ON ASYMPTOTIC SPEED OF SOLUTIONS TO LEVEL-SET MEAN CURVATURE FLOW EQUATIONS WITH DRIVING AND SOURCE TERMS

We investigate a model equation in the crystal growth, which is described by a level-set mean curvature ow equation with driving and source terms. We establish the well-posedness of solutions, and study the asymptotic speed. Interestingly, a new type of nonlinear phenomena in terms of asymptotic speed of solutions appears, which is very sensitive to the shapes of source terms

A random projection method for sharp phase boundaries in lattice Boltzmann simulations

Existing lattice Boltzmann models that have been designed to recover a macroscopic description of immiscible liquids are only able to make predictions that are quantitatively correct when the interface that exists between the fluids is smeared over several nodal points. Attempts to minimise the thickness of this interface generally leads to a phenomenon known as lattice pinning, the precise cause of which is not well understood. This spurious behaviour is remarkably similar to that associated with the numerical simulation of hyperbolic partial differential equations coupled with a stiff source term. Inspired by the seminal work in this field, we derive a lattice Boltzmann implementation of a model equation used to investigate such peculiarities. This implementation is extended to different spacial discretisations in one and two dimensions. We shown that the inclusion of a quasi-random threshold dramatically delays the onset of pinning and facetting

Numerical solution of gravitational dynamics in asymptotically anti-de Sitter spacetimes

A variety of gravitational dynamics problems in asymptotically anti-de Sitter (AdS) spacetime are amenable to efficient numerical solution using a common approach involving a null slicing of spacetime based on infalling geodesics, convenient exploitation of the residual diffeomorphism freedom, and use of spectral methods for discretizing and solving the resulting differential equations. Relevant issues and choices leading to this approach are discussed in detail. Three examples, motivated by applications to non-equilibrium dynamics in strongly coupled gauge theories, are discussed as instructive test cases. These are gravitational descriptions of homogeneous isotropization, collisions of planar shocks, and turbulent fluid flows in two spatial dimensions.Comment: 70 pages, 19 figures; v4: fixed minus sign typo in last term of eqn. (3.47

Three basic issues concerning interface dynamics in nonequilibrium pattern formation

These are lecture notes of a course given at the 9th International Summer School on Fundamental Problems in Statistical Mechanics, held in Altenberg, Germany, in August 1997. In these notes, we discuss at an elementary level three themes concerning interface dynamics that play a role in pattern forming systems: (i) We briefly review three examples of systems in which the normal growth velocity is proportional to the gradient of a bulk field which itself obeys a Laplace or diffusion type of equation (solidification, viscous fingers and streamers), and then discuss why the Mullins-Sekerka instability is common to all such gradient systems. (ii) Secondly, we discuss how underlying an effective interface description of systems with smooth fronts or transition zones, is the assumption that the relaxation time of the appropriate order parameter field(s) in the front region is much smaller than the time scale of the evolution of interfacial patterns. Using standard arguments we illustrate that this is generally so for fronts that separate two (meta)stable phases: in such cases, the relaxation is typically exponential, and the relaxation time in the usual models goes to zero in the limit in which the front width vanishes. (iii) We finally summarize recent results that show that so-called ``pulled'' or ``linear marginal stability'' fronts which propagate into unstable states have a very slow universal power law relaxation. This slow relaxation makes the usual ``moving boundary'' or ``effective interface'' approximation for problems with thin fronts, like streamers, impossible.Comment: 48 pages, TeX with elsart style file (included), 9 figure