Local error estimation by doubling

Abstract

ZusammenfassungLocal Error Estimation by Doubling. Doubling, or Richardson extrapolation, is a general principle for the estimation of the local error made by a one-step method for the numerical solution of the initial value problem for a system of ordinary differential equations. Some contributions are made to the theory of doubling. Principles of comparing explicit Runge-Kutta formulas are reviewed and illustrated in a balanced appraisal of doubling in the context of fourth order formulas. Some apparently contradictory comparisons in the literature are explained.

Embedded Runge–Kutta scheme for step-size control in the interaction picture method

“…In [1][2][3] it is solved by a Fourier spectral method. For the GNLSE, problem (16) where Ω = R can be solved by a direct use of Fourier transforms [4,14]. Moreover the cost of the evaluation of the 4 non-linear terms N(ϕ) is strongly dependent to the physical application.…”

“…The idea behind the step doubling method (also known as Richardson extrapolation method) for estimation of the local error is the following [16]. The local error ℓ [4] k+1 for the RK4 method (12) at grid point s k+1 is given by (24).…”

“…However, since in the ERK4(3) method the local error for each step is computed faster than in the SD method the total CPU time of the computation is much lower. The global error at the fiber end (z = L) is smaller when the IP method is used in conjunction with the SD method since it is not the solution computed with the RK4 method that is propagated but the more accurate one computed with the half step-size in the SD method [16].…”

“…Some attempts for using an adaptive step-size control strategy in conjunction with the RK4-IP method have however been made. The well known step-doubling approach (or Richardson extrapolation) is the more common way for local error estimation and step-size control [16]. However its computational over cost may be considered as prohibitive.…”

Convergence of a step-doubling Galerkin method for parabolic problems

“…This single step method consists of taking two half steps of backward Euler over a time interval to obtain one approximate solution and then a single step over that same interval to obtain another [8,12]. Having both approximations gives us two things: the two approximations can be compared to get an estimate of the size of the time-integration error; and the two firstorder correct approximations can be extrapolated to obtain a second-order correct approximate solution.…”