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On the order of accuracy of the immersed boundary method: higher order convergence rates for sufficiently smooth problems. (English) Zbl 1115.76386
Summary: The immersed boundary method is both a mathematical formulation and a numerical scheme for problems involving the interaction of a viscous incompressible fluid and a (visco-)elastic structure. In [M.-C. Lai, Simulations of the flow past an array of circular cylinders as a test of the immersed boundary method, Ph.D. thesis, Courant Institute of Mathematical Sciences, New York University, 1998; Lai and Peskin, J. Comput. Phys. 160, 705–719 (2000; Zbl 0954.76066)], M.-C. Lai and C.S. Peskin introduced a formally second order accurate immersed boundary method, but the convergence properties of their algorithm have only been examined computationally for problems with nonsmooth solutions. Consequently, in practice only first order convergence rates have been observed. In the present work, we describe a new formally second order accurate immersed boundary method and demonstrate its performance for a prototypical fluid-structure interaction problem, involving an immersed viscoelastic shell of finite thickness, studied over a broad range of Reynolds numbers. We consider two sets of material properties for the viscoelastic structure, including a case where the material properties of the coupled system are discontinuous at the fluid-structure interface. For both sets of material properties, the true solutions appear to possess sufficient smoothness for the method to converge at a second order rate for fully resolved computations.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
74D99 Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials)
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76D05 Navier-Stokes equations for incompressible viscous fluids
Software:
hypre; PETSc; SAMRAI
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[1] M.-C. Lai, Simulations of the flow past an array of circular cylinders as a test of the immersed boundary method, Ph.D. thesis, Courant Institute of Mathematical Sciences, New York University, 1998
[2] Lai, M.-C.; Peskin, C.S., An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, J. comput. phys., 160, 2, 705-719, (2000) · Zbl 0954.76066
[3] Peskin, C.S.; McQueen, D.M., A three-dimensional computational method for blood flow in the heart. I. immersed elastic fibers in a viscous incompressible fluid, J. comput. phys., 81, 2, 372-405, (1989) · Zbl 0668.76159
[4] McQueen, D.M.; Peskin, C.S., A three-dimensional computational method for blood flow in the heart. II. contractile fibers, J. comput. phys., 82, 2, 289-297, (1989) · Zbl 0701.76130
[5] Peskin, C.S.; McQueen, D.M., Fluid dynamics of the heart and its valves, (), 309-337
[6] Bottino, D.C.; Fauci, L.J., A computational model of ameboid deformation and locomotion, Eur. biophys. J., 27, 5, 532-539, (1998)
[7] Wang, N.T.; Fogelson, A.L., Computational methods for continuum models of platelet aggregation, J. comput. phys., 151, 2, 649-675, (1999) · Zbl 0981.92005
[8] McQueen, D.M.; Peskin, C.S., A three-dimensional computer model of the human heart for studying cardiac fluid dynamics, Comput. graphics, 34, 1, 56-60, (2000)
[9] Peskin, C.S., The immersed boundary method, Acta numer., 11, 479-517, (2002) · Zbl 1123.74309
[10] Leveque, R.J.; Li, Z., Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM J. sci. comput., 18, 3, 709-735, (1997) · Zbl 0879.76061
[11] Cortez, R.; Minion, M.L., The blob projection method for immersed boundary problems, J. comput. phys., 161, 2, 428-453, (2000) · Zbl 0962.74078
[12] Cortez, R., The method of regularized stokeslets, SIAM J. sci. comput., 23, 4, 1204-1225, (2001) · Zbl 1064.76080
[13] Lee, L.; Leveque, R.J., An immersed interface method for incompressible Navier-Stokes equations, SIAM J. sci. comput., 25, 3, 832-856, (2003) · Zbl 1163.65322
[14] Gottlieb, S.; Shu, C.-W.; Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM rev., 43, 1, 89-112, (2001) · Zbl 0967.65098
[15] Twizell, E.H.; Gumel, A.B.; Arigu, M.A., Second-order, L0-stable methods for the heat equation with time-dependent boundary conditions, Adv. comput. math., 6, 3-4, 333-352, (1996) · Zbl 0872.65084
[16] McCorquodale, P.; Colella, P.; Johansen, H., A Cartesian grid embedded boundary method for the heat equation on irregular domains, J. comput. phys., 173, 2, 620-635, (2001) · Zbl 0991.65099
[17] Colella, P., Multidimensional upwind methods for hyperbolic conservation laws, J. comput. phys., 87, 1, 171-200, (1989) · Zbl 0694.65041
[18] Minion, M.L., On the stability of Godunov-projection methods for incompressible flow, J. comput. phys., 123, 2, 435-449, (1996) · Zbl 0848.76050
[19] Minion, M.L., A projection method for locally refined grids, J. comput. phys., 127, 1, 158-178, (1996) · Zbl 0859.76047
[20] Chorin, A.J., Numerical solution of the Navier-Stokes equations, Math. comput., 22, 104, 745-762, (1968) · Zbl 0198.50103
[21] Chorin, A.J., On the convergence of discrete approximations to the Navier-Stokes equations, Math. comput., 23, 106, 341-353, (1969) · Zbl 0184.20103
[22] Bell, J.B.; Colella, P.; Glaz, H.M., A second-order projection method for the incompressible Navier-Stokes equations, J. comput. phys., 85, 2, 257-283, (1989) · Zbl 0681.76030
[23] Almgren, A.S.; Bell, J.B.; Crutchfield, W.Y., Approximate projection methods: part I. inviscid analysis, SIAM J. sci. comput., 22, 4, 1139-1159, (2000) · Zbl 0995.76059
[24] McQueen, D.M.; Peskin, C.S., Shared-memory parallel vector implementation of the immersed boundary method for the computation of blood flow in the beating Mammalian heart, J. supercomput., 11, 3, 213-236, (1997)
[25] Jung, E.N.; Peskin, C.S., Two-dimensional simulations of valveless pumping using the immersed boundary method, SIAM J. sci. comput., 23, 1, 19-45, (2001) · Zbl 1065.76156
[26] Zhu, L.D.; Peskin, C.S., Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method, J. comput. phys., 179, 2, 452-468, (2002) · Zbl 1130.76406
[27] Roma, A.M.; Peskin, C.S.; Berger, M.J., An adaptive version of the immersed boundary method, J. comput. phys., 153, 2, 509-534, (1999) · Zbl 0953.76069
[28] Almgren, A.S.; Bell, J.B.; Szymczak, W.G., A numerical method for the incompressible Navier-Stokes equations based on an approximate projection, SIAM J. sci. comput., 17, 2, 358-369, (1996) · Zbl 0845.76055
[29] M.F. Lai, A projection method for reacting flow in the zero Mach number limit, Ph.D. thesis, University of California at Berkeley, 1993
[30] Martin, D.F.; Colella, P., A cell-centered adaptive projection method for the incompressible Euler equations, J. comput. phys., 163, 2, 271-312, (2000) · Zbl 0991.76052
[31] J.M. Stockie, Analysis and computation of immersed boundaries, with application to pulp fibres, Ph.D. thesis, Institute of Applied Mathematics, University of British Columbia, 1997
[32] Tornberg, A.-K.; Engquist, B., Numerical approximations of singular source terms in differential equations, J. comput. phys., 200, 2, 462-488, (2004) · Zbl 1115.76392
[33] Brown, D.L.; Cortez, R.; Minion, M.L., Accurate projection methods for the incompressible Navier-Stokes equations, J. comput. phys., 168, 2, 464-499, (2001) · Zbl 1153.76339
[34] SAMRAI: Structured adaptive mesh refinement application infrastructure, <http://www.llnl.gov/CASC/SAMRAI>
[35] Hornung, R.D.; Kohn, S.R., Managing application complexity in the SAMRAI object-oriented framework, Concurrency comput.: pract. ex., 14, 5, 347-368, (2002) · Zbl 1008.68527
[36] S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, H. Zhang, PETSc Web page, <http://www.mcs.anl.gov/petsc>
[37] S. Balay, K. Buschelman, V. Eijkhout, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, H. Zhang, PETSc Users Manual, Technical Report ANL-95/11 - Revision 2.1.5, Argonne National Laboratory, 2004
[38] Balay, S.; Eijkhout, V.; Gropp, W.D.; McInnes, L.C.; Smith, B.F., Efficient management of parallelism in object oriented numerical software libraries, (), 163-202 · Zbl 0882.65154
[39] hypre: High performance preconditioners, <http://www.llnl.gov/CASC/hypre> · Zbl 1056.65046
[40] Falgout, R.D.; Yang, U.M., hypre: a library of high performance preconditioners, (), 632-641, also available as LLNL Technical Report UCRL-JC-146175 · Zbl 1056.65046
[41] Briggs, W.L.; Henson, V.E.; McCormick, S.F., A multigrid tutorial, (2000), Society for Industrial and Applied Mathematics Philadelphia, PA · Zbl 0958.65128
[42] W.J. Rider, Filtering nonsolenoidal modes in numerical solutions of incompressible flows, Technical Report LA-UR-3014, Los Alamos National Laboratory, 1994
[43] Crockett, R.K.; Colella, P.; Fisher, R.T.; Klein, R.I.; McKee, C.F., An unsplit, cell-centered Godunov method for ideal MHD, J. comput. phys., 203, 2, 422-448, (2005) · Zbl 1143.76599
[44] Harlow, F.H.; Welch, J.E., Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. fluids, 8, 12, 2182-2189, (1965) · Zbl 1180.76043
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