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32
An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries
 J. Comp. Phys
, 2006
"... We present an immersed interface method for the incompressible NavierStokes equations capable of handling rigid immersed boundaries. The immersed boundary is represented by a set of Lagrangian control points. In order to guarantee that the noslip condition on the boundary is satisfied, singular fo ..."
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Cited by 37 (3 self)
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We present an immersed interface method for the incompressible NavierStokes equations capable of handling rigid immersed boundaries. The immersed boundary is represented by a set of Lagrangian control points. In order to guarantee that the noslip condition on the boundary is satisfied, singular forces are applied on the fluid. The forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are interpolated using cubic splines. The strength of the singular forces is determined by solving a small system of equations iteratively at each time step. The NavierStokes equations are discretized on a staggered Cartesian grid by a second order accurate projection method for pressure and velocity. Keywords: Immersed interface method, NavierStokes equations, Cartesian grid method, finite difference, fast Poisson solvers, irregular domains.
Composite Finite Elements for 3D Image Based Computing
 COMPUTING AND VISUALIZATION IN SCIENCE 12
, 2009
"... We present an algorithmical concept for modeling and simulation with partial differential equations (PDEs) in image based computing where the computational geometry is defined through previously segmented image data. Such problems occur in applications from biology and medicine where the underlying ..."
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Cited by 28 (4 self)
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We present an algorithmical concept for modeling and simulation with partial differential equations (PDEs) in image based computing where the computational geometry is defined through previously segmented image data. Such problems occur in applications from biology and medicine where the underlying image data has been acquired through, e.g. computed tomography (CT), magnetic resonance imaging (MRI) or electron microscopy (EM). Based on a levelset description of the computational domain, our approach is capable of automatically providing suitable composite finite element functions that resolve the complicated shapes in the medical/biological data set. It is efficient in the sense that the traversal of the grid (and thus assembling matrices for finite element computations) inherits the efficiency of uniform grids away from complicated structures. The method’s efficiency heavily depends on precomputed lookup tables in the vicinity of the domain boundary or interface. A suitable multigrid method is used for an efficient solution of the systems of equations resulting from the composite finite element discretization. The paper focuses on both algorithmical and implementational details. Scalar and vector valued model problems as well as real applications underline the usability of our approach.
Logically Rectangular Grids and Finite Volume Methods for PDEs in Circular and Spherical Domains
 In preparation; http://www.amath. washington.edu/~rjl/pubs/circles
, 2005
"... Abstract. We describe a class of logically rectangular quadrilateral and hexahedral grids for solving PDEs in circular and spherical domains, including grid mappings for the circle, the surface of the sphere and the threedimensional ball. The grids are logically rectangular and the computational do ..."
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Cited by 22 (6 self)
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Abstract. We describe a class of logically rectangular quadrilateral and hexahedral grids for solving PDEs in circular and spherical domains, including grid mappings for the circle, the surface of the sphere and the threedimensional ball. The grids are logically rectangular and the computational domain is a single Cartesian grid. Compared to alternative approaches based on a multiblock data structure or unstructured triangulations, this approach simplifies the implementation of numerical methods and the use of adaptive refinement. A more general domain with a smooth boundary can be gridded by composing one of the mappings from this paper with another smooth mapping from the circle or sphere to the desired domain. Although these grids are highly nonorthogonal, we show that the highresolution wavepropagation algorithm implemented in clawpack can be effectively used to approximate hyperbolic problems on these grids. Since the ratio between the largest and smallest grid is below 2 for most of our grid mappings, explicit finite volume methods such as the wave propagation algorithm do not suffer from the center or pole singularities that arise with polar or latitudelongitude grids. Numerical test calculations illustrate the potential use of these grids for a variety of applications including Euler equations, shallow water equations, and acoustics in a heterogeneous medium. Pattern formation from a reactiondiffusion equation on the sphere is also considered. All examples are implemented in the clawpack software package and full source code is available on the web, along with matlab routines for the various mappings.
A new highorder immersed interface method for solving elliptic equations with imbedded interface of discontinuity
 J. Comput. Phys
, 2007
"... Abstract This paper presents a new highorder immersed interface method for elliptic equations with imbedded interface of discontinuity. Compared with the original secondorder immersed interface method of [R.J. LeVeque, Z. Li. The immersed interface method for elliptic equations with discontinuous ..."
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Cited by 13 (1 self)
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Abstract This paper presents a new highorder immersed interface method for elliptic equations with imbedded interface of discontinuity. Compared with the original secondorder immersed interface method of [R.J. LeVeque, Z. Li. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31 (1994) 100125], the new method achieves arbitrarily highorder accuracy for derivatives at an irregular grid point by imposing only two physical jump conditions together with a wider set of grid stencils. The new interface difference formulas are expressed in a general explicit form so that they can be applied to different multidimensional problems without any modification. The new interface algorithms of up to O(h 4 ) accuracy have been derived and tested on several one and twodimensional elliptic equations with imbedded interface. Compared to the standard secondorder immersed interface method, the test results show that the new fourthorder immersed interface method leads to a significant improvement in accuracy of the numerical solutions. The proposed method has potential advantages in the application to twophase flow because of its highorder accuracy and simplicity in applications.
Systematic derivation of jump conditions for the immersed interface method in threedimensional flow simulation
 SIAM J. Sci. Comput
"... Abstract. In this paper, we systematically derive jump conditions for the immersed interface ..."
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Cited by 12 (3 self)
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Abstract. In this paper, we systematically derive jump conditions for the immersed interface
Derivation of highorder compact finite difference schemes for nonuniform grid using polynomial interpolation,
 J. Comput. Phys.
, 2005
"... Abstract This article presents a family of very highorder nonuniform grid compact finite difference schemes with spatial orders of accuracy ranging from 4th to 20th for the incompressible NavierStokes equations. The highorder compact schemes on nonuniform grids developed in Shukla and Zhong [R ..."
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Cited by 10 (0 self)
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Abstract This article presents a family of very highorder nonuniform grid compact finite difference schemes with spatial orders of accuracy ranging from 4th to 20th for the incompressible NavierStokes equations. The highorder compact schemes on nonuniform grids developed in Shukla and Zhong [R.K. Shukla, X. Zhong, Derivation of highorder compact finite difference schemes for nonuniform grid using polynomial interpolation, J. Comput. Phys. 204 (2005) 404] for linear model equations are extended to the full NavierStokes equations in the vorticity and streamfunction formulation. Two methods for the solution of Helmholtz and Poisson equations using highorder compact schemes on nonuniform grids are developed. The schemes are constructed so that they maintain a highorder of accuracy not only in the interior but also at the boundary. Secondorder semiimplicit temporal discretization is achieved through an implicit Backward Differentiation scheme for the linear viscous terms and an explicit AdamBashforth scheme for the nonlinear convective terms. The boundary values of vorticity are determined using an influence matrix technique. The resulting discretized system with boundary closures of the same highorder as the interior is shown to be stable, when applied to the twodimensional incompressible NavierStokes equations, provided enough grid points are clustered at the boundary. The resolution characteristics of the highorder compact finite difference schemes are illustrated through their application to the onedimensional linear wave equation and the twodimensional driven cavity flow. Comparisons with the benchmark solutions for the twodimensional driven cavity flow, thermal convection in a square box and flow past an impulsively started cylinder show that the highorder compact schemes are stable and produce extremely accurate results on a stretched grid with more points clustered at the boundary.
Convergence of a compact scheme for the pure streamfunction formulation of the unsteady NavierStokes system
 SIAM J. Numer. Anal
"... Abstract. This paper is devoted to the analysis of a new compact scheme for the Navier– Stokes equations in pure streamfunction formulation. Numerical results using that scheme have been reported in [M. BenArtzi et al., J. Comput. Phys., 205 (2005), pp. 640–664]. The scheme discussed here combines ..."
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Cited by 5 (3 self)
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Abstract. This paper is devoted to the analysis of a new compact scheme for the Navier– Stokes equations in pure streamfunction formulation. Numerical results using that scheme have been reported in [M. BenArtzi et al., J. Comput. Phys., 205 (2005), pp. 640–664]. The scheme discussed here combines the Stephenson scheme for the biharmonic operator and ideas from boxscheme methodology. Consistency and convergence are proved for the full nonlinear system. Instead of customary periodic conditions, the case of boundary conditions is addressed. It is shown that in one dimension the truncation error for the biharmonic operator is O(h4) at interior points and O(h) at nearboundary points. In two dimensions the truncation error is O(h2) at interior points (due to the crossterms) and O(h) at nearboundary points. Hence the scheme is globally of order four in the onedimensional periodic case and of order two in the twodimensional periodic case, but of order 3/2 for one and twodimensional nonperiodic boundary conditions. We emphasize in particular that there is no special treatment of the boundary, thus allowing robust use of the scheme. The finite element analogy of the finite difference schemes is invoked at several stages of the proofs in order to simplify their verifications.
An Embedded Boundary Integral Solver for the Unsteady Incompressible NavierStokes Equations
, 2002
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