Yossi Farjoun

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.

Aggregation according to classical kinetics: from nucleation to coarsening.

A previous paper [Y. Farjoun and J. C. Neu, Phys. Rev. E 78, 051402 (2008)] presents a simple kinetic model of the initial creation transient, starting from pure monomer. During this transient the majority of clusters are created, and the distribution of cluster sizes that emerges from it is predicted to be discontinuous at the largest cluster size. It is well known that the further evolution according to the Lifshitz-Slyozov model of coarsening preserves this discontinuity. The result is at odds with the original proposal of Lifshitz and Slyozov, that the physical late-stage coarsening distribution is the smooth one. The current paper presents an analytic-numerical solution of the Lifshitz-Slyozov equations, starting from the discontinuous creation distribution. Of course, this analysis selects the discontinuous late-stage coarsening distribution, but there is much more. It resolves the intermediate stages between the creation transient and late-state coarsening and provides specific scales of time and cluster size that characterize the onset of coarsening.

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 prpagate 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.

A rarefaction-tracking method for hyperbolic conservation laws

A numerical method for scalar conservation laws in one space dimension is presented. 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. Numerical examples are presented, and specific applications and extensions of the approach outlined.

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.

A Thin Cantilever Beam in a Flow

We study the small vibrations of a thin flexible beam immersed in a laminar flow, in which we assume the dominant restoring force in most of the domain is tension due to the shear stress, while bending elasticity plays a small but non‐negligible role. A linearized description is considered, which is reduced to an eigenvalue problem. The resulting singularly‐perturbed problem is solved asymptotically up to the first modification of the eigenvalue.

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.