statistical theory for the stochastic burgers equation in the inviscid limit a statistical theory is developed for the stochastic burgers equation in the inviscid limit master equations for the probability density functions of velocity velocity difference and velocity gradient are derived no closure assumptions are made instead closure is achieved through a dimension reduction process namely the unclosed terms are expressed in terms of statistical quantities for the singular structures of the velocity field here the shocks master equations for the environment of the shocks are further expressed in terms of the statistics of singular structures on the shocks namely the points of shock generation and collisions the scaling laws of the structure functions are derived through the analysis of the master equations rigorous bounds on the decay of the tail probabilities for the velocity gradient are obtained using realizability constraints we also establish that the probability density function q of the velocity gradient decays as jj 7 2 as 1 c 2000 john wiley sons inc

statistics of shocks in solutions of inviscid burgers equation the purpose of this paper is to analyze statistical properties of discontinuities of solutions of the inviscid burgers equation having a typical realizationb y of the brownian motion as an initial datum this case was proposed and studied numerically in the companion paper by she aurell and frisch the description of the statistics is given in terms of the behavior of the convex hull of the random process w y int limits 0 y b eta eta d eta the hausdorff dimension of the closed set of thosey where the convex hull coincides withw is also studied

the pointwise estimates of solutions for a model system of the radiating gas in multi dimensions in this paper we study the time asymptotic behavior of solutions for a hyperbolic elliptic coupled system known as a model system of the radiating gas in rn when the initial perturbation corresponding to a positive constant state is sufficiently small in hs rn the global existence and pointwise estimates of the solution are obtained by using the method of the green function combined with some energy estimates furthermore we obtain the lp 1 p convergence rate of the solution

asymptotic behavior of solutions to a hyperbolic elliptic coupled system in multi dimensional radiating gas this paper is concerned with the asymptotic behavior of solutions to the cauchy problem of a hyperbolic elliptic coupled system in the multi dimensional radiating gas ut a u2 divq 0 divq q u 0 with initial data u x1 xn 0 u0 x1 xn u x1 first for the case with the same end states u u 0 we prove the existence and uniqueness of the global solutions to the above cauchy problem by combining some a priori estimates and the local existence based on the continuity argument then lp convergence rates of solutions are respectively obtained by applying l2 energy method for n 1 2 3 and lp energy method for 3 n 8 and interpolation inequality furthermore by semigroup argument we obtain the decay rates to the diffusion waves for 1 n 8 secondly for the case with the different end states u u our main concern is that the corresponding cauchy problem in n dimensional space n 1 2 3 behaviors like planar rarefaction waves its convergence rate is also obtained by l2 energy method and l1 estimate

a non local elliptic hyperbolic system related to the short pulse equation abstract in this paper we prove the well posedness of a non local elliptic hyperbolic system related to the short pulse equation it is a model which describes the evolution of the electrical field of linearly polarized continuum spectrum pulses in optical waveguides including fused silica telecommunication type or photonic crystal fibers as well as hollow capillaries filled with transparent gases

mean field approximation and a small parameter in turbulence theory numerical and physical experiments on two dimensional 2d turbulence show that the differences of transverse components of velocity field are well described by gaussian statistics and kolmogorov scaling exponents in this case the dissipation fluctuations are irrelevant in the limit of small viscosity in general one can assume the existence of a critical space dimensionality d d c at which the energy flux and all odd order moments of velocity difference change sign and the dissipation fluctuations become dynamically unimportant at dl d c the flow can be described by the mean field theory leading to the observed gaussian statistics and kolmogorov scaling of transverse velocity differences it is shown that in the vicinity of d d c the ratio of the relaxation and translation characteristic times decreases to zero thus giving rise to a small parameter of the theory the expressions for pressure and dissipation contributions to the exact equation for the generating function of transverse velocity differences are derived in the vicinity of d d c the resulting equation describes experimental data on two dimensional turbulence and demonstrates the onset of intermittency as d ensuremath d c g0 and r stackrel ensuremath rightarrow l 0 in three dimensional flows in close agreement with experimental data in addition some exact relations between correlation functions of velocity differences are derived it is also predicted that the single point probability density function of transverse velocity components in developing as well as in the large scale stabilized two dimensional turbulence is a gaussian

the small dispersion limit of the korteweg de vries equation iii in parts i and ii we have derived explicit formulas for the distribution limit u of the solution of the kdv equation as the coefficient of uxxx tends to zero this formula contains n parameters 1 n whose values as well as whose number depends on x and t in section 4 we have shown that for t tb n 1 and the value of was determined in section 5 we have shown that the parameters i satisfy a nonlinear system of partial differential equations in part iii section 6 we show that for t large n 3 and we determine the asymptotic behavior of 1 2 3 and of u and u2 for t large the explicit formulas show that u and u2 are o t 1 and o t 2 respectively see formulas 6 2 and 6 24 in section 7 we study initial data whose value tends to zero as x and to 1 as x if one accepts some plausible guesses about the behavior of solutions with such initial data we derive an explicit formula for the solution and determine the large scale asymptotic behavior of the solution the function s is expressible in terms of complete elliptic integrals a similar formula is derived for u2 in section 8 we indicate how to extend the treatment of this series of papers to multihumped but still negative initial data

on crank nicolson adams bashforth timestepping for approximate deconvolution models in two dimensions we consider a crank nicolson adams bashforth temporal discretization together with a finite element spatial discretization for efficiently computing solutions to approximate deconvolution models of incompressible flow in two dimensions we prove a restriction on the timestep that will guarantee stability and provide several numerical experiments that show the proposed method is very effective at finding accurate coarse mesh approximations for benchmark flow problems

short note viscous and inviscid regularizations in a class of evolutionary partial differential equations we investigate solution properties of a class of evolutionary partial differential equations pdes with viscous and inviscid regularization an equation in this class of pdes can be written as an evolution equation involving only first order spatial derivatives coupled with the helmholtz equation a recently developed two step iterative method p h chiu l lee t w h sheu a dispersion relation preserving algorithm for a nonlinear shallow water wave equation j comput phys 228 2009 8034 8052 is employed to study this class of pdes the method is in principle superior for pde s in this class as it preserves their physical dispersive features in particular we focus on a leray type regularization h s bhat r c fetecau a hamiltonian regularization of the burgers equation j nonlinear sci 16 2006 615 638 of the hopf equation proposed in alternative to the classical burgers viscous term we show that the regularization effects induced by the alternative model can be vastly different from those induced by burgers viscosity depending on the smoothness of initial data in the limit of zero regularization we validate our numerical scheme by comparison with a particle method which admits closed form solutions further effects of the interplay between the dispersive terms comprising the leray regularization are illustrated by solutions of equations in this class resulting from regularized burgers equation by selective elimination of dispersive terms

on the application of congruent upwind discretizations for large eddy simulations upwind schemes were judged inappropriate for performing accurate large eddy simulations of turbulent flow owing to the artificial dissipation that is present at high wavenumbers of the energy content such a conclusion has been drawn also from some results obtained by adopting finite difference schemes the present paper illustrates the performances of some new finite volume upwind discretization of the convective terms in the case of the 1 d burgers model equation while studying the effects of numerical discretization on several sub grid scales turbulence models starting from the classical static and dynamic eddy viscosity models through the recent deconvolution based ones basing on previously published papers large eddy simulations along with a deconvolution based procedure for de filtering the evolving variable have been originally developed and applied it will be shown how the coherent application of the procedure allows us to develop high order accurate finite volume upwind schemes which maintain a good spectral resolution in the entire range of resolved scale such schemes can be candidate for performing accurate simulations of real turbulence

on the accuracy of the finite element method plus time relaxation if u denotes a local spatial average of u then u u u is the associated fluctuation consider a time relaxation term added to the usual finite element method the simplest case for the model advection equation u t a u x t is u h t a u h v h u h v h f x t v h we analyze the error in this and more importantly higher order extensions and show that the added time relaxation term not only suppresses excess energy in marginally resolved scales but also increases the accuracy of the resulting finite element approximation

the regularized chapman enskog expansion for scalar conservation laws rosenao r has recently proposed a regularized version of the chapmanenskog expansion of hydrodynamics this regularized expansion resembles the usual navier stokes viscosity terms at low wave numbers but unlike the latter it has the advantage of being a bounded macroscopic approximation to the linearized collision operator in this paper we study the behavior of the rosenau regularization of the chapman enskog expansion r c e in the context of scalar conservation laws we show that this r c e model retains the essential properties of the usual viscosity approximation e g existence of travelling waves monotonicity upper lipschitz continuity etc and at the same time it sharpens the standard viscous shock layers we prove that the regularized r c e approximation converges to the underlying inviscid entropy solution as its mean free path e 0 and we estimate the convergence rate

bv estimates on lax friedrichs scheme for a radiating gas model with nonlinear radiative inhomogeneity abstract in this paper we obtain bv estimates and l 1 stability of solutions to a radiating gas model with nonlinear radiative inhomogeneity by using lax friedrichs scheme when a modified c f l courant friedrichs levy condition is satisfied our result is a natural extension to that of the radiating gas model with linear radiative inhomogeneity so that it provides an understanding on the balance between nonlinear convection and nonlinear radiative inhomogeneity the method for case of linear radiative inhomogeneity to obtain the key estimate on uniform bound does not work for the case of nonlinear radiative inhomogeneity the difficulty is overcome by employing a simple but effective monotonic technic

the riemann problem for the leray burgers equation abstract for riemann data consisting of a single decreasing jump we find that the leray regularization captures the correct shock solution of the inviscid burgers equation however for riemann data consisting of a single increasing jump the leray regularization captures an unphysical shock this behavior can be remedied by considering the behavior of the leray regularization with initial data consisting of an arbitrary mollification of the riemann data as we show for this case the leray regularization captures the correct rarefaction solution of the inviscid burgers equation additionally we prove the existence and uniqueness of solutions of the leray regularized equation for a large class of discontinuous initial data all of our results make extensive use of a reformulation of the leray regularized equation in the lagrangian reference frame the results indicate that the regularization works by bending the characteristics of the inviscid burgers equation and thereby preventing their finite time crossing

fujita s conjecture and frobenius amplitude we prove a version of fujita s conjecture in arbitrary characteristic generalizing results of k e smith our methods use the frobenius morphism but avoid tight closure theory we also obtain versions of fujita s conjecture for coherent sheaves with certain ampleness properties

a dynamic multiscale viscosity method for the spectral approximation of conservation laws we consider the spectral approximation of a conservation law in the limit of small or vanishing viscosities in this regime the continuous solution of the problem is known to develop sharp spatial and temporal gradients referred to as shocks also the standard fourier galerkin solution is known to break down if the mesh parameter is larger than the shock width in this paper we propose a new dynamic multiscale viscosity method which enables the solution of such systems with relatively coarse discretizations the key features of this method are 1 separate viscosities are applied to the coarse and the fine scale equations 2 these viscosities are determined as a part of the calculation dynamically from a consistency condition which must be satisfied if the resulting numerical solution is optimal in a user defined sense in this paper we develop these conditions and demonstrate how they may be used to determine the numerical viscosities we apply the proposed method to the one dimensional burgers equation and note that it yields results that compare favorably with the vanishing spectral viscosity solution 2005 elsevier b v all rights reserved

instantons in the burgers equation the instanton solution for the forced burgers equation is found this solution describes the exponential tail of the probability distribution function of velocity differences in the region where shock waves are absent that is for large positive velocity differences the results agree with the one found recently by polyakov who used the operator product conjecture if this conjecture is true then our wkb asymptotics of the wyld functional integral should be exact to all orders of perturbation expansion around the instanton solution we also generalized our solution for the arbitrary dimension of the burgers kpz equation as a result we found the asymptotics of the angular dependence of the velocity difference probability distribution function textcopyright 1996 the american physical society

semistable sheaves in positive characteristic we prove maruyama s conjecture on the boundedness of slope semistable sheaves on a projective variety defined over a noetherian ring our approach also gives a new proof of the boundedness for varieties defined over a characteristic zero field this result implies that in mixed characteristic the moduli spaces of gieseker semistable sheaves are projective schemes of finite type the proof uses a new inequality bounding slopes of the restriction of a sheaf to a hypersurface in terms of its slope and the discriminant this inequality also leads to effective restriction theorems in all characteristics improving earlier results in characteristic zero

stabilized approximation to degenerate transport equations via filtering abstract we analyze a stabilization technique for degenerate transport equations of particular interest are coupled parabolic hyperbolic problems when the diffusion coefficient is zero in part of the domain the unstabilized computed approximations of these problems are highly oscillatory and several techniques have been proposed and analyzed to mitigate the effects of the sub grid errors that contribute to the oscillatory behavior in this paper we modify a time relaxation algorithm proposed in 1 and further studied in 10 our modification introduces the relaxation operator as a post processing step the operator is not time dependent so the discrete relaxation system need only be factored once we provide convergence analysis for our algorithm along with numerical results for several model problems

truncation of scales by time relaxation abstract we study a time relaxation regularization of flow problems proposed and tested extensively by stolz and adams the aim of the relaxation term is to drive the unresolved fluctuations in a computational simulation to zero exponentially fast by an appropriate and often problem dependent choice of its coefficient this relaxation term is thus intermediate between a tunable numerical stabilization and a continuum modeling term our aim herein is to understand how this term by itself acts to truncate solution scales and to use this understanding to give insight into parameter selection

computations of slowly moving shocks computations of slowly moving shocks by shock capturing schemes may generate oscillations that appear as a wavy tail attached to the shock front these oscillations are generated already by first order schemes but become more pronounced in higher order schemes due to their lower dissipation we focus on two first order schemes which seem to exhibit different behaviors i the first order upwind uw scheme which generates strong oscillations and ii the lax friedrichs scheme which appears not to generate any disturbances at all a key observation is that in the uw case the numerical viscosity in the shock family vanishes inside the slow shock layer simple scaling arguments show that third order effects on the solution may no longer be neglected we derive the third order modified equation for the uw scheme and regard the oscillatory solution as a traveling wave solution of the parabolic modified equation plus a small perturbation we then look at the governing equation for the perturbation which points to a plausible mechanism by which postshock oscillations are generated it contains a third order source term that becomes significant inside the shock layer and a nonlinear coupling term which projects the perturbation on all characteristic fields including those not associated with the shock family

a regularization of the burgers equation using a filtered convective velocity this paper examines the properties of a regularization of the burgers equation in one and multiple dimensions using a filtered convective velocity which we have dubbed as the convectively filtered burgers cfb equation a physical motivation behind the filtering technique is presented an existence and uniqueness theorem for multiple dimensions and a general class of filters is proven multiple invariants of motion are found for the cfb equation which are shown to be shared with the viscous and inviscid burgers equations traveling wave solutions are found for a general class of filters and are shown to converge to weak solutions of the inviscid burgers equation with the correct wave speed numerical simulations are conducted in 1d and 2d cases where the shock behavior shock thickness and kinetic energy decay are examined energy spectra are also examined and are shown to be related to the smoothness of the solutions this approach is presented with the hope of being extended to shock regularization of compressible euler equations

singular limits for a parabolic elliptic regularization of scalar conservation laws abstract we consider scalar hyperbolic conservation laws with a nonconvex flux in one space dimension then weak solutions of the associated initial value problems can contain undercompressive shock waves we regularize the hyperbolic equation by a parabolic elliptic system that produces undercompressive waves in the hyperbolic limit regime moreover we show that in another limit regime called capillarity limit we recover solutions of a diffusive dispersive regularization which is the standard regularization used to approximate undercompressive waves in fact the new parabolic elliptic system can be understood as a low order approximation of the third order diffusive dispersive regularization thus sharing some similarities with the relaxation approximations a study of the traveling waves for the parabolic elliptic system completes the paper

on a leray α model of turbulence in this paper we introduce and study a new model for three dimensional turbulence the leray model this model is inspired by the lagrangian averaged navier stokes model of turbulence also known navier stokes model or the viscous camassa holm equations as in the case of the lagrangian averaged navier stokes model the leray model compares successfully with empirical data from turbulent channel and pipe flows for a wide range of reynolds numbers we establish here an upper bound for the dimension of the global attractor the number of degrees of freedom of the leray model of the order of l l d 12 7 where l is the size of the domain and l d is the dissipation length scale this upper bound is much smaller than what one would expect for three dimensional models i e l l d 3 this remarkable result suggests that the leray model has a great potential to become a good sub grid scale large eddy simulation model of turbulence we support this observation by studying analytically and computationally the energy spectrum and show that in addition to the usual k 5 3 kolmogorov power law the inertial range has a steeper power law spectrum for wavenumbers larger than 1 finally we propose a prandtl like boundary layer model induced by the leray model and show a very good agreement of this model with empirical data for turbulent boundary layers

a high resolution differential filter for large eddy simulation a high resolution low pass differential filter for numerical simulations on unstructured grids is proposed the finite element discretization of the filter equation on a structured grid results in a discrete compact filter the proposed filter is significantly less dissipative than germano s differential filter while completely suppresses fluctuations at the grid cut off frequency manufactured solutions were used to verify the performance of the proposed filter the results suggest that the proposed filter will be very effective for explicit filtering and approximate deconvolution modeling in large eddy simulations

legendre pseudospectral viscosity method for nonlinear conservation laws in this paper the legendre spectral viscosity sv method for the approximate solution of initial boundary value problems associated with nonlinear conservation laws is studied the authors prove that by adding a small amount of sv bounded solutions of the legendre sv method converge to the exact scalar entropy solution the convergence proof is based on compensated compactness arguments and therefore applies to certain 2 times 2 systems finally numerical experiments for scalar as well as the one dimensional system of gas dynamics equations are presented which confirm the convergence of the legendre sv method moreover these numerical experiments indicate that by post processing the sv approximation one can recover the entropy solution within spectral accuracy

a stabilized semi implicit galerkin scheme for navier stokes equations by introducing a time relaxation term for the time derivative of higher frequency components we proposed a stabilized semi implicit galerkin scheme for evolutionary navier stokes equations in this paper analysis shows that such a scheme has weaker stability conditions than that of a classical semi implicit galerkin scheme and when a suitable relaxation parameter s is chosen it generates an approximate solution with the same accuracy as the classical one that means the proposed scheme might use a larger time step to generate a bounded approximate solution thus it is more suitable for long time simulations

commutator errors in large eddy simulation commutator errors arise in large eddy simulation of incompressible turbulent flow from the application of non uniform filters to the continuity and navier stokes equations for non uniform high order filters with bounded moments the magnitude of the commutator errors is shown to be of the same order as that of the turbulent stress fluxes consequently one cannot reduce the size of the commutator errors independently of the turbulent stress terms by any judicious construction of such filter operators independent control over the commutator errors compared to the turbulent stress fluxes can instead be obtained by appropriately restricting the spatial variations of the filter width and filter skewness for situations in which the dynamical consequences of the commutator errors are significant e g near solid boundaries explicit similarity modelling for the commutator errors is proposed including the application of leray regularization the performance of this commutator error parametrization is illustrated for the one dimensional burgers equation the leray approach is found to capture the filtered flow with higher accuracy than conventional similarity modelling which is particularly relevant for large filter width variations

on the uniqueness of discontinuous solutions to the degasperis procesi equation abstract we prove uniqueness within a class of discontinuous solutions to the nonlinear and third order dispersive degasperis procesi equation t u t x x 3 u 4 u x u 3 x u x x 2 u u x x x 3 u in a recent paper g m coclite k h karlsen on the well posedness of the degasperis procesi equation j funct anal 233 2006 60 91 we proved for this equation the existence and uniqueness of l 1 bv weak solutions satisfying an infinite family of kru kov type entropy inequalities the purpose of this paper is to replace the kru kov type entropy inequalities by an ole nik type estimate and to prove uniqueness via a nonlocal adjoint problem an implication is that a shock wave in an entropy weak solution to the degasperis procesi equation is admissible only if it jumps down in value like the inviscid burgers equation

probability density and scaling exponents of the moments of longitudinal velocity difference in strong turbulence we consider a few cases of homogeneous and isotropic turbulence differing by the mechanisms of turbulence generation the advective terms in the navier stokes and burgers equations are similar it is proposed that the longitudinal structure functions s n r in homogeneous and isotropic three dimensional turbulence are governed by a one dimensional 1d equation of motion resembling the 1d burgers equation with the strongly nonlocal pressure contributions accounted for by galilean invariance breaking terms the resulting equations not involving parameters taken from experimental data give both scaling exponents and amplitudes of the structure functions in an excellent agreement with experimental data the derived probability density function p ensuremath delta u r ensuremath ne p ensuremath ensuremath delta u r but p ensuremath delta u r p ensuremath ensuremath delta u ensuremath r in accord with the symmetry properties of the navier stokes equations with decrease of the displacement r the probability density which cannot be represented in a scale invariant form shows smooth variation from the gaussian at the large scales to close to exponential function thus demonstrating onset of small scale intermittency it is shown that accounting for the subdominant contributions to the structure functions s n r ensuremath propto r ensuremath xi n is crucial for a derivation of the amplitudes of the moments of the velocity difference

two dimensional turbulence in the inverse cascade range numerical and physical experiments on forced two dimensional navier stokes equations show that transverse velocity differences are described by normal kolmogorov scaling proportional r 2n 3 and obey gaussian statistics since nontrivial scaling is a sign of the strong nonlinearity of the problem these two results seem to contradict each other a theory explaining these observations is presented in this paper the derived self consistent expression for the pressure gradient contributions leads to the conclusion that small scale transverse velocity differences are governed by a linear langevin like equation stirred by a nonlocal universal solution dependent gaussian random force this explains the experimentally observed gaussian statistics of transverse velocity differences and their kolmogorov scaling the solution for the pdf of longitudinal velocity differences is based on the numerical smallness of the energy flux in two dimensional turbulence the theory makes a few quantitative predictions that can be tested experimentally

a spectral vanishing viscosity method for large eddy simulations a new simulation approach for high reynolds number turbulent flows is developed combining concepts of monotonicity in nonlinear conservation laws with concepts of large eddy simulation the spectral vanishing viscosity svv first introduced by e tadmor siam j numer anal 26 30 1989 is incorporated into the navier stokes equations for controlling high wavenumber oscillations unlike hyperviscosity kernels the svv approach involves a second order operator which can be readily implemented in standard finite element codes in the work presented here discretization is performed using hierarchical spectral hp methods accommodating effectively an ab initio intrinsic scale separation the key result is that monotonicity is enforced via svv leading to stable discretizations without sacrificing the formal accuracy i e exponential convergence in the proposed discretization several examples are presented to demonstrate the effectiveness of the new approach including a comparison with eddy viscosity spectral les of turbulent channel flow in its current implementation the svv approach for controlling the small scales is decoupled from the large scales but a procedure is proposed that will provide coupling similar to the classical les formulation

velocity difference probability density functions for burgers turbulence in this paper the polyakov equation phys rev e bold 52 6183 1995 for the velocity difference probability density functions with the random gaussian external force with the correlation function kappa y approximately 1 minus y sup alpha is analyzed solutions for the cases alpha l brace 2 1 2 1 r brace are found which agree very well with available numerical results it is also argued that the stationary regime of burgers turbulence can depend not only on the distribution of the external force but also on the dissipative regularization copyright ital 1997 ital the american physical society

numerical analysis of filter based stabilization for evolution equations we consider filter based stabilization for evolution equations in general and for the navier stokes equations in particular the first method we consider is to advance in time one time step by a given method and then to apply an uncoupled and modular filter to get the approximation at the new time level this filter based stabilization although algorithmically appealing is viewed in the literature as introducing far too much numerical dissipation to achieve a quality approximate solution we show that this is indeed the case we then consider a modification evolve one time step filter deconvolve then relax to get the approximation at the new time step we give a precise analysis of the numerical diffusion and error in this process and show it has great promise confirmed in several numerical experiments

large eddy simulation of shock turbulence interaction the approximate deconvolution model adm for large eddy simulation les is adapted for use with a central fourth order finite volume jameson type scheme widely applied for aerodynamics calculations of industrial interest with adm an approximation of the unfiltered solution is obtained by repeated filtering and given a good approximation of the unfiltered solution the nonlinear terms of the filtered navier stokes equation are computed directly the effect of scales not represented on the numerical grid is modeled by a relaxation regularization involving a secondary filter operation and a relaxation parameter the supersonic turbulent boundary layer along a compression ramp at a freestream mach number of m 3 and a reynolds number based on free stream quantities and the mean momentum thickness at inflow of re 1685 is computed to evaluate the ability of adm to represent discontinuities and turbulent flow regions it was observed that a unified modeling of discontinuities and turbulence required a local adaptation of the secondary filter used for the relaxation regularization the les results are compared with corresponding filtered direct numerical simulation dns data very good agreement between the filtered dns and the les is observed for the mean fluctuating and averaged wall quantities

fourth order finite difference p sv seismograms i describe the properties of a fourth order accurate space second order accurate time two dimensional p sk finite difference scheme based on the madariagavirieux staggered grid formulation the numerical scheme is developed from the first order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses the madariaga virieux staggered grid scheme has the desirable quality that it can correctly model any variation in material properties including both large and small poisson s ratio materials with minimal numerical dispersion and numerical anisotropy dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints wavelength the scheme can be used to accurately simulate wave propagation in mixed acoustic elastic media making it ideal for modeling marine problems explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free surface or within a layer and to satisfy free surface boundary conditions benchmark comparisons of finite difference and analytical solutions to lamb s problem are almost identical as are comparisons of finite difference and reflectivity solutions for elastic elastic and acoustic elastic layered models

sailing the deep blue sea of decaying burgers turbulence we study lagrangian trajectories and scalar transport statistics in decaying burgers turbulence we choose velocity fields solutions of the inviscid burgers equation whose probability distributions are specified by kida s statistics they are time correlated and neither time reversal invariant nor gaussian we discuss in some detail the effect of shocks on trajectories and transport equations we derive the inviscid limit of these equations using a formalism of operators localized on shocks we compute the probability distribution functions of the trajectories although they do not define markov processes as physically expected these trajectories are statistically well defined but collapse with probability one at infinite time we point out that the advected scalars enjoy inverse energy cascades we also make a few comments on the connection between our computations and persistence problems

canonical phase space approach to the noisy burgers equation probability distributions we present a canonical phase space approach to stochastic systems described by langevin equations driven by white noise mapping the associated fokker planck equation to a hamilton jacobi equation in the nonperturbative weak noise limit we invoke a principle of least action for the determination of the probability distributions we apply the scheme to the noisy burgers and kardar parisi zhang equations and discuss the time dependent and stationary probability distributions in one dimension we derive the long time skew distribution approaching the symmetric stationary gaussian distribution in the short time region we discuss heuristically the nonlinear soliton contributions and derive an expression for the distribution in accordance with the directed polymer replica and asymmetric exclusion model results we also comment on the distribution in higher dimensions

velocity fluctuations in forced burgers turbulence burgers turbulence the tails of the velocity gradient distribution in the regions where there are no shocks we evaluate the right tail through a rather simple computation and compare it to the more sophisticated approaches developed recently 7 3 6 we then give a conjecture on the left tail which is based on a plausible argument requiring the system to reach a stationary state we shall first discuss the one dimensional case then turn to higher dimensions and compare our results with the previously available ones

burgers poisson a nonlinear dispersive model equation a dispersive model equation is considered which has been proposed by whitham linear and nonlinear waves john wiley sons new york 1974 as a shallow water model and which can also be seen as a caricature of two species euler poisson problems a number of formal properties as well as similarities to other dispersive equations are derived a travelling wave analysis and some numerical tests are carried out the equation features wave breaking in finite time a local existence result for smooth solutions and a global existence result for weak entropy solutions are proved finally a small dispersion limit is carried out for situations where the solution of the limiting equation is smooth

an examination of the homentropic euler equations with averaged characteristics abstract this paper examines the properties of the homentropic euler equations when the characteristics of the equations have been spatially averaged the new equations are referred to as the characteristically averaged homentropic euler cahe equations an existence and uniqueness proof for the modified equations is given the speed of shocks for the cahe equations are determined the riemann problem is examined and a general form of the solutions is presented finally numerically simulations on the homentropic euler and cahe equations are conducted and the behaviors of the two sets of equations are compared

dimension reduction method for ode fluid models we develop a new dimension reduction method for large size systems of ordinary differential equations odes obtained from a discretization of partial differential equations of viscous single and multiphase fluid flow the method is also applicable to other large size classical particle systems with negligibly small variations of particle concentration we propose a new computational closure for mesoscale balance equations based on numerical iterative deconvolution to illustrate the computational advantages of the proposed reduction method we use it to solve a system of smoothed particle hydrodynamic odes describing single phase and two phase layered poiseuille flows driven by uniform and periodic in space body forces for the single phase poiseuille flow driven by the uniform force the coarse solution was obtained with the zero order deconvolution for the single phase flow driven by the periodic body force and for the two phase flows the higher order the first and second order deconvolutions were necessary to obtain a sufficiently accurate solution

dithering of absorbers for efficient finite difference modeling of viscoelastic wave propagation viscoelastic modeling traditionally uses multiple relaxation mechanisms absorbers per wave type in each cell of the numerical grid we show that one absorber per cell suffices when the absorbers of different relaxation frequencies and absorption strengths are dithered in a supercell i e a very small area of several nearby cells the relaxation frequencies used in a supercell can be preselected to cover the bandwidth required in the modeling and distributed locally in a random manner to minimize numerical scattering the absorption strengths however must be found iteratively to yield an average absorption that matches the desired medium over the bandwidth we have tested the dithered absorber method for modeling a variety of 2 d media using a staggered grid finite difference scheme for a uniform medium our result agrees well with the exact analytic solution for inhomogeneous models the dithered absorber solution agrees well with the finite difference solution computed with five absorbers per cel

turbulence without pressure we develop exact field theoretic methods to treat turbulence when the effect of pressure is negligible we find explicit forms of certain probability distributions demonstrate that the breakdown of galilean invariance is responsible for intermittency and establish the operator product expansion we also indicate how the effects of pressure can be turned on perturbatively

probability distribution functions of derivatives and increments for decaying burgers turbulence a lagrangian method is used to show that the power law with a ensuremath 7 2 exponent in the negative tail of the probability distribution function pdf of the velocity gradient and of velocity increments predicted by e et al phys rev lett 78 1904 1997 for forced burgers turbulence is also present in the unforced case the theory is extended to the second order space derivative whose pdf has power law tails with exponent ensuremath 2 at both large positive and negative values and to the time derivatives pdf s of space and time derivatives have the same asymptotic functional forms this is interpreted in terms of a random taylor hypothesis

intermittency of height fluctuations in stationary state of the kardar parisi zhang equation with infinitesimal surface tension in 1 1 dimensions the kardar parisi zhang kpz equation with infinitesimal surface tension dynamically develops sharply connected valley structures within which the height derivative is not continuous we discuss the intermittency issue in the problem of stationary state forced kpz equation in 1 1 dimensions it is proved that the moments of height increments c a behave as x 1 x 2 xi a with xi a a for length scales x 1 x 2 sigma the length scale sigma is the characteristic length of the forcing term we have checked the analytical results by direct numerical simulation

modeling of a constant q methodology and algorithm for an efficient and optimally inexpensive viscoelastic technique linear anelastic phenomena in wave propagation problems can be well modeled through a viscoelastic mechanical model consisting of standard linear solids in this paper we present a method for modeling of constant q as a function of frequency based on an explicit closed formula for calculation of the parameter fields several standard linear solids connected in parallel can be tuned through a single parameter to yield an excellent constant q approximation the proposed method enables substantial savings in computations and memory requirements experiments show that the new method also yields higher accuracy in the modeling of q than e g the pade approximant method

existence of mori fibre spaces for 3 folds in char p abstract we prove the following results for projective klt pairs of dimension 3 over an algebraically closed field of characteristic p 5 the cone theorem the base point free theorem the contraction theorem finiteness of minimal models termination with scaling existence of mori fibre spaces etc

the inviscid burgers equation with initial data of brownian type the solutions to burgers equation in the limit of vanishing viscosity are investigated when the initial velocity is a brownian motion or fractional brownian motion function i e a gaussian process with scaling exponent 0 h 1 typea or the derivative thereof with scaling exponent 1 h 0 typeb largesize numerical experiments are performed helped by the fact that the solution is essentially obtained by performing a legendre transform the main result is obtained for typea and concerns the lagrangian functionx a which gives the location at timet 1 of the fluid particle which started at the locationa it is found to be a complete devil s staircase the cumulative probability of lagrangian shock intervals a also the distribution of shock amplitudes follows a a h law for small a the remaining regular lagrangian locations form a cantor set of dimensionh in eulerian coordinates the shock locations are everywhere dense the scaling properties of various statistical quantities are also found heuristic interpretations are provided for some of these results rigorous results for the case of brownian motion are established in a companion paper by ya sinai for typeb initial velocities e g white noise there are very few small shocks and shock locations appear to be isolated finally it is shown that there are universality classes of random but smooth non scaling initial velocities such that the long time large scale behavior is after rescaling the same as for typea orb

on different notions of tameness in arithmetic geometry the notion of a tamely ramified covering is canonical only for curves several notions of tameness for coverings of higher dimensional schemes have been used in the literature we show that all these definitions are essentially equivalent furthermore we prove finiteness theorems for the tame fundamental groups of arithmetic schemes

discrete models of fluids spatial averaging closure and model reduction we consider semidiscrete ode models of single phase fluids and two fluid mixtures in the presence of multiple fine scale heterogeneities the size of these ode systems can be very large spatial averaging is then a useful tool for reducing computational complexity of the problem the averages satisfy exact balance equations of mass momentum and energy these equations do not form a satisfactory continuum model because evaluation of stress and heat flux requires solving the underlying odes to produce continuum equations that can be simulated without resolving microscale dynamics we recently proposed a closure method based on the use of regularized deconvolution here we continue the investigation of deconvolution closure with the long term objective of developing consistent computational upscaling for multiphase particle methods the structure of the fine scale particle solvers is reminiscent of molecular dynamics for this reason we use nonlinear averaging introduced for atomistic systems by noll ha

incorporation of attenuation into time domain computations of seismic wave fields the only numerically tractable way yet found to incorporate attenuation into numerical time domain computations of seismic wave fields is to approximate the viscoelastic modulus by a low order rational function of frequency the coefficients of this function can be determined by the pade approximation our test computations show however that this approximation generally is of poor quality therefore we suggest a new approach which is based on the rheological model of the generalized maxwell body which has a modulus of the desired rational form we choose the relaxation frequencies logarithmically equidistant in the frequency band of interest and determine the weight factors by simple numerical curve fitting to an arbitrary q law this approach is superior to the method above both in accuracy and in computational efficiency for most practical applications approximations of orders 2 or 3 are sufficient the computing time and memory requirements for a finite difference calculation are then approximately twice those of a purely elastic calculation as a first application of the method we compute sh channel waves in discontinuous coal seams with q 50 within the coal the results show that the high frequency airy phase is strongly attenuated this indicates that care has to be taken in comparing the results of purely elastic model calculations of the propagation of seam waves with experimental data

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

canonical phase space approach to the noisy burgers equation presenting a general phase space approach to stochastic processes we analyze in particular the fokker planck equation for the noisy burgers equation and discuss the time dependent and stationary probability distributions in one dimension we derive the long time skew distribution approaching the symmetric stationary gaussian distribution in the short time regime we discuss heuristically the nonlinear soliton contributions and derive an expression for the distribution in accordance with the directed polymer replica model and asymmetric exclusion model results

temporally regularized direct numerical simulation abstract experience with fluid flow simulation suggests that in some instances under resolved direct numerical simulation dns without a residual stress model per se but with artificial damping of small scales to account for energy lost in the cascade from resolved to unresolved scales may be as reliable as simulations based on more complex models of turbulence one efficient and versatile manner to selectively damp under resolved spatial scales is by a relaxation regularization e g stolz and adams s stolz n a adams an approximate deconvolution procedure for large eddy simulation phys fluids ii 1999 1699 1701 we consider the analogous approach based on time scales time filtering and damping of under resolved temporal features the paper explores theoretical and practical aspects of temporally damped fluid flow simulations we prove existence of solutions to the resulting continuum model we also establish the effect of the damping of under resolved temporal features as the energy balance and dissipation and prove that the time fluctuations 0 in a precise sense the method is then demonstrated to obtain both steady state and time dependent coarse grid solutions of the navier stokes equations

conservation laws with singular nonlocal sources abstract several physical phenomena are modeled by conservation laws with fluxes or sources that are singular in the origin here an integro differential regularization of those problems is proposed the existence of positive solutions with finite total variation is proved

ample filters of invertible sheaves abstract let x be a scheme proper over a commutative noetherian ring a we introduce the concept of an ample filter of invertible sheaves on x and generalize the most important equivalent criteria for ampleness of an invertible sheaf we also prove the theorem of the base for x and generalize serre s vanishing theorem we then generalize results for twisted homogeneous coordinate rings which were previously known only when x was projective over an algebraically closed field specifically we show that the concepts of left and right ampleness are equivalent and that the associated twisted homogeneous coordinate ring must be noetherian

convergence tests for the piecewise parabolic method and navier stokes solutions for homogeneous compressible turbulence the need to simulate fully developed turbulence with wide range of scales led us to use the piecewise parabolic method ppm to solve the euler equations of motions to obtain data for 3 d homogeneous compressible decaying turbulence numerical simulations were performed on computational meshes of up to 10243 zones these data were compared with data obtained by solving the navier stokes ns equations results of studying the kinetic energy enstrophy and the energy power spectra with different resolutions are presented for both the ppm and ns data the results of the comparison show convergence of the ppm and ns solutions to the same limit

well posedness of a reduced order approximate deconvolution turbulence model abstract we prove well posedness for a recently proposed turbulence model and derive an estimate for its modeling error the model is derived from the fourth order representation of the stolz adams approximate deconvolution model by making an additional deconvolution approximation to reduce the model to second order which also allows the model to be computed with efficient and unconditionally stable algorithms

decay rates to the planar rarefaction waves for a model system of the radiating gas in n dimensions abstract in this paper we study the asymptotic decay rates to the planar rarefaction waves to the cauchy problem for a hyperbolic elliptic coupled system called as a model system of the radiating gas in r n n 3 4 5 if the initial perturbations corresponding to the planar rarefaction waves are sufficiently small in h 2 l 1 w 2 6 r n the analysis is based on the l p energy method and several special interpolation inequalities

on the abundance problem for 3 folds in characteristic p 5 in this article we prove two cases of the abundance conjecture for 3 folds in characteristic p 5 i x delta is klt and kappa x k x delta 1 and ii x delta is klt k x delta equiv 0 and x is not uniruled

global well posedness and relaxation limits of a model for radiating gas abstract we study the initial value problem for a hyperbolic elliptic coupled system with arbitrary large discontinuous initial data we prove existence and uniqueness for that model by means of l1 contraction and comparison properties moreover after suitable scalings we study both the hyperbolic parabolic and the hyperbolic hyperbolic relaxation limits for that system

probability distribution functions for the random forced burgers equation statistical properties of solutions of the random forced burgers equation have been a subject of intensive studies recently see refs 1 6 of particular interest are the asymptotic properties of probability distribution functions associated with velocity gradients and velocity increments aside from the fact that such issues are of direct interest to a large number of problems such as the growth of random surfaces 1 it is also hoped that the field theoretic techniques developed for the burgers equation will eventually be useful for understanding more complex phenomena such as turbulence in this paper we propose a new and direct approach for analyzing the scaling properties of the various distribution functions for the random forced burgers equation we will consider the problem

generic positivity and foliations in positive characteristic abstract we prove generic semipositivity of the tangent bundle of a non uniruled calabi yau variety in positive characteristic we also construct an example of a nef line bundle in characteristic zero whose each reduction to positive characteristic is not nef

viscoelastic finite difference modeling real earth media disperse and attenuate propagating mechanical waves this anelastic behavior can be described well by a viscoelastic model we have developed a finite difference simulator to model wave propagation in viscoelastic media the finite difference method was chosen in favor of other methods for several reasons finite difference codes are more portable than for example pseudospectral codes moreover finite difference schemes provide a convenient environment in which to define complicated boundaries a staggered scheme of second order accuracy in time and fourth order accuracy in space appears to be optimally efficient because of intrinsic dispersion no fixed grid points per wavelength rule can be given instead we present tables which enable a choice of grid parameters for a given level of accuracy since the scheme models energy absorption natural and efficient absorbing boundaries may be implemented merely by changing the parameters near the grid boundary the viscoelastic scheme is

explicit filtering and exact reconstruction of the sub filter stresses in large eddy simulation explicit filtering has the effect of reducing numerical or aliasing errors near the grid scale in large eddy simulation les we use a differential filter namely the inverse helmholtz operator which is readily applied to unstructured meshes the filter is invertible which allows the sub filter scale sfs stresses to be exactly reconstructed in terms of the filtered solution unlike eddy viscosity models the method of filtering and reconstruction avoids making any physical assumptions and is therefore valid in any flow regime the sub grid scale sgs stresses are not recoverable by reconstruction but the second order finite element method used here is an adequate source of numerical dissipation in lieu of an sgs model results for incompressible turbulent channel flow at re 180 are presented which show that explicit filtering and exact sfs reconstruction is a significant improvement over the standard les approach of implicit filtering and eddy viscosity sgs modelling

a subgrid scale deconvolution approach for shock capturing we develop a method for the modeling of flow discontinuities which can arise as weak solutions of inviscid conservation laws due to its similarity with recently proposed approximate deconvolution models for large eddy simulation the method potentially allows for a unified treatment of flow discontinuities and turbulent subgrid scales a filtering approach is employed since for the filtered evolution equations the solution is smooth and can be solved for by standard central finite difference schemes without special consideration of discontinuities a sufficiently accurate representation of the filtered nonlinear combination of discontinuous solution components which arise from the convection term can be obtained by a regularized deconvolution applied to the filtered solution for stable integration the evolution equations are supplemented by a relaxation regularization based on a secondary filter operation and a relaxation parameter an estimate for the relaxation parameter is provided the method is related to the spectral vanishing viscosity method and the regularized chapman enskog expansion method for conservation laws we detail the approach and demonstrate its efficiency with the inviscid and viscous burgers equations the isothermal shock problem and the one dimensional euler equations

three dimensional forced burgers turbulence supplemented with a continuity equation we investigate turbulent limit of the forced burgers equation supplemented with a continuity equation in three dimensions the scaling exponent of the conditional two point correlation function of density i e r x1 r x2 udu ux12x2u 2a 3 is calculated self consistently in the nonuniversal region from which we obtain a 353 also we derive an equation governing the evolution of the probability density function pdf of longitudinal velocity increments in length scale from which a possible mechanism for the dependence of the inertial pdf to one point urms is developed

convergence of spectral methods for nonlinear conservation laws we discuss the convergence of fourier methods for scalar nonlinear conservation laws that exhibit spontaneous shock discontinuities numerical tests indicate that the convergence may and in fact in some cases we prove it must fail with or without post processing of the numerical solution instead we introduce here a new kind of spectrally accurate vanishing viscosity to augment the fourier approximation of such nonlinear conservation laws using compensated compactness arguments augmented by the assumption of l infty stability we show for the inviscid burgers model equation that this spectral viscosity method prevents oscillations and convergence to the unique entropy solution follows

simply connected projective manifolds in characteristic p 0 have no nontrivial stratified bundles we show that simply connected projective manifolds in characteristic p 0 have no nontrivial stratified bundles this gives a positive answer to a conjecture by d gieseker ann sc norm super pisa 4 ser 2 1 1 31 1975 the proof uses hrushovski s theorem on periodic points