Results 1  10
of
10
Wave Propagation Algorithms for Multidimensional Hyperbolic Systems
 JOURNAL OF COMPUTATIONAL PHYSICS
, 1997
"... ..."
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 ..."
Abstract

Cited by 57 (22 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 46 (4 self)
 Add to MetaCart
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 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 ..."
Abstract

Cited by 16 (8 self)
 Add to MetaCart
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...
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 ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
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.
WaveletBased Subgrid Modeling: 1. Principles and Scalar Equations
"... A systematic technique for the derivation of subgrid scale models in the numerical solution of partial differential equations, is described. The technique is based on wavelet projections of the discrete operator followed by a sparse approximation. The resulting numerical method will accurately repre ..."
Abstract
 Add to MetaCart
A systematic technique for the derivation of subgrid scale models in the numerical solution of partial differential equations, is described. The technique is based on wavelet projections of the discrete operator followed by a sparse approximation. The resulting numerical method will accurately represent subgrid scale phenomena on a coarse grid. Applications to numerical homogenization and wave propagation in materials with subgrid inhomogeneities are presented. 1 Introduction In the numerical simulation of partial differential equations, the existence of subgrid scale phenomena poses considerable difficulties. With subgrid scale phenomena, we mean processes which should influence the solution on the computational grid but which have length scales shorter than the grid size. Fine scales in the initial values may for example interact with fine scales in the material properties and produce coarse scale contributions to the solution. There are many traditional ways to deal with this probl...
Reactive Autophobic Spreading of Drops
, 2002
"... this paper, we formulate a model, based on Greenspan's lubrication theory approximation [14], that describes the coupling between the motion of a drop and the deposition of a surfactant monolayer. The model consists of a Poisson equation for the drop height on a moving domain, together with conditio ..."
Abstract
 Add to MetaCart
this paper, we formulate a model, based on Greenspan's lubrication theory approximation [14], that describes the coupling between the motion of a drop and the deposition of a surfactant monolayer. The model consists of a Poisson equation for the drop height on a moving domain, together with conditions for the velocity of the domain boundary that incorporate the effect of surfactant deposition. We develop a numerical scheme to compute solutions of the model, using an immersed interface method to solve the Poisson equation, and a levelset method to evolve the moving domain. The numerical 00219991/02 $35.00 c All rights reserved. August 27, 2002 13:50 APJ/Journal of Computational Physics JCPH7168 2 HUNTER, LI, AND ZHAO solutions include traveling drops that are qualitatively similar to those observed in experiments
JOURNAL OF COMPUTATIONAL PHYSICS 147, 60–85 (1998) ARTICLE NO. CP985965 A Cartesian Grid Embedded Boundary Method for Poisson’s Equation on Irregular Domains 1
, 1997
"... We present a numerical method for solving Poisson’s equation, with variable coefficients and Dirichlet boundary conditions, on twodimensional regions. The approach uses a finitevolume discretization, which embeds the domain in a regular Cartesian grid. We treat the solution as a cellcentered quan ..."
Abstract
 Add to MetaCart
We present a numerical method for solving Poisson’s equation, with variable coefficients and Dirichlet boundary conditions, on twodimensional regions. The approach uses a finitevolume discretization, which embeds the domain in a regular Cartesian grid. We treat the solution as a cellcentered quantity, even when those centers are outside the domain. Cells that contain a portion of the domain boundary use conservative differencing of secondorder accurate fluxes on each cell volume. The calculation of the boundary flux ensures that the conditioning of the matrix is relatively unaffected by small cell volumes. This allows us to use multigrid iterations with a simple point relaxation strategy. We have combined this with an adaptive mesh refinement (AMR) procedure. We provide evidence that the algorithm is secondorder accurate on various exact solutions and compare the adaptive and nonadaptive calculations. c ○ 1998 Academic Press 1.