM J Numer Anal, 11 (1974), 21 1 [6] W H Enright and J D Pryce, Two Fortran packages for assessing initial value methods, ACM Trans Math, Software, 1 (1987), 1 27 [7] J Field and R M Noyes, Oscillations in chemical systems IV: Limit cycle behavior in a model of a real chemical reaction, J Chem Phys, 60 (1974), [8] G Galimberti and V Pereyra, Solving confluent Vandermonde systems of Hermite type, Numer Math, 18 (1971), [9] C W Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice- Hall, Englewood Cliffs, NJ, 1971

24 CMDE Vol 5, No 4, 2017, pp [10] E Hairer and G Wanner, Solving Ordinary Differential Equations II Stiff and Differential- Algebraic Problems, corr sec print, Springer-Verlag, Berlin, 2002 [11] A C Hindmarsh, Avoiding BDF stability barriers in the MOL solution of advection dominated problems, Appl Numer Math, 17 (1995), [12] G Hojjati, M Y Rahimi and S M Hosseini, New second derivative multistep methods for stiff systems, Appl Math Model, 0 (2006), [1] T E Hull, W H Enright, B M Fellen and A E Sedgwick, Comparing numerical methods for ordinary differential equations, SIAM J Numer Anal, 9 (1972), [14] G Ismail and I Ibrahim, New efficient second derivative multistep methods for stiff systems, Appl Math Model, 2 (1999), [15] M M Khalsaraei, N N Oskuyi and G Hojjati, A class of second derivative multistep methods for stiff systems, Acta Universitatis Apulensis, 0 (2012), [16] F T Krogh, Changing stepsize in the integration of differential equations using modified divided differences, in Proc Conf on the Numerical Solution of Ordinary Differential Equations, University of Texas at Austin 1972 (Ed DG Bettis), Lecture Notes in Mathematics No 62, Springer-Verlag, Berlin, (1974), [17] J D Lambert, Numerical Methods for Ordinary Differential Systems, Wiley, Chichester UK, 1991 [18] T Nguyen-Ba, E Kengne and R Vaillancourt, One-step 4-stage Hermite Birkhoff Taylor ODE solver of order 12, Can Appl Math Q, 16(1) (2008), [19] A Nordsieck, On numerical integration of ordinary differential equations, Math Comp, 16 (1962), [20] N Obrechkoff, Neue Quadraturformeln, Abh Preuss Akad Wiss Math Nat Kl, 4 (1940), 1 20 [21] L R Petzold, A description of DASSL: A differential/algebraic system solver, In Proceedings of IMACS World Congress, Montréal, Canada, 1982 [22] P W Sharp, Numerical comparison of explicit Runge Kutta pairs of orders four through eight, ACM Trans Math Software, 17 (1991), [2] R F Warming and M Beam, An extension of A-stability to alternating direction implicit methods, BIT, 19 (1979),