Showing papers by "Stanley Osher published in 1998"
TL;DR: A level set method for capturing the interface between two fluids is combined with a variable density projection method to allow for computation of a two-phase flow where the interface can merge/break and the flow can have a high Reynolds number.
825 citations
TL;DR: The procedure is used to construct a new family of nonstaggered, central schemes for hyperbolic conservation laws by converting the family of staggered central schemes recently introduced, which retain the desirable properties of simplicity and high resolution and yield Riemann-solver-free recipes which avoid dimensional splitting.
Abstract: We present a general procedure to convert schemes which are based on staggered spatial grids into nonstaggered schemes. This procedure is then used to construct a new family of nonstaggered, central schemes for hyperbolic conservation laws by converting the family of staggered central schemes recently introduced in [H. Nessyahu and E. Tadmor, J. Comput. Phys., 87 (1990), pp. 408--463; X. D. Liu and E. Tadmor, Numer. Math., 79 (1998), pp. 397--425; G. S. Jiang and E. Tadmor, SIAM J. Sci. Comput., 19 (1998), pp. 1892--1917]. These new nonstaggered central schemes retain the desirable properties of simplicity and high resolution, and in particular, they yield Riemann-solver-free recipes which avoid dimensional splitting. Most important, the new central schemes avoid staggered grids and hence are simpler to implement in frameworks which involve complex geometries and boundary conditions.
186 citations
TL;DR: In this article, Tadmor and Nessyahu proposed a second order cell-averages-based extension of the Lax?Friedrichs central scheme, using component-wise rather than field-by-field limiting.
173 citations
TL;DR: In this article, Zhao et al. reproduce the general behavior of complicated bubble and droplet motions using the variational level set formulation introduced by the authors earlier, and they use the Lagrange multiplier associated with this constraint.
82 citations
TL;DR: In this paper, a model for epitaxial phenomena based on the motion of island boundaries is described by the level-set method, which treats the growing film as a continuum in the lateral direction, but retains atomistic discreteness in the growth direction.
Abstract: We introduce a model for epitaxial phenomena based on the motion of island boundaries, which is described by the level-set method. Our model treats the growing film as a continuum in the lateral direction, but retains atomistic discreteness in the growth direction. An example of such an ‘‘island dynamics’’ model using the level-set method is presented and compared with the corresponding rate equation description. Extensions of our methodology to more general settings are then discussed. @S1063-651X~98!50212-7#
75 citations
TL;DR: This work introduces a new formulation for the motion of curves in R2 (easily extendable to themotion of surfaces in R3), when the original motion generally corresponds to an ill-posed problem such as the Cauchy--Riemann equations, and presents an analysis of curvature regularizations and some other theoretical justification.
Abstract: We introduce a new formulation for the motion of curves in R2 (easily extendable to the motion of surfaces in R3), when the original motion generally corresponds to an ill-posed problem such as the Cauchy--Riemann equations. This is, in part, a generalization of our earlier work in [6], where we applied similar ideas to compute flows with highly concentrated vorticity, such as vortex sheets or dipoles, for incompressible Euler equations. Our new formulation involves extending the level set method of [12] to problems in which the normal velocity is not intrinsic. We obtain a coupled system of two equations, one of which is a level surface equation. This yields a fixed-grid, Eulerian method which regularizes the ill-posed problem in a topological fashion. We also present an analysis of curvature regularizations and some other theoretical justification. Finally, we present numerical results showing the stability properties of our approach and the novel nature ofthe regularization, including the development o...
34 citations
TL;DR: A complementary projection technique can be used to formulate upwind differencing without specifying a basis, and for systems with eigenvalues of high multiplicity, this approach simplifies the analytical and programming effort and reduces the computational cost.
19 citations
01 Jan 1998
TL;DR: 1D numerical methods for treating an interface separating a liquid drop and a high speed gas flow and has direct, although algorithmically complicated, extensions to 2nd and 3rd order Runge Kutta methods.
Abstract: We develop 1D numerical methods for treating an interface separating a liquid drop and a high speed gas flow. The droplet is an incompressible Navier-Stokes fluid. The gas is a compressible, multi-species, chemically reactive Navier-Stokes fluid (Fedkiw et al., 1996; Fedkiw, 1996). The interface is followed with a marker particle, although the level set method will be used for the eventual 2D extension (Sussman, 1995). Away from the interface, we solve the equations with TVD Runge Kutta schemes in time and conservative finite difference ENO schemes in space (Shu and Osher, 1988). Near the interface, we cannot apply this discretization, since the equations differ in both number and type across the interface. Instead we use the interface location for domain decomposition, and apply a moving control volume formulation nearby. This is done in a conservative framework, compatible with the outer finite difference scheme. Full details are given for a simple forward Euler time stepping scheme, and this has direct, although algorithmically complicated, extensions to 2nd and 3rd order Runge Kutta methods. Future work will focus on the extension to 2D, and simplifications of the higher order time stepping algorithms.
11 citations