Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T21:21:28.515Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of Finite Difference Discretizations of Multi-Physics Interface Conditions

Published online by Cambridge University Press:  03 June 2015

Björn Sjögreen  and
Jeffrey W. Banks
Get access

Abstract

We consider multi-physics computations where the Navier-Stokes equations of compressible fluid flow on some parts of the computational domain are coupled to the equations of elasticity on other parts of the computational domain. The different subdomains are separated by well-defined interfaces. We consider time accurate computations resolving all time scales. For such computations, explicit time stepping is very efficient. We address the issue of discrete interface conditions between the two domains of different physics that do not lead to instability, or to a significant reduction of the stable time step size. Finding such interface conditions is non-trivial.

We discretize the problem with high order centered difference approximations with summation by parts boundary closure. We derive L2 stable interface conditions for the linearized one dimensional discretized problem. Furthermore, we generalize the interface conditions to the full non-linear equations and numerically demonstrate their stable and accurate performance on a simple model problem. The energy stable interface conditions derived here through symmetrization of the equations contain the interface conditions derived through normal mode analysis by Banks and Sjögreen in [8] as a special case.

Keywords

65M0665M1274F10Fluid-structure interactionfinite difference methodsummation by partsmultiphysics interface condition
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 2 , February 2013 , pp. 386 - 410
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Abarbanel, S. and Gottlieb, D., Optimal time splitting for two- and three-dimensional Navier-Stokes equations with mixed derivatives, J. Comput. Phys., 41 (1981), 1–33.Google Scholar
[2]Banks, J. W., Henshaw, W. D. and Schwendeman, D. W., Deforming composite grids for solving fluid structure problems, J. Comput. Phys., 231 (2012), 3518–3547.Google Scholar
[3]Belytschko, T., Liu, W. K. and Moran, B., Nonlinear Finite Elements for Continua and Structures, John Wiley & Sons Ltd., 2000.Google Scholar
[4]Carpenter, M. H., Gottlieb, D. and Abarbanel, S., Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes, J. Comput. Phys., 111 (1994), 220–236.Google Scholar
[5]Chesshire, G. S. and Henshaw, W. D., Composite overlapping meshes for the solution of partial differential equations, J. Comput. Phys., 90(1) (1990), 1–64.CrossRefGoogle Scholar
[6]Halpern, L., Artificial boundary conditions for incompletely parabolic perturbations of hyperbolic systems, SIAM J. Math. Anal., 22 (1991), 1256–1283.Google Scholar
[7]Henshaw, W. D. and Chand, K. K., A composite grid solver for conjugate heat transfer in fluid-structure systems, J. Comput. Phys., 228 (2009), 3708–3741.Google Scholar
[8]Banks, J. W. and Sjögreen, B., Stability of numerical interface conditions for fluid/structure interaction, Commun. Comput. Phys., 10 (2011), 279–304.CrossRefGoogle Scholar
[9]Kreiss, H.-O. and Petersson, N. A., A second order accurate embedded boundary method for the wave equation with Dirichlet data, SIAM J. Sci. Comput., 27 (2006), 1141–1167.Google Scholar
[10]Kreiss, H.-O., Petersson, N. A. and Ystrom, J., Difference approximations of the Neumann problem for the second order wave equation, SIAM J. Numer. Anal., 42 (2004), 1292–1323.CrossRefGoogle Scholar
[11]Nilsson, S., Petersson, N. A., Sjögreen, B. and Kreiss, H.-O., Stable finite difference approximations for the elastic wave equation in second order formulation, SIAM J. Numer. Anal., 45 (2007), 1902–1936.Google Scholar
[12]Olsson, P., Summation by parts, projections and stability I, Math. Comput., 64 (1995), 1035–1065.CrossRefGoogle Scholar
[13]Sjögreen, B. and Petersson, N. A., A fourth order accurate finite difference scheme for the elastic wave equation in second order formulation, J. Sci. Comput., 2011, LLNL-JRNL-483427.Google Scholar
[14]Sjögreen, B. and Yee, H. C., On skew-symmetric splitting and entropy conservaton schemes for the euler equations, in Proceedings of ENUMATH09, Uppsala, Sweden, June 29-July 3, 2009.Google Scholar
[15]Sjögreen, B. and Yee, H. C., Multiresolution wavelet based adaptive numerical dissipation control for shock-turbulence computation, J. Sci. Comput., 20 (2004), 211–255.Google Scholar
[16]Strand, B., Summation by parts for finite difference approximations for d/dx, J. Comput. Phys., 110 (1994), 47–67.Google Scholar
[17]Wilkins, M. L., Computer Simulation of Dynamic Phenomena, Springer, 1999.Google Scholar