Fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains

A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered. By applying the energy method, well-posed boundary conditions for the continuous problem are derived. Summation-by-Parts (SBP...

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Bibliographic Details
Published in:Journal of computational physics 2015, Vol.291, p.82-98
Main Authors: Nikkar, Samira, Nordström, Jan
Format: Article
Language:English
Subjects:
Approximation
Boundaries
Boundary conditions
Coefficients
Conservation
Finite difference method
Mathematical analysis
Mathematical models
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Summary:A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered. By applying the energy method, well-posed boundary conditions for the continuous problem are derived. Summation-by-Parts (SBP) operators for the space and time discretization, together with a weak imposition of boundary and initial conditions using Simultaneously Approximation Terms (SATs) lead to a provable fully-discrete energy-stable conservative finite difference scheme. We show how to construct a time-dependent SAT formulation that automatically imposes boundary conditions, when and where they are required. We also prove that a uniform flow field is preserved, i.e. the Numerical Geometric Conservation Law (NGCL) holds automatically by using SBP-SAT in time and space. The developed technique is illustrated by considering an application using the linearized Euler equations: the sound generated by moving boundaries. Numerical calculations corroborate the stability and accuracy of the new fully discrete approximations.
ISSN:0021-9991
1090-2716
1090-2716
DOI:10.1016/j.jcp.2015.02.027