Abstract
We here generalize the embedded boundary method that was developed for boundary discretizations of the wave equation in second order formulation in Kreiss et al. (SIAM J. Numer. Anal. 40(5):1940–1967, 2002) and for the Euler equations of compressible fluid flow in Sjögreen and Peterson (Commun. Comput. Phys. 2:1199–1219, 2007), to the compressible Navier-Stokes equations. We describe the method and we implement it on a parallel computer. The implementation is tested for accuracy and correctness. The ability of the embedded boundary technique to resolve boundary layers is investigated by computing skin-friction profiles along the surfaces of the embedded objects. The accuracy is assessed by comparing the computed skin-friction profiles with those obtained by a body fitted discretization.
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Abarbanel, S., Ditkowski, A.: Asymptotically stable fourth-order accurate schemes for the diffusion equation on complex shapes. J. Comput. Phys. 133, 279–288 (1997)
Abarbanel, S., Ditkowski, A.: Multi-dimensional asymptotically stable finite difference schemes for the advection-diffusion equation. Comput. Fluids 28, 481–510 (1999)
Abarbanel, S., Ditkowski, A., Yefet, A.: Bounded error schemes for the wave equation on complex domains. J. Sci. Comput. 26, 67–81 (2006)
Berger, M.J., Helzel, C., LeVeque, R.J.: H-box methods for the approximation of hyperbolic conservation laws on irregular grids. SIAM J. Numer. Anal. 41, 893–918 (2003)
Calhoun, D., LeVeque, R.J.: A Cartesian grid finite-volume method for the advection-diffusion equation in irregular geometries. J. Comput. Phys. 157, 143–180 (2000)
Coirier, W.J., Powell, K.G.: An accuracy assessment of Cartesian-mesh approaches for the Euler equations. J. Comput. Phys. 117, 121–131 (1995)
Colella, P., Graves, D.T., Keen, B.J., Modiano, D.: A Cartesian grid embedded boundary method for hyperbolic conservation laws. J. Comput. Phys. 211, 347–366 (2006)
Ferm, L., Lötstedt, P.: Accurate and stable grid interfaces for finite volume methods. Appl. Numer. Math. 49, 407–224 (2004)
Forrer, H., Jeltsch, R.: A higher-order boundary treatment for Cartesian-grid methods. J. Comput. Phys. 140, 259–277 (1998)
Gustafsson, B., Kreiss, H.-O., Sundström, A.: Stability theory of difference approximations for mixed initial boundary value problems. ii. Math. Comput. 26(119), 649–686 (1972)
Helzel, C.: Accurate methods for hyperbolic problems on embedded boundary grids. In: Asakura, F., Aiso, H., Kawashima, S., Matsumura, A., Nishibata, S., Nishihara, K. (eds.) Hyperbolic Problems, Theory, Numerics and Applications. Yokohama Publishers, Yokohama (2006)
Helzel, C., Berger, M.J., LeVeque, R.J.: A high-resolution rotated grid method for conservation laws with embedded geometries. SIAM J. Sci. Comput. 26, 785–809 (2005)
Knuth, D.E.: Sorting and Searching, 2nd edn. The Art of Computer Programming, vol. 3. Addison-Wesley, Reading (1998)
Kreiss, H.-O., Peterson, N.A.: A second order accurate embedded boundary method for the wave equation with Dirichlet data. SIAM J. Sci. Comput. 27, 1141–1167 (2006)
Kreiss, H.-O., Peterson, N.A., Yström, J.: Difference approximations for the second order wave equation. SIAM J. Numer. Anal. 40(5), 1940–1967 (2002)
Kreiss, H.-O., Peterson, N.A., Yström, J.: Difference approximations of the Neumann problem for the second order wave equation. SIAM J. Numer. Anal. 42, 1292–1323 (2004)
Michelson, D.: Convergence theorem for difference approximations of hyperbolic quasi-linear initial-boundary value problems. Math. Comput. 49, 445–459 (1987)
Mittal, R., Iaccarino, G.: Immersed boundary methods. Annu. Rev. Fluid. Mech. 37, 239–261 (2005)
Olsen, O.K.: Embedded boundary method for Navier-Stokes equations. Master’s Thesis, Royal Institute of Technology (2005)
Pember, R.B., Bell, J.B., Colella, P., Crutchfield, W.Y., Welcome, M.L.: An adaptive Cartesian grid method for unsteady compressible flow in irregular regions. J. Comput. Phys. 120, 278–304 (1995)
Peskin, C.S.: The immersed boundary method. Acta Numer. 11, 479–517 (2002)
Petersson, A.: http://www.andrew.cmu.edu/user/sowen/software/xcog.html
Quirk, J.J.: An alternative to unstructured grids for computing gas dynamic flows around arbitrarily complex two-dimensional bodies. Comput. Fluids 23(1), 125–142 (1994)
Sjögreen, B., Yee, H.C.: Multiresolution wavelet based adaptive numerical dissipation control for high order methods. J. Sci. Comput. 20(2), 211–255 (2004)
Sjögreen, B., Peterson, N.A.: A Cartesian embedded boundary method for hyperbolic conservation laws. Commun. Comput. Phys. 2, 1199–1219 (2007)
Sjögreen, B., Yee, H.: Variable high order multiblock overlapping grid methods for mixed steady and unsteady multiscale viscous flows. Commun. Comput. Phys. 5, 730–744 (2009)
Skogqvist, P.: High order adaptive difference methods for combustible flows. Ph.D. Thesis, Royal Institute of Technology, Stockholm, Sweden (2001)
van Albada, G., van Leer, B., Roberts, J.W.W.: A comparative study of computational methods in cosmic gas dynamics. Astron. Astrophys. 108, 76–84 (1982)
Yee, H., Sjögreen, B.: Development of low dissipative high order filter schemes for multiscale Navier-Stokes/MHD systems. J. Comput. Phys. 225(1), 910–934 (2007)
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This work performed under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. LLNL-JRNL-402504.
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Kupiainen, M., Sjögreen, B. A Cartesian Embedded Boundary Method for the Compressible Navier-Stokes Equations. J Sci Comput 41, 94–117 (2009). https://doi.org/10.1007/s10915-009-9289-x
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DOI: https://doi.org/10.1007/s10915-009-9289-x