Showing papers in "Journal of Scientific Computing in 2003"

Modeling Textures with Total Variation Minimization and Oscillating Patterns in Image Processing

Abstract: This paper is devoted to the modeling of real textured images by functional minimization and partial differential equations. Following the ideas of Yves Meyer in a total variation minimization framework of L. Rudin, S. Osher, and E. Fatemi, we decompose a given (possible textured) image f into a sum of two functions u+v, where u∈BV is a function of bounded variation (a cartoon or sketchy approximation of f), while v is a function representing the texture or noise. To model v we use the space of oscillating functions introduced by Yves Meyer, which is in some sense the dual of the BV space. The new algorithm is very simple, making use of differential equations and is easily solved in practice. Finally, we implement the method by finite differences, and we present various numerical results on real textured images, showing the obtained decomposition u+v, but we also show how the method can be used for texture discrimination and texture segmentation.

An Eulerian Formulation for Solving Partial Differential Equations Along a Moving Interface

Abstract: In this paper we study an Eulerian formulation for solving partial differential equations (PDE) on a moving interface. A level set function is used to represent and capture the moving interface. A dual function orthogonal to the level set function defined in a neighborhood of the interface is used to represent some associated quantity on the interface and evolves according to a PDE on the moving interface. In particular we use a convection diffusion equation for surfactant concentration on an interface passively convected in an incompressible flow as a model problem. We develop a stable and efficient semi-implicit scheme to remove the stiffness caused by surface diffusion.

Semi-Implicit Level Set Methods for Curvature and Surface Diffusion Motion

Abstract: In this paper we introduce semi-implicit methods for evolving interfaces by mean curvature flow and surface diffusion using level set methods.

A Level Set Approach for the Numerical Simulation of Dendritic Growth

Abstract: In this paper, we present a level set approach for the modeling of dendritic solidification. These simulations exploit a recently developed second order accurate symmetric discretization of the Poisson equation, see l12r. Numerical results indicate that this method can be used successfully on complex interfacial shapes and can simulate many of the physical features of dendritic solidification. We apply this algorithm to the simulation of the dendritic crystallization of a pure melt and find that the dendrite tip velocity and tip shapes are in excellent agreement with solvability theory. Numerical results are presented in both two and three spatial dimensions.

Overapproximating Reachable Sets by Hamilton-Jacobi Projections

Abstract: In earlier work, we showed that the set of states which can reach a target set of a continuous dynamic game is the zero sublevel set of the viscosity solution of a time dependent Hamilton-Jacobi-Isaacs (HJI) partial differential equation (PDE). We have developed a numerical tool—based on the level set methods of Osher and Sethian—for computing these sets, and we can accurately calculate them for a range of continuous and hybrid systems in which control inputs are pitted against disturbance inputs. The cost of our algorithm, like that of all convergent numerical schemes, increases exponentially with the dimension of the state space. In this paper, we devise and implement a method that projects the true reachable set of a high dimensional system into a collection of lower dimensional subspaces where computation is less expensive. We formulate a method to evolve the lower dimensional reachable sets such that they are each an overapproximation of the full reachable set, and thus their intersection will also be an overapproximation of the reachable set. The method uses a lower dimensional HJI PDE for each projection with a set of disturbance inputs augmented with the unmodeled dimensions of that projection's subspace. We illustrate our method on two examples in three dimensions using two dimensional projections, and we discuss issues related to the selection of appropriate projection subspaces.

High Order Fluctuation Schemes on Triangular Meshes

Abstract: We develop a new class of schemes for the numerical solution of first-order steady conservation laws. The schemes are of the residual distribution, or fluctuation-splitting type. These schemes have mostly been developed in the context of triangular or tetrahedral elements whose degrees of freedom are their nodal values. We work here with more general elements that allow high-order accuracy. We introduce, for an arbitrary number of degrees of freedom, a simple mapping from a low-order monotone scheme to a monotone scheme that is as accurate as the degrees of freedom will allow. Proofs of consistency, convergence and accuracy are presented, and numerical examples from second, third and fourth-order schemes.

Formulations for Numerically Approximating Hyperbolic Systems Governing Sediment Transport

Abstract: This paper investigates the accurate numerical solution of the equations governing bed-load sediment transport. Two approaches: a steady and an unsteady approach are discussed and five different formulations within these frameworks are derived. A flux-limited version of Roe's scheme is used with the different formulations on a channel test problem and the results compared.

Regularization Techniques for Numerical Approximation of PDEs with Singularities

Abstract: The rate of convergence for numerical methods approximating differential equations are often drastically reduced from lack of regularity in the solution. Typical examples are problems with singular source terms or discontinuous material coefficients. We shall discuss the technique of local regularization for handling these problems. New numerical methods are presented and analyzed and numerical examples are given. Some serious deficiencies in existing regularization methods are also pointed out.

TV Based Image Restoration with Local Constraints

Abstract: The problem of recovering an image that has been blurred and corrupted with additive noise is ill-posed Among the methods that have been proposed to solve this problem, one of the most successful ones is that of constrained Total Variation (TV) image restoration, proposed by L Rudin, S Osher, and E Fatemi In its original formulation, to ensure the satisfaction of constraints, TV restoration requires the estimation of a global parameter λ (a Lagrange multiplier) We observe that if λ is global, the constraints of the method are also satisfied globally, but not locally The effect is that the restoration is better achieved in some regions of the image than in others To avoid this, we propose a variant of the TV restoration model including, instead of a single constraint λ, a set of constraints λi, each one corresponding to a region Oi of the image We discuss the existence and uniqueness of solutions of the proposed model and display some numerical experiments

Shock-Vortex Interactions at High Mach Numbers

Abstract: We perform a computational study of the interaction of a planar shock wave with a cylindrical vortex. We use a particularly robust High Resolution Shock Capturing scheme, Marquina's scheme, to obtain high quality, high resolution numerical simulations of the interaction. In the case of a very-strong shock/vortex encounter, we observe a severe reorganization of the flow field in the downstream region, which seems to be due mainly to the strength of the shock. The numerical data is analyzed to study the driving mechanisms for the production of vorticity in the interaction.

Boundary Procedures for Summation-by-Parts Operators

Abstract: Four different methods of imposing boundary conditions for the linear advection-diffusion equation and a linear hyperbolic system are considered. The methods are analyzed using the energy method and the Laplace transform technique. Numerical calculations are done, considering in particular the case when the initial data and boundary data are inconsistent.

Multidomain WENO Finite Difference Method with Interpolation at Subdomain Interfaces

Abstract: High order finite difference WENO methods have the advantage of simpler coding and smaller computational cost for multi-dimensional problems, compared with finite volume WENO methods of the same order of accuracy. However a main restriction is that conservative finite difference methods of third and higher order of accuracy can only be used on uniform rectangular or smooth curvilinear meshes. In order to overcome this difficulty, in this paper we develop a multidomain high order WENO finite difference method which uses an interpolation procedure at the subdomain interfaces. A simple Lagrange interpolation procedure is implemented and compared to a WENO interpolation procedure. Extensive numerical examples are shown to indicate the effectiveness of each procedure, including the measurement of conservation errors, orders of accuracy, essentially non-oscillatory properties at the domain interfaces, and robustness for problems containing strong shocks and complex geometry. Our numerical experiments have shown that the simple and efficient Lagrange interpolation suffices for the subdomain interface treatment in the multidomain WENO finite difference method, to retain essential conservation, full high order of accuracy, essentially non-oscillatory properties at the domain interfaces even for strong shocks, and robustness for problems containing strong shocks and complex geometry. The method developed in this paper can be used to solve problems in relatively complex geometry at a much smaller CPU cost than the finite volume version of the same method for the same accuracy. The method can also be used for high order finite difference ENO schemes and an example is given to demonstrate a similar result as that for the WENO schemes.

A Fourth Order Scheme for Incompressible Boussinesq Equations

Abstract: A fourth order finite difference method is presented for the 2D unsteady viscous incompressible Boussinesq equations in vorticity-stream function formulation. The method is especially suitable for moderate to large Reynolds number flows. The momentum equation is discretized by a compact fourth order scheme with the no-slip boundary condition enforced using a local vorticity boundary condition. Fourth order long-stencil discretizations are used for the temperature transport equation with one-sided extrapolation applied near the boundary. The time stepping scheme for both equations is classical fourth order Runge–Kutta. The method is highly efficient. The main computation consists of the solution of two Poisson-like equations at each Runge–Kutta time stage for which standard FFT based fast Poisson solvers are used. An example of Lorenz flow is presented, in which the full fourth order accuracy is checked. The numerical simulation of a strong shear flow induced by a temperature jump, is resolved by two perfectly matching resolutions. Additionally, we present benchmark quality simulations of a differentially-heated cavity problem. This flow was the focus of a special session at the first MIT conference on Computational Fluid and Solid Mechanics in June 2001.

A Simple Scheme for Volume-Preserving Motion by Mean Curvature

Abstract: In this article, we present a diffusion-generated approach for evolving volume-preserving motion by mean curvature. Our algorithm alternately diffuses and sharpens characteristic functions to produce a normal velocity which equals the mean curvature minus the average mean curvature. This simple algorithm naturally treats topological mergings and breakings and can be made very fast even when the volume constraint is enforced to double precision (or more). Two dimensional numerical studies are provided to demonstrate the convergence of the method for smooth and nonsmooth problems.

Strong Stability Preserving Properties of Runge–Kutta Time Discretization Methods for Linear Constant Coefficient Operators

Abstract: Strong stability preserving (SSP) high order Runge–Kutta time discretizations were developed for use with semi-discrete method of lines approximations of hyperbolic partial differential equations, and have proven useful in many other applications. These high order time discretization methods preserve the strong stability properties of first order explicit Euler time stepping. In this paper we analyze the SSP properties of Runge Kutta methods for the ordinary differential equation uteLu where L is a linear operator. We present optimal SSP Runge–Kutta methods as well as a bound on the optimal timestep restriction. Furthermore, we extend the class of SSP Runge–Kutta methods for linear operators to include the case of time dependent boundary conditions, or a time dependent forcing term.

A Slowness Matching Eulerian Method for Multivalued Solutions of Eikonal Equations

Abstract: Traveltime, or geodesic distance, is locally the solution of the eikonal equation of geometric optics. However traveltime between sufficiently distant points is generically multivalued. Finite difference eikonal solvers approximate only the viscosity solution, which is the smallest value of the (multivalued) traveltime (“first arrival time”). The slowness matching method stitches together local single-valued eikonal solutions, approximated by a finite difference eikonal solver, to approximate all values of the traveltime. In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation, so that the eikonal equation may be viewed as an evolution equation in one of the spatial directions. This paraxial assumption simplifies both the efficient computation of local traveltime fields and their combination into global multivalued traveltime fields via the slowness matching algorithm. The cost of slowness matching is on the same order as that of a finite difference solver used to compute the viscosity solution, when traveltimes from many point sources are required as is typical in seismic applications. Adaptive gridding near the source point and a formally third order scheme for the paraxial eikonal combine to give second order convergence of the traveltime branches.

An Introduction to Eulerian Geometrical Optics (1992–2002)

Abstract: This document is an attempt at introducing the different “Eulerian” numerical methods which have recently been developed for the simulation of geometric optics and related models.

Estimation of 3D Surface Shape and Smooth Radiance from 2D Images: A Level Set Approach

Abstract: We cast the problem of shape reconstruction of a scene as the global region segmentation of a collection of calibrated images. We assume that the scene is composed of a number of smooth surfaces and a background, both of which support smooth Lambertian radiance functions. We formulate the problem in a variational framework, where the solution (both the shape and radiance of the scene) is a minimizer of a global cost functional which combines a geometric prior on shape, a smoothness prior on radiance and a data fitness score. We estimate the shape and radiance via an alternating minimization: The radiance is computed as the solutions of partial differential equations defined on the surface and the background. The shape is estimated using a gradient descent flow, which is implemented using the level set method. Our algorithm works for scenes with smooth radiances as well as fine homogeneous textures, which are known challenges to traditional stereo algorithms based on local correspondence.

AMR vs High Order Schemes

Abstract: Adaptive Mesh Refinement (AMR) schemes are generally considered promising because of the ability of the scheme to place grid points or computational degrees of freedom at the location in the flow where truncation errors are unacceptably large. For a given order, AMR schemes can reduce work. However, for the computation of turbulent or non-turbulent mixing when compared to high order non-adaptive methods, traditional 2nd order AMR schemes are computationally more expensive. We give precise estimates of work and restrictions on the size of the small scale grid and show that the requirements on the AMR scheme to be cheaper than a high order scheme are unrealistic for most computational scenarios.

Numerical Simulation of Pulse Detonation Engine Phenomena

Abstract: This computational study examines transient, reactive compressible flow phenomena associated with the pulse detonation wave engine. The PDWE is an intermittent combustion engine that relies on unsteady detonation wave propagation for combustion and compression elements of the propulsive cycle. The present computations focus on high order numerical simulations of the generic PDWE configuration with simplified reaction kinetics, so that rapid, straightforward estimates of engine performance may be made. Both one- and two-dimensional simulations of the high speed reactive flow phenomena are performed and compared to determine the applicability of 1D simulations for performance characterization. Examination of the effects of the combustion reaction mechanism and the use of a pressure relaxation length for 1D simulations is made. Characteristic engine performance parameters, in addition to engine noise estimates within and external to the detonation tube, are presented.

High Order Finite Difference Approximations of Electromagnetic Wave Propagation Close to Material Discontinuities

Abstract: Electromagnetic wave propagation close to a material discontinuity is simulated by using summation by part operators of second, fourth and sixth order accuracy The interface conditions at the discontinuity are imposed by the simultaneous approximation term procedure Stability is shown and the order of accuracy is verified numerically

A Level Set Algorithm for Tracking Discontinuities in Hyperbolic Conservation Laws II: Systems of Equations

Abstract: A level set algorithm for tracking discontinuities in hyperbolic conservation laws is presented. The algorithm uses a simple finite difference approach, analogous to the method of lines scheme presented in l36r. The zero of a level set function is used to specify the location of the discontinuity. Since a level set function is used to describe the front location, no extra data structures are needed to keep track of the location of the discontinuity. Also, two solution states are used at all computational nodes, one corresponding to the “real” state, and one corresponding to a “ghost node” state, analogous to the “Ghost Fluid Method” of l12r. High order pointwise convergence was demonstrated for scalar linear and nonlinear conservation laws, even at discontinuities and in multiple dimensions in the first paper of this series l3r. The solutions here are compared to standard high order shock capturing schemes, when appropriate. This paper focuses on the issues involved in tracking discontinuities in systems of conservation laws. Examples will be presented of tracking contacts and hydrodynamic shocks in inert and chemically reacting compressible flow.

Computations of Explosive Boiling in Microgravity

Abstract: Dynamics of the explosive growth of a vapor bubble in microgravity is investigated by direct numerical simulation. A front tracking/finite difference technique is used to solve for the velocity and the temperature field in both phases and to account for inertia, viscosity, and surface deformation. The method is validated by comparison of the numerical results with the available analytical formulations such as the evaporation of a one-dimensional liquid/vapor interface, frequency of oscillations of capillary waves, and other numerical results. Evolution of a three-dimensional vapor nucleus during explosive boiling is followed and a fine scale structure similar to experimental results is observed. Two-dimensional simulations yield a similar qualitative instability growth.

A Discontinuous Spectral Element Method for the Level Set Equation

Abstract: Level set methodology is crucially pertinent to tracking moving singular surfaces or thin fronts with steep gradients in the numerical solutions of partial differential equations governing complex flow fields. This methodology must be consistent with the basic solution technique for the partial differential equations. To this end, a discontinuous spectral element approach is developed for level set advection and reinitialization as these methods are becoming increasingly popular for the solution of the fluid dynamic problems. Example computations are provided, which demonstrate the high order accuracy of the method.

Understanding the Shu–Osher Conservative Finite Difference Form

Abstract: Shu and Osher introduced a conservative finite difference discretization for hyperbolic conservation laws using nodal values rather than the traditional cell averages. Their form was obtained by introducing mathematical relations that simplify the resulting numerical methods. Here we instead “derive” their form from the standard cell average approach. In the process, we clarify the origin of their relations and the properties of this formulation. We also investigate the extension of their form to non-uniform grids. We show the strict conservation form only extends to grids with quadratic or exponential stretching. However, a slight generalization can be applied to all smoothly stretched grids with no loss of essential properties.

The Computation of Martensitic Microstructure with Piecewise Laminates

Abstract: We present a numerical method for the approximation of microstructure in martensitic crystals by piecewise laminates, and we give computational results for several three-dimensional models of martensitic microstructure by using piecewise second-order laminates.

Convection of Concentrated Vortices and Passive Scalars as Solitary Waves

Abstract: A new version of a computational method, Vorticity Confinement, is described. Vorticity Confinement has been shown to efficiently treat thin features in multi-dimensional incompressible fluid flow, such as vortices and streams of passive scalars, and to convect them over long distances with no spreading due to numerical errors. Outside the features, where the flow is irrotational or the scalar vanishes, the method automatically reduces to conventional discretized finite difference fluid dynamic equations. The features are treated as a type of weak solution and, within the features, a nonlinear difference equation, as opposed to finite difference equation, is solved that does not necessarily represent a Taylor expansion discretization of a simple partial differential equation (PDE). The approach is similar to artificial compression and shock capturing schemes, where conservation laws are satisfied across discontinuities. For the features, the result of this conservation is that integral quantities such as total amplitude and centroid motion are accurately computed. Basically, the features are treated as multi-dimensional nonlinear discrete solitary waves that live on the computational lattice. These obey a “confinement” relation that is a generalization to multiple dimensions of 1-D discontinuity capturing schemes. A major point is that the method involves a discretization of a rotationally invariant operator, rather than a composition of separate 1-D operators, as in conventional discontinuity capturing schemes. The main objective of this paper is to introduce a new formulation of Vorticity Confinement that, compared to the original formulation, is simpler, allows more detailed analysis, and exactly conserves momentum for vortical flow. First, a short critique of conventional methods for these problems is given. The basic new method is then described. Finally, analysis of the new method and initial results are presented.

Concepts and Application of Time-Limiters to High Resolution Schemes

Abstract: A new class of implicit high-order non-oscillatory time integration schemes is introduced in a method-of-lines framework. These schemes can be used in conjunction with an appropriate spatial discretization scheme for the numerical solution of time dependent conservation equations. The main concept behind these schemes is that the order of accuracy in time is dropped locally in regions where the time evolution of the solution is not smooth. By doing this, an attempt is made at locally satisfying monotonicity conditions, while maintaining a high order of accuracy in most of the solution domain. When a linear high order time integration scheme is used along with a high order spatial discretization, enforcement of monotonicity imposes severe time-step restrictions. We propose to apply limiters to these time-integration schemes, thus making them non-linear. When these new schemes are used with high order spatial discretizations, solutions remain non-oscillatory for much larger time-steps as compared to linear time integration schemes. Numerical results obtained on scalar conservation equations and systems of conservation equations are highly promising.

An Improved Quadrature Rule for the Flux-Computation in Staggered Central Difference Schemes in Multidimensions

Abstract: We present a new second-order, nonoscillatory, central difference scheme on two-dimensional, staggered, Cartesian grids for systems of conservation laws. The scheme uses a new, carefully designed integration rule for the flux computations and thereby takes more propagation directions into account. This effectively reduces grid orientation effects produced for two-dimensional radially symmetric gas flows and improves the accuracy for smooth solutions.

Degenerate Two-Phase Incompressible Flow IV: Local Refinement and Domain Decomposition

Abstract: This is the fourth paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic-parabolic partial differential system which describes the flow of two incompressible, immiscible fluids in porous media. In this paper we describe a finite element approximation for this system on locally refined grids. This adaptive approximation is based on a mixed finite element method for the elliptic pressure equation and a Galerkin finite element method for the degenerate parabolic saturation equation. Both discrete stability and sharp a priori error estimates are established for this approximation. Iterative techniques of domain decomposition type for solving it are discussed, and numerical results are presented.