Global Error Estimation with Runge--Kutta Methods II

An analysis of global error estimation using the Zadunaiskytechnique with Runge—Kutta methods is presented. Threeforms of interpolant which can lead to valid asymptotic estimationare considered. Test results indicate that the Hermite formcoupled with special Runge—Kutta formulae is to be preferred,particularly when two-term global error estimation can be obtained.Very reliable estimation can be achieved and it is suggestedthat the technique could form the basis of a production code.     

Global Error Estimation with Runge--Kutta Methods

An analysis of global error estimation for Runge—Kuttasolutions of ordinary differential equations is presented. Thebasic technique is that of Zadunaisky in which the global erroris computed from a numerical solution of a neighbouring problemrelated to the main problem by some method of interpolation.It is shown that Runge—Kutta formulae which permit validglobal error estimation using low-degree interpolation can bedeveloped, thus leading to more accurate and computationallyconvenient algorithms than was hitherto expected. Some specialRunge—Kutta processes up to order 4 are presented togetherwith numerical results.     

High-Order Embedded Runge-Kutta-Nystrom Formulae

New efficient embedded Runge-Kutta-Nystrom processes of orders8(6) and 12(10) are presented for the numerical solution ofthe special second-order differential equation y'(x) = f[x,y(x)]. Test results indicate their improved efficiency relativeto other RKN formulae in current use.     

Families of Runge-Kutta-Nystrom Formulae

Criteria to be satisfied by efficient embedded Runge-Kutta-Nystromformulae are presented, and new families are derived. Test resultsindicate their improved efficiency relative to other RKN formulaein current use.     

High-Order Embedded Runge-Kutta-Nystrom Formulae

Two typographical errors were contained in Table 1 on page 425.Use of the values for a₇₃ and b₇' will invalidate the RKN8(6)9FMformula pair. The correct values for these parameters are The authors are grateful to Steve Stalos of Laurel, MD, USA,for reporting these errors.