Contributed Talk
Error Inhibiting Methods with Post-Processing for Ordinary Differential Equations
School of Mathematical Sciences, Tel Aviv University, Israel
Abstract
Efficient high-order numerical methods for propagating the solution of ordinary differential equations
are essential to numerical simulations of models in science and engineering. Several popular classes
of numerical time-propagating schemes exist, such as the Runge-Kutta, linear multi-step, and general linear methods.
An important characteristic of numerical schemes is the truncation error, loosely speaking,
the error accumulated in one iteration, normalized by the time step. This truncation error governs
the accuracy of these schemes. In all classical schemes, the final error is of the same order as the truncation error.
Our research concerns the interplay between the truncation error and the scheme that generates the final error.
Understanding this interplay enabled us to construct error-inhibiting schemes that impede the accumulation
of the local truncation error over time, resulting in a global error, which is one order higher than expected
from the local truncation error. In this talk, we present this interplay and specify the conditions in which
we can specify the exact form of the leading error term. We use this form to generate a post-processing filter
that enables us to recover a solution that is two orders higher than expected from truncation error analysis.
Several new explicit and implicit methods with this property are given and tested on various ordinary
and partial differential equations, including strong stability preserving (SSP), implicit-explicit (IMEX),
and variable time-step methods. We show that these methods provide a solution that is two orders higher than
expected from truncation error analysis.
This is a joint work with Sigal Gottlieb, Zachary J. Grant, and Guy Rothmann.