Abstract
A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations is considered. We use the energy method to derive well-posed boundary conditions for the continuous problem. Summation-by-Parts (SBP) operators together with a weak imposition of the boundary and initial conditions using Simultaneously Approximation Terms (SATs) guarantee energy-stability of the fully discrete scheme. We construct a time-dependent SAT formulation that automatically imposes the boundary conditions, and show that the numerical Geometric Conservation Law (GCL) holds. Numerical calculations corroborate the stability and accuracy of the approximations. As an application we study the sound propagation in a deforming domain using the linearized Euler equations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
S. Abrabanel, D. Gottleib, Optimal time splitting for two- and three-dimensional Navier-Stokes equations with mixed derivatives. J. Comput. Phys. 41, 1–43 (1981)
Y. Abe, N. Izuka, T. Nonomura, K. Fuji, Symmetric-conservative metric evaluations for higher-order finite difference scheme with the GCL identities on the three-dimensional moving and deforming mesh, in ICCFD7, (2012)
C. Farhat, P. Geuzaine, C. Grandmont, The discrete geometric conservation law and the nonlinear stability of ALE schemes for the solution of flow problems on moving grids. J. Comput. Phys. 174, 669–694 (2001)
T. Lundquist, J. Nordström, The SBP-SAT technique for initial value problems. J. Comput. Phys. 270, 86–104 (2014)
S. Nikkar, J. Nordström, Fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains, LiTH- MAT-R, 2014:15, Department of Mathematics, Linköping University, 2014
J. Nordström, Conservative finite difference formulations, variable coefficients, energy estimates and artificial dissipation. J. Sci. Comput. 29, 375–404 (2006)
J. Nordström, H. Carpenter, High-order finite difference methods, multidimensional linear problems and curvilinear coordinates. J.Comput. Phys. 173, 149–174 (2001)
J. Nordström, T. Lundquist, Summation-by-parts in time. J. Comput. Phys. 251, 487–499 (2013)
K. Salari, Code verification by the method of manufactured solutions. doi:10.2172/759450
B. Strand, Summation by parts for finite difference approximations of d/dx. J. Comput. Phys. 110, 47–67 (1994)
M. Svärd, J. Nordström, A stable high-order finite difference scheme for the compressible Navier Stokes equations: no-slip wall boundary conditions. J. Comput. Phys. 227(10), 4805–4824 (2008)
P.D. Thomas, C.K. Lombard, Geometric conservation law and its application to flow computations on moving grids. AIAA J. 17, 1030–1037 (1979).
E. Turkel, Symmetrization of the fluid dynamics matrices with applications. Math. Comput. 27, 729–736 (1973)
C.F. Van Loan, The ubiquitous Kronecker product. J. Comput. Appl. Math. 123, 85–100 (2000)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Nikkar, S., Nordström, J. (2015). Fully Discrete Energy Stable High Order Finite Difference Methods for Hyperbolic Problems in Deforming Domains. In: Kirby, R., Berzins, M., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014. Lecture Notes in Computational Science and Engineering, vol 106. Springer, Cham. https://doi.org/10.1007/978-3-319-19800-2_35
Download citation
DOI: https://doi.org/10.1007/978-3-319-19800-2_35
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-19799-9
Online ISBN: 978-3-319-19800-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)