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Runge-Kutta and Adams methods are the most popular codes to solve numerically nonstiff ODEs. The Adams methods are useful to reduce the number of function calls, but they usually require more CPU time than the Runge-Kutta methods. In this work we develop a numerical study of a variable step length Adams implementation, which can only take preassigned step-size ratios. Our aim is the reduction of the CPU time of the code by means of the precalculation of some coefficients. We present several numerical tests that show the behaviour of the proposed implementation.

The Adams methods (Bashforth and Adams [

When the stepsize is constant, the explicit

The formulation of the implicit

The

There are lots of different ways to implement a variable-stepsize variable-order Adams code (actually one different for each developer). For example, codes as DEABM [

Krogh developed a way to compute the Adams formulas in terms of modified divided differences. Following the notation of Hairer et al. [

The coefficients can be calculated easily by the recurrences

The coefficients “

Let us study the number of ratios needed for each coefficient.

For

We only need to apply the relation (with

Let us fix only

As the ratios

The importance of Proposition

For the predictor (

Again from Proposition

The “

A variable-stepsize algorithm needs an estimator of the error to change the stepsize. This estimator can be computed in different ways. We follow Lambert's [

As the

In practice the variable-stepsize Adams method is also endowed with the capability of changing its order, so it is necessary to obtain some local error estimators for decreasing or increasing the order. By modifying the previous equation we obtain the estimators

The “

The CPU time used by the code will increase drastically if we let the order grow without control. For this reason it is necessary to impose a maximum order. In double precision (8 bytes per data, 15-16 decimal digits) the maximum order is usually

The “

The amount of RAM needed to store and read the coefficients can be an important disadvantage of a

Usually, the implementation of the variable-stepsize Adams method includes a restriction in the ratios of the kind of

The fixed ratios implementation that we propose selects the ratio as the largest

For the numerical study we programmed a code, denoted as VSVOABM, with a classical implementation. VSVOABM uses (

The authors presented in [

Five revolutions of the classical

An

The

A fictitious seven-body problem, denoted

In the figures we compare the decimal logarithm of the

The main conclusions were the following.

Ratios

Coefficients “

FRABM methods with

We present some results of five revolutions of a highly eccentric orbit

Decimal logarithm of the stepsizes in a highly eccentric two-body problem.

Ratios in a highly eccentric two-body problem.

We have tested some other eccentricities in the Two-Body problem. As the eccentricity reduces to 0, the changes in the stepsizes in Figure

In Table

Percentiles 25, 50, and 75 of the ratios for the highly eccentric

Tolerance | |||
---|---|---|---|

0.94 | 1.04 | 1.17 | |

0.96 | 1.11 | 1.14 | |

0.95 | 1.07 | 1.11 | |

0.93 | 1.04 | 1.09 | |

0.93 | 1.03 | 1.07 | |

0.94 | 1.02 | 1.06 | |

0.95 | 1.02 | 1.04 | |

0.95 | 1.02 | 1.04 | |

0.96 | 1.01 | 1.03 | |

0.97 | 1.01 | 1.03 | |

0.97 | 1.01 | 1.02 |

Relevant percentiles of the ratios of accepted steps in a global sample.

0.94 | 0.96 | 0.97 | 0.98 | 0.98 | 1.00 | 1.01 | 1.02 | 1.03 | 1.03 | 1.04 | 1.04 | 1.09 |

We must conclude that a good choice of a finite number of prefixed ratios must select them near to 1, even for problems with big changes in the stepsize.

The coefficients “

We present different FRABM integrations for the Arenstorf problem, all of them with

In all the tests we made, both

FRABM, computing versus reading “

Now we will develop some concrete FRABM algorithms. It is impossible to reproduce all of the experiments we made in [

First of all, we looked for the minimum and maximum ratios

In our experiments there were no clear winners, but maybe the pair

With the addition of the ratio

In the vast majority of occasions the stepsize is accepted, not rejected. That is why we decided to keep

Two different

We tested several different strategies for

For no specific reason, we add the ratio 1.05 to develop a

RAM cost in terms of the number of prefixed ratios

Number of coefficients | RAM cost | |
---|---|---|

3 | 29523 | 0.23 MB |

4 | 349524 | 2.67 MB |

5 | 2441405 | 18.63 MB |

6 | 12093234 | 92.26 MB |

FRABM

In Figure

We cannot conclude yet that the best method is FRABM5, because when

We divided the methods into three categories in terms of their RAM requirements (see Table

RAM cost in terms of the number of prefixed ratios

RAM | Category | ||
---|---|---|---|

3 | 13 | 2.03 MB | Low cost |

4 | 11 | 2.67 MB | Low cost |

5 | 10 | 3.73 MB | Low cost |

4 | 12 | 10.67 MB | Middle cost |

5 | 11 | 18.63 MB | Middle cost |

4 | 13 | 42.67 MB | High cost |

5 | 12 | 93.13 MB | High cost |

We rejected the case

FRABM5, maximum orders

The behaviour of the methods shown in Figure

Figures

Low cost, FRABM3 with

High cost, FRABM3 with

In this section we compare the pair of selected FRABM5 methods (green line with stars for

Error versus CPU time. FRABM5

Error versus evaluations. FRABM5

Let us interpret the figures. At a point where a pair of methods has the same

Taking into account only the Adams methods and

When comparing the

It is well known that Runge-Kutta methods, like DOPRI853, can hardly compete with multistep methods when comparing the number of evaluations. This can clearly be seen in all of the experiments. However, as Runge-Kutta methods do not need to recalculate their coefficients in each step, they are very superior in CPU time unless the function

The function

The Pleiades problem is expensive to evaluate because it is formed by 28

A discretized

A discretized hanging

The comparison in number of function calls in the Brusselator problem does not show any new significant conclusion. However, in the Rope examples FRAMB5,

In the Brusselator problem, DOPRI853 clearly beats all FRABM5 methods in CPU time, which corresponds to the indication of Hairer, Nørsett, and Wanner. In the Rope problem we can also see the negative influence of the dimension in the behaviour of the multistep methods. With 8 first-order equations FRABM5 methods are more efficient than DOPRI853, specially with

VSVOABM is not far from FRABM5 methods in the Brusselator and Rope problems, but it only beats

We sum up the conclusions of this efficacy section.

When the CPU time is used to measure the efficacy, DOPRI853 is the best choice when

When comparing the number of evaluations,