One-step 5-stage Hermite–Birkhoff–Taylor ODE solver of order 12

Nguyen‐Ba, Truong; Han, Hao; Yagoub, Hemza; Vaillancourt, Rémi

doi:10.1016/j.amc.2009.01.043

Cited by 6 publications

(6 citation statements)

References 17 publications

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“…, y k−1 . The starting values for HBO(d, k, p) are calculated by the one-step, 4-stage, Hermite-Birkhoff-Taylor method of order d + 3 using y to y (d) with appropriate small step sizes [19]. The d derivatives, y to y (d) , of the Taylor series are calculated at each integration step by known recurrence formulae (see, for example, [9, pp.…”

Section: Numerical Resultsmentioning

confidence: 99%

On contractivity preserving 4- to 7-step predictor-corrector HBO series for ODEs

Nguyen‐Ba¹,

Giordano²,

Nguyen-Thu³

et al. 2017

J. of Mod. Meth. in Numer. Math.

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The contractivity-preserving 2-and 3-step predictor-corrector series methods for ODEs (T. Nguyen-Ba, A. Alzahrani, T. Giordano and R. Vaillancourt, On contractivity-preserving 2-and 3-step predictor-corrector series for ODEs, J. Mod. Methods Numer. Math. 8:1-2 (2017), pp. 17-39. doi:10.20454/jmmnm.2017.1130) are expanded into new optimal, contractivity-preserving (CP), d-derivative, k-step, predictor-corrector, Hermite-Birkhoff-Obrechkoff series methods, denoted by HBO(d, k, p), k = 4, 5, 6, 7, with nonnegative coefficients for solving nonstiff first-order initial value problems y = f (t, y), y(t 0 ) = y 0 . The main reason for considering this class of formulae is to obtain a set of methods which have larger regions of stability and generally higher upper bound p u of order p of HBO(d, k, p) for a given d. Their stability regions have generally a good shape and grow generally with decreasing p − d. A selected CP HBO method: 6-derivative 4-step HBO of order 14, denoted by HBO(6,4,14) which has maximum order 14 based on the CP conditions compares satisfactorily with Adams-Cowell of order 13 in PECE mode, denoted by AC(13), in solving standard N-body problems over an interval of 1000 periods on the basis of the relative error of energy as a function of the CPU time. HBO(6,4,14) also compares well with AC(13) in solving standard N-body problems on the basis of the growth of relative positional error, relative energy error and 10000 periods of integration. The coefficients of HBO(6,4,14) are listed in the appendix.

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Section: Numerical Resultsmentioning

confidence: 99%

On contractivity preserving 4- to 7-step predictor-corrector HBO series for ODEs

Nguyen‐Ba¹,

Giordano²,

Nguyen-Thu³

et al. 2017

J. of Mod. Meth. in Numer. Math.

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“…5.5174421509757043e-01 9.9999999999999978e-01 1.0000000000000000e+00 a 22 5.5174421509757043e-01 6.6666666666666652e-01 5.4545454545454553e-01 α 20 1.0 1.3333333333333333e+00 1.6363636363636365e+00 α 21 -3.3333333333333326e-01 -8.1818181818181823e-01 α 22 1.8181818181818182e-01 c 3 2.4362881788049254e-01 1.3900272722515278e-01 3.2993104524169897e-01 a 32 -3.0811539721707792e-01 -2.3122638363300693e-01 -1.2058808235627541e-01 α 30 1.0 7.0356244419149327e-01 8.5264981456489419e-01 α 31 2.9643755580850684e-01 1.9976495301364053e-01 α 32 -5.2414767578534707e-02 c 4 1.0000000000000000e+00 1.1332868919448371e+00 1.0627316813082199e+00 a 43 7.9334000563492402e-01 1.0438074153541037e+00 9.4729711406669859e-01 a 42 -3.4508422073249445e-01 -3.6469378491528104e-01 -3.4384938036290208e-01 α 40 1.0000000000000000e+00 7.8750659483934782e-01 8.8707824364921162e-01 α 41 2.1249340516065218e-01 1.3967291485145458e-01 α 42 -2.6751158500666193e-02 b 4 -3.4508422073249451e-01 -5.2429845046824930e-01 -4.2620226004797684e-01 b 3 4.6859311278708565e-01 5.4584881154184672e-01 6.7230267871607263e-01 b 2 3.2474689284783831e-01 3.3835393825928867e-01 1.9264155566555125e-01 α 0 1.0 9.7342903400044756e-01 1.0167342380881887e+00 α 1 2.6570965999552467e-02 -1.7664995964569472e-02 α 2 9.3075787638099094e-04 problems. For HBDF methods, we have used a Matlab code of Newton-Krylov solver with generalized minimum residual (GMRES), i.e.…”

Section: Comparing Cpu Time Of Hb(p) P = 9 10 and Hbdf(p) P = 10 12mentioning

confidence: 98%

“…These elementary matrix functions are used, first, to find the solution u , = 1, 2, 4, 5 in elementary matrix functions form and, then, to construct fast Algorithms 3, 4, 5 and 4, in Appendix A, to solve systems (18), (20), (22) (whose matrix M 3 = M 2 of (20)), (23) and (26) at each integration step.…”

Section: Fast Solution Of Vandermonde-type Systems For Particular Hb(p)mentioning

confidence: 99%

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On variable step Hermite–Birkhoff solvers combining multistep and 4-stage DIRK methods for stiff ODEs

Nguyen‐Ba

2015

Numer Algor

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Variable-step (VS) 4-stage k-step Hermite-Birkhoff (HB) methods of order p = (k + 2), p = 9, 10, denoted by HB(p), are constructed as a combination of linear k-step methods of order (p − 2) and a diagonally implicit one-step 4-stage Runge-Kutta method of order 3 (DIRK3) for solving stiff ordinary differential equations. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep and Runge-Kutta type order conditions which are reorganized into linear confluent Vandermonde-type systems. This approach allows us to develop L(α)-stable methods of order up to 11 with α > 63 • . Fast algorithms are developed for solving these systems in O(p 2 ) operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The stepsizes of these methods are controlled by a local error estimator. HB(p) of order p = 9 and 10 compare favorably with existing Cash modified extended backward differentiation formulae of order 7 and 8, MEBDF(7-8) and Ebadi et al. hybrid backward differentiation formulae of order 10 and 12, HBDF(10-12) in solving problems often used to test higher order stiff ODE solvers on the basis of CPU time and error at the endpoint of the integration interval.

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“…. , 9, in (18) and (19) with the corresponding righthand sides of (18) (12) A complex number λh is in R if r s satisfies the root condition: |r s | 1 (see [12, pp. 378-380]).…”

Section: Region Of Absolute Stabilitymentioning

confidence: 99%

Nine-stage multi-derivative Runge-Kutta method of order 12

Nguyen‐Ba

Bozic

Kengne

et al. 2009

Publ Inst Math (Belgr)

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Abstract. A nine-stage multi-derivative Runge-Kutta method of order 12, called HBT(12)9, is constructed for solving nonstiff systems of first-order differential equations of the form y = f (x, y), y(x 0 ) = y 0 . The method uses y and higher derivatives y (2) to y (6) as in Taylor methods and is combined with a 9-stage Runge-Kutta method. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution leads to order conditions which are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The stepsize is controlled by means of the derivatives y (3) to y (6) . The new method has a larger interval of absolute stability than Dormand-Prince's DP(8,7)13M and is superior to DP(8,7)13M and Taylor method of order 12 in solving several problems often used to test high-order ODE solvers on the basis of the number of steps, CPU time, maximum global error of position and energy. Numerical results show the benefits of adding high-order derivatives to Runge-Kutta methods.

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One-step 5-stage Hermite–Birkhoff–Taylor ODE solver of order 12

Cited by 6 publications

References 17 publications

On contractivity preserving 4- to 7-step predictor-corrector HBO series for ODEs

On contractivity preserving 4- to 7-step predictor-corrector HBO series for ODEs

On variable step Hermite–Birkhoff solvers combining multistep and 4-stage DIRK methods for stiff ODEs

Nine-stage multi-derivative Runge-Kutta method of order 12

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