a bgk type flux vector splitting scheme for the ultrarelativistic euler equations a gas kinetic solver is developed for the ultrarelativistic euler equations the scheme is based on the direct splitting of the flux function of the euler equations with inclusion of particle collisions in the transport process consequently the artificial dissipation in the new scheme is much reduced in comparison with the usual kinetic flux vector splitting kfvs schemes which are based on the free particle transport at the cell interfaces in the gas evolution stage although in a usual kfvs scheme the free particle transport gives robust solutions it gives a smeared solution at the contact discontinuities the new bgk type kfvs scheme solves this problem and gives robust and reliable solutions as well as good resolution at the contact discontinuity in contrast to the classical kinetic theory we have only a finite domain of dependence due to the structure of the light cone the scheme is naturally multidimensional and is extended to the two dimensional case in a usual dimensionally split manner that is the formulae for the fluxes can be used along each coordinate direction the high order resolution of the scheme is achieved by using a muscl type initial reconstruction in the numerical case studies the results obtained from the bgk type kfvs schemes are compared with the exact riemann solution kfvs schemes upwind schemes and central schemes

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

dirichlet inhomogeneous boundary value problem for the n 1 complex ginzburg landau equation abstract we study the following complex ginzburg landau equation with cubic nonlinearity on r n for t 0 u t a i u b i u 2 u a b 0 under initial and dirichlet boundary conditions u x 0 h x for x u x t q x t on where h q are given smooth functions under suitable conditions we prove the existence of a global solution in h1 further this solution approaches to the solution of the nls limit under identical initial and boundary data as a b 0

gas kinetic schemes for compressible flow of real gases abstract this paper extends gas kinetic schemes to the euler equations for real gases in the current scheme the maxwell boltzmann gas distribution function is modified to recover macroscopic flow equations more specifically the internal degree of freedom of the gas distribution function becomes a function of flow variables according to the general equation of state the numerical results confirm the accuracy and robustness of the gas kinetic approach

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

a neumann boundary value problem for a generalized ginzburg landau equation we study the following generalized 1d ginzburg landau equation on w 0 x 0 u t 1 i m u x x a 1 ib 1 u 2u x a 2 ib 2 u 2u x 1 i n u 4u with initial and neumann boundary conditions u x 0 h x u x 0 t p t under suitable conditions we prove that there is a unique weak solution that exists for all time

a new perturbation analysis for signal enumeration in rotational invariance techniques the esprit algorithm is a subspace based high resolution method used in source localization and spectral analysis which provides very accurate estimates of the signal parameters however the underlying theory assumes a known model order which is usually not the case in many applications in particular it is well known that underevaluating the model order biases the estimation in this paper we analyze the perturbation induced by an erroneous model order and we present an error bound for the estimated parameters based on this theoretical framework we propose a new method for selecting an appropriate modeling order which consists in minimizing the error bound this approach is applied to both synthetic and musical signals and its performance is compared to that of existing methods such as the information theoretic criteria

finite dimensional behaviour for the derivative ginzburg landau equation in two spatial dimensions abstract in this paper we study the long time behaviour of the solutions for the generalized derivative ginzburg landau equation in two spatial dimensions we show the existence of the global attractor for this equation and establish the estimates of the upper bounds of hausdorff and fractal dimensions for the global attractor

a local space time conservation scheme and its application in shock wave propagation abstract in this paper a local space time conservation scheme based on non staggered grids is introduced which is a variation of space time conservation element and solution element ce se scheme it inherits most features and advantages of ce se method including unified treatment of space and time and high accuracy resolution of hyperbolic conservation equations moreover riemann solvers are not needed to capture shocks and dimensional splitting methods are not needed in the multi dimensional schemes the stability of the present scheme is verified through von neumann analysis moreover several shock wave problems including one two and three dimensional cases are simulated by the present scheme by carefully comparing the present scheme s numerical results with exact solutions experimental results original ce se scheme s numerical results and third order eno scheme s numerical results it can be conclude that the present scheme is efficient and accurate

exact maximum likelihood parameter estimation of superimposed exponential signals in noise a unified framework for the exact maximum likelihood estimation of the parameters of superimposed exponential signals in noise encompassing both the time series and the array problems is presented an exact expression for the ml criterion is derived in terms of the linear prediction polynomial of the signal and an iterative algorithm for the maximization of this criterion is presented the algorithm is equally applicable in the case of signal coherence in the array problem simulation shows the estimator to be capable of providing more accurate frequency estimates than currently existing techniques the algorithm is similar to those independently derived by kumaresan et al in addition to its practical value the present formulation is used to interpret previous methods such as prony s pisarenko s and modifications thereof

a generalized weighted linear predictor frequency estimation approach for a complex sinusoid based on linear prediction and weighted least squares three simple iterative algorithms for frequency estimation of a complex sinusoid in additive white noise are devised the proposed approach which utilizes the first order as well as higher order linear prediction terms simultaneously but does not require phase unwrapping can be considered as a generalized version of the weighted linear predictor frequency estimator in particular convergence as well as mean and variance analysis of the most computationally efficient frequency estimator namely gwlp 2 are provided computer simulations are included to contrast the performance of the proposed algorithms with several conventional computationally attractive frequency estimators and crame spl acute r rao lower bound for different frequencies observation lengths and signal to noise ratios

on the convergence of interior reflective newton methods for nonlinear minimization subject to bounds we consider a new algorithm an interior reflective newton approach for the problem of minimizing a smooth nonlinear function of many variables subject to upper and or lower bounds on some of the variables this approach generatesstrictly feasible iterates by using a new affine scaling transformation and following piecewise linear paths reflection paths the interior reflective approach does not require identification of an activity set in this paper we establish that the interior reflective newton approach is globally and quadratically convergent moreover we develop a specific example of interior reflective newton methods which can be used for large scale and sparse problems

variance of least squares estimators for a damped sinusoidal process the variance of least squares estimators for the parameter estimation of a damped sinusoidal process is analyzed based on first order perturbation analytical expressions for the variances of the frequency damping factor amplitude and phase estimators are derived explicit expressions are given for both damped and undamped single mode cases the effect of mode separation on the accuracy is investigated through the two mode case the dependence of the variances on number of data points model order signal to noise ratio and mode separation is investigated both analytically and numerically for practical applications extensive monte carlo simulation results are given to verify enhance and illustrate the analytical expressions

esprit estimation of signal parameters via rotational invariance techniques an approach to the general problem of signal parameter estimation is described the algorithm differs from its predecessor in that a total least squares rather than a standard least squares criterion is used although discussed in the context of direction of arrival estimation esprit can be applied to a wide variety of problems including accurate detection and estimation of sinusoids in noise it exploits an underlying rotational invariance among signal subspaces induced by an array of sensors with a translational invariance structure the technique when applicable manifests significant performance and computational advantages over previous algorithms such as mem capon s mlm and music

fourier spectral approximation to long time behaviour of the derivative three dimensional ginzburg landau equation in this paper we consider a derivative ginzburg landau equation with periodic initial value condition in three dimensional space a fully discrete galerkin fourier spectral approximation scheme is constructed and then the dynamical behaviour of the discrete system is analysed firstly the existence of global attractors an of the discrete system are proved by a priori estimate of the discrete solution next the convergence of approximate attractors is proved by error estimates of the discrete solution furthermore the long time convergence as n and 0 simultaneously as well as the numerical long time stability of the discrete scheme are obtained

stable manifolds to bounded solutions in possibly ill posed pdes abstract we prove several results establishing existence and regularity of stable manifolds for different classes of special solutions for evolution equations these equations may be ill posed a single specific solution an invariant torus filled with quasiperiodic orbits or more general manifolds of solutions in the later cases which include several orbits we also establish the invariant manifolds of an orbit depend smoothly on the orbit analytically in the case of quasi periodic orbits and finitely differentiably in the case of more general families we first establish a general abstract theorem which under suitable spectral non degeneracy analyticity assumptions on the linearized equation establishes the existence of the desired manifold related results appear in the literature but our results allow that the nonlinearity is unbounded and we obtain smoothness of the invariant manifolds this makes the results in this paper applicable to some several models of current interest that could not be treated otherwise we discuss in detail the boussinesq equation of water waves similar phenomena happen in other long wave approximations and complex ginzburg landau equation more recently we observed 11 that our results also apply to mean field games since the equations we consider may be ill posed part of the requirements for the stable manifold is that one can define the forward dynamics on them note also that the methods that are based in the existence of dynamics such as graph transform do not apply to ill posed equation we use the methods based on integral equations perron method associated with the partial dynamics but we need to take advantage of smoothing properties of the partial dynamics note that even if the families of solutions we started with are finite dimensional the stable manifolds may be infinite dimensional

stable and chaotic solutions of the complex ginzburg landau equation with periodic boundary conditions abstract we study analytically and numerically the dynamical behavior of the solutions of the complex ginzburg landau equation with diffraction but without diffusion which governs the spatial evolution of the field in an active nonlinear laser cavity accordingly the solutions are subject to periodic boundary conditions the analysis reveals regions of stable stationary solutions in the model s parameter space and a wide range of oscillatory and chaotic behaviors close to the first bifurcation destabilizing the spatially uniform solution a stationary single humped solution is found in an asymptotic analytical form which turns out to be in very good agreement with the numerical results simulations reveal a series of stable stationary multi humped solutions

a new parametric adaptive nonstationarity detector and application techniques for hypothesis testing can be used to solve a broad class of nonstationarity detection problems which is a key issue in a variety of applications to achieve lower complexity and to deal with real time detection in practical applications we develop a new adaptive nonstationarity detector by exploiting a parametric model a weighted maximum a posteriori map estimator is developed to estimate the parameters associated with the parametric model we then derive a regularized wald test from the weighted map estimate which is adaptively implemented by a regularized recursive least squares rls algorithm several important issues are discussed including model order selection forgetting factor and regularization parameter selection for rls and numerically stable implementation using qr decomposition which are intrinsic parts of the proposed parametric adaptive detector simulation results are presented to illustrate the efficiency of the proposed nonstationarity detector with adaptive estimation and automatic model selection especially for slowly varying type of nonstationarity such as time varying spectrums and speeches

weak and strong solutions of the complex ginzburg landau equation abstract the generalized complex ginzburg landau equation t a ra 1 iv a 1 i a 2 a has been proposed and studied as a model for turbulent dynamics in nonlinear partial differential equations it is a particularly interesting model in this respect because it is a dissipative version of the hamiltonian nonlinear schrodinger equation possessing solutions that form localized singularities in finite time in this paper we investigate existence and regularity of solutions to this equation subject to periodic boundary conditions in various spatial dimensions appropriately defined weak solutions are established globally in time and unique strong solutions are found locally a new collection of a priori estimates are presented and we discuss the relationship of our results for the complex ginzburg landau equation to analogous issues for fluid turbulence described by the incompressible navier stokes equations

the probability of a subspace swap in the svd we extend the work of tufts kot and vaccaro tkv published in 1980 to improve the analytical characterization of threshold breakdown in svd methods our results sharpen the tkv results by lower bounding the probability of a subspace swap in the svd our key theoretical result is the characteristic function for a random variable whose probability of exceeding zero bounds the probability of a threshold breakdown

kinetic node pair formulation for two dimensional flows from continuum to transitional regime a hybrid finite element finite volume node pair discretization of conservation laws is reformulated in terms of a bhatnagar gross krook kinetic scheme to address flows from the continuum up to the transitional regime in a seamless fashion integrals of the particle distribution function from the kinetic theory of gases are adopted to compute the numerical fluxes along the boundary of each control volume flow features typical of the transitional regime like velocity and temperature slip condition at solid walls are automatically assured by the kinetic formulation of the node pair boundary conditions exemplary two dimensional numerical experiments ranging from continuum flows up to the transitional regime are presented

on the initial value problem for the generalized two dimensional ginzburg landau equation abstract the ginzburg landau type complex partial differential equations are simplified mathematical models for various pattern formation systems in mechanics physics and chemistry most work so far concentrates on ginzburg landau type equations with one spatial dimension 1d in this paper the authors study a complex generalized ginzburg landau equation with two spatial dimensions 2d sufficient conditions for the existence and uniqueness of global solutions for the initial value problem of the generalized 2d ginzburg landau equation are obtained this rigorously establishes the foundation for further investigation of this type of model

a shock capturing sph scheme based on adaptive kernel estimation here we report a method that converts standard smoothed particle hydrodynamics sph into a working shock capturing scheme without relying on solutions to the riemann problem unlike existing adaptive sph simulations the present scheme is based on an adaptive kernel estimation of the density which combines intrinsic features of both the kernel and nearest neighbor approaches in a way that the amount of smoothing required in low density regions is effectively controlled symmetrized sph representations of the gas dynamic equations along with the usual kernel summation for the density are used to guarantee variational consistency implementation of the adaptive kernel estimation involves a very simple procedure and allows for a unique scheme that handles strong shocks and rarefactions the same way since it represents a general improvement of the integral interpolation on scattered data it is also applicable to other fluid dynamic models when the method is applied to supersonic compressible flows with sharp discontinuities as in the classical one dimensional shock tube problem and its variants the accuracy of the results is comparable and in most cases superior to that obtained from high quality godunov type methods and sph formulations based on riemann solutions the extension of the method to two and three space dimensions is straightforward in particular for the two dimensional cylindrical noh s shock implosion and sedov point explosion problems the present scheme produces much better results than those obtained with conventional sph codes

exponential attractor for the 3d ginzburg landau type equation abstract in this paper we consider a complex ginzburg landau type equation with periodic initial value condition in three spatial dimensions sufficient conditions for existence and uniqueness of global solutions are obtained by uniform a priori estimates of solutions furthermore the existence of a global attractor with finite hausdorff and fractal dimensions is proved finally the existence of the exponential attractor is proved

regular article a new discrete velocity method for navier stokes equations the relation between the lattice boltzmann method which has recently become popular and the kinetic schemes which are routinely used in computational fluid dynamics is explored a new discrete velocity method for the numerical solution of navier stokes equations for incompressible fluid flow is presented by combining both the approaches the new scheme can be interpreted as a pseudo compressibility method and for a particular choice of parameters this interpretation carries over to the lattice boltzmann method

dimension of the attractors associated to the ginzburg landau partial differential equation abstract we study the long time behavior of solutions to the ginzburg landau partial differential equation it is shown that a finite dimensional attractor captures all the solutions an upper bound on this dimension is given in terms of physical quantities by estimating the lyapunov exponents on the trajectories finally using the well known side band instabilities of an exact time dependent solution stokes solution we derive lower bounds on the dimension of the universal attractor moreover the lower and upper bounds agree

on the possibility of soft and hard turbulence in the complex ginzburg landau equation abstract we present analytical methods which predict the occurrence of both soft weak and hard strong turbulence in the complex ginzburg landau cgl equation a t ra 1 i a 1 i a a 2 on a periodic domain 0 1 d in d spatial dimensions hard turbulence is characterised by large fluctuations away from spatial and temporal averages with a cascade of energy to small scales this form of hard turbulence appears to occur not in 1d but only in 2d and 3d in parameter regions which are bounded by hyperbolic curves in the second and fourth quadrants of the planes where the system is modulationally unstable 1 2n f n n 1 a 2 n a 2n dx for n 0 for large times and large r upper bounds exist for the infinite set of fn s constructed from the hierarchy of differential inequalities fn 2nr cn a 2 fn bnf2n fn 1 for cn bn 0 f0 1 estimates for the bottom rung f2 give upper bounds for the whole ladder long time upper bounds on f2 and a 2 and hence all fn are well controlled in the soft region but become much larger in the hard region whereas spati al and temporal averages remain comparatively small when the nonlinearity is a a 2q the critical case qd 2 gives parallel results

linearized adi schemes for two dimensional space fractional nonlinear ginzburg landau equation abstract space and time approximations for two dimensional space fractional complex ginzburg landau equation are examined the schemes under consideration are discreted by the second order backward differential formula bdf2 in time and two classes of the fractional centered finite difference methods in space a linearized technique is employed by the extrapolation we prove the unique solvability and stability for both numerical methods the convergence of both numerical methods is analyzed at length utilizing the energy argument and the convergence orders under the optimal step size ratio are o 2 h 2 and o 2 h 4 in the sense of the discrete l 2 norm where is the time step size h max h x h y and h x h y are spatial grid sizes in the x direction and y direction respectively in addition we construct a multistep alternating direction implicit adi scheme and a multistep compact adi scheme based on bdf2 for the efficiently numerical implementation finally numerical examples are carried out to verify our theoretical results

cartesian grid method for gas kinetic scheme on irregular geometries a cartesian grid method combined with a simplified gas kinetic scheme is presented for subsonic and supersonic viscous flow simulation on complex geometries under the cartesian mesh the boundaries are represented by a set of direction oriented boundary points and the computational grid points are classified into four different categories the fluid point the solid point the drop point and the interpolation point a constrained weighted least square method is employed to evaluate the physical quantities at the interpolation points different boundary conditions including isothermal boundary adiabatic boundary and euler slip boundary are presented by different interpolation strategies we adopt a simplified gas kinetic scheme as the flux solver for both subsonic and supersonic flow computations the methodology of constructing a simplified kinetic flux function can be extended to other flow systems a few numerical examples are used to validate the cartesian grid method and the simplified flux solver the reconstruction scheme for recovering the boundary conditions of compressible viscous and heat conducting flow with a cartesian mesh can provide a smooth distribution of physical quantities at solid boundary and present an accurate solution for the flow study with complex geometry

parametric modeling for damped sinusoids from multiple channels the problem of parametric modeling for noisy damped sinusoidal signals from multiple channels is addressed utilizing the shift invariance property of the signal subspace the number of distinct sinusoidal poles in the multiple channels is first determined with the estimated number the distinct frequencies and damping factors are then computed with the multi channel weighted linear prediction method the estimated sinusoidal poles are then matched to each channel according to the extreme value theory of distribution of random fields simulations are performed to show the performance advantages of the proposed multi channel sinusoidal modeling methodology compared with existing methods

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

asymptotics for the generalized two dimensional ginzburg landau equation abstract in this paper the authors have studied a generalized ginzburg landau equation in two spatial dimensions 2d they have shown that this equation under periodic boundary conditions has the maximal attractor with finite hausdorff dimension this rigorously establishes the foundation for further investigation of this type of model

sharper global existence for the generalized 1d nonhomogeneous ginzburg landau equation we study the following generalized 1d ginzburg landau equation on 0 0 ut 1 i uxx a1 ib1 u 2ux a2 ib2 u2ux 1 i u 4u with initial and dirichlet boundary conditions u x 0 h x u 0 t q t based on detail analysis the sharper existence and uniqueness of global solutions are obtained under sufficient conditions

a robust and accurate led bgk solver on unstructured adaptive meshes starting from the bgk model of the boltzmann equation we develop a robust and accurate finite volume gas kinetic scheme on unstructured triangular meshes the proposed numerical approach is composed of two steps an initial reconstruction step and a gas evolution step in the initial reconstruction step an unstructured version of the local extremum diminishing interpolation is applied to the conservative variables and to compute left and right states along a node edge in the gas evolution step the local integral solution of the bgk model is used to compute numerical fluxes at a cell interface this approach provides an alternative to riemann solvers and yields numerical schemes which possess many desirable properties that may not be found in godunov type schemes a classich refinement adaptive procedure is implemented to increase the spatial resolution of high speed unsteady flow characteristics such as shock waves contact discontinuities or expansion waves with minimal computational costs and memory overheads this procedure involves mesh enrichment coarsening steps to either insert nodes on an edge center in high gradient regions or delete nodes in over resolved regions numerical results of several test cases for unsteady compressible inviscid flows are presented in order to verify the accuracy and robustness of the current numerical approach the computed results are compared with analytical solutions experimental data the results of structured mesh calculations and the results obtained by widely used flux splitting methods

simulations of compressible flows with strong shocks by an adaptive lattice boltzmann model an adaptive lattice boltzmann model for compressible flows is presented the particle velocity set is so large that the mean flow may have a high velocity the support set of the equilibrium distribution function is quite small and varies with the mean velocity and internal energy the adaptive nature of this support set permits the mean flows to have high mach number meanwhile it makes the model simple and practicable the model is suitable for perfect gases with an arbitrary specific heat ratio navier stokes equations are derived by the chapman enskog method from the bgk boltzmann equation when the viscous terms and the diffusion terms are considered as a discretion error this system becomes an inviscid euler system several simulations of flows with strong shocks including the forward facing step test double mach reflection test and a strong shock of mach number 5 09 diffracting around a corner were carried out on hexagonal lattices showing the model s capability of simulating the propagation of strong shock waves

low speed flow simulation by the gas kinetic scheme this paper extends the gas kinetic bgk type scheme to low mach number flows and thus shows that incompressible flow solutions are accurately obtained from the bgk scheme in the low mach number limit the influence of boundary conditions internal molecular degrees of freedomk and the flow mach numbermon the accuracy of the solutions of incompressible or nearly incompressible flow problems is quantitatively evaluated the gas kinetic scheme is tested carefully in two numerical examples namely the cavity flow problem and the flow passing a backward facing step problem for the cavity flow problem the numerical results from the gas kinetic scheme under different reynolds numbers compare well with ghia s data for the backward step problem the numerical results are compared accurately with previously published experimental data

a robust low diffusive kinetic scheme for the navier stokes euler equations a new kinetic scheme based on the equilibrium flux method efm and modified using osher intermediate states is proposed this new scheme called efmo combines the robustness of the equilibrium flux method and the accuracy of flux difference splitting schemes the original efm scheme is expressed in terms of simple wave decomposition in which only the linearly degenerate subpath is calculated from osher numerical flux while nonlinear waves are still evaluated from the regular efm splitting owing to its capability of withstanding intense nonlinear waves and yet exactly resolving contact discontinuities efmo is particularly well suited for the resolution of the navier stokes equations as demonstrated by a series of severe test cases including the high speed viscous flow around a cone a shock boundary layer interaction problem a vacuum apparition problem the hypersonic flow around a circular cylinder at mach 100 and the forward facing step at mach 3

rkdg finite element method combined with bgk scheme for solving fluid dynamics system this paper presents a new type of second order scheme for solving the system of euler equations which combines the runge kutta discontinuous galerkin rkdg finite element method and the bhatnagar gross krook bgk scheme we first discretize the euler equations in space with the dg method and then the resulting system will be discretized using an rk method finally a second order bgk method is used to construct the numerical flux the proposed scheme keeps the main advantages of the dg finite element method including its flexibility in handling irregular solution domains and in parallelization moreover with the bgk approach the proposed schemes avoid using riemann solvers the efficiency and effectiveness of the proposed method are illustrated by several numerical examples in one and two dimensional spaces

length scales in solutions of the complex ginzburg landau equation abstract we generalise and in certain cases improve on previous a priori estimates of sobolev norms of solutions to the generalised complex ginzburg landau equation a set of dynamic length scales based on ratios of these norms is defined we are able to derive lower bounds for time averages and long time limits of these length scales the bounds scale like the inverses of our l bounds

gas kinetic finite volume methods flux vector splitting and artificial diffusion abstract in order to improve our understanding of boltzmann type schemes which have been recently proposed for the euler and navier stokes equations by prendergast and xu we have modified the gas kinetic approach and applied it to simple hyperbolic conservation laws we found that the gas kinetic discretization can be explained in terms of standard central difference and upwind schemes artificial viscosity concepts are reviewed and linked to the grid size and the physical length scale of the discontinuity also a new three dimensional gas kinetic scheme for the numerical navier stokes equations whose solution satisfies the entropy condition is presented two numerical limits of the scheme are obtained the first one is the one step lax wendroff scheme and the second one is the kinetic flux vector splitting scheme a new relaxation scheme for steady state calculations is also formulated and implemented in the multigrid time stepping technique of jameson when applied to the euler equations the resulting method yields high accuracy and fast convergence to a steady state