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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.

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
hypre; PETSc; SAMRAI
Full Text: DOI
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