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14
Wave Propagation Algorithms for Multidimensional Hyperbolic Systems
 JOURNAL OF COMPUTATIONAL PHYSICS
, 1997
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The immersed interface method for the Navier–Stokes equations with singular forces
 J. Comput. Phys
"... Peskin’s Immersed Boundary Method has been widely used for simulating many fluid mechanics and biology problems. One of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain. However, t ..."
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Cited by 83 (5 self)
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Peskin’s Immersed Boundary Method has been widely used for simulating many fluid mechanics and biology problems. One of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain. However, the Immersed Boundary Method is known to be firstorder accurate and usually smears out the solutions. In this paper, we propose an immersed interface method for the incompressible Navier–Stokes equations with singular forces along one or several interfaces in the solution domain. The new method is based on a secondorder projection method with modifications only at grid points near or on the interface. From the derivation of the new method, we expect fully secondorder accuracy for the velocity and nearly secondorder accuracy for the pressure in the maximum norm including those grid points near or on the interface. This has been confirmed in our numerical experiments. Furthermore, the computed solutions are sharp across the interface. Nontrivial numerical results are provided and compared with the Immersed Boundary Method. Meanwhile, a new version of the Immersed Boundary Method using the level set representation of the interface is also proposed in this paper. c ○ 2001 Academic Press Key Words: Navier–Stokes equations; interface; discontinuous and nonsmooth solution; immersed interface method; immersed boundary method; projection method; level set method. 1.
A Hybrid Method for Moving Interface Problems with Application to the HeleShaw Flow
, 1997
"... this paper, a hybrid approach which combines the immersed interface method with the level set approach is presented. The fast version of the immersed interface method is used to solve the differential equations whose solutions and their derivatives may be discontinuous across the interfaces due to t ..."
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Cited by 82 (23 self)
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this paper, a hybrid approach which combines the immersed interface method with the level set approach is presented. The fast version of the immersed interface method is used to solve the differential equations whose solutions and their derivatives may be discontinuous across the interfaces due to the discontinuity of the coefficients or/and singular sources along the interfaces. The moving interfaces then are updated using the newly developed fast level set formulation which involves computation only inside some small tubes containing the interfaces. This method combines the advantage of the two approaches and gives a secondorder Eulerian discretization for interface problems. Several key steps in the implementation are addressed in detail. This newapproach is then applied to HeleShaw flow, an unstable flow involving two fluids with very different viscosity. 1997 Academic Press L
A Numerical Study of ElectroMigration Voiding by Evolving Level Set Functions on a Fixed Cartesian Grid
 J. Comput. Phys
, 1999
"... this paper involves the evolution of voids under electromigration in a conducting metal line where the driving forces for diffusion are the gradient of the curvature and the electric potential along the void boundary. The normal velocity of the void surface is given by the partial differential equa ..."
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Cited by 28 (11 self)
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this paper involves the evolution of voids under electromigration in a conducting metal line where the driving forces for diffusion are the gradient of the curvature and the electric potential along the void boundary. The normal velocity of the void surface is given by the partial differential equation (PDE) U n = # s (C 1 # + C 2 #), (1.1) where # s is the surface Laplacian, # is the potential function associated with an applied electric field, and # is the mean curvature along the boundary; for a circle, the curvature is a positive constant. The coefficients C 1 , C 2 are related to the physical constants C 1 = eD s # 1/3 Z # k B T k , C 2 = D s # 4/3 # s k B T k , (1.2) where e is the charge of an electron, # is the atomic volume, Z # is a phenomenological constant related to the effective valence of an atom, k B is Boltzmann's constant, T k is the temperature, # s is the surface energy. The constant D s is defined as D s = D # s # s k B T k e Q s /k B T k , (1.3) where # s is the thickness of the diffusion layer, D # s e Q /k B T k is the surface diffusion coefficient, and Q s is the activation energy for surface diffusion. The electric potential # satisfies the Laplace equation ## = 0, with noflux boundary condition ## #n =0 on the void boundary as well as other appropriate boundary conditions on the computational boundary. For a void bounded by a closed surface, it can be shown from the divergence theorem that the void conserves its volume (or area) during surface diffusion. In Eq. (1.1), the first term, # s #, is a nonlocal driving force which tends to drift the void along with the electric current. The second term, # s # , is a fourth order nonlinear term which only depends on the local geometry. The boundary evolution governed...
WaveletBased Numerical Homogenization with Applications
 Lecture Notes in Computational Science and Engineering
, 2002
"... Classical homogenization is an analytic technique for approximating multiscale differential equations. The numbers of scales are reduced and the resulting equations are easier to analyze or numerically approximate. The class of problems that classical homogenization applies to is quite restricted. W ..."
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Cited by 19 (1 self)
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Classical homogenization is an analytic technique for approximating multiscale differential equations. The numbers of scales are reduced and the resulting equations are easier to analyze or numerically approximate. The class of problems that classical homogenization applies to is quite restricted. We shall describe a numerical procedure for homogenization, which starts from a discretization of the multiscale differential equation. In this procedure the discrete operator is represented in a wavelet space and projected onto a coarser subspace. The wavelet homogenization applies to a wider class of problems than classical homogenization. The projection procedure is general and we give a presentation of a framework in Hilbert space, which also applies to the differential equation directly. The wavelet based homogenization technique is applied to discretizations of the Helmholtz equation. In one problem from electromagnetic compatibility a subgrid scale geometrical detail is represented on a coarser grid. In another a waveguide filter is efficiently approximated in a lower dimension. The technique is also applied to the derivation of effective equations for a nonlinear problem and to the derivation of coarse grid operators in multigrid. These multigrid methods work very well for equations with highly oscillatory or discontinuous coefficients.
A Level Set Method for Interfacial Flows with Surfactant
"... A levelset method for the simulation of fluid interfaces with insoluble surfactant is presented in twodimensions. The method can be straightforwardly extended to threedimensions and to soluble surfactants. The method couples a semiimplicit discretization for solving the surfactant transport equa ..."
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Cited by 17 (1 self)
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A levelset method for the simulation of fluid interfaces with insoluble surfactant is presented in twodimensions. The method can be straightforwardly extended to threedimensions and to soluble surfactants. The method couples a semiimplicit discretization for solving the surfactant transport equation recently developed by Xu and Zhao [62] with the immersed interface method originally developed by LeVeque and Li and [31] for solving the fluid flow equations and the LaplaceYoung boundary conditions across the interfaces. Novel techniques are developed to accurately conserve component mass and surfactant mass during the evolution. Convergence of the method is demonstrated numerically. The method is applied to study the effects of surfactant on single drops, dropdrop interactions and interactions among multiple drops in Stokes flow under a steady applied shear. Due to Marangoni forces and to nonuniform Capillary forces, the presence of surfactant results in larger drop deformations and more complex dropdrop interactions compared to the analogous cases for clean drops. The effects of surfactant are found to be most significant in flows with multiple drops. To our knowledge, this is the first time that the levelset method has been used to simulate fluid interfaces with surfactant.
NEW FORMULATIONS FOR INTERFACE PROBLEMS IN POLAR COORDINATES ∗
"... Abstract. In this paper, numerical methods are proposed for some interface problems in polar or Cartesian coordinates. The new methods are based on a formulation that transforms the interface problem with a nonsmooth or discontinuous solution to a problem with a smooth solution. The new formulation ..."
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Abstract. In this paper, numerical methods are proposed for some interface problems in polar or Cartesian coordinates. The new methods are based on a formulation that transforms the interface problem with a nonsmooth or discontinuous solution to a problem with a smooth solution. The new formulation leads to a simple second order finite difference scheme for the partial differential equation and a new interpolation scheme for the normal derivative of the solution. In conjunction with the fast immersed interface method, a fast solver has been developed for the interface problems with piecewise constant but discontinuous coefficient using the new formulation in polar coordinate system.
Simulation of a waveguide filter using waveletbased numerical homogenization
 Journal of Computational Physics
, 2001
"... We apply waveletbased numerical homogenization to the simulation of an optical waveguide filter. We use the method to derive approximate onedimensional models and subgrid models of the filter. Numerical examples of the technique are presented, and the computational gains are investigated. c ° 2001 ..."
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We apply waveletbased numerical homogenization to the simulation of an optical waveguide filter. We use the method to derive approximate onedimensional models and subgrid models of the filter. Numerical examples of the technique are presented, and the computational gains are investigated. c ° 2001 Academic Press Key Words: wavelets; numerical homogenization; waveguide; filter; subgrid model; multiresolution analysis; Helmholtz equation.
IMMERSED INTERFACE METHODS FOR STOKES FLOW WITH ELASTIC BOUNDARIES OR SURFACE TENSION
"... Abstract. A secondorder accurate interface tracking method for the solution of incompressible Stokes flow problems with moving interfaces on a uniform Cartesian grid is presented. The interface may consist of an elastic boundary immersed in the fluid or an interface between two dierent fluids. The ..."
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Abstract. A secondorder accurate interface tracking method for the solution of incompressible Stokes flow problems with moving interfaces on a uniform Cartesian grid is presented. The interface may consist of an elastic boundary immersed in the fluid or an interface between two dierent fluids. The interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed. The Stokes equations are then discretized using the secondorder accurate nite dierence methods for elliptic equations with singular sources developed in our previous paper [SIAM J. Numer. Anal., 31(1994), pp. 1019{1044]. The resulting velocities are interpolated to the interface to determine the motion of the interface. An implicit quasiNewton method is developed that allows reasonable time steps to be used. Key words. Stokes flow, creeping flow, interface tracking, discontinuous coecients, immersed interface methods, Cartesian grids, bubbles AMS subject classications. 76M20, 65M06, 76D07 PII. S1064827595282532 1. Introduction. In