Research reports

Years: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

An Accuracy Barrier for Stable Three-Time-Level DifferenceSchemes for Hyperbolic Equations

by R. Jeltsch and R. A. Renaut and J. H. Smit

(Report number 1995-01)

Abstract
We consider three-time-level difference schemes for the linear constant coefficient advection equation $emu_t=cu_x/em$. In 1985 it was conjectured that the barrier to the local order emp/em of schemes which are stable is given by $[emp \le 2\min/em{emR,S/em}]$. Here emR/em and emS/em denote the number of downwind and upwind points, respectively, in the difference stencil with respect to the characteristic of the differential equation through the update point. Here we prove the conjecture for a class of explicit and implicit schemes of maximal accuracy. In order to prove this result, the existing theory on order stars has to be generalized to the extent where it is applicable to an order star on the Riemann surface of the algebraic function associated with a difference scheme. Proof of the conjecture for all schemes relies on an additional conjecture about the geometry of the order star.

Keywords: scalar advection equation, difference scheme,accuracy, stability, order star, algebraic function, Riemann surface

@Techreport{JRS95_169,
  author = {R. Jeltsch and R. A. Renaut and J. H. Smit},
  title = {An Accuracy Barrier for Stable Three-Time-Level DifferenceSchemes for Hyperbolic Equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {1995-01},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports1995/1995-01.pdf },
  year = {1995}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).