Global Error versus Tolerance for Explicit Runge-Kutta Methods

Abstract

Initial value solvers typically input a problem specification and an error tolerance, and output an approximate solution. Faced with this situation many users assume, or hope for, a linear relationship between the global error and the tolerance. In this paper we examine the potential for such 'tolerance proportionality' in existing explicit Runge-Kutta algorithms. We take account of recent developments in the derivation of high-order formulae, defect control strategies, and interpolants for continuous solution and first derivative approximations. Numerical examples are used to verify the theoretical predictions. The analysis draws on the work of Stetter, and the numerical testing makes use of the nonstiff DETEST package of Enright and Pryce.

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

“…This property is known in the literature as TP. Later investigations by Higham [10,11] extended Stetter's results to more general adaptive methods that possess a TP dependence of type ge n = v(t n ) r + o( r ), → 0 + , n = 1, 2, . .…”

“…In the above-mentioned TP theory developed by Higham [10][11][12]2] a continuous variation of the step-size along the integration interval was assumed in the sense that the step-size h n+1 in advancing from t n to t n+1 =t n +h n+1 satisfies an asymptotic relation h n+1 = (t n ) 1/q + o( 1/q ) with a fixed integer q 1 and a continuous problem-depending function (t) * > 0. Such a relation implies that, in general, the step-size will vary from step to step but this does not affect negatively the performance of explicit RK codes in which the computational cost is independent of the size of the step.…”

“…In general, these estimations are costly, fail in problems with stability difficulties or when the numerical method is near the limit of accuracy and they are not included in standard integrators. An alternative less demanding is to make use of the TP property exhibited for many adaptive codes, in which case the existence of an asymptotic expansion on the tolerance ( ), allows to derive an estimate of the global error [10,11,16,19,20]. This estimation is based on global extrapolation when considering two independent integrations for different tolerances.…”

Parallel defect control

“…The defect (residual) in a continuous approximation to the solution of an initial value problem is a useful theoretical tool [17,26,28], and Enright [6] recently suggested using a defect sample as the basis of an error control mechanism. Such an approach is intuitively reasonable -if the defect is small then a "nearby" problem has been solved, Furthermore, standard differential inequalities [13~ page 56] show that the global error satisfies a method-independent bound and hence, in this sense, routines that control the defect can be thought of as interchangeable black boxes.…”

“…Such an approach is intuitively reasonable -if the defect is small then a "nearby" problem has been solved, Furthermore, standard differential inequalities [13~ page 56] show that the global error satisfies a method-independent bound and hence, in this sense, routines that control the defect can be thought of as interchangeable black boxes. Defect control is also one way to ensure tolerance proportionality; that is, an asymptotically linear relationship between the global error and the accuracy tolerance [17,26].…”