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High-order methods for parabolic equations using a space-time flux reconstruction approach

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conference contribution
posted on 2022-03-08, 09:46 authored by Thomas A McCaughtry, Rob WatsonRob Watson, Marco Geron
A novel implementation of the flux reconstruction (FR) approach featuring a hyperbolic reformulation of the governing equations in space-time is presented for viscous linear and non-linear flow problems in both one and two spatial dimensions. The procedure generates high-order accurate schemes — in both space and time — that can analyse diffusion-type equations by recasting second-order equations as first-order systems. Conventional high-order accurate analysis of parabolic equations is severely restricted by limits on time step, which can be avoided by reformulation into a system of hyperbolic equations, with the caveat that only steady solutions may be considered. However, the computation of the resulting system within the space-time FR framework permits the high-order accuracy analysis of unsteady flows with rapid convergence to the steady state, in the pseudo-time sense, within a procedure that can be implemented in a straightforward manner for diffusion-type problems. Eigendecomposition is used to demonstrate that the new systems are hyperbolic in nature for both the 1D and 2D Advection-Diffusion Equations. The development and successful implementation of first-order space-time FR schemes for the 1D and 2D Diffusion Equations is illustrated. It is also verified that the target order-of-accuracy (OOA) is achieved for schemes involving both one and two spatial dimensions. An application of the space-time flux reconstruction approach to the Euler Equations is presented and discussed, with a view to future implementation to the Unsteady Navier-Stokes Equations with similar hyperbolic reformulation of viscous terms.

History

School

  • Aeronautical, Automotive, Chemical and Materials Engineering

Department

  • Aeronautical and Automotive Engineering

Published in

AIAA AVIATION 2021 FORUM

Source

AIAA AVIATION 2021 FORUM

Publisher

American Institute of Aeronautics and Astronautics, Inc.

Version

  • AM (Accepted Manuscript)

Rights holder

© The Authors

Publisher statement

This is the accepted version of a paper presented at the AIAA AVIATION 2021 FORUM. The definitive published version is available at https://doi.org/10.2514/6.2021-2736

Publication date

2021-07-28

Copyright date

2021

ISBN

9781624106101

Language

  • en

Location

Virtual

Event dates

2nd August 2021 - 6th August 2021

Depositor

Dr Rob Watson. Deposit date: 7 March 2022

Article number

AIAA 2021-2736

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