metastability of the cahn hilliard equation in one space dimension abstract we establish metastability of the one dimensional cahn hilliard equation for initial data that is order one in energy and order one in h 1 away from a point on the so called slow manifold with n well separated layers specifically we show that for such initial data on a system of lengthscale there are three phases of evolution 1 the solution is drawn after a time of order 2 into an algebraically small neighborhood of the n layer branch of the slow manifold 2 the solution is drawn after a time of order 3 into an exponentially small neighborhood of the n layer branch of the slow manifold 3 the solution is trapped for an exponentially long time exponentially close to the n layer branch of the slow manifold the timescale in phase 3 is obtained with the sharp constant in the exponential

slow motion manifolds far from the attractor in multistable reaction diffusion equations abstract we consider a scalar reaction diffusion equation with multistable nonlinearity with a particular symmetry by reduction to a family of transmission problems in r and by contraction arguments a manifold close to an invariant manifold formed by functions exhibiting a pattern of transition layers is constructed an approximation for the associated vector field is also provided this shows that the motion on those manifolds is exponentially slow as in the well known case of the bistable equation however in opposition to the bistable case some of these manifolds are far from the attractor since these manifolds correspond to metastable patterns this shows the importance of the transient motion toward the attractor and the importance of these manifolds in organizing that motion it is also shown that by a suitable perturbation we can obtain new equilibria on those manifolds

adaptive localized replay an efficient integration scheme for accurate simulation of coarsening dynamical systems coarsening dynamical systems cds of ordinary differential equations odes have as a defining feature the presence of discontinuities in the governing equations due to the removal of vanishing elements from the system simple commonly used integration strategies for odes neglect these discontinuities resulting in large errors in contrast available methods to resolve discontinuities exhibit total running times that scale like o n 2 as the number of elements n the resulting need to choose between either speed or accuracy frustrates attempts to gather well converged statistical data from large ensembles of elements a common goal in studies of coarsening in this work we introduce a numerical scheme called adaptive localized replay alr which overcomes this dilemma provided the underlying system has sparse neighbor dependence by analyzing the system after each timestep and then selectively re simulating only those small parts of the system that were affected by discontinuities we attain high order accuracy while maintaining running times of o n using a reference implementation of our algorithm applied to a sample problem from the field of faceted surface evolution we obtain convergence and runtime results for the alr method which compares favorably with existing methods

multidomain spectral solution of the euler gas dynamics equations abstract we present interface treatments for computing the euler gas dynamics equations by a multidomain chebyshev spectral collocation method these treatments are suitable for use at the corners of subdomains and for patched or overlapping subdomains they can be applied to interfaces placed in subsonic supersonic or transonic regions of a flow computed results indicate that the solutions are spectrally accurate can be more accurate and more efficient than a single domain calculation and that reflections of waves at interfaces are not significant

local representation groups group cohomology techniques are used to derive conditions for a group g to be a local splitting group for g as a first step the exactness of the inflation restriction sequence gives a characterisation of factor systems arising in locally operating representations of transitive transformation groups some examples and applications of the theory are also given

metastable patterns for the cahn hilliard equation part ii layer dynamics and slow invariant manifold abstract in this paper we study the dynamics of the 1 dimensional cahn hilliard equation u t 2 u xx w u xx in a finite interval in a neighborhood of an equilibrium with n 1 transition layers where is a small parameter and w is a double well energy density function with equal minima the lower bound of the layer motion speed is given explictly and the layer motion directions are described precisely if a solution of the cahn hilliard equation starts outside a neighborhood of the equilibrium of size o ln 1 it is proved that there is an n dimensional unstable invariant manifold which is a smooth graph over the approximate manifold constructed in j differential equations 111 1994 421 457 with its global lipschitz constant exponentially small and this unstable invariant manifold attracts solutions exponentially fast uniformly in

numerical simulation of transitional flow past a wing using flux reconstruction method in this paper the capabilities of a high order flux reconstruction scheme for the implicit large eddy simulations of a transitional flow at a low reynolds number of 60 000 around the sd7003 wing are examined this is the case posed in the first and second international workshops on the high order computational fluid dynamics method held in 2012 and 2013 the correction function used in the flux reconstruction scheme is the radau polynomial which is equivalent to the discontinuous galerkin scheme a time accurate implicit lower upper symmetric gauss seidel solution algorithm for the application of the flux reconstruction scheme to complex unsteady flows is developed and it is found to be able to produce comparable results to the explicit runge kutta scheme while achieving better computational efficiency simulations are carried out at 4 and 8 deg with polynomials of degree p 1 2 and 3 resulting in second third and fourth order accurate flux reconstruction schemes respectively two structured he

locally operating realizations of transformation lie groups using the mackey theory of induced representations a systematic study of the locally operating multiplier realizations of a connected lie group g that acts transitively on a space time manifold is presented we obtain a mathematical characterization of the locally operating multiplier realizations and a reduction of the problem of multiplier locally operating realizations to linear ones via a splitting group g m for g in this way the locally operating multiplier realizations are obtained by induction from finite dimensional linear representations of a well determined subgroup of g some examples such as the two dimensional euclidean group the galilei group and the one dimensional newton hooke group are given

slow dynamics for the cahn hilliard equation in higher space dimensions the motion of bubbles it is known that the van der waals cahn hilliard w c h dynamics can be approximated by a quasi static stefan problem with surface tension it turns out that the stefan problem has a manifold of equilibria equal in dimension to that of the domain any sphere of fixed radius with interface contained in the domain is an equilibrium indistinguishable from the point of view of the perimeter functional we resolve this degeneracy by showing that at the w c h level this manifold is replaced by a quasi invariant stable manifold on which the typical solution preserves its bubble like shape until it reaches the boundary moreover we show that the bubble moves superslowly we also obtain an equation that determines those special spheres that correspond to equilibria at the w c h level our work establishes the phenomenon of superslow motion in higher space dimensions in the class of single interface solutions

an efficient correction procedure via reconstruction for simulation of viscous flow on moving and deforming domains in this paper we report the development of a new parallel solver using the correction procedure via reconstruction cpr for viscous flows on moving and deforming grids by employing an accurate treatment of flux derivatives for moving and deforming unstructured grids consisting of all quadrilateral cells it is found that the geometric conservation law is not explicitly required the free stream preservation is automatically satisfied the cpr code is verified using a benchmark case for a moving inviscid vortex on moving and deforming grids the optimal orders of accuracy are obtained it is subsequently employed to study viscous flows on moving and deforming grids the cpr method is faster than and nearly as accurate as the sd method for solving viscous flow problems with moving boundaries

metastability and stability of patterns in a convolution model for phase transitions where e is a small positive parameter and f is bistable having only three zeros 1 a 1 and is balanced i e r 1 1 f u du 1 4 0 see fig 1 for a typical initial datum j which crosses a several times at the beginning the solution evolves as if the diffusion term euxx were not there and in the time scale o ej n ej the solution gets close to 1 or 1 in intervals separated by thin layers of thickness at the order of o ej n ej 17 28 at this moment a spatial pattern is generated it persists on a time scale o ece but is eventually destroyed typically u x t 1 u x t 1 uniformly for x on 1 20 1 as t 1 it is possible that u x t converges to a or a nonconstant steady state as t 1 but this is an unlikely event because all these steady states are unstable 16 27 research supported in part by the national science foundation and bor leqsf

slow motion in higher order systems and g convergence in one space dimension the slow motion in higher order systems and convergence in one space dimension was studied some of the theorems and computations that support and proves the stability of the equations are presented

stable boundary conditions and difference schemas for navier stokes equations the navier stokes equations at high reynolds numbers can be viewed as an incompletely elliptic perturbation of the euler equations by using the entropy function for the euler equations as a measure of energy for the navier stokes equations we are able to obtain nonlinear energy estimates for the mixed initial boundary value problem these estimates are used to derive boundary conditions that guarantee l 2 boundedness even when the reynolds number tends to infinity finally we propose a new difference scheme for modeling the navier stokes equations in multidimensions for which we are able to obtain discrete energy estimates exactly analogous to those we obtained for the differential equation

a high order solver for unsteady incompressible navier stokes equations using the flux reconstruction method on unstructured grids with implicit dual time stepping we report development of a high order compact flux reconstruction method for solving unsteady incompressible flow on unstructured grids with implicit dual time stepping the method falls under the class of methods now referred to as flux reconstruction correction procedure via reconstruction the governing equations employ chorin s classic artificial compressibility formulation with dual time stepping to solve unsteady flow problems an implicit non linear lower upper symmetric gauss seidel scheme with backward euler discretization is used to efficiently march the solution in pseudo time while a second order backward euler discretization is used to march in physical time we verify and validate implementation of the high order method coupled with our implicit time stepping scheme using both steady and unsteady incompressible flow problems the current implicit time stepping scheme is proven effective in satisfying the divergence free constraint on the velocity field in the artificial compressibility formulation within the context of the high order flux reconstruction method this compact high order method is very suitable for parallel computing and can easily be extended to moving and deforming grids

a staggered grid multidomain spectral method for the compressible navier stokes equations we present a flexible non conforming staggered grid chebyshev spectral multidomain method for the solution of the compressible navier stokes equations in this method subdomain corners are not included in the approximation thereby simplifying the subdomain connectivity to allow for local refinement of the polynomial order or subdomain subdivision non conforming subdomains are treated by a mortar method spectral accuracy is shown in one and two dimensional linear and non linear problems application is made to four compressible flow problems the couette flow a steady boundary layer over a flat plate steady transonic flow in a nozzle and unsteady flow over a cylinder at a reynolds number of 75

dispersive and friction induced stabilization of the cahn hilliard inverse cascade we discuss the stabilization of the inverse cascade in the large scale instability of the kolmogorov flow described by the complete cahn hilliard equation with inclusion of effect large scale friction and deformation radius the friction and the values halting the inverse cascade at the various possible intermediate states are calculated by means of singular perturbation techniques and compared to the values resulting from numerical simulation of the complete cahn hilliard equation the excellent agreement validates the theory our main result is that the critical values of friction or halting the inverse cascade scale exponentially as a function of the jet separation in the final flow contrary to previous theories and phenomenological approach 2002 elsevier science b v all rights reserved pacs 47 10 g 92 60 ek 47 27 i 47 27 ak 47 35 i

local realizations of kinematical groups with a constant electromagnetic field i the relativistic case this paper is devoted to the study of the description of elementary physical systems interacting with an external constant electromagnetic field and the construction of their differential wave equations from a group theoretical point of view in this context certain local realizations of the poincare group are studied the linearization of this problem is carried out by building the associated representation group that turns out to be the well known maxwell group in this way the usual method concerning local realizations that has been employed in studying free systems to the interacting case is extended

vector bundle representations of groups in quantum physics the theory of vector bundle representations of lie groups g is developed induced and locally operating linear representations of g are shown to be generated by vector bundle representations and gauge equivalence of locally operating linear representations is shown to correspond to equivalence of vector bundle representations it is pointed out that every vector bundle representation is equivalent to an induced one some applications to the quantum systems associated with monopole and instanton gauge field configurations are also discussed

spinodal decomposition for the cahn hilliard equation in higher dimensions part i probability and wavelength estimate this paper is the first in a series of two papers addressing the phenomenon of spinodal decomposition for the cahn hilliard equation where is a bounded domain with sufficiently smooth boundary and f is cubic like for example f u u u3 we will present the main ideas of our approach and explain in what way our method differs from known results in one space dimension due to grant 26 furthermore we derive certain probability and wavelength estimates the probability estimate is needed to understand why in a neighborhood of a homogeneous equilibrium u0 of the cahn hilliard equation with mass in the spinodal region a strongly unstable manifold has dominating effects this is demonstrated for the linearized equation but will be essential for the nonlinear setting in the second paper 37 as well moreover we introduce the notion of a characteristic wavelength for the strongly unstable directions

a conservative staggered grid chebyshev multidomain method for compressible flows ii a semi structured method we present a chebyshev multidomain method that can solve systems of hyperbolic equations in conservation form on an unrestricted quadrilateral subdivision of a domain within each subdomain the solutions and fluxes are approximated by a staggered grid chebyshev method thus the method is unstructured in terms of the subdomain decomposition but strongly structured within the subdomains communication between subdomains is done by a mortar method in such a way that the method is globally conservative the method is applied to both linear and nonlinear test problems and spectral accuracy is demonstrated

locally operating realizations of nonconnected transformation lie groups a systematic study of locally operating multiplier realizations of nonconnected lie groups of transformations is presented that generalizes previous results on connected groups the semilinear locally operating multiplier realizations of a nonconnected group g are those obtained through an induction process from the finite dimensional semilinear representations of a given subgroup of a representation group g for g

coarsening dynamics of the one dimensional cahn hilliard model the dynamics of one dimensional cahn hilliard model is studied the stationary and particle type solutions the bubbles are perused as a function of initial conditions boundary conditions and system size we characterize the bubble solutions which are involved in the coarsening dynamics and establish the bifurcation scenarios of the system a set of ordinary differential equation permits us to describe the coarsening dynamics in very good agreement with numerical simulations we also compare these dynamics with the bubble dynamics deduced from the classical kink interaction computation where our model seems to be more appropriated in the case of two bubbles we deduce analytical expressions for the bubble s position and the bubble s width besides a simple description of the ulterior dynamics is presented

a stable penalty method for the compressible navier stokes equations i open boundary conditions the purpose of this paper is to present asymptotically stable open boundary conditions for the numerical approximation of the compressible navier stokes equations in three spatial dimensions the treatment uses the conservation form of the navier stokes equations and utilizes linearization and localization at the boundaries based on these variables the proposed boundary conditions are applied through a penalty procedure thus ensuring correct behavior of the scheme as the reynolds number tends to infinity the versatility of this method is demonstrated for the problem of a compressible flow past a circular cylinder

dynamics as a mechanism preventing the formation of finer and finer microstructure we study the dynamics of pattern formation in the one dimensional partial differential equation u u w u x x u xxt u 0 text u u x t text x in 0 1 text t 0 proposed recently by ball holmes james pego swart bhjps as a mathematical cartoon for the dynamic formation of microstructures observed in various crystalline solids here w is a double well potential like 1 4 ux 2 1 2 what makes this equation interesting and unusual is that it possesses as a lyapunov function a free energy consisting of kinetic energy plus a nonconvex elastic energy but no interfacial energy contribution which does not attain a minimum but favours the formation of finer and finer phase mixtures e u u t int limits 0 1 frac u t 2 2 w u x frac u 2 2 dx our analysis of the dynamics confirms the following surprising and striking difference between statics and dynamics conjectured in bhjps on the basis of numerical simulations of swart holmes sh while minimizing the above energy predicts infinitely fine patterns mathematically weak but not strong convergence of all minimizing sequences unvn of e u v in the sobolev space w1p 0 1 l2 0 1 solutions to the evolution equation of ball et al typically develop patterns of small but finite length scale mathematically strong convergence in w1p 0 1 l2 0 1 of all solutions u t ut t with low initial energy as time t

the keller segel model with logistic sensitivity function and small diffusivity the keller segel model is the classical model for chemotaxis of cell populations it consists of a drift diffusion equation for the cell density coupled to an equation for the chemoattractant here a variant of this model is studied in one dimensional position space where the chemotactic drift is turned off for a limiting cell density by a logistic term and where the chemoattractant density solves an elliptic equation modeling a quasi stationary balance of reaction and diffusion with production of the chemoattractant by the cells the case of small cell diffusivity is studied by asymptotic and numerical methods on a time scale characteristic for the convective effects convergence of solutions to weak entropy solutions of the limiting nonlinear hyperbolic conservation law is proven numerical and analytic evidence indicates that solutions of this problem converge to irregular patterns of cell aggregates separated by entropic shocks from vacuum regions as time tends to infinity close to each of these p

multidomain spectral solution of compressible viscous flows abstract we develop a nonoverlapping multidomain spectral collocation method to solve compressible viscous flows at the interfaces the advection terms are treated with a characteristic correction method the diffusion terms are treated with a penalty method spectral accuracy is demonstrated on linear model problems in one and two space dimensions the method is applied to a subsonic and a supersonic flow over a flat plate the results are compared to solutions of the boundary layer equations which show that two digit accuracy in the adiabatic plate temperature is obtained with 16 points in the boundary layer for a freestream mach number of two a second application is to a transonic flow in a two dimensional converging diverging nozzle where the computed results are compared to experimental data

a strategy for the efficient simulation of viscous compressible flows using a multi domain pseudospectral method abstract a new multi domain pseudospectral method is developed to simulate viscous compressible flow in a quasi one dimensional nozzle the flow variables at the interface points are advanced in time using the same second order time marching scheme as the interior points the spatial derivatives of the inviscid flux vector are evaluated at the interface points using a spectrally accurate modification of van leer s flux vector splitting method the derivatives of the viscous flux vector are evaluated alternately from neighboring sub domains the scheme is found to be spectrally accurate shock waves are resolved without oscillations for moderate reynolds numbers

a conservative staggered grid chebyshev multidomain method for compressible flows we present a new multidomain spectral collocation method that uses a staggered grid for the solution of compressible flow problems the solution unknowns are defined at the nodes of a gauss quadrature rule the fluxes are evaluated at the nodes of a gauss lobatto rule the method is conservative free stream preserving and exponentially accurate a significant advantage of the method is that subdomain corners are not included in the approximation making solutions in complex geometries easier to compute

short note a comparison of computational efficiencies of spectral difference method and correction procedure via reconstruction we report computational efficiencies of two types of numerical solvers the first type uses the spectral difference sd method and the second one uses the correction procedure via reconstruction cpr in this paper we employ the lumped g 2 scheme proposed by huynh as an example of the cpr approach the solvers deal with both inviscid euler equations and navier stokes equations on 2d unstructured grids which are comprised of all quadrilateral cells both types of solvers are programmed using fortran 90 with similar management of data structures we employ identical time marching schemes for both sd and cpr methods spatial 3rd and 4th order of accuracy for both methods is demonstrated by a study of a moving inviscid vortex the comparisons were directed to measure the computational efficiency of both sd and cpr methods in spatial discretization with respect to the fourth order methods cpr is 27 faster than sd for inviscid flow and more promisingly cpr is over 40 faster than sd for viscous flow

locally operating realizations of groups and superequivalence of factor systems the unitary multiplier locally operating realizations of a transitive group of transformations g are identified with those obtained by an induction process from linear finite dimensional representations of a group related with the isotopy group of the given action in this induction process each pseudoequivalence class of linear representations of splits into several gauge equivalence classes of unitary multiplier locally operating realizations of g the concept of superequivalence of factor systems appears here in a natural way

a stable penalty method for the compressible navier stokes equations iii multidimensional domain decomposition schemes this paper concluding the trilogy develops schemes for the stable solution of wave dominated unsteady problems in general three dimensional domains the schemes utilize a spectral approximation in each subdomain and asymptotic stability of the semidiscrete schemes is established the complex computational domains are constructed by using nonoverlapping quadrilaterals in the two dimensional case and hexahedrals in the three dimensional space to illustrate the ideas underlying the multidomain method a stable scheme for the solution of the three dimensional linear advection diffusion equation in general curvilinear coordinates is developed the analysis suggests a novel yet simple stable treatment of geometric singularities like edges and vertices the theoretical results are supported by a two dimensional implementation of the scheme the main part of the paper is devoted to the development of a spectral multidomain scheme for the compressible navier stokes equations on conservation form and a unified approach for dealing with the open boundaries and subdomain boundaries is presented well posedness and asymptotic stability of the semidiscrete scheme is established in a general curvilinear volume with special attention given to a hexahedral domain the treatment includes a stable procedure for dealing with boundary conditions at a solid wall the efficacy of the scheme for the compressible navier stokes equations is illustrated by obtaining solutions to subsonic and supersonic boundary layer flows with various types of boundary conditions the results are found to agree with the solution of the compressible boundary layer equations