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A Flexible Boundary Procedure for Hyperbolic Problems: Multiple Penalty Terms Applied in a Domain

Published online by Cambridge University Press:  03 June 2015

Jan Nordström ,
Qaisar Abbas ,
Brittany A. Erickson  and
Hannes Frenander
Jan Nordström*
Affiliation:
Department of Mathematics, Division of Computational Mathematics, Linköping University, SE-581 83 Linköping, Sweden
Qaisar Abbas*
Affiliation:
Department of Information Technology, Division of Scientific Computing, Uppsala University, SE-751 05 Uppsala, Sweden
Brittany A. Erickson*
Affiliation:
Department of Geological Sciences, San Diego State University, San Diego, CA 92182-1020, USA
Hannes Frenander*
Affiliation:
Department of Mathematics, Division of Computational Mathematics, Linköping University, SE-581 83 Linköping, Sweden
*
Corresponding author.Email:jan.nordstrom@liu.se
Email:qaisar.abbas@it.uu.se
Email:berickson@projects.sdsu.edu
Email:hannes.frenander@liu.se
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Abstract

A new weak boundary procedure for hyperbolic problems is presented. We consider high order finite difference operators of summation-by-parts form with weak boundary conditions and generalize that technique. The new boundary procedure is applied near boundaries in an extended domain where data is known. We show how to raise the order of accuracy of the scheme, how to modify the spectrum of the resulting operator and how to construct non-reflecting properties at the boundaries. The new boundary procedure is cheap, easy to implement and suitable for all numerical methods, not only finite difference methods, that employ weak boundary conditions. Numerical results that corroborate the analysis are presented.

Keywords

35L0535L6535Q3565M0665M1265Z05Summation-by-partsweak boundary conditionspenalty techniquehigh-order accuracyfinite difference schemesstabilitysteady-statenon-reflecting boundary conditions.
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 2 , August 2014 , pp. 345 - 358
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Abarbanel, S. and Gottlieb, D.Optimal time splitting for two- and three-dimensional Navier- Stokes equations with mixed derivatives. Journal of Computational Physics, 41(1):1–33, 1981.Google Scholar
[2]Abbas, Q. and Nordström, J.Weak versus strong no-slip boundary conditions for the Navier- Stokes equations. Engineering Applications of Computational Fluid Mechanics, 4(1):29–38, 2010.CrossRefGoogle Scholar
[3]Berg, J. and Nordström, J.Stable robin solid wall boundary conditions for the Navier-Stokes equations. Journal of Computational Physics, 230(19): 7519–7532, 2011.Google Scholar
[4]Bodony, D. J.Analysis of sponge zones for computational fluid mechanics. Journal of Computational Physics, 212(2): 681–702, 2006.Google Scholar
[5]Bodony, D. J.Accuracy of the simultaneous-approximation-term boundary condition for time-dependent problems. Journal of Scientific Computing, 43(1): 118–133, 2010.CrossRefGoogle Scholar
[6]Carpenter, M. H., Nordström, J., and Gottlieb, D.Revisiting and extending interface penalties for multi-domain summation-by-parts operators. Journal of Scientific Computing, 45(1-3): 118–150, 2010.Google Scholar
[7]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. Journal of Computational Physics, 111(2): 220–236, 1994.Google Scholar
[8]Carpenter, M. H., Nordström, J., and Gottlieb, D.A stable and conservative interface treatment of arbitrary spatial accuracy. Journal of Computational Physics, 148: 341–365, 1999.Google Scholar
[9]Colonius, T.Modeling artificial boundary conditions for compressible flow. Annual Review of Fluid Mechanics, 336: 315–345, 2004.CrossRefGoogle Scholar
[10]Davies, H. C.A lateral boundary formulation for multilevel prediction models. Quart. J.R. Met. Soc., 102: 405–418, 1976.Google Scholar
[11]Dieterich, J.Time-dependent friction and the mechanics of stick-slip. Pure appl. Geophys., 116: 790–806, 1978.Google Scholar
[12]Dieterich, J. H. Modeling of rock friction, 1, experimental results and constitutive equations. J. Geophy. Res., 84:2161–2168,1979a.Google Scholar
[13]Dieterich, J. H. and Kilgore, B. D.Direct observation of frictional contacts: new insights for state dependent properties. Pure Appl. Geophys., 143: 283–302, 1994.Google Scholar
[14]Erickson, B. A. and Nordström, J.Stable, high order accurate adaptive schemes for long time, highly intermittent geophysics problems. Journal of Computational and Applied Mathematics, 271: 328–338, 2014.CrossRefGoogle Scholar
[15]Fisher, T. C., Carpenter, M. H., Nordström, J., Yamaleev, N. K., and Swanson, C.Discretely conservative finite-difference formulations for nonlinear conservation laws in split form: Theory and boundary conditions. Journal of Computational Physics, 234: 353–375, 2013.Google Scholar
[16]Gong, J. and Nordström, J.Interface procedures for finite difference approximations of the advectiondiffusion equation. Journal of Computational and Applied Mathematics, 236(5): 602–620, 2011.CrossRefGoogle Scholar
[17]Gustafsson, B.Far-field boundary conditions for time-dependent hyperbolic systems. SIAM Journal on Scientific and Statistical Computing, 9(3): 812–828, 1988.Google Scholar
[18]Gustafsson, B. and Kreiss, H. O.Boundary conditions for time dependent problems with an artificial boundary. Journal of Computational Physics, 30(3): 333–351, 1979.Google Scholar
[19]Gustafsson, B., Kreiss, H. O., and Oliger, J.Time dependent Problems and Difference Methods. Wiley-Interscience, New York, 1995.Google Scholar
[20]Gustafsson, B.High Order Difference Methods for Time Dependent PDE. Number 38 in Springer Series in Computational Mathematics. Springer-Verlag, 2008.Google Scholar
[21]Hagstrom, T.Radiation boundary conditions for the numerical simulation of waves. Acta Numerica, 8: 47–106, 1999.Google Scholar
[22]Hicken, J. E. and Zingg, D. W.Parallel Newton-Krylov solver for the Euler equations discretized using simultaneous-approximation terms. AIAA Journal, 46(11): 2773–2786, 2008.Google Scholar
[23]Huan, X., Hicken, J. E., and Zingg, D. W.Interface and boundary schemes for high-order methods. In The 39th AIAA Fluid Dynamics Conference, San Antonio, USA, 22-25 June 2009, June 2009. AIAA Paper No. 2009-3658.Google Scholar
[24]Kållberg, P.Test of a lateral boundary relaxation scheme in a barotropic model. Report 3, European Centre for Medium Range Weather Forecasts, Bracknell, United Kingdom, 1977.Google Scholar
[25]Kozdon, J. E., Dunham, E. M., and Nordström, J.Simulation of dynamic earthquake ruptures in complex geometries using high-order finite difference methods. Journal of Scientific Computing, 55(1): 92–124, 2013.Google Scholar
[26]Kozdon, J. E., Dunham, E. M., and Nordström, J.Interaction of waves with frictional interfaces using summation-by-parts difference operators: Weak enforcement of nonlinear boundary conditions. J. Sci. Comput., 50: 341–367, 2012.Google Scholar
[27]Kreiss, H. O. and Oliger, J.Comparison of accurate methods for the integration of hyperbolic equations. Tellus, 24: 199–215, 1972.Google Scholar
[28]Kreiss, H. O. and Scherer, G.Finite element and finite difference methods for hyperbolic partial differential equations, Mathematical Aspects of Finite Elements in Partial Differential Equations. Academic Press, Inc., 1974.Google Scholar
[29]Kreiss, H. O. and Scherer, G.On the existence of energy estimates for difference approximations for hyperbolic systems. Technical report, Department of Scientific Computing, Uppsala University, 1977.Google Scholar
[30]Marone, C.Laboratory-derived friction laws and their application to seismic faulting. Annu. Rev. Earth Planet. Sci., 26: 643–696, 1998.Google Scholar
[31]Mattsson, K.Boundary procedures for summation-by-parts operators. Journal of Scientific Computing, 18(1): 133–153, 2003.Google Scholar
[32]Nordström, J.Conservative finite difference formulations, variable coefficients, energy estimates and artificial dissipation. 29(3):375–404, 2006.Google Scholar
[33]Nordström, J.Linear and nonlinear boundary conditions for wave propagation problems, Recent developments in the numerics of nonlinear hyperbolic conservation laws, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, No. 120, 283–299, 2013.CrossRefGoogle Scholar
[34]Nordström, J. and Carpenter, M. H.Boundary and interface conditions for high-order finite- difference methods applied to the Euler and Navier-Stokes equations. Journal of Computational Physics, 148(2): 621–645, 1999.Google Scholar
[35]Nordström, J. and Carpenter, M. H.High-order finite difference methods, multidimensional linear problems, and curvilinear coordinates. Journal of Computational Physics, 173(1):149–174, 2001.Google Scholar
[36]Nordström, J. and Eriksson, S.Fluid structure interaction problems: The necessity of a well posed, stable and accurate formulation. Communications in Computational Physics, 8(5): 1111–1138, 2010.Google Scholar
[37]Nordström, J., Gong, J., van der Weide, E., and Svärd, M.A stable and conservative high order multi-block method for the compressible Navier-Stokes equations. Journal of Computational Physics, 228(24):9020–9035, 2009.Google Scholar
[38]Nordström, J., Nordin, N., and Henningson, D.The fringe region technique and the fourier method used in the direct numerical simulation of spatially evolving viscous flows. SIAM Journal of Scientific Computing, 20(4): 1365–1393, 1999.Google Scholar
[39]Nordström, J. and Svärd, M.Well posed boundary conditions for the Navier-Stokes equations. SIAM Journal of Numerical Analysis, 43(3): 1231–1255, 2005.Google Scholar
[40]Olsson, P.Summation by parts, projections, and stability, i. Mathematics of Computation, 64(211): 1035–1065, 1995.Google Scholar
[41]Olsson, P.Summation by parts, projections, and stability, ii. Mathematics of Computation, 64(212): 1473–1493, 1995.Google Scholar
[42]Strand, B.Summation by parts for finite difference approximations for d/dx. Journal of Computational Physics, 110(1): 47–67, 1994.Google Scholar
[43]Svärd, M.On coordinate transformations for summation-by-parts operators. Journal of Scientific Computing, 20(1): 29–42, 2004.Google Scholar
[44]Svärd, M., Carpenter, M. H., and Nordström, J.A stable high-order finite difference scheme for the compressible Navier-Stokes equations: far-field boundary conditions. Journal of Computational Physics, 225(1): 1020–1038, 2007.Google Scholar
[45]Svärd, M. and Nordström, J.On the order of accuracy for difference approximations of initial-boundary value problems. Journal of Computational Physics, 218(1): 333–352, 2006.Google Scholar
[46]Svärd, M. and Nordström, J.A stable high-order finite difference scheme for the compressible Navier-Stokes equations: no-slip wall boundary conditions. Journal of Computational Physics, 227(10): 4805–4824, 2007.Google Scholar
[47]Tsynkov, S. V.Numerical solution of problems on unbounded domains. a review. Applied Numerical Mathematics, 27(4): 465–532, 1998.Google Scholar
[48]Zingg, D. W.Comparison of high-accuracy finite-difference methods for linear wave propagation. SIAM Journal on Scientific Computing, 22(2): 476–502, 2001.Google Scholar