Global error estimation based on the tolerance proportionality for some adaptive Runge–Kutta codes

Abstract

Modern codes for the numerical solution of Initial Value Problems (IVPs) in ODEs are based in adaptive methods that, for a user supplied tolerance , attempt to advance the integration selecting the size of each step so that some measure of the local error is . Although this policy does not ensure that the global errors are under the prescribed tolerance, after the early studies of Stetter [Considerations concerning a theory for ODE-solvers, in: R. The reliability of standard local error control algorithms for initial value ordinary differential equations, in: Proceedings: The Quality of Numerical Software: Assessment and Enhancement, IFIP Series, Springer, Berlin, 1997], it has been proved that in many existing explicit Runge-Kutta codes the global errors behave asymptotically as some rational power of . This step-size policy, for a given IVP, determines at each grid point t n a new step-size h n+1 = h(t n ; ) so that h(t; ) is a continuous function of t.In this paper a study of the tolerance proportionality property under a discontinuous step-size policy that does not allow to change the size of the step if the step-size ratio between two consecutive steps is close to unity is carried out. This theory is applied to obtain global error estimations in a few problems that have been solved with the code Gauss2 [S. Gonzalez-Pinto, R. Rojas-Bello, Gauss2, a Fortran 90 code for second order initial value problems, http://www.netlib.org/ode/ ], based on an adaptive two stage Runge-Kutta-Gauss method with this discontinuous step-size policy.

A Transformation Method for Solving Conservative Nonlinear Two-Degree-of-Freedom Systems

“…whose exact solutions are similar to those given by (6). We shall next explore the applicability of the nonlinear transformation method in obtaining the equivalent representation form of (1) and, then, we will compare the numerical integration solutions of five dynamics systems, having nonlinear restoring forces with rational or irrational terms, with respect to their equivalent representation forms [13][14][15][16][17][18][19][20]. First, let us consider the case for which the restoring forces are of the cubic type.…”

“…This situation increases the RMSE errors as listed in Table 1. In an attempt to further quantify the accuracy of our proposed procedure, we next introduce the global error estimation based on a technique similar to the ones developed in [14][15][16] in which the system global error, GE, can be determined from the following expression:…”

Stepwise Global Error Control in an Explicit Runge-Kutta Method Using Local Extrapolation with High-Order Selective Quenching

“…wherein we have implicitly defined coefficients b ••• q and appropriate increment functions. It transpires that DP853 has b H,V q = 0 for q ∈ [2, 3, 4, 5] and b L q = 0 for q ∈ [2, 3,4,5,6,7,8,10,11] .…”

“…which follows from (6). The second term on the RHS is the contribution to ∆ L i+1 due to the global error of the input µ H i or µ L i .…”

A Singly Diagonally Implicit Two-Step Peer Triple with Global Error Control for Stiff Ordinary Differential Equations

“…Many issues related to the theory and practice of global error estimation and control have already been addressed in literature. At present, we are familiar with good global error estimation methods and error control strategies developed in onestep, multistep, Nordsieck, peer, and many other classes of schemes [1,4,5,6,7,8,9,10,12,13,16,17,21,22,23,27,28,29,30,34,35,36,41,42,44,45,46,47,49,50]. The purpose of our paper is to design an efficient and accurate adaptive ODE solver for stiff and very stiff problems, which arise in various areas of applied research, as those listed above.…”