a direct approach to the exit problem this paper considers the problem of exit for a dynamical system driven by small white noise from the domain of attraction of a stable state a direct singular perturbation analysis of the forward equation is presented based on kramers approach in which the solution to the stationary fokker planck equation is constructed for a process with absorption at the boundary and a source at the attractor in this formulation the boundary and matching conditions fully determine the uniform expansion of the solution without resorting to external selection criteria for the expansion coefficients such as variational principles or the lagrange identity as in our previous theory the exit density and the mean first passage time to the boundary are calculated from the solution of the stationary fokker planck equation as the probability current density and as the inverse of the total flux on the boundary respectively as an application a uniform expansion is constructed for the escape rate in kramers problem o

a staggered grid lagrangian eulerian remap code for 3 d mhd simulations abstract in this paper an approach to multidimensional magnetohydrodynamics mhd which correctly handles shocks but does not use an approximate riemann solver is proposed this approach is simple and is based on control volume averaging with a staggered grid the method builds on the older and often overlooked technique of on each step taking a fully 3 d lagrangian step and then conservatively remapping onto the original grid at the remap step gradient limiters are applied so that the scheme is monotonicity preserving for euler s equations this technique combined with an appropriately staggered grid and wilkins artificial viscosity can give results comparable to those from approximate riemann solvers we show how this can be extended to include a magnetic field maintaining the divergence free condition and pressure positivity and then present numerical test results where possible a comparison with other shock capturing techniques is presented and the advantages and disadvantages of the proposed scheme are clearly explained

sensitivity analysis of the stochastically and periodically forced brusselator abstract the problem of sensitivity of nonlinear system limit cycle with respect to small stochastic and periodic disturbances is considered sensitivity analysis on the basis of quasipotential function is performed the quasipotential is used widely in statistical physics for instance by graham for analysis of nonequilibrium thermodynamics problem we consider an application of quasipotential technique to sensitivity problem for the plane orbit case an approximation of quasipotential is expressed by some scalar function this function sensitivity function is introduced as a base tool of a quantitative description for a system response on the external disturbances new cycle characteristics sensitivity factor parameter of stiffness are considered the analysis of the forced brusselator based on sensitivity function is shown from this analysis the critical value of brusselator parameter is found the dynamics of forced brusselator for this critical value is investigated for small stochastic disturbances the burst of response amplitude is shown for small periodic disturbances the period doubling regime of the transition to chaos scenario is demonstrated

improved fct algorithm for shock hydrodynamics the new ydfct algorithm is presented for the numerical solution of hydrodynamic flow problems with steep gradients this algorithm is an explicit finite difference scheme based on the flux corrected transport technique and it is an improvement of the etbfct and the xdfct algorithms the erosion of sharp edges the phase distortion of curved profiles and the terracing effect and velocity oscillations are eliminated or significantly reduced these improvements are obtained by following action of the limiter at edges more detailed linear numerical analysis of the difference scheme and considering the velocity gradient in the diffusion and antidiffusion coefficients properties of the new algorithm are also demonstrated by the solution of three test problems

polar decomposition in e rings an e ring is a generalization of the ring of bounded linear operators on a hilbert space together with the subset consisting of all effect operators on that space associated with an e ring is a partially ordered abelian group called its directed group that generalizes the additive group of bounded hermitian operators on the hilbert space we prove that every element of the directed group of an e ring has a polar decomposition if and only if every element has a carrier projection and is split by a projection into a positive and a negative part

solution of the boundary value problem for optimal escape in continuous stochastic systems and maps topologies of invariant manifolds and optimal trajectories are investigated in stochastic continuous systems and maps a topological method is introduced that simplifies the solution of boundary value problems the activation energy is calculated as a function of a set of parameters characterizing the initial conditions of the escape path the method is applied explicitly to compute the optimal escape path and the activation energy for a variety of dynamical systems and maps

a linearity preserving nodal variation limiting algorithm for continuous galerkin discretization of ideal mhd equations abstract in this work a stabilized continuous galerkin cg method for magnetohydrodynamics mhd is presented ideal compressible inviscid mhd equations are discretized in space on unstructured meshes using piecewise linear or bilinear finite element bases to get a semi discrete scheme stabilization is then introduced to the semi discrete method in a strategy that follows the algebraic flux correction paradigm this involves adding some artificial diffusion to the high order semi discrete method and mass lumping in the time derivative term the result is a low order method that provides local extremum diminishing properties for hyperbolic systems the difference between the low order method and the high order method is scaled element wise using a limiter and added to the low order scheme the limiter is solution dependent and computed via an iterative linearity preserving nodal variation limiting strategy the stabilization also involves an optional consistent background high order dissipation that reduces phase errors the resulting stabilized scheme is a semi discrete method that can be applied to inviscid shock mhd problems and may be even extended to resistive and viscous mhd problems to satisfy the divergence free constraint of the mhd equations we add parabolic divergence cleaning to the system various time integration methods can be used to discretize the scheme in time we demonstrate the robustness of the scheme by solving several shock mhd problems

pasting of lattice ordered effect algebras in this study we present techniques for constructing a lattice ordered effect algebra using a given family of mv effect algebras first we introduce the greechie diagrams of the pastings of a family of mv effect algebras as application of the greechie diagrams we provide some sufficient conditions for pasting a lattice ordered effect algebra using a family of mv effect algebras especially we illustrate a kind of technique of how to obtain a lattice ordered effect algebra by substituting the atoms of an orthomodular lattice with the lattice ordered effect algebras finally we prove a version of the loop lemma for mv effect algebras pasting

hybrid block amr in cartesian and curvilinear coordinates mhd applications abstract we present a novel hybrid block adaptive scheme for use in solving sets of near conservation laws in general orthogonal coordinate systems the adaptive mesh refinement amr scheme is block based i e individual grids have a pre fixed number of grid cells and is implemented for any dimensionality d its hybrid character relaxes the common approach where a block that needs refinement triggers 2 d subblocks when grids are refined by a factor of 2 this introduces incomplete families in the grid hierarchy but approaches the optimal fit to developing flow features inherent in the original patch based amr strategy our hybrid block amr approach is compared with a patch based amr one both exploiting openmp parallelism the implementation is able to handle general curvilinear coordinates for which restriction and prolongation formulae are presented along with boundary treatments at singular boundaries demonstrative examples cover hydro and magnetohydrodynamic mhd applications including tests on a 2d polar grid a 2 5d spherical and a 3d cylindrical configuration the applications cover classical up to relativistic mhd simulations of particular relevance for astrophysical magnetized jet and stellar wind studies

connectives and fuzziness for classical effects abstract this article investigates various constructs on a set of classical effects measurable fuzzy sets or fuzzy events we begin by studying the properties of a connective and its dual connective which seem to have been neglected in the literature the importance of these connectives for fuzzy probability theory is pointed out we then introduce and investigate the properties of a commutator which can be employed to define a degree of relative fuzziness a special case of this commutator defines a degree of fuzziness the expectation of the commutator provides a relative entropy whose properties are delineated and the expectation of the special case gives an entropy function that has been previously studied as a measure of fuzziness

singularities of fluctuational paths for an overdamped two well system driven by white noise abstract noise induced escape from a metastable state in an overdamped double well system is investigated with the wkb approximation employed the pattern of fluctuational paths is given by using the action plot which gives rise to mpeps the pattern is divided into various regions where fluctuational paths inside show qualitatively distinguished characteristics the global quasi potential is calculated and its value at the saddle on the boundary agrees with the one obtained by action plot we show a contour and 3d plot of the quasi potential to illustrate the relative stabilities of metastable states the bifurcations of singularities as the system parameter varies are analyzed by the second order term of the wkb expansion in the vicinity of x axis its asymptotic behavior determines the positions of cusps and furthermore its evolution turns out to delineate the twisting structure of the lagrangian manifold

crossing the quasi threshold manifold of a noise driven excitable system we consider the noise induced escapes in an excitable system possessing a quasi threshold manifold along which there exists a certain point of minimal quasi potential in the weak noise limit the optimal escaping path turns out to approach this particular point asymptotically making it analogous to an ordinary saddle numerical simulations are performed and an elaboration on the effect of small but finite noise is given which shows that the ridges where the prehistory probability distribution peaks are located mainly within the region where the quasi potential increases gently the cases allowing anisotropic noise are discussed and we found that varying the noise term in the slow variable would dramatically raise the whole level of quasi potentials leading to significant changes in both patterns of optimal paths and exit locations

constructive analysis of noise induced transitions for coexisting periodic attractors of the lorenz model we study the stochastically forced lorenz model in the parameter zone admitting two coexisting limit cycles under the transition to chaos via period doubling bifurcations noise induced transitions between both different parts of the single attractor and two coexisting separate attractors are demonstrated the effects of structural stabilization and noise symmetrization are discussed we suggest a stochastic sensitivity function technique for the analysis of noise induced transitions between two coexisting limit cycles this approach allows us to construct the dispersion ellipses of random trajectories for any poincare sections possibilities of our descriptive geometric method for a detailed analysis of noise induced transitions between two periodic attractors of lorenz model are demonstrated

topological features of large fluctuations to the interior of a limit cycle received 29 october 1996 we investigate the pattern of optimal paths along which a dynamical system driven by weak noise moves with overwhelming probability when it fluctuates far away from a stable state our emphasis is on systems that perform self sustained periodic vibrations and have an unstable focus inside a stable limit cycle we show that in the vicinity of the unstable focus the flow field of optimal paths generically displays a pattern of singularities in particular it contains a switching line that separates areas to which the system arrives along optimal paths of topologically different types the switching line spirals into the focus and has a self similar structure depending on the behavior of the system near the focus it may be smooth or have finite length branches our results are based on an analysis of the topology of the lagrangian manifold for an auxiliary purely dynamical problem that determines the optimal paths we illustrate our theory by studying both theoretically and numerically a van der pol oscillator driven by weak white noise s1063 651x 97 02703 7 many nonlinear dissipative systems display stable periodic oscillations i e their state spaces contain stable limit cycles 1 examples include lasers 2 radiofrequency generators 1 a chemical 3 a and biological 3 b systems in many cases the vibrations are nonsinusoidal as in a multimode laser with strong mode coupling 4 passive optically bistable elements 5 and various sorts of engineering structure 6 fluctuations in periodically oscillating systems play an extremely important role they eventually destroy periodicity of the self sustained oscillations so that the oscillation phase becomes random and a stationary probability distribution in the state space of the system is formed 7 8 this distribution reaches a maximum at the cycle does not vary much along the cycle and falls off rapidly in directions transverse to the cycle the shape of the maximum depends on the character of small transverse fluctuations away from the cycle if the fluctuations are due to the system being perturbed by weak noise of intensity d the transverse falloff will usually be approximately gaussian 7 9 with standard deviation proportional to d 1 2 as d 0

a new optimized fct algorithm for shock wave problems abstract the new xdfct algorithm is presented for the solution of hydrodynamic flow problems with steep gradients this algorithm is an explicit finite difference scheme based on the fluxcorrected transport technique and it is a modification of existing etbfct algorithm the use of different diffusive and antidiffusive fluxes enables us to use twice the time step with similar accuracy this is shown by linear numerical analysis and verified by solution of three test problems

notes on the eigensystem of magnetohydrodynamics the eigenstructure of the equations governing one dimensional ideal magnetohy drodynamics is examined motivated by the wish to exploit it for construction of high resolution computational algorithms the results are given in simple forms that avoid indeterminacy or degeneracy whenever possible the unavoidable indeterminacy near the magnetosonic or triple umbilic state is analysed and shown to cause no difficulty in evaluating a numerical flux function the structure of wave paths close to this singularity is obtained and simple expressions are presented for the structure coefficients that govern wave steepening

use of hamiltonian mechanics in systems driven by colored noise the evaluation of the path integral representation for stochastic processes in the weak noise limit shows that these systems are governed by a set of equations which are those of a classical dynamics we show that even when the noise is colored these may be put into a hamiltonian form which leads to better insights and improved numerical treatments we concentrate on solving hamilton s equations over an infinite time interval in order to determine the leading order contribution to the mean escape time for a bistable potential the paths may be oscillatory and inherently unstable in which case one must use a multiple shooting numerical technique over a truncated time period in order to calculate the infinite time optimal paths to a given accuracy we look at two systems in some detail the underdamped langevin equation driven by external exponentially correlated noise and the overdamped langevin equation driven by external quasi monochromatic noise we deduce that caustics focusing and bifurcation of the optimal path are general features of all but the simplest stochastic processes

stationary probability distribution near stable limit cycles far from hopf bifurcation points we obtain analytic results for the stationary probability distribution in the vicinity of a stable limit cycle for markov systems described by a fokker planck equation or a birth death master equation the results apply best for ranges of parameters removed from hopf bifurcation points as a by product we demonstrate that there holds a liouville like theorem for the stationary probability distribution the product of the velocity along the limit cycle times the area of the cross section of the probability distribution transverse to the cycle is a constant a numerical simulation of a chemical model system with a limit cycle shows good agreement with the analytic results

on godunov type methods near low densities abstract when the energy of a flow is largely kinetic many conservative differencing schemes may fail by predicting non physical states with negative density or internal energy we describe as positively conservative the subclass of schemes that by contrast always generate physical solutions from physical data and show that the godunov method is positively conservative it is also shown that no scheme whose interface flux derives from a linearised riemann solution can be positively conservative classes of data that will bring about the failure of such schemes are described however the harten lax van leer hlle scheme is positively conservative under certain conditions on the numerical wavespeeds and this observation allows the linearised schemes to be rescued by modifying the wavespeeds employed

stochastic analysis of a hopf bifurcation master equation approach the effect of inhomogeneous fluctuations in a reaction diffusion system exhibiting a hopf bifurcation is analyzed using the master equation approach a taylor expansion of the logarithm of the stationary probability known as the stochastic potential is calculated this procedure displays marked analogies with the theory of normal forms the critical potential reduced to its local expansion around an arbitrary point of the limit cycle brings out the essential role played by the phase of the oscillating variables a comparison with the langevin analysis of walgraefet al j chem phys 78 6 3043 1983 is performed

flux corrected transport techniques for multidimensional compressible magnetohydrodynamics a prescription is given for conservatively integrating generalized hydromagnetic equations using flux corrected transport fct techniques by placing the magnetic field components at the interface locations of the finite difference grid the field is kept divergence free to within machine roundoff error the use of fct techniques allows an integration scheme of high accuracy to be employed while the numerical ripples associated with large dispersion errors are avoided the method is particularly well suited for problems involving magnetohydrodynamic shocks and other discontinuities

escape problem for irreversible systems the problem of noise induced escape from a metastable state arises in physics chemistry biology systems engineering and other areas the problem is well understood when the underlying dynamics of the system obey detailed balance when this assumption fails many of the results of classical transition rate theory no longer apply and no general method exists for computing the weak noise asymptotic behavior of fundamental quantities such as the mean escape time in this paper we present a general technique for analyzing the weak noise limit of a wide range of stochastically perturbed continuous time nonlinear dynamical systems we simplify the original problem which involves solving a partial differential equation into one in which only ordinary differential equations need be solved this allows us to resolve some old issues for the case when detailed balance holds when it does not hold we show how the formula for the asymptotic behavior of the mean escape time depends on the dynamics of the system along the most probable escape path we also present results on short time behavior and discuss the possibility of ital focusing along the escape path

the behavior of flux difference splitting schemes near slowly moving shock waves abstract an investigation of the behavior of shock capturing schemes which compute the numerical flux from a solution of riemann s problem is performed the schemes of godunov roe and osher are examined for a one dimensional model problem consisting of a nearly stationary shock both scalar and systems of equations are examined it is found that for slow shocks there is a significant error generated when solving systems of equations while the scalar results are well behaved this error consists of a long wavelength noise in the downstream running wave families that is not effectively damped by the dissipation of the scheme the source of this error is shown and the implications for the performance of these schemes are considered this error may contribute to the slow convergence to steady state reported by many researchers

finite difference time domain algorithm for electromagnetic problems involving material movement the finite difference time domain fdtd method adapted for magnetic field diffusion problems is used to study the electromagnetic induction in moving materials by including motional emf in standard fdtd electromagnetic equations the material movement is implemented by continuously changing material properties in each computational cell consistent to material advection the flux corrected transport fct algorithm is used to transport magnetic field in a fixed eulerian cell a higher time step is achieved by artificially increasing permittivity of the medium this new approach is validated with standard analytical solutions for planar magnetic flux compression system and magnetic field diffusion in moving conductors with a non relativistic velocity to our knowledge this is the first approach to use fdtd method for electromagnetic problems involving material motion

comparison of some flux corrected transport and total variation diminishing numerical schemes for hydrodynamic and magnetohydrodynamic problems two versions of flux corrected transport and two versions of total variation diminishing schemes are tested for several one and two dimensional hydrodynamic and magnetohydrodynamic problems two of the schemes ydfct and tvdlf are tested extensively for the first time the results give an insight into the limitations of the methods their relative strengths and weaknesses some subtle points of the algorithms and the effects of selecting different options for certain methods are emphasised

a scaling theory of bifurcations in the symmetric weak noise escape problem we consider two dimensional overdamped double well systems perturbed by white noise in the weak noise limit the most probable fluctuational path leading from either point attractor to the separatrix the most probable escape path or mpep must terminate on the saddle between the two wells however as the parameters of a symmetric double well system are varied a unique mpep may bifurcate into two equally likely mpeps at the bifurcation point in parameter space the activation kinetics of the system become non arrhenius we quantify the non arrhenius behavior of a system at the bifurcation point by using the maslov wkb method to construct an approximation to the quasistationary probability distribution of the system that is valid in a boundary layer near the separatrix the approximation is a formal asymptotic solution of the smoluchowski equation our construction relies on a new scaling theory which yields critical exponents describing weak noise behavior at the bifurcation point near the saddle

non chaotic response of non linear oscillators under combined deterministic and weak stochastic excitations non linear oscillators under harmonic and or weak stochastic excitations are considered in this paper under harmonic excitations alone an analytical technique based on a set of exponential transformations followed by harmonic balancing is proposed to solve for a variety of one periodic orbits the stability boundaries for such orbits in the associated parameter space are constructed using the floquet theory under a combination of harmonic and weak stochastic excitations a stochastic perturbation approach around the deterministic orbit is adopted to obtain response statistics in terms of the evolving moment functions in the present study the stochastic perturbation is assumed to be an additive white noise process and equations for the evolving moments are derived using ito differential rule a fifth order cumulant neglect closure is used to close the infinite hierarchy of moment equations limited numerical results are presented to illustrate the implementation of the proposed scheme the method is found to be quite versatile and admits ready extensions to md o f systems under combined harmonic and white or non white multiplicative or additive random excitations

flux corrected transport this paper describes a class of explicit eulerian finite difference algorithms for solving the continuity equation which are built around a technique called flux correction these flux corrected transport fct algorithms are of indeterminate order but yield realistic accurate results in addition to the mass conserving property of most conventional algorithms the fct algorithms strictly maintain the positivity of actual mass densities so steep gradients and inviscid shocks are handled particularly well this first paper concentrates on a simple one dimensional version of fct utilizing shasta a new transport algorithm for the continuity equation which is described in detail

limiting and divergence cleaning for continuous finite element discretizations of the mhd equations abstract this work introduces a new type of constrained algebraic stabilization for continuous piecewise linear finite element approximations to the equations of ideal magnetohydrodynamics mhd at the first step of the proposed flux corrected transport fct algorithm the galerkin element matrices are modified by adding graph viscosity proportional to the fastest characteristic wave speed at the second step limited antidiffusive corrections are applied and divergence cleaning is performed for the magnetic field the limiting procedure developed for this stage is designed to enforce local maximum principles as well as positivity preservation for the density and thermodynamic pressure additionally it adjusts the magnetic field in a way which penalizes divergence errors without violating conservation laws or positivity constraints numerical studies for 2d test problems are performed to demonstrate the ability of the proposed algorithms to accomplish this task in applications to ideal mhd benchmarks

scalar hyperbolic pde simulations and coupling strategies we report on grid adaptive multi dimensional simulations for hyperbolic pdes with a deliberate focus on the analytically tractable scalar case motivated by recent efforts towards multi physics simulation strategies we investigate a variety of coupling strategies for numerically solving hyperbolic partial differential equations pdes we use adaptive mesh refinement in combination with shock capturing spatio temporal discretizations and first present accuracy and validation tests on both smooth and shock dominated evolutions to investigate the feasibility of coupling means for multi physics simulations we then introduce new reference tests where spatially different flux prescriptions require coupling strategies across the domains where local advection burgers or nonconvex behavior is imposed for these nonlinear single scalar equations we can illustrate and analytically explain the evolutions obtained when handling cases where fluxes differ in spatially non overlapping regions we discuss both conservative and non conservative ways of coupling across the region boundaries when coupling scalar conservation laws where the characteristic speeds change discontinuously across the regional boundaries these two strategies yield differing but mathematically sound solution behaviors their relevance for multi physics simulations where one couples systems of hyperbolic pdes is discussed

a high resolution tvd finite volume scheme for the euler equations in conservation form abstract a finite volume numerical algorithm using nonuniformly distributed skewed quadrilateral cells is developed to solve the euler equations in conservation form fluxes across the cell boundaries are computed using the total variation diminishing tvd methodology developed by harten proof of the tvd property for nonuniform grid systems and a truncation error analysis are presented along with a discussion of the treatment of boundary conditions four test cases are given to examine the efficacy of the present scheme using a system of skewed and nonuniform cells the excellent agreement of the numerical results with exact solutions and carefully designed experiments demonstrates the ability of the present scheme to accurately resolve complicated wave developments and interactions using highly nonuniform multiblock zonal cell distributions

an fct finite element scheme for ideal mhd equations in 1d and 2d abstract this paper presents an implicit finite element fe scheme for solving the equations of ideal magnetohydrodynamics in 1d and 2d the continuous galerkin approximation is constrained using a flux corrected transport fct algorithm the underlying low order scheme is constructed using a rusanov type artificial viscosity operator based on scalar dissipation proportional to the fast wave speed the accuracy of the low order solution can be improved using a shock detector which makes it possible to prelimit the added viscosity in a monotonicity preserving iterative manner at the fct correction step the changes of conserved quantities are limited in a way which guarantees positivity preservation for the density and thermal pressure divergence free magnetic fields are extracted using projections of the fct predictor into staggered finite element spaces forming exact sequences in the 2d case the magnetic field is projected into the space of raviart thomas finite elements numerical studies for standard test problems are performed to verify the ability of the proposed algorithms to enforce relevant constraints in applications to ideal mhd flows