Error Inhibiting Block One-step Schemes for Ordinary Differential Equations

Abstract

The commonly used one step methods and linear multi-step methods all have a global error that is of the same order as the local truncation error (as defined in [6,13,1,8,15]). In fact, this is true of the entire class of general linear methods. In practice, this means that the order of the method is typically defined solely by order conditions which are derived by studying the local truncation error. In this work we investigate the interplay between the local truncation error and the global error, and develop a methodology which defines the construction of explicit error inhibiting block one-step methods (alternatively written as explicit general linear methods [2]). These error inhibiting schemes are constructed so that the accumulation of the local truncation error over time is controlled, which results in a global error that is one order higher than the local truncation error. In this work, we delineate how to carefully choose the coefficient matrices so that the growth of the local truncation error is inhibited. We then use this theoretical understanding to construct several methods that have higher order global error than local truncation error, and demonstrate their enhanced order of accuracy on test cases. These methods demonstrate that the error inhibiting concept is realizable. Future work will further develop new error inhibiting methods and will analyze the computational efficiency and linear stability properties of these methods.

Explicit and implicit error inhibiting schemes with post-processing

“…Using a selection of the IMEX-EIS+ methods we developed, we evolve this in time to T f = 0.5, for ∆t = T f N for various values of N . In Table 3 we show the convergence rates of the IMEX-EIS+ methods and the pIMEX-EIS+ methods (3,3), (3,4), and (4,5) for values of N from 210 to 1440, and the IMEX-EIS+ (5,6) method for N from 60 to 280 (due to the high accuracy of the solutions). Convergence plots shown in Figure 3.…”

“…Solid line is before post-processing and dotted line after post-processing. Right: non-parallelizable methods IMEX-EIS+ (3,3) in blue, IMEX-EIS+ (3,4) in red, and IMEX-EIS+ (4,5) in green. Solid line is before post-processing and dotted line after post-processing.…”

“…The methods considered usually have a a global error that is of the same order as the local truncation error. More recently, methods with global errors that are one order higher than predicted by the local truncation error have been devised [18,28,5]. In prior work [6,7] we investigated the interplay between the local truncation error and the global error to construct explicit and implicit error inhibiting schemes that control the accumulation of the local truncation error over time, resulting in a global error that is one order higher than expected from the local truncation error, and which can be post-processed to obtain a solution which is two orders higher than expected.…”

“…These methods have truncation errors that are of order p but have global errors of order p + 1. In [5] Ditkowski and Gottlieb derived sufficient conditions for a class of general linear methods (GLMs) under which one can control the accumulation of the local truncation error over time evolution and produce methods that are one order higher than expected from the truncation error alone.…”

“…In recent work, Schneider and colleagues [23,24] produced super-convergent IMEX Peer methods (a Peer method is a GLM where each stage is of the same order). These methods satisfy error inhibiting conditions similar to those in [5] and so produce order p + 1 although their truncation error is of order p, so we shall refer to them here as IMEX-EIS schemes. In this work, we extend the error inhibiting with post-processing theory in [6] and [7] to IMEX methods.…”