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Yossi Farjoun

Publications and source records attributed to Yossi Farjoun.

15 recordsLinked to original sources
arXiv2012

Charge transport in a superlattice: a numerical study using moment methods

A semiclassical model of charge transport in a semiconductor superlattice is solved, using moments in the wavenumber direction and finite elements in the spatial direction (first order). The selection of numerical methods guarantees the conservation of current while allowing for high accuracy results. When a dc voltage bias is held between the ends of the sample, self-sustaining oscillations of the current through the superlattice are observed in a narrow range of voltages. the calculated solution displayed the expected accuracy: Spectral convergence in the number of moments used, and first-order convergence in the number of grid-cells. This result paves the way for higher-order methods (in the spatial direction) and the numerical solution of more complex models of charge transport including quantum models based on the Wigner function.

cond-mat.mes-hall↗
arXiv2012

A characteristic particle method for traffic flow simulations on highway networks

A characteristic particle method for the simulation of first order macroscopic traffic models on road networks is presented. The approach is based on the method "particleclaw", which solves scalar one dimensional hyperbolic conservations laws exactly, except for a small error right around shocks. The method is generalized to nonlinear network flows, where particle approximations on the edges are suitably coupled together at the network nodes. It is demonstrated in numerical examples that the resulting particle method can approximate traffic jams accurately, while only devoting a few degrees of freedom to each edge of the network.

math.NA↗
arXiv2012

The sound of an evolving floating sculpture

Commissioned by MIT's in-house artist Jane Philbrick, we evolve an abstract 2D surface (resembling Marta Pan's 1961 "Sculpture Flottante I") under mean curvature, all the while calculating the eigenmodes and eigenvalues of the Laplace-Beltrami operator on the resulting shapes. These are then synthesized into a sound-wave embodying the "swan song" of the surfaces as the evolve to points and vanish. The surface is approximated by a triangulation, and we present a robust approach to approximate the normal directions and the mean curvature. The resulting video and sound-track were parts in the Jane Philbrick's exhibition "Everything Trembles" in Lund, Sweden, 2009.

math.NA↗
arXiv2010

The hanging thin rod: A singularly perturbed eigenvalue problem

We study the vibrations of a hanging thin flexible rod, in which the dominant restoring force in most of the domain is tension due to the weight of the rod, while bending elasticity plays a small but non-negligible role. We consider a linearized description, which we may reduce to an eigenvalue problem. We solve the resulting singularly perturbed problem asymptotically up to the first modification of the eigenvalue. On the way, we illustrate several important problem-solving techniques: modeling, nondimensionalization, scaling, and especially use of asymptotic series.

math.CA↗
arXiv2010

An exact particle method for scalar conservation laws and its application to stiff reaction kinetics

An "exact" method for scalar one-dimensional hyperbolic conservation laws is presented. The approach is based on the evolution of shock particles, separated by local similarity solutions. The numerical solution is defined everywhere, and is as accurate as the applied ODE solver. Furthermore, the method is extended to stiff balance laws. A special correction approach yields a method that evolves detonation waves at correct velocities, without resolving their internal dynamics. The particle approach is compared to a classical finite volume method in terms of numerical accuracy, both for conservation laws and for an application in reaction kinetics.

math.NA↗
arXiv2009

A rarefaction-tracking method for hyperbolic conservation laws

We present a numerical method for scalar conservation laws in one space dimension. The solution is approximated by local similarity solutions. While many commonly used approaches are based on shocks, the presented method uses rarefaction and compression waves. The solution is represented by particles that carry function values and move according to the method of characteristics. Between two neighboring particles, an interpolation is defined by an analytical similarity solution of the conservation law. An interaction of particles represents a collision of characteristics. The resulting shock is resolved by merging particles so that the total area under the function is conserved. The method is variation diminishing, nevertheless, it has no numerical dissipation away from shocks. Although shocks are not explicitly tracked, they can be located accurately. We present numerical examples, and outline specific applications and extensions of the approach.

math.NA↗
arXiv2009

An exactly conservative particle method for one dimensional scalar conservation laws

A particle scheme for scalar conservation laws in one space dimension is presented. Particles representing the solution are moved according to their characteristic velocities. Particle interaction is resolved locally, satisfying exact conservation of area. Shocks stay sharp and propagate at correct speeds, while rarefaction waves are created where appropriate. The method is variation diminishing, entropy decreasing, exactly conservative, and has no numerical dissipation away from shocks. Solutions, including the location of shocks, are approximated with second order accuracy. Source terms can be included. The method is compared to CLAWPACK in various examples, and found to yield a comparable or better accuracy for similar resolutions.

math.NA↗
arXiv2009

Aggregation According to Classical Kinetics--From Nucleation to Coarsening

We solve the standard Lifshitz-Slyozov (LS) model with conservation of total particles in the limit of small super-saturation. The new element is an effective initial condition that follows from the initial exhaustion of nucleation as described in a previous paper [Farjoun and Neu, Phys. Rev. E 78]. The effective initial condition is characterized by a narrow distribution of cluster-sizes, all much larger than critical. In the subsequent solution, one of the LS similarity solutions emerges as the long-time limit, as expected. But our solution tells more. In particular, there is a "growth" era prior to what is usually called "coarsening". During "growth" the clusters (all of nearly the same size much larger than critical) eventually exhaust the super-saturation (the exhaustion of nucleation in the previous era results from only a small decrease in super-saturation). This allows the critical size to catch up to the clusters, and the traditional "coarsening" begins: Subcritical clusters dissolve and fuel the growth of the remaining super-critical clusters. Our analysis tracks the evolution of cluster sizes during growth and coarsening by complimentary use of asymptotic and numerical methods. We establish characteristic times and cluster sizes associated with growth and coarsening from physical parameters and the initial super-saturation. The emerging distribution is discontinuous at the largest cluster size, and our model selects the discontinuous LS similarity solution as the long time limit. There are strong indications that the smooth similarity solution proposed in the original LS paper emerges on a, yet longer, late-coarsening time-scale.

cond-mat.mtrl-sci↗
arXiv2008

Nucleation, Growth, and Coarsening -- A Global View on Aggregation

We present a new model of homogeneous aggregation that contains the essential physical ideas of the classical predecessors, the Becker-Doring and Lifshitz-Slyovoz models. These classical models, which give different predictions, are asymptotic limits of the new model at small and large cluster sizes (respectively). Since the new theory is valid for large and small clusters, it allows for a complete description of the nucleation process; predicting the Zeldovich nucleation rate, and the diffusion limited growth of large clusters. By retaining the physically valid ingredients from both models, we can explain the seeming incompatibilities and arbitrary choices of the classical models. We solve the equations of our new model asymptotically in the small super-saturation limit. The solution exhibits three successive `eras': nucleation, growth, and coarsening, each with its specific scales of time and cluster size. During the nucleation era, the bulk of the clusters are formed by favorable fluctuations over a free energy barrier, according to the analysis by Zeldovich. During the Growth era no new clusters are created, and the expansion of the existing ones continues. Eventually the coarsening era begins. During this competitive attrition process, smaller clusters dissolve and fuel the further growth of the larger survivors. By resolving the preceding creation and growth eras, our analysis gives explicitly the characteristic time and cluster size of the coarsening era, and a unique selection of the long time, self-similar cluster size distribution.

cond-mat.stat-mech↗
arXiv2008

Creation of Clusters via a Thermal Quench

The nucleation and growth of clusters in a progressively cooled vapor is studied. The chemical-potential of the vapor increases, resulting in a rapidly increasing nucleation rate. The growth of the newly created clusters depletes monomers, and counters the increase in chemical-potential. Eventually, the chemical potential reaches a maximum and begins to decrease. Shortly thereafter the nucleation of new clusters effectively ceases. Assuming a slow quench rate, asymptotic methods are used to convert the non-linear advection equation of the cluster-size distribution into a fourth-order differential equation, which is solved numerically. The distribution of cluster-sizes that emerges from this creation era of the quench process, and the total amount of clusters generated are found.

cond-mat.mtrl-sci↗
arXiv2008

The Tallest Column -- A Dynamical System Approach using a Similarity Solution

A classic problem, the design of the tallest column, is solved again using a different method. By the use of a similarity solution the equations are transformed and the difficult singularity at the endpoint is peeled away. The resulting autonomous system has a critical point and the solution must be on its stable manifold. The solution is found by starting near the critical point in the direction of the stable manifold, and solving backwards numerically. This removes the need for an iterative integration method that was previously used. The method is shown to work for clamped or hinged boundary condition and can also be used for other problems involving singularities at the endpoints.

math-ph↗
arXiv2008

Solving One Dimensional Scalar Conservation Laws by Particle Management

We present a meshfree numerical solver for scalar conservation laws in one space dimension. Points representing the solution are moved according to their characteristic velocities. Particle interaction is resolved by purely local particle management. Since no global remeshing is required, shocks stay sharp and propagate at the correct speed, while rarefaction waves are created where appropriate. The method is TVD, entropy decreasing, exactly conservative, and has no numerical dissipation. Difficulties involving transonic points do not occur, however inflection points of the flux function pose a slight challenge, which can be overcome by a special treatment. Away from shocks the method is second order accurate, while shocks are resolved with first order accuracy. A postprocessing step can recover the second order accuracy. The method is compared to CLAWPACK in test cases and is found to yield an increase in accuracy for comparable resolutions.

math.NA↗
arXiv2007

The Optimal Shape of a Javelin

The problem of finding the optimal tapering of a free (non-supported) javelin is described and solved. For the optimal javelin, the lowest mode of vibration has the highest possible frequency. With this tapering inner damping will lead to the cessation of the vibration at the fastest possible rate. The javelin is modeled as a beam of uniform material. The differential equations governing the vibration and the tapering of the beam are derived. These equations have a difficult singularity at the tips of the beam. A procedure using a similarity solution is used to solve this singular system, and the solution is found. The maximal frequency is found to be almost 5 times larger than the frequency of a cylindrical rod.

math-ph↗
arXiv2007

Asymptotics of the Euler number of bipartite graphs

We define the Euler number of a bipartite graph on $n$ vertices to be the number of labelings of the vertices with $1,2,...,n$ such that the vertices alternate in being local maxima and local minima. We reformulate the problem of computing the Euler number of certain subgraphs of the Cartesian product of a graph $G$ with the path $P_m$ in terms of self adjoint operators. The asymptotic expansion of the Euler number is given in terms of the eigenvalues of the associated operator. For two classes of graphs, the comb graphs and the Cartesian product $P_2 \Box P_m$, we numerically solve the eigenvalue problem.

math.CO↗
arXiv2006

Resolving Paradoxes of Classical Nucleation Theory

We present a new model of homogeneous aggregation that contains the essential physical ideas of the classical predecessors, the Becker-Doring and Lifshitz-Slyovoz models. These classical models, which give different predictions, are asymptotic limits of the new model at small (BD) and large (LS) cluster sizes. Since the new theory is valid for large and small clusters, it allows for a complete description of the nucleation process; one that can predict the creation of super-critical clusters at the Zeldovich nucleation rate, and the diffusion limited growth of large clusters during coarsening. By retaining the physically valid ingredients from both models, we explain the seeming incompatibilities and arbitrary choices of the classical models.

cond-mat.mtrl-sci↗