wavelet based fractal analysis of dna sequences abstract the fractal scaling properties of dna sequences are analyzed using the wavelet transform mapping nucleotide sequences onto a dna walk produces fractal landscapes that can be studied quantitatively by applying the so called wavelet transform modulus maxima method this method provides a natural generalization of the classical box counting techniques to fractal signals the wavelets playing the role of generalized oscillating boxes from the scaling behavior of partition functions that are defined from the wavelet transform modulus maxima this method allows us to determine the singularity spectrum of the considered signal and thereby to achieve a complete multifractal analysis moreover by considering analyzing wavelets that make the wavelet transform microscope blind to patches of different nucleotide composition that are observed in genomic sequences we demonstrate and quantify the existence of long range correlations in the noncoding regions although the fluctuations in the patchy landscape of the dna walks reconstructed from both noncoding and protein coding regions are found homogeneous with gaussian statistics our wavelet based analysis allows us to discriminate unambiguously between the fluctuations of the former which behave like fractional brownian motions from those of the latter which cannot be distinguished from uncorrelated random brownian walks we discuss the robustness of these results with respect to various legitimate codings of the dna sequences finally we comment about the possible understanding of the origin of the observed long range correlations in noncoding dna sequences in terms of the nonequilibrium dynamical processes that produce the isochore structre of the genome

a symmetric formulation for computing transient shallow water flows abstract the shallow water equations fall into the category of symmetric advective diffusive systems with source terms these types of equations can be very effectively solved using stabilized methods such as supg and gls a semi discrete finite element method based on these ingredients is presented for the computation of transient shallow water flows special care has been taken in the design of the operators in order to improve the performance for unsteady calculations the solution is advanced in time via a predictor multi corrector algorithm which includes as special cases the second order trapezoidal rule a first order explicit method and a first order implicit method equivalent to the constant in time element of the space time formulation

hyperbolic conservation laws with relaxation the effect of relaxation is important in many physical situations it is present in the kinetic theory of gases elasticity with memory gas flow with thermo non equilibrium water waves etc the governing equations often take the form of hyperbolic conservation laws with lower order terms in this article we present and analyze a simple model of hyperbolic conservation laws with relaxation effects dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified stability criteria for diffusion waves expansion waves and traveling waves are found and justified nonlinearly time asymptotic expansion and the energy method are used in the analysis for dissipative waves the expansion is similar in spirit to the chapman enskog expansion in the kinetic theory for shock waves however a different approach is needed

convergence to equilibrium for the relaxation approximations of conservation laws we study the cauchy problem for 2 2 semilinear and quasilinear hyperbolic systems with a singular relaxation term special comparison and compactness properties are established by assuming the subcharacteristic condition therefore we can prove the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to 0 this research was strongly motivated by the recent numerical investigations of s jin and z xin on the relaxation schemes for conservation laws 1996 john wiley sons inc

second order boltzmann schemes for compressible euler equations in one and two space dimensions a class of second order numerical schemes for the compressible euler equations is described and their l 1 stability i e rho geqq 0 t geqq 0 is proved following van leer s approach the solution rho u sqrt t here is represented as piecewise linear functions the necessity of a slope limitation appears naturally in the derivation of the schemes but it can be less strict than the slope reconstructions usually used these schemes are written in terms of explicit flux splitting formula and are naturally multidimensional in space the upwinding is obtained through a very generalized notion of characteristics the kinetic one

discrete velocity models and relaxation schemes for traffic flows we present simple discrete velocity models for traffic flows the novel feature in the corresponding relaxation system is the presence of nonnegative velocities only we show that in the small relaxation limit the discrete models reduce to the lighthill whitham richards equation in addition we propose second order schemes combined with implicit explicit time integrators as proper discretization of the relaxation type system numerical tests are carried out on various situations in traffic flow the results show that the proposed models are able to describe correctly the formation of backward waves induced by traffic jam

weak entropy boundary conditions for isentropic gas dynamics via kinetic relaxation abstract we consider a kinetic bgk model relaxing to isentropic gas dynamics previously introduced by the authors but with dirichlet boundary condition on the incoming velocities we pass to the limit as the relaxation parameter tends to zero by compensated compactness inside the domain and obtain that the limit satisfies entropy inequalities on the boundary involving weak traces of entropy fluxes our method is very general and could be applied to any entropy satisfying bgk model as soon as we have strong compactness of the macroscopic variables inside the domain

materials with internal variables and relaxation to conservation laws the theory of materials with internal state variables of coleman gurtin cg provides a natural framework to investigate the structure of relaxation approximations of conservation laws from the viewpoint of continuum thermomechanics after reviewing the requirements imposed on constitutive theories by the principle of consistency with the clausius duhem inequality we pursue two specific theories pertaining to stress relaxation and relaxation of internal energy they each lead to a relaxation framework for the theory of thermoelastic non conductors of heat equipped with globally defined entropy functions for the associated relaxation process next we consider a semilinear model problem of stress relaxation we discuss uniform stability and compactness for solutions of the relaxation system in the zero relaxation limit and establish convergence to the system of isothermal elastodynamics by using compensated compactness finally we prove a strong dissipation estimate for the relaxation approximations proposed in jin xin jx when the limit system is equipped with a strictly convex entropy

global bv solutions to a p system with relaxation in this paper we study the global existence of bv solutions to a p system with a relaxation source term using the glimm s scheme the pressure p is given by a law with 1 by a suitable choice of the measure for the strength of the shock waves we show that the total strength of the waves and the total variation of the solutions are uniformly bounded with respect to the relaxation parameter furthermore when the relaxation parameter tends to zero we show that the sequence of bv solutions eventually converges to a weak solution to the equilibrium equation 2000 academic press

the third order relaxation schemes for hyperbolic conservation laws a class of semi discrete third order relaxation schemes are presented for relaxation systems which approximate systems of hyperbolic conservation laws these schemes for the scalar conservation law are shown to satisfy the property of total variation diminishing tvd in the zero relaxation limit a third order tvd runge kutta splitting method is developed for the temporal discretization of the semi discrete schemes numerical results are given illustrating these schemes on one dimensional nonlinear problems

on the symmetric form of systems of conservation laws with entropy abstract this paper reviews the symmetrizability of systems of conservation laws which possess entropy functions symmetric formulations in conservation form for the equations of gas dynamics are presented

the thermodynamics of fractals revisited with wavelets abstract the multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform this new approach is based on the definition of partition functions from the wavelet transform modulus maxima we demonstrate that very much like thermodynamic functions the generalized fractal dimensions d q and the f singularity spectrum can be readily determined from the scaling behavior of these partition functions we show that this method provides a natural generalization of the classical box counting techniques to fractal signals the wavelets playing the role of generalized boxes we illustrate our theoretical considerations on pedagogical examples e g devil s staircases and fractional brownian motions we also report the results of some recent application of the wavelet transform modulus maxima method to fully developed turbulence data that we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic informations about the scaling properties of fractal objects in particular we show that a dynamical system which leaves invariant such an object can be uncovered form the space scale arrangement of its wavelet transform modulus maxima we elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem this step towards a statistical mechanics of fractals is illustrated on discrete period doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos finally we apply our technique to analyze the fractal hierarchy of dla azimuthal cantor sets defined by intersecting the inner frozen region of large mass off lattice diffusion limited aggregates dla wit a circle this study clearly lets out the existence of an underlying multiplicative process that is likely to account for the fibonacci structural ordering recently discovered in the apparently disordered arborescent dla morphology

multiscale analysis for interacting particles relaxation systems and scalar conservation laws we investigate the derivation of semilinear relaxation systems and scalar conservation laws from a class of stochastic interacting particle systems these systems are markov jump processes set on a lattice they satisfy detailed mass balance but not detailed balance of momentum and are equipped with multiple scalings using a combination of correlation function methods with compactness and convergence properties of semidiscrete relaxation schemes we prove that at a mesoscopic scale the interacting particle system gives rise to a semilinear hyperbolic system of relaxation type while at a macroscopic scale it yields a scalar conservation law rates of convergence are obtained in both scalings

existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in eulerian and lagrangian coordinates we prove the existence and compactness stability of entropy solutions for the hyperbolic systems of conservation laws corresponding to the isentropic gas dynamics where the pressure and density are related by a law for any 1 our results considerably extend and simplify the program initiated by diperna and provide a complete existence proof our methods are based on the compensated compactness and the kinetic formulation of systems of conservation laws 1996 john wiley sons inc

variants of relaxed schemes and two dimensional gas dynamics we present a relaxed scheme with more precise information about local speeds of propagation and a multidimensional construction of the cell averages the physical domain of dependence is simulated correctly and high resolution is maintained relaxation schemes have advantages that include high resolution simplicity and explicitly no approximate riemann solvers and no characteristic decomposition is necessary performance of the scheme is illustrated by tests on two dimensional euler equations of gas dynamics

on asymptotic equivalence of perturbed linear systems of differential and difference equations abstract standard results on asymptotic integration of systems of linear differential equations give sufficient conditions which imply that a system is strongly asymptotically equivalent to its principal diagonal part these involve certain dichotomy conditions on the diagonal part as well as growth conditions on the off diagonal perturbation terms here we study perturbations with a triangularly induced structure and see that growth conditions can be substantially weakened in addition we give results for not necessarily triangular perturbations which in some sense interpolate between the classical theorems of levinson and hartman wintner some analogous results for systems of linear difference equations are also given

construction of bgk models with a family of kinetic entropies for a given system of conservation laws we introduce a general framework for kinetic bgk models we assume to be given a system of hyperbolic conservation laws with a family of lax entropies and we characterize the bgk models that lead to this system in the hydrodynamic limit and that are compatible with the whole family of entropies this is obtained by a new characterization of maxwellians as entropy minimizers that can take into account the simultaneous minimization problems corresponding to the family of entropies we deduce a general procedure to construct such bgk models and we show how classical examples enter the framework we apply our theory to isentropic gas dynamics and full gas dynamics and in both cases we obtain new bgk models satisfying all entropy inequalities

on the barenblatt model for non equilibrium two phase flow in porous media we prove global existence and uniqueness of solutions to a quasilinear goursat problem which was proposed by g i barenblatt to describe non equilibrium two phase fluid flow in permeable porous media when the equilibrium relaxation time tends to zero the solution is shown to converge to the entropy solution of the corresponding initial boundary value problem for the classical buckley leverett equation

higher entropy conservation and numerical stability of compressible turbulence simulations we present a numerical formulation for the treatment of nonlinear instabilities in shock free compressible turbulence simulations the formulation is high order and contains no artificial dissipation numerical stability is enhanced through semi discrete satisfaction of global conservation properties stemming from the second law of thermodynamics and the entropy equation the numerical implementation is achieved using a conservative skew symmetric splitting of the nonlinear terms the robustness of the method is demonstrated by performing unresolved numerical simulations and large eddy simulations of compressible isotropic turbulence at a very high reynolds number results show the scheme is capable of capturing the statistical equilibrium of low mach number compressible turbulent fluctuations at infinite reynolds number comparisons with the entropy splitting technique j comput phys 162 2000 33 j comput phys 178 2002 307 staggered method j comput phys 191 2 2003 392 and skew symmetric like schemes j comput phys 161 2000 114 confirm the superiority of the current approach we also discuss a flaw in the skew symmetric splitting implemented in the literature very good results are obtained based on the proper splitting

entropy stable wall boundary conditions for the three dimensional compressible navier stokes equations non linear entropy stability and a summation by parts framework are used to derive entropy stable wall boundary conditions for the three dimensional compressible navier stokes equations a semi discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator the data at the boundary are weakly imposed using a penalty flux approach and a simultaneous approximation term penalty technique although discontinuous spectral collocation operators on unstructured grids are used herein for the purpose of demonstrating their robustness and efficacy the new boundary conditions are compatible with any diagonal norm summation by parts spatial operator including finite element finite difference finite volume discontinuous galerkin and flux reconstruction correction procedure via reconstruction schemes the proposed boundary treatment is tested for three dimensional subsonic and supersonic flows the numerical computations corroborate the non linear stability entropy stability and accuracy of the boundary conditions

the lip stability and error estimates for a relaxation scheme we show the discrete lip stability for a relaxation scheme proposed by jin and xin comm pure appl math 48 1995 pp 235 277 to approximate convex conservation laws equipped with the lip stability we obtain global error estimates in the spaces w s p for 1 leq s leq 1 p 1 leq p leq infty and pointwise error estimates for the approximate solution obtained by the relaxation scheme the proof uses the framework introduced by nessyahu and tadmor siam j numer anal 29 1992 pp 1505 1519 we also show a maximum principle for the relaxation scheme when the initial data are in an equilibrium state

hyperbolic conservation laws with stiff relaxation terms and entropy we study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms reduced systems inviscid and viscous local conservation laws and weakly nonlinear limits are derived through asymptotic expansions an entropy condition is introduced for n n systems that ensures the hyperbolicity of the reduced inviscid system the resulting characteristic speeds are shown to be interlaced with those of the original system moreover the first correction to the reduced system is shown to be dissipative a partial converse is proved for 2 2 systems this structure is then applied to study the convergence to the reduced dynamics for the 2 2 case 1994 john wiley sons inc

alternating evolution schemes for hyperbolic conservation laws the alternating evolution ae system of liu j hyperbolic differ equ 5 2008 pp 421 447 partial t u partial x f v frac 1 epsilon v u partial t v partial x f u frac 1 epsilon u v serves as a refined description of systems of hyperbolic conservation laws partial t phi partial x f phi 0 phi x 0 phi 0 x the solution of conservation laws is precisely captured when two components take the same initial value as phi 0 or is approached by two components exponentially fast when epsilon downarrow 0 if two initial states are sufficiently close this nice property enables us to construct novel shock capturing schemes by sampling the ae system on alternating grids in this paper we develop a class of local ae schemes by taking advantage of the ae system our approach is based on an average of the ae system over a hypercube centered at x with vertices at x pm delta x the numerical scheme is then constructed by sampling the averaged system over alternating grids higher order accuracy is achieved by a combination of high order nonoscillatory polynomial reconstruction from the obtained averages and a matching runge kutta solver in time discretization local ae schemes are made possible by letting the scale parameter epsilon reflect the local distribution of nonlinear waves the ae schemes have the advantage of easier formulation and implementation and efficient computation of the solution for the first and second order local ae schemes applied to one dimensional scalar conservation laws we prove the numerical stability in the sense of satisfying the maximum principle and total variation diminishing property the formulation procedure of ae schemes in multiple dimensions is given followed by both the first and second order ae schemes for two dimensional conservation laws numerical experiments for both scalar conservation laws and compressible euler equations are presented to demonstrate the high order accuracy and capacity of these ae schemes

a convex entropy for a hyperbolic system with relaxation abstract we explicitly construct a convex entropy function for a hyperbolic system with relaxation this entropy is defined in the sense of chen levermore and liu which resembles boltzman s h theorem for kinetic equations this construction follows the idea of suliciu s energy function for a rate type mixed hyperbolic elliptic system such an entropy will be useful in proving the entropy property of the relaxation schemes introduced by jin and xin for conservation laws

entropy satisfying flux vector splittings and kinetic bgk models we establish forward and backward relations between entropy satisfying bgk relaxation models such as those introduced previously by the author and the first order flux vector splitting numerical methods for general systems of conservation laws classically to a kinetic bgk model that is compatible with some family of entropies we can associate an entropy flux vector splitting we prove that the converse is true any entropy flux vector splitting can be interpreted by a kinetic model and we obtain an explicit characterization of entropy satisfying flux vector splitting schemes we deduce a new proof of discrete entropy inequalities under a sharp cfl condition that generalizes the monotonicity criterion in the scalar case in particular this gives a stability condition for numerical kinetic methods with noncompact velocity support a new interpretation of general kinetic schemes is also provided via approximate riemann solvers we deduce the construction of finite velocity relaxation systems for gas dynamics and obtain a hllc scheme for which we are able to prove positiveness of density and internal energy and discrete entropy inequalities

discrete velocity models of the boltzmann equation a survey on the mathematical aspects of the theory discrete velocity models of the boltzmann equation are of considerable conceptual interest in the kinetic theory of gases and at the same time a fascinating mathematical subject the last decade has seen a lot of work on this subject which is reviewed here we put an emphasis on mathematical aspects in particular the global existence question qualitative properties of solutions explicit solutions and the fluid dynamical limit the mechanical background is discussed more concisely in part i numerous contributions concerning other questions are outlined we try to give a complete reference list up to and including 1985

bifractality of human dna strand asymmetry profiles results from transcription we use the wavelet transform modulus maxima method to investigate the multifractal properties of strand asymmetry dna walk profiles in the human genome this study reveals the bifractal nature of these profiles which involve two competing scale invariant up to repeat masked distances ensuremath lesssim 40 phantom rule 0 3em 0ex mathrm kbp components characterized by h older exponents h 1 0 78 and h 2 1 respectively the former corresponds to the long range correlated homogeneous fluctuations previously observed in dna walks generated with structural codings the latter is associated with the presence of jumps in the original strand asymmetry noisy signal s we show that a majority of upward downward jumps colocate with gene transcription start end sites here 7228 human gene transcription start sites from the refgene database are found within 2 phantom rule 0 3em 0ex mathrm kbp from an upward jump of amplitude ensuremath delta s ensuremath geqslant 0 1 which suggests that about 36 of annotated human genes present significant transcription induced strand asymmetry and very likely high expression rate

the fluid dynamic limit of the broadwell model of the nonlinear boltzmann equation in the presence of shocks this paper studies the asymptotic equivalence of the broadwell model of the nonlinear boltzmann equation to its corresponding euler equation of compressible gas dynamics in the limit of small mean free path e it is shown that the fluid dynamical approximation is valid even if there are shocks in the fluid flow although there are thin shock layers in which the convergence does not hold more precisely by assuming that the fluid solution is piecewise smooth with a finite number of noninteracting shocks and suitably small oscillations we can show that there exist solutions to the broadwell equations such that the broadwell solutions converge to the fluid dynamical solutions away from the shocks at a rate of order e as the mean free path e goes to zero for the proof we first construct a formal solution for the broadwell equation by matching the truncated hilbert expansion and shock layer expansion then the existence of broadwell solutions and its convergence to the fluid dynamic solution is reduced to the stability analysis for the approximate solution we use an energy method which makes full use of the inner structure of time dependent shock profiles for the broadwell equations

systems of ordinary differential equations which generate an order preserving flow a survey of results this article consists of a survey of results concerning the qualitative behavior of solutions of systems of ordinary differential equations which generate an order preserving flow we restrict our consideration to partial orderings on r n induced by any one of its orthants a flow preserves ordering if any two solutions x t and y t are ordered x t leqq y t for all t 0 whenever x 0 leqq y 0 many of the important results for such systems have only recently been obtained principally by m w hirsch who pointed out the tendency of their solutions to converge to equilibrium less well known are some global geometric constraints on the stable manifold of an equilibrium and the existence of heteroclinic orbits connecting ordered equilibria a particularly striking result for this class of systems is the easily computable necessary and sufficient condition for stability of an equilibrium one of our main goals is to show that by allowing partial orderings on r n generated by orthants

a discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws abstract we present a new relaxation approximation to scalar conservation laws in several space variables by means of semilinear hyperbolic systems of equations with a finite number of velocities under a suitable multidimensional generalization of the whitham relaxation subcharacteristic condition we show the convergence of the approximated solutions to the unique entropy solution of the equilibrium cauchy problem

long range correlation properties of coding and noncoding dna sequences genbank analysis an open question in computational molecular biology is whether long range correlations are present in both coding and noncoding dna or only in the latter to answer this question we consider all 33301 coding and all 29453 noncoding eukaryotic sequences each of length larger than 512 base pairs bp in the present release of the genbank to dtermine whether there is any statistically significant distinction in their long range correlation properties standard fast fourier transform fft analysis indicates that coding sequences have practically no correlations in the range from 10 bp to 100 bp spectral exponent beta 0 00 0 04 where the uncertainty is two standard deviations in contrast for noncoding sequences the average value of the spectral exponent beta is positive 0 16 0 05 which unambiguously shows the presence of long range correlations we also separately analyze the 874 coding and the 1157 noncoding sequences that have more than 4096 bp and find a larger region of power law behavior we calculate the probability that these two data sets coding and noncoding were drawn from the same distribution and we find that it is less than 10 10 we obtain independent confirmation of these findings using the method of detrended fluctuation analysis dfa which is designed to treat sequences with statistical heterogeneity such as dna s known mosaic structure patchiness arising from the nonstationarity of nucleotide concentration the near perfect agreement between the two independent analysis methods fft and dfa increases the confidence in the reliability of our conclusion

provably unconditionally stable second order time accurate mixed variational methods for phase field models we introduce provably unconditionally stable mixed variational methods for phase field models our formulation is based on a mixed finite element method for space discretization and a new second order accurate time integration algorithm the fully discrete formulation inherits the main characteristics of conserved phase dynamics namely mass conservation and nonlinear stability with respect to the free energy we illustrate the theory with the cahn hilliard equation but our method may be applied to other phase field models we also propose an adaptive time stepping version of the new time integration method we present some numerical examples that show the accuracy stability and robustness of the new method

a new perspective on kinetic schemes compared to standard numerical methods for hyperbolic systems of conservation laws kinetic schemes model propagation of information by particles instead of waves in this article the wave and the particle concept are shown to be closely related moreover a general approach to the construction of kinetic schemes for hyperbolic conservation laws is given that summarizes several approaches discussed by other authors

global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy we consider the cauchy problem for a general one dimensional n n hyperbolic symmetrizable system of balance laws it is well known that in many physical examples for instance for the isentropic euler system with damping the dissipation due to the source term may prevent the shock formation at least for smooth and small initial data our main goal is to find a set of general and realistic sufficient conditions to guarantee the global existence of smooth solutions and possibly to investigate their asymptotic behavior for systems which are entropy dissipative a quite natural generalization of the kawashima condition for hyperbolic parabolic systems can be given in this paper we first propose a general framework for this kind of problem by using the so called entropy variables then we go on to prove some general statements about the global existence of smooth solutions under different sets of conditions in particular the present approach is suitable for dealing with most of the physical examples of systems with a relaxation extension our main tools will be some refined energy estimates and the use of a suitable version of the kawashima condition

discrete kinetic schemes for multidimensional systems of conservation laws we present here some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations which includes the relaxation schemes by s jin and z xin these schemes have a simple formulation even in the multidimensional case and do not need the solution of the local riemann problems for these approximations we give a suitable multidimensional generalization of the whitham s stability subcharacteristic condition in the scalar multidimensional case we establish the rigorous convergence of the approximated solutions to the unique entropy solution of the equilibrium cauchy problem

convergence rates for relaxation schemes approximating conservation laws in this paper we prove a global error estimate for a relaxation scheme approximating scalar conservation laws to this end we decompose the error into a relaxation error and a discretization error including an initial error omega ep we obtain the rate of convergence of sqrt ep in l1 for the relaxation step the estimate here is independent of the type of nonlinearity in the discretization step a convergence rate of sqrt del x in l1 is obtained these rates are independent of the choice of initial error omega ep thereby we obtain the order 1 2 for the total error

finite volume relaxation schemes for multidimensional conservation laws we consider finite volume relaxation schemes for multidimensional scalar conservation laws these schemes are constructed by appropriate discretization of a relaxation system and it is shown to converge to the entropy solution of the conservation law with a rate of h 1 4 in l 0 t l loc 1 r d

asymptotic stability of riemann solutions in bgk approximations to certain multidimensional systems of conservation laws abstract we prove the asymptotic stability of nonplanar two states riemann solutions in bgk approximations of a class of multidimensional systems of conservation laws the latter consists of systems whose flux functions in different directions share a common complete system of riemann invariants the level surfaces of which are hyperplanes the asymptotic stability to which the main result refers is in the sense of the convergence as t in l loc 1 of the space of directions x t that is the solution z t x of the perturbed cauchy problem for the corresponding bgk system satisfies x z t t d r as t in l loc 1 r n where r is the self similar entropy solution of the two states nonplanar riemann problem for the system of conservation laws

asymptotic integration of linear differential equations abstract we establish a new result on the asymptotic integration of linear differential systems of the form y r y where is diagonal and r is small more specifically we allow for different scales of asymptotic behaviour of the individual matrix elements of and r we also discuss an application to a self adjoint fourth order equation

hydrodynamic limits for kinetic equations and the diffusive approximation of radiative transport for acoustic waves we consider a class of kinetic equations equipped with a single conservation law which generate l 1 contractions we discuss the hydrodynamic limit to a scalar conservation law and the diffusive limit to a possibly degenerate parabolic equation the limits are obtained in the dissipative sense equivalent to the notion of entropy solutions for conservation laws which permits the use of the perturbed test function method and allows for simple proofs a general compactness framework is obtained for the diffusive scaling in l 1 the radiative transport equations satisfied by the wigner function for random acoustic waves present such a kinetic model that is endowed with conservation of energy the general theory is used to validate the diffusive approximation of the radiative transport equation

zero relaxation and dissipation limits for hyperbolic conservation laws we are interested in hyperbolic systems of conservation laws with relaxation and dissipation particularly the zero relaxation limit such a limit is of interest in several physical situations including gas flow near thermo equilibrium kinetic theory with small mean free path and viscoelasticity with vanishing memory in this article we study hyperbolic systems of two conservation laws with relaxation for the stable case where the equilibrium speed is subcharacteristic with respect to the frozen speeds we illustrate for a model in viscoelasticity that no oscillation develops for the nonlinear system in the zero relaxation limit for the marginally stable case where the equilibrium speed may equal one of the frozen speeds we show for a model in phase transitions that no oscillation arises when the dissipation is present and goes to zero more slowly than the relaxation our analysis includes the construction of suitable entropy pairs to derive energy estimates we need such energy estimates not only for the compactness properties but also for the deviation from the equilibrium of the solutions for the relaxation systems the theory of compensated compactness is then applied to study the oscillation in the zero relaxation limit 1993 john wiley sons inc

a discrete bgk approximation for strongly degenerate parabolic problems with boundary conditions abstract we consider a class of bgk systems with a finite number of velocities depending on a positive relaxation parameter that approximate strongly degenerate hyperbolic parabolic equations with initial boundary conditions we prove a priori estimates for the solutions of the systems showing that these functions converge towards the entropy solutions of strongly degenerate problems when the relaxation parameter goes to zero

skew selfadjoint form for systems of conservation laws abstract hyperbolic systems of conservation laws augumented with an entropy inequality are studied it is shown that such systems can be written in a quasilinear skew selfadjoint form centered differencing of such a form under the smooth regime ends up with a systematic recipe for constructing quasiconservative schemes where the global entropy conservation is recovered employing the above formulation in bounded regions under the nonsmooth regime as well a local entropy decay estimate is also concluded examples of the shallow water and the full gasdynamics equations are explicitly treated

explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems we design numerical schemes for nonlinear degenerate parabolic systems with possibly dominant convection these schemes are based on discrete bgk models where both characteristic velocities and the source term depend singularly on the relaxation parameter general stability conditions are derived and convergence is proved to the entropy solutions for scalar equations

a relaxation scheme for conservation laws with a discontinuous coefficient lwe study a relaxation scheme of the jin and xin type for conservation laws with a flux function that depends discontinuously on the spatial location through a coefficient k x if k e bv we show that the relaxation scheme produces a sequence of approximate solutions that converge to a weak solution the murat tartar compensated compactness method is used to establish convergence we present numerical experiments with the relaxation scheme and comparisons are made with a front tracking scheme based on an exact 2 x 2 riemann solver

kinetic formulation of the isentropic gas dynamics and p systems we consider the 2 x 2 hyperbolic system of isentropic gas dynamics in both eulerian or lagrangian variables also called the p system we show that they can be reformulated as a kinetic equation using an additional kinetic variable such a formulation was first obtained by the authors in the case of multidimensio nal scalar conservation laws a new phenomenon occurs here namely that the advection velocity is now a combination of the macroscopic and kinetic velocities various applications are given we recover the invariant regions deduce new l estimates using moments lemma and prove l w stability for 7 3

critical thresholds in hyperbolic relaxation systems article history received 8 august 2008 available online 18 april 2009 critical threshold phenomena in one dimensional 2 2q uasi linear hyperbolic relaxation systems are investigated assuming both the subcharacteristic condition and genuine nonlinearity of the flux we prove global in time regularity and finite time singularity formation of solutions simultaneously by showing the critical threshold phenomena associated with the underlying relaxation systems our results apply to the well known isentropic euler system with damping within the same framework it is also shown that the solution of the semi linear relaxation system remains smooth for all time provided the subcharacteristic condition is satisfied

global existence to the bgk model of boltzmann equation abstract we present an existence and a stability proof for solutions to the bgk model of boltzmann equation f v x f f m f t 0 x r n v r n m f p 2 t n 2 exp v u 2 2t p pu p u 2 pt 1 x r n 1 v v 2 f t x v dv it relies on the strong compactness of u t and on a new estimate on the third moment of v dv we also prove the entropy relation for 1

numerical passage from kinetic to fluid equations a numerical way to pass from the bhatnagar gross krook model of the boltzmann equation to compressible euler equations is presented in order to do so a stable discretization of the kinetic equati

asymptotic solutions and error estimates for linear systems of difference and differential equations abstract classical results concerning the asymptotic behavior solutions of systems of linear differential or difference equations lead to formulas containing factors that are asymptotically constant i e k o 1 as t tends to infinity here we are interested in more precise information about the o 1 terms specifically how they depend precisely on certain perturbation terms in the equation results along these lines were given by gel fond and kubenskaya for scalar difference equations and we will both extend and generalize one of them as well as provide some corresponding results for differential equations

resolution d equations d evolution quasilineaires en dimension n d espace a l aide d equations lineaires en dimension n 1 on montre que l on peut calculer les solutions de l equation d evolution quasi lineaire dans r n d t u div b u a u 0 u x t r x r n t 0 et en particulier de l equation d ordre 1 d t u div b u 0 u x t r x r n t 0 a l aide d une equation d evolution lineaire dans r n r

kinetic decomposition for kinetic models of bgk type abstract in this paper we consider kinetic models of bgk type which describe the scalar conservation law at the microscopic scale we use new technique developed in comm partial differential equations 27 2002 1229 in order to get the convergence first we obtain the approximate transport equation for the given kinetic models of bgk type then using the averaging lemma we obtain the convergence this paper shows how to relate the given kinetic model with the averaging lemma to get the convergence

numerical algorithms for simulations of a traffic model on road networks we introduce a simulation algorithm based on a fluid dynamic model for traffic flows on road networks which are considered as graphs composed by arcs that meet at some junctions the approximation of scalar conservation laws along arcs is made by three velocities kinetic schemes with suitable boundary conditions at junctions here we describe the algorithm and we give an example

a kinetic construction of global solutions of first order quasilinear equations on considere le probleme de cauchy pour des equations quasilineaires d ordre 1 de type conservation dans la region t 0 u t i 1 u a i u x i 0 u x o u o x ou x x 1 x n u u x t et a i i 1 n sont des fonctions continument differentiables a valeurs reelles d une seule variable reelle

alternating evolution discontinuous galerkin methods for hamilton jacobi equations in this work we propose a high resolution alternating evolution discontinuous galerkin aedg method to solve hamilton jacobi equations the construction of the aedg method is based on an alternating evolution system of the hamilton jacobi equation following the previous work liu et al 2013 31 on ae schemes for hamilton jacobi equations a semi discrete aedg scheme derives directly from a sampling of this system on alternating grids higher order accuracy is achieved by a combination of high order polynomial approximation near each grid and a time discretization with matching accuracy the aedg methods have the advantage of easy formulation and implementation and efficient computation of the solution for the linear equation we prove the l 2 stability of the method numerical experiments for a set of hamilton jacobi equations are presented to demonstrate both accuracy and capacity of these aedg schemes

alternating evolution methods for static hamilton jacobi equations abstract we design analyze and numerically validate alternating evolution schemes for solving static hamilton jacobi equations this class of high resolution numerical schemes is based on the alternating evolution reformulation of the original hamilton jacobi equation stability and convergence properties are proven for first order schemes numerical implementation of up to third order schemes is carried out for a set of widely used examples further application of the proposed method to the kinetic hamilton jacobi equation is given while the hamiltonian is determined by an integral in phase space

hyperbolic systems with supercharacteristic relaxations and roll waves we study a distinguished limit for general 2 2 hyperbolic systems with relaxation which is valid in both the subcharacteristic and supercharacteristic cases this is a weakly nonlinear limit which leads the underlying relaxation systems into a burgers equation with a source term the sign of the source term depends on the characteristic interleaving condition in the supercharacteristic case the problem admits a periodic solution known as the roll wave generated by a small perturbation around equilibrium constants such a limit is justified in the presence of artificial viscosity using the energy method

alternating evolution discontinuous galerkin methods for convection diffusion equations in this work we propose a high order alternating evolution discontinuous galerkin aedg method to solve convection diffusion equations the difficulties related to numerical fluxes of dg methods for diffusion problems have been a major issue of investigation in the literature the aedg scheme presented here is based on an alternating evolution system for convection diffusion equations and therefore no numerical fluxes are needed in the scheme formulation moreover the method is shown to be consistent conservative and stable numerical experiments are provided to show the goodness of the proposed method

l1nonlinear stability of traveling waves for a hyperbolic system with relaxation where x t r r the unknowns u v belong to r the function f f u is in c r and 0 is a fixed constant relaxation systems often arise in many physical situations let us recall for example gases not in thermodynamic equilibrium vk kinetic theory ce cip pi chromatography raa river flows traffic flows and more general waves wh more recently the study of a special class of hyperbolic systems with relaxation was developed in view of the numerical approximation of discontinuous solutions of conservation laws jx an the 2 2 relaxation hyperbolic systems of conservation laws were first analyzed by liu li2 who justified some nonlinear stability criteria for diffusion waves expansion waves and traveling waves article no 0180

kinetic approximation of a boundary value problem for conservation laws we design numerical schemes for systems of conservation laws with boundary conditions these schemes are based on relaxation approximations taking the form of discrete bgk models with kinetic boundary conditions the resulting schemes are riemann solver free and easily extendable to higher order in time or in space for scalar equations convergence is proved we show numerical examples including solutions of euler equations

entropy stable discontinuous interfaces coupling for the three dimensional compressible navier stokes equations non linear entropy stability and a summation by parts sbp framework are used to derive entropy stable interior interface coupling for the semi discretized three dimensional 3d compressible navier stokes equations a complete semidiscrete entropy estimate for the interior domain is achieved combining a discontinuous entropy conservative operator of any order 1 2 with an entropy stable coupling condition for the inviscid terms and a local discontinuous galerkin ldg approach with an interior penalty ip procedure for the viscous terms the viscous penalty contributions scale with the inverse of the reynolds number re so that for re their contributions vanish and only the entropy stable inviscid interface penalty term is recovered this paper extends the interface couplings presented 1 2 and provides a simple and automatic way to compute the magnitude of the viscous ip term the approach presented herein is compatible with any diagonal norm summation by parts sbp spatial operator including finite element finite volume finite difference schemes and the class of high order accurate methods which include the large family of discontinuous galerkin discretizations and flux reconstruction schemes this note relies on the formalism introduced in 1 3 and complements the new class of interior entropy stable sbp operators of any order for the 3d compressible navier stokes equations on unstructured grids that was proposed in 1 2 to keep the notation as simple as possible a uniform cartesian grid is considered in the derivation however the extension to generalized curvilinear coordinates and unstructured grids follows immediately if the transformation from computational to physical space preserves the semi discrete geometric conservation 4 the proposed interface coupling technique has been successfully combined with a high order entropy stable discretization for the simulation of two dimensional 2d and 3d viscous subsonic and supersonic flows presented in 1 5 6

a kinetic equation with kinetic entropy functions for scalar conservation laws we construct a nonlinear kinetic equation and prove that it is welladapted to describe general multidimensional scalar conservation laws in particular we prove that it is well posed uniformly in e the microscopic scale we also show that the proposed kinetic equation is equipped with a family of kinetic entropy functions analogous to boltzmann s microscopich function such that they recover krushkov type entropy inequality on the macroscopic scale finally we prove by both bv compactness arguments in the multidimensional case and by compensated compactness arguments in the one dimensional case that the local density of kinetic particles admits a continuum limit as it converges strongly with e 0 to the unique entropy solution of the corresponding conservation law

boltzmann type schemes for gas dynamics and the entropy property let h be a convex function a boltzmann equation is built following the b g k model for which h is an entropy in the fluid limit the compressible euler equations are obtained with a convex entropy naturally associated to h the motivation is to define numerical boltzmann schemes with finite speed of propagation corresponding to an equilibrium density function with bounded support the classical scheme obtained with h f f log f is based on the equilibrium density function exp v 2 2 which has infinite support

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

an entropy stable nodal discontinuous galerkin method for the resistive mhd equations part i theory and numerical verification abstract the first paper of this series presents a discretely entropy stable discontinuous galerkin dg method for the resistive magnetohydrodynamics mhd equations on three dimensional curvilinear unstructured hexahedral meshes compared to other fluid dynamics systems such as the shallow water equations or the compressible navier stokes equations the resistive mhd equations need special considerations because of the divergence free constraint on the magnetic field for instance it is well known that for the symmetrization of the ideal mhd system as well as the continuous entropy analysis a non conservative term proportional to the divergence of the magnetic field typically referred to as the powell term must be included as a consequence the mimicry of the continuous entropy analysis in the discrete sense demands a suitable dg approximation of the non conservative terms in addition to the ideal mhd terms this paper focuses on the resistive mhd equations our first contribution is a proof that the resistive terms are symmetric and positive definite when formulated in entropy space as gradients of the entropy variables which enables us to show that the entropy inequality holds for the resistive mhd equations this continuous analysis is the key for our dg discretization and guides the path for the construction of an approximation that discretely mimics the entropy inequality typically termed entropy stability our second contribution is a detailed derivation and analysis of the discretization on three dimensional curvilinear meshes the discrete analysis relies on the summation by parts property which is satisfied by the dg spectral element method dgsem with legendre gauss lobatto lgl nodes although the divergence free constraint is included in the non conservative terms the resulting method has no particular treatment of the magnetic field divergence errors which might pollute the solution quality our final contribution is the extension of the standard resistive mhd equations and our dg approximation with a divergence cleaning mechanism that is based on a generalized lagrange multiplier glm as a conclusion to the first part of this series we provide detailed numerical validations of our dgsem method that underline our theoretical derivations in addition we show a numerical example where the entropy stable dgsem demonstrates increased robustness compared to the standard dgsem