Two FORTRAN packages for assessing initial value methods

Enright, W. H.; Pryce, John D.

doi:10.1145/23002.27645

Cited by 146 publications

(88 citation statements)

References 3 publications

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“…Table 1 lists the number of steps (NS) and the maximum global error (GE) of HBT(12)9 and T12 related to the step control for the DETEST problems [11] of class A, B, and E over the time interval [0, 20] with set tolerance (TOL). Thus we can compare the step controls of HBT(12)9 and T12.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The considered problems are Kepler's, Hénon-Heiles' and the equatorial main problems over the time interval [0, t f ] Table 1. For some test problems of [11], time interval [0,20] and LT = log 10 (TOL), the table lists the number of steps (NS) and the maximum global error (GE) for HBT(12)9 (left column) and T12 (right column).…”

Section: Numerical Results Related To the Step Controlmentioning

confidence: 99%

“…where r = r(1 : 9) has components The equations for r(7), r (8) and r(9) correspond to equations (10), (11) and (9), respectively.…”

Section: 5mentioning

confidence: 99%

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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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Section: Numerical Resultsmentioning

confidence: 99%

Section: Numerical Results Related To the Step Controlmentioning

confidence: 99%

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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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“…To derive an appropriate SDC CRK formula we will assume we have an RDC CRK formula, u i (x) satisfying (8). For example, for the particular fifth, sixth and eighth order CRKs we will implement and investigate in the next section, the corresponding RDC CRKs we begin with have been identified and discussed in [5].…”

Section: Deriving An Sdc Crkmentioning

confidence: 99%

“…For a p th order discrete RK formula, we are primarily interested in optimal order (i.e., the corresponding defect is O(h p i ) ) local interpolants satisfying (8) or (9). Interpolants with lower order defects can be used effectively in some applications but, when defect error control is used, such methods will not generally be as efficient.…”

Section: Introductionmentioning

confidence: 99%

The reliability/cost trade-off for a class of ODE solvers

Enright

2009

Numer Algor

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In the numerical solution of ODEs, it is now possible to develop efficient techniques that will deliver approximate solutions that are piecewise polynomials. The resulting methods can be designed so that the piecewise polynomial will satisfy a perturbed ODE with an associated defect (or residual) that is directly controlled in a consistent fashion. We will investigate the reliability/cost trade off that one faces when implementing and using such methods, when the methods are based on an underlying discrete Runge-Kutta formula.In particular we will identify a new class of continuous Runge-Kutta methods with a very reliable defect estimator and a validity check that reflects the credibility of the estimate. We will introduce different measures of the "reliability" of an approximate solution that are based on the accuracy of the approximate solution; the maximum magnitude of the defect of the approximate solution; and how well the method is able to estimate the maximum magnitude of the defect of the approximate solution. We will also consider how methods can be implemented to detect and cope with special difficulties such as the effect of round-off error (on a single step) or the ability of a method to estimate the magnitude of the defect when the stepsize is large (as might happen when using a high-order method at relaxed accuracy requests).Numerical results on a wide selection of problems will be summarized for methods of orders five, six and eight. It will be shown that a modest increase in the cost per step can lead to a significant improvement in the quality of the approximate solutions and the reliability of the method. For example, the numerical results demonstrate that, if one is willing to increase the cost per step by 50%, then a method can deliver approximate solutions where the reported estimated maximum defect is within 1% of its true value on 95% of the steps.

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Handling effectively the rejected stages in Runge–Kutta pairs implementation

Xiang,

Lin,

Simos

et al. 2024

Math Methods in App Sciences

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Runge–Kutta (RK) pairs are widely used for numerically solving initial value problems (IVPs), but dealing with step rejections during integration is a common occurrence. Conventionally, when a step is rejected, all calculations made during that step are discarded, and a completely new set of computations is initiated. In our research, we propose a method to address this inefficiency by repurposing the previously computed RK stages from rejected steps. Our primary focus is on the renowned RKF45 pair, consisting of fifth‐ and fourth‐order methods. When a step rejection occurs, we leverage the stages computed in prior steps and introduce just three additional stages. These stages are then used to evaluate the results with a smaller step size. This approach effectively reduces computational costs in various challenging IVPs where RK algorithms with different step sizes encounter difficulties

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Two FORTRAN packages for assessing initial value methods

Cited by 146 publications

References 3 publications

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

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

The reliability/cost trade-off for a class of ODE solvers

Handling effectively the rejected stages in Runge–Kutta pairs implementation

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