on lp x norms the relation between a banach function space x and its subspaces formed by i those functions with absolutely continuous norm ii those with continuous norm and iii the closure of the set of

on the spaces lp x ω and wm p x ω in this paper we present some basic results on the generalized lebesgue spaces l p x omega and generalized lebesgue sobolev spaces w m w p x omega these results provide the necessary framework for the study of variational problems and elliptic equations with non standard p x growth conditions c 2001 academic press

eigenvalue problems for anisotropic elliptic equations an orlicz sobolev space setting abstract the paper studies a class of anisotropic eigenvalue problems involving an elliptic nonhomogeneous differential operator on a bounded domain from r n with a smooth boundary some results regarding the existence or non existence of eigenvalues are obtained in each case the competition between the growth rates of the anisotropic coefficients plays an essential role in the description of the set of eigenvalues

on variational inequalities with maximal monotone operators and multivalued perturbing terms in sobolev spaces with variable exponents abstract we are concerned in this paper with variational inequalities of the form a u v u f u v u l v u v k u k where a is a maximal monotone operator f is an integral multivalued lower order term and k is a closed convex set in a sobolev space of variable exponent we study both coercive and noncoercive inequalities in the noncoercive case a sub supersolution approach is followed to obtain the existence and some other qualitative properties of solutions between sub and supersolutions

goodness of fit problem and scanning innovation martingales this paper is mainly devoted to the following statistical problem in the case of random variables of any finite dimension and both simple or parametric hypotheses how to construct convenient empirical processes which could provide the basis for goodness of fit tests more or less in the same way as the uniform empirical process does in the case of simple hypothesis and scalar random variables the solution of this problem is connected here with the theory of multiparameter martingales and the theory of function parametric processes namely for the limiting gaussian processes some kind of filtration is introduced and so called scanning innovation processes are constructed the adapted standard wiener processes in one to one correspondence with initial gaussian processes this is done for the function parametric versions of the processes

regular article on a class of pade finite volume methods a class of pade finite volume methods providing an improved spectral resolution is presented and compared with well known methods the formulation is based on the sliding averages of the variables and allows the computation of derivatives of all orders using the fourier analysis these methods are examined with respect to i order of accuracy ii spectral resolution iii boundary conditions and iv stability

existence and multiplicity of solutions for quasilinear nonhomogeneous problems an orlicz sobolev space setting abstract we study the boundary value problem div log 1 u q u p 2 u f u in u 0 on where is a bounded domain in r n with smooth boundary we distinguish the cases where either f u u p 2 u u r 2 u or f u u p 2 u u r 2 u with p q 1 p q min n r and r n p n p n p in the first case we show the existence of infinitely many weak solutions for any 0 in the second case we prove the existence of a nontrivial weak solution if is sufficiently large our approach relies on adequate variational methods in orlicz sobolev spaces

eigenvalues of p x laplacian dirichlet problem abstract this paper studies the eigenvalues of the p x laplacian dirichlet problem div u p x 2 u u p x 2 u in u 0 on where is a bounded domain in rn and p x is a continuous function on such that p x 1 we show that the set of eigenvalues is a nonempty infinite set such that sup we present some sufficient conditions for inf 0 and for inf 0 respectively

a compact finite difference scheme on a non equidistant mesh abstract a fourth order compact finite difference scheme is presented for second order partial differential equations the scheme is derived for a one dimensional non equidistant mesh which makes it particularly useful in problems with sharp boundary layers the inclusion of general boundary conditions does not reduce the order of the scheme for the boundary points the scheme is tested on a representative model equation we shortly discuss its stability in the simplest time dependent heat flow problem and its use in more dimensions

optimization of time integration schemes coupled to spatial discretization for use in caa applications in this paper errors arising from spatial and temporal discretization are discussed a new formulation to optimize time integration schemes is proposed the new optimization also takes the errors coming from the spatial discretization into account which leads to a minimization of the total errors it is applicable to central and upwind type spatial discretizations a six stage runge kutta scheme is optimized for central and upwind spatial discretizations its efficiency is demonstrated on a one dimensional convection equation and on a linearized euler problem the results indicate an important improvement

on a sub supersolution method for variational inequalities with leray lions operators in variable exponent spaces abstract in this paper we consider a sub supersolution method for variational inequalities with leray lions operators in sobolev spaces with variable exponents existence and qualitative properties of solutions between sub and supersolutions are established

regularity results for a class of functionals with non standard growth we consider the integral functional under non standard growth assumptions that we call p x type namely we assume that a relevant model case being the functional under sharp assumptions on the continuous function p x 1 we prove regularity of minimizers energies exhibiting this growth appear in several models from mathematical physics

a multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids the goal of this correction is to correct a mistake that appears in the proof of both lemma 3 9 and 3 10 in the article mih ilescu r dulescu 2006 1 the mistake in the proof of lemma 3 9 and 3 10 is mainly owing to the fact that we wrongly computed the expression of function g x t

a high order low dispersion symmetry preserving finite volume method for compressible flow on curvilinear grids a new high order finite volume method is presented that preserves the skew symmetry of convection for the compressible flow equations the method is intended for large eddy simulations les of compressible turbulent flows in particular in the context of hybrid rans les computations the method is fourth order accurate and has low numerical dissipation and dispersion due to the finite volume approach mass momentum and total energy are locally conserved furthermore the skew symmetry preservation implies that kinetic energy sound velocity and internal energy are all locally conserved by convection as well the method is unique in that all these properties hold on non uniform curvilinear structured grids due to the conservation of kinetic energy there is no spurious production or dissipation of kinetic energy stemming from the discretization of convection this enhances the numerical stability and reduces the possible interference of numerical errors with the subgrid scale model by minimizing the numerical dispersion the numerical errors are reduced by an order of magnitude compared to a standard fourth order finite volume method

multiple solutions for a nonhomogeneous dirichlet problem in orlicz sobolev spaces abstract in this paper moving within the framework of orlicz sobolev spaces we guarantee through variational arguments the existence of three weak solutions to the nonhomogeneous boundary value problem div a u x u x f x u g x u in u 0 on with bounded domain in r n with smooth boundary real parameters f g r r caratheodory functions and the function t a t t odd increasing homeomorphism from r onto r applications and comparisons are also presented in particular we improve a result for an eigenvalue problem established by mih ilescu and repovs in 15

strong approximation of semimartingales and statistical processes as an application of general convergence results for semimartingales exposed in their book limit theorems for stochastic processes jacod and shiryaev obtained a fundamental result on the convergence of likelihood ratio processes to a gaussian limit we strengthen this result in a quantitative sense and show that versions of the likelihood ratio processes can be defined on the space of the limiting experiment such that we get pathwise almost sure approximations with respect to the uniform metric the approximations are considered under both sequences of measures the hypothesisp n and the alternative overline p n a consequence is e g an estimate for the speed of convergence in the prohorov metric new approximation techniques for stochastic processes are developed

a high order finite volume remapping scheme for nonuniform grids a hierarchy of one dimensional high order remapping schemes is presented and their performance with respect to accuracy and convergence rate investigated the schemes are also compared based on remapping experiments in closed domains the piecewise quartic method pqm is presented based on fifth order accurate piecewise polynomials and is motivated by the need to significantly improve hybrid coordinate systems of ocean climate models which require the remapping to be conservative monotonic and highly accurate a limiter for this scheme is fully described that never decreases the polynomial degree except at the location of extrema we assess the use of high order explicit and implicit i e compact estimates for the edge values and slopes needed to build the piecewise polynomials in both piecewise parabolic method ppm and pqm it is shown that all limited pqm schemes perform significantly better than limited ppm schemes and that pqm schemes are much more cost effective

perturbed kirchhoff type neumann problems in orlicz sobolev spaces the aim of this paper is to establish the existence of infinitely many solutions for perturbed kirchhoff type non homogeneous neumann problems involving two parameters to be precise we prove that an appropriate oscillating behaviour of the nonlinear term even under small perturbations ensures the existence of infinitely many solutions our approach is based on recent variational methods for smooth functionals defined on orlicz sobolev spaces

quasilinear elliptic non homogeneous dirichlet problems through orlicz sobolev spaces abstract in this paper we are interested in the existence of infinitely many weak solutions for a non homogeneous eigenvalue dirichlet problem by using variational methods in an appropriate orlicz sobolev setting we determine intervals of parameters such that our problem admits either a sequence of non negative weak solutions strongly converging to zero provided that the non linearity has a suitable behaviour at zero or an unbounded sequence of non negative weak solutions if a similar behaviour occurs at infinity

domain decomposition technique for solution of acoustic wave scattering a novel domain decomposition technique is applied that couples nonlinear and linear euler solvers to model acoustic scattering of waves in the vicinity of high velocity gradients and large amplitude waves to reduce numerical resources the linear solver is applied in domains in which the wave amplitude is small and linearization is applicable whereas the nonlinear solver is used in the vicinity of high wave amplitudes and stagnation points three numerical models are compared in this paper 1 a linear euler 2 a nonlinear euler and 3 the domain decomposition technique for all three models the finite difference dispersion relation preserving scheme is applied to all spatial derivatives and the low dissipation dispersion runge kutta scheme is applied for the temporal integration a perfectly matched layer is applied at the edges of the computational domain to absorb outgoing waves and minimize reflections to include complex bodies and geometries an overset mesh is applied after model validation t

eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent in this paper we are concerned with a new class of anisotropic quasilinear elliptic equations with a power like variable reaction term one of the main features of our work is that the differential operator involves partial derivatives with different variable exponents so that the functional analytic framework relies upon anisotropic sobolev and lebesgue spaces existence and nonexistence results are deeply influenced by the competition between the growth rates of the anisotropic coefficients our main results point out some striking phenomena related to the existence of a continuous spectrum in several distinct situations

remarks on eigenvalue problems involving the p x laplacian abstract this paper deals with the eigenvalue problem involving the p x laplacian of the form div u p x 2 u u q x 2 u in u 0 on where is a bounded domain in r n p c 0 inf x p x 1 q l 1 q x q x e p x for x e is a positive constant p x is the sobolev critical exponent it is shown that for every t 0 the problem has at least one sequence of solutions u n t n t such that 1 p x u n t p x t and n t as n the principal eigenvalues for the problem in several important cases are discussed especially the similarities and the differences in the eigenvalue problem between the variable exponent case and the constant exponent case are exposed some known results on the eigenvalue problem are extended

optimized compact difference based finite volume schemes for linear wave phenomena high order compact difference based finite volume schemes are developed analyzed and implemented for linear wave propagation phenomena with particular emphasis on computational electromagnetics in the time domain the formulation combines the primitive function approach with five point spatially sixth and fourth order methods optimization in the semi discrete case is achieved by minimizing dispersion and isotropy error the fully discrete scheme is examined by adopting the classical fourth order runge kutta technique stability bounds are established and the coefficients for the spatial discretization are readjusted for optimum performance the scheme is then formally extended to multiple dimensions consistent boundary conditions are presented for the reconstruction operator as well as flux specification several calculations in one and three dimensions confirm the properties of the method

propagation and spurious reflection in finite element approximations of hyperbolic equations abstract there are a variety of spurious phenomena in the numerical simulation of wave propagation which manifest themselves by the appearance of short wavelength oscillations near discontinuities and computational boundaries the appropriate tool for the description of those phenomena is that given by fourier analysis in particular introduction in the analysis of the concepts of energy propagation and group velocity from mathematical physics produces a set of particularly interesting results the corresponding mathematics have been reasonably well described in the recent literature for usual finite difference schemes but several new aspects present themselves in the numerical schemes based on finite element methods we will in this paper examine wave propagation in the numerical approximation of hyperbolic equations obtained with the linear finite element galerkin method pointing out where the corresponding properties differ from those of finite differences in particular it will be shown that the energy conserved in finite element schemes is not the usual square of the l 2 norm as it is with finite differences the expression of this energy is given by a modified form of parseval s relation group velocities of rightgoing and leftgoing solutions corresponding to the same frequency are not equal illustration of spurious reflection at downwind boundaries upwind boundaries and at interfaces in mesh refinement are given together with their mathematics

weighted sobolev theorem with variable exponent for spatial and spherical potential operators abstract in s g samko b g vakulov weighted sobolev theorem with variable exponent for spatial and spherical potential operators j math anal appl 310 2005 229 246 sobolev type p q theorems were proved for the riesz potential operator i in the weighted lebesgue generalized spaces l p r n with the variable exponent p x and a two parameter power weight fixed to an arbitrary finite point x 0 and to infinity under an additional condition relating the weight exponents at x 0 and at infinity we show in this note that those theorems are valid without this additional condition similar theorems for a spherical analogue of the riesz potential operator in the corresponding weighted spaces l p s n on the unit sphere s n in r n 1 are also improved in the same way

the number of solutions for a class of nonlocal nonhomogeneous gradient operator equations abstract this paper deals with the number of solutions for a class of nonlocal nonhomogeneous gradient operator equations of the form a i u i u f where i c 1 x r x is a reflexive banach space i 0 0 i is even and strictly convex i u u as u i x x is a bounded homeomorphism but is not necessarily homogeneous a 0 r is continuous f x 0 some properties and examples of such a functional i are given some results on the number of solutions of the nonlocal equation are obtained

a finite volume method for one dimensional reaction diffusion problems abstract a truly conservative finite volume method for one dimensional reaction diffusion equations is presented the method is based on piecewise second degree polynomials approximations that result in linear fluxes a symmetric mass matrix and a semi discrete system of nonlinear ordinary differential equations in time it is shown that the semi discrete method is linearly stable provided that the problem is mathematically well posed it is also shown that implicit discretizations of the time variable are stable whereas the explicit one is conditionally stable and has a smaller domain of linear stability than the standard explicit finite difference method it is also illustrated that upon lumping the mass matrix the method coincides with a finite difference discretization of the original problem a detailed formulation of the dirichlet neumann and robin boundary conditions is presented such a formulation makes use of control volumes which are here referred to as incomplete ones although the finite volume technique presented here uses a node centered formulation it can be easily extended to cell centered ones the technique is also developed for multidimensional problems

on statistics of markov step processes representation of log likelihood ratio processes in filtered local models observe over a long time the trajectory of a markov step processx x t t 0 whose generator depends on an unknown parameter consider the corresponding statistical model locally over small neighbourhoods of some fixed we prove a decomposition of log likelihood ratio processes in the local model of type h t m vartheta n tfrac 1 2 h t left langle m vartheta n right rangle h remainder terms whereh is the local parameter andm n is the score function martingale suitably normed and scaled weak convergence of filtered local models at is reduced to a problem of weak convergence of the score function martingale this can be solved in a variety of cases as a consequence we obtain laq lamn or lan at under fixed time sampling as well as under a certain class of sequential sampling plans

sobolev embedding theorems for spaces wk p x ω this paper gives a sobolev type embedding theorem for the generalized lebesgue sobolev space wk p x where is an open domain in rn n 2 with cone property and p x is a lipschitz continuous function defined on satisfying 1 p p nk the main result can be stated as follows for any measurable function q x x formula with p x q x p x np x n kp x there exists a continuous embedding from wk p x to lq x

invariance theorems concerning reflection at numerical boundaries abstract downwind computational boundaries in the numerical approximation of hyperbolic equations are in general not transparent and they create spurious reflection a useful measure of this is given by the energy or usual sum of squares of the reflected solution in response to an arbitrary solution which originates from within the computing domain we prove in that respect a somewhat unexpected property namely that for those full discretizations which are obtained by applying to a space discretization of the equations an energy conservative discrete time marching method the energy reflected at the boundary is independent of the value of t and is strictly equal to the energy reflected in the semidiscrete case this is verified in numerical experiments optimal boundary equations may be defined in the semidiscrete case of those which maximize the rate of convergence to zero of the reflected energy when h 0 a corollary of the preceding result is that those boundary equations remain optimal in the same sense when used in an energy conservative full discretization moreover this convergence result continues to hold when a nonconservative but stable i e dissipative time discretization is used this is verified numerically with the first order adams predictor corrector method the results of this paper are derived with the mathematics of the simple three point central difference discretization of spatial derivatives obvious generalizations to other cases are mentioned at the end of the paper

compact finite volume schemes on boundary fitted grids the paper focuses on the development of a framework for high order compact finite volume discretization of the three dimensional scalar advection diffusion equation in order to deal with irregular domains a coordinate transformation is applied between a curvilinear non orthogonal grid in the physical space and the computational space advective fluxes are computed by the fifth order upwind scheme introduced by pirozzoli s pirozzoli conservative hybrid compact weno schemes for shock turbulence interaction j comp phys 178 2002 81 while the coupled derivative scheme m h kobayashi on a class of pade finite volume methods j comp phys 156 1999 137 is used for the discretization of the diffusive fluxes numerical tests include unsteady diffusion over a distorted grid linear free surface gravity waves in a irregular domain and the advection of a scalar field the proposed methodology attains high order formal accuracy and shows very favorable resolution characteristics for the simulation of problems with a wide range of length scales

compact fourth order finite volume method for numerical solutions of navier stokes equations on staggered grids the development of a compact fourth order finite volume method for solutions of the navier stokes equations on staggered grids is presented a special attention is given to the conservation laws on momentum control volumes a higher order divergence free interpolation for convective velocities is developed which ensures a perfect conservation of mass and momentum on momentum control volumes three forms of the nonlinear correction for staggered grids are proposed and studied the accuracy of each approximation is assessed comparatively in fourier space the importance of higher order approximations of pressure is discussed and numerically demonstrated fourth order accuracy of the complete scheme is illustrated by the doubly periodic shear layer and the instability of plane channel flow the efficiency of the scheme is demonstrated by a grid dependency study of turbulent channel flows by means of direct numerical simulations the proposed scheme is highly accurate and efficient at the same level of accuracy the fourth order scheme can be ten times faster than the second order counterpart this gain in efficiency can be spent on a higher resolution for more accurate solutions at a lower cost

solutions for neumann boundary value problems involving p x laplace operators abstract in this paper we study the nonlinear neumann boundary value problem of the following form p div u p x 2 u u p x 2 u f x u in u p x 2 u g x u on using the variational method under appropriate assumptions on f and g we obtain a number of results on existence and multiplicity of solutions also what is also interesting is that we obtain some results which can be considered as extensions of the classical result named combined effects of concave and convex nonlinearities moreover the positive solution of the problem is considered

on the sub supersolution method for p x laplacian equations abstract this paper deals with the sub supersolution method for the p x laplacian equations a sub supersolution principle for the dirichlet problems involving the p x laplacian is established it is proved that the local minimizers in the c 1 topology are also local minimizers in the w 1 p x topology for given energy functionals a strong comparison theorem for the p x laplacian equations is presented some applications of the abstract theorems obtained in this paper to the eigenvalue problems for the p x laplacian equations are given

development of a non reflecting boundary condition for multidimensional nonlinear duct acoustic computation the quality of any time domain nonlinear cfd simulation of silencers is determined by the modeling procedure used for the anechoic termination at the boundary end of the computational domain in this study a novel anechoic boundary condition based on the characteristic theory combined with the linear relaxation method has been developed in an open source cfd code wave propagation in ducts and mufflers has been analyzed and the effect of the damping properties of the boundary condition has been studied the study shows that the explicit damping of the solution vector produces good performance at all the frequencies with acoustic waves having varying incidence to the boundary also results indicate that numerical reflection of acoustic waves with high incident angles is reduced if the damping is applied at the boundary within the semi implicit solution of the governing equations finally the boundary condition has been applied together with a nonlinear solver to evaluate the acoustic attenuation performance of circular flow reversing chambers and single plug perforated mufflers with and without mean flow velocity of the gas code validation has been carried out against experimental data

existence of solutions for p x laplacian dirichlet problem this paper presents several sufficient conditions for the existence of solutions for the dirichlet problem of p x laplacian div u p x 2 u f x u x u 0 x especially an existence criterion for infinite many pairs of solutions for the problem is obtained the discussion is based on the theory of the spaces lp x and w01 p x

curvilinear finite volume schemes using high order compact interpolation during the last years the need of high fidelity simulations on complex geometries for aeroacoustics predictions has grown most of high fidelity numerical schemes in terms of low dissipative and low dispersive effects lie on finite difference fd approach but for industrial applications fd schemes are less robust compared to finite volume fv ones thus the present study focuses on the development of a new compact fv scheme for two and three dimensional applications the proposed schemes are formulated in the physical space and not in the computational space as it is the case in most of the known works therefore they are more appropriate for general grids they are based on compact interpolation to approximate interface averaged field values using known cell averaged values for each interface the interpolation coefficients are determined by matching taylor series expansions around the interface center two types of schemes can be distinguished the first one uses only the curvilinear abscissa along a mesh line to derive a sixth order compact interpolation formulae while the second more general uses coordinates in a spatial three dimensional frame well chosen this latter is formally sixth order accurate in a preferred direction almost orthogonal to the interface and at most fourth order accurate in transversal directions for non linear problems different approaches can be used to keep the high order scheme however in the present paper a muscl like formulation was sufficient to address the presented test cases all schemes have been modified to treat multiblock and periodic interfaces in such a way that high order accuracy stability good spectral resolution conservativeness and low computational costs are guaranteed this is a first step to insure good scalability of the schemes although parallel performances issues are not addressed as high frequency waves badly resolved could be amplified and then destabilize the scheme compact filtering operators have been used numerous test cases as the linear convection of a gaussian wave the convection of a lamb oseen vortex and the diffraction of an acoustic wave on a plane have been realized to validate the schemes the most efficient schemes are shown to be at least fifth order accurate on linear and non linear convection problems they are also less dissipative and less dispersive on non uniform curvilinear grids than schemes using implicit interpolation with constant coefficients of the same order on uniform cartesian grids

a finite volume formulation of compact central schemes on arbitrary structured grids in the present paper a formulation allowing the use of compact schemes in the finite volume context on arbitrary meshes is presented the proposed formulation is based on the use of an implicit formula to evaluate the fluxes on the cell faces a series of numerical experiments for a 2d model convection equation a flat plate a subsonic vortical problem as well as the les simulation of channel flow has been carried out the results indicate an important improvement in obtained accuracy compared to a standard central finite volume formulation

an accuracy and stability study of the 2d spectral volume method in this article the accuracy and the stability of 2d spectral volume schemes are studied by means of an analysis of the wave propagation properties it is shown that several sv partitions suffer from a weak instability stable schemes with lower dispersion and diffusion errors are proposed numerical tests show an important improvement in the accuracy of the fourth order scheme

mountain pass and ekeland s principle for eigenvalue problem with variable exponent in this article we study the boundary value problem div u p x 2 u u x 2 u u q x 2 u in u 0 on where is a smooth bounded domain in n and p q are continuous functions on we show that for any 0 there exists infinitely many weak solutions respectively if 0 and small enough then there exists a non negative non trivial weak solution our approach relies on the variable exponent theory of generalized lebesgue sobolev spaces combined with a 2 symmetric version for even functionals of the mountain pass theorem respectively on simple variational arguments based on ekeland s variational principle this article is devoted to the special issue vaxjo conference 2008

two non trivial solutions for a non homogeneous neumann problem an orlicz sobolev space setting in this paper we study a non homogeneous neumann type problem which involves a nonlinearity satisfying a non standard growth condition by using a recent variational principle of ricceri we establish the existence of at least two non trivial solutions in an appropriate orlicz sobolev space

a fourth order accurate finite volume compact method for the incompressible navier stokes solutions abstract this paper presents a finite volume fourth order accurate compact scheme for discretization of the incompressible navier stokes equations in primitive variable formulation the numerical method of integrating the navier stokes equations comprises a compact finite volume formulation of the average convective and diffusive fluxes the pressure velocity coupling is achieved via the coupled solution of the resulting system of equations the solution of the coupled set of equations is performed with an implicit newton krylov matrix free method for stationary problems for simulation of unsteady flows a standard fourth order runge kutta method was used for temporal discretization and the velocity pressure coupling was ensured at each stage also using the matrix free method several incompressible viscous steady and unsteady flow problems have been computed to assess the robustness and accuracy of the proposed method

higher order kfvs algorithms using compact upwind difference operators a family of high order accurate compact upwind difference operators have been used together with the split fluxes of the kfvs kinetic flux vector splitting scheme to obtain high order semidiscretizations of the 2d euler equations of inviscid gas dynamics in general coordinates a tvd multistage runge kutta time stepping scheme is used to compute steady states for selected transonic supersonic flow problems which indicate the higher accuracy and low diffusion realizable in such schemes

high order numerical sommerfeld boundary conditions theory and experiments abstract we develop high order non reflecting boundary equations for a semidiscrete approximation of the simple hyperbolic advection equation u 1 cu x 0 these boundary equation are based on a discrete interpretation of sommerfeld s radiation condition for a second order wave equation which is associated with the semi discrete equation the performance of these schemes is expressed by an exact measure of the energy reflected at the boundary for low order cases the discrete sommerfeld boundary equations are identical with the standard finite difference equations but for high orders of approximation starting with 4 points the discrete sommerfeld schemes differ from standard finite differences it is shown and verified experimentally that the discrete sommerfeld schemes are optimal in the sense that they produce the least amount of reflected energy moreover it is known theoretically and we verify experimentally that the reflected energy remains invariat when the semi discrete equation are time discretized with the trapezoidal crank nicolson method the corresponding fully discrete boundary equations are thus also optimal in the sense that they minimize the reflected energy

optimized compact finite difference schemes with maximum resolution direct numerical simulations and computational aeroacoustics require an accurate finite difference scheme that has a high order of truncation and high resolution characteristics in the evaluation of spatial derivatives compact finite difference schemes are optimized to obtain maximum resolution characteristics in space for various spatial truncation orders an analytic method with a systematic procedure to achieve maximum resolution characteristics is devised for multidiagonal schemes based on the idea of the minimization of dispersive phase errors in the wave number domain and these are applied to the analytic optimization of multidiagonal compact schemes actual performances of the optimized compact schemes with a variety of truncation orders are compared by means of numerical simulations of simple wave convections and in this way the most effective compact schemes are found for tridiagonal and pentadiagonal cases respectively from these comparisons the usefulness of an optimized high order tridiagonal compact scheme that is more efficient than a pentadiagonal scheme is discussed for the optimized high order spatial schemes the feasibility of using classical high order runge kutta time advancing methods is investigated

existence results for nonlinear elliptic equations with leray lions operator and dependence on the gradient the existence of positive solutions for nonlinear elliptic problems under dirichlet boundary condition is studied as well as the compactness and directness of the solution set the main novelties consist in the presence of a leray lions operator in the differential part and in the dependence of the reaction term on the gradient of the solution our approach relies on the method of sub supersolution nonlinear regularity theory and strong maximum principle

solutions for p x laplacian dirichlet problems with singular coefficients using the critical point theory we investigate the existence and multiplicity of solutions for p x laplacian dirichlet problem with singular coefficients

existence of three solutions for a non homogeneous neumann problem through orlicz sobolev spaces abstract the aim of this paper is to establish a multiplicity result for an eigenvalue non homogeneous neumann problem which involves a nonlinearity fulfilling a nonstandard growth condition precisely a recent critical points result for differentiable functionals is exploited in order to prove the existence of a determined open interval of positive eigenvalues for which the problem admits at least three weak solutions in an appropriate orlicz sobolev space

eigenvalues of the p x laplacian neumann problems abstract we study the eigenvalues of the p x laplacian operator with zero neumann boundary condition on a bounded domain where p x is a continuous function defined on the domain with p x 1 we show that similarly to the p laplacian case the smallest eigenvalue of the problem is 0 and it is simple and the supremum of all the eigenvalues is infinity however unlike the p laplacian case for very general variable exponent p x the first eigenvalue is not isolated that is the infimum of all positive eigenvalues of the problem is 0 we also study some properties of the set of functions having p x average value zero

on the use of higher order finite difference schemes on curvilinear and deforming meshes this study enables the use of very high order finite difference schemes for the solution of conservation laws on stretched curvilinear and deforming meshes to illustrate these procedures we focus on up to 6th order pade type spatial discretizations coupled with up to 10th order low pass filters these are combined with explicit and implicit time integration methods to examine wave propagation and wall bounded flows described by the navier stokes equations it is shown that without the incorporation of the filter application of the high order compact scheme to nonsmooth meshes results in spurious oscillations which inhibit their applicability inclusion of the discriminating low pass high order filter restores the advantages of high order approach even in the presence of large grid discontinuities when three dimensional curvilinear meshes are employed the use of standard metric evaluation procedures significantly degrades accuracy since freestream preservation is violated to overcome this problem a simple technique is adopted which ensures metric cancellation and thus ensures freestream preservation even on highly distorted curvilinear meshes for dynamically deforming grids an effective numerical treatment is described to evaluate expressions containing the time varying transformation metrics with these techniques metric cancellation is guaranteed regardless of the manner in which grid speeds are defined the efficacy of the new procedures is demonstrated by solving several model problems as well as by application to flow past a rapidly pitching airfoil and past a flexible panel

dispersion and dissipation properties of the 1d spectral volume method and application to a p multigrid algorithm in this article the wave propagation properties of the 1d spectral volume method are studied through analysis of the fourier footprint of the schemes a p multigrid algorithm for the spectral volume method is implemented restriction and prolongation operators are discussed and the efficiency of the smoothing operators is analyzed the results are verified for simple 1d advection problems and for a quasi 1d euler flow it is shown that a significant decrease in computational effort is possible with the p multigrid algorithm

derivation of high order compact finite difference schemes for non uniform grid using polynomial interpolation in this paper simple polynomial interpolation is used to derive arbitrarily high order compact schemes for the first derivative and tridiagonal compact schemes for the second derivative consisting of three second derivative nodes in the interior and two on the boundary on non uniform grids boundary and near boundary schemes of the same order as the interior are also developed using polynomial interpolation and for a general compact scheme on a non uniform grid it is shown that polynomial interpolation is more efficient than the conventional method of undetermined coefficients for finding coefficients of the scheme the high order non uniform schemes along with boundary closure of up to 14th order thus obtained are shown to be stable on a non uniform grid with appropriate stretching so that more grid points are clustered near the boundary the stability and resolution properties of the high order non uniform grid schemes are studied and the results of three numerical tests on stability and accuracy properties are also presented

compact disturbance equations for aeroacoustic simulations a set of compact disturbance equations is developed for high accuracy and efficient aeroacoustic simulations the compact disturbance equations in their complete form are an exact rearrangement of the navier stokes equations but incorporate various linear and nonlinear disturbance equations such as the linearized euler equations and the linearized navier stokes equations their attractive mathematical properties facilitate the implementations of the full compact disturbance equations and the reduced equations in essentially the same form as in existing computational fluid dynamics solvers with minor modifications a high resolution unsteady simulation in a reduced domain can be embedded inside a less expensive reynolds averaged navier stokes solution of flow in a larger possibly very complex configuration this results in a hybrid reynolds averaged navier stokes large eddy simulation method with reduced meshing difficulties and computational costs but improved grid qualities and more accurate boundary

a new flux vector splitting compact finite volume scheme a new high resolution finite volume method is proposed here that uses flux vector splitting as the building block to represent the physical processes the high accuracy projection in space is obtained here inter alia by using some compact schemes of sengupta et al 1 and lele 2 that have extremely high accuracy in spectral plane this produces a new class of finite volume fv scheme for spatial discretization that has been analyzed for spectral accuracy numerical stability and dispersion relation preservation drp property following the full domain analysis method of 1 using two different time integration strategies these combination of spatial and temporal methods have been tested for two linear wave problems reported in 1 3 and the solution of burgers equation is compared with the analytical results presented in adams and shariff 4 the shock capturing method is based on the proposed technique in 5 that does not require the usage of any nonlinear differencing techniques or limiters to show that the proposed method can also handle irregular grids the burgers equation is solved using non uniform grids also to demonstrate the utility of the proposed scheme to solve practical problems the benchmark problem of nonlinear wave propagation in a one dimensional shock tube has been solved with the initial data as given in 23 by solving the euler equation the analysis and comparison of this flux vector splitting scheme with other well known methods clearly demonstrate the superior scale resolution of the proposed method at the same time providing a neutrally stable scheme when used with four stage runge kutta scheme also the same scheme displays extended ranges of wave numbers and circular frequencies over which drp property is valid the computed results of three test cases and their comparison with analytical results clearly reveal that the presented method can be used for practical problems

multiple solutions for a nonlinear and non homogeneous problem in orlicz sobolev spaces abstract we study a non homogeneous boundary value problem in a smooth bounded domain in r n we prove the existence of at least two non negative and non trivial weak solutions our approach relies on orlicz sobolev spaces theory combined with adequate variational methods and a variant of mountain pass lemma

density of smooth functions in wk p x ω kovacik rakosnik investigated the spaces l p x omega of functions which are integrable with variable power p x and the corresponding counterparts of the sobolev spaces w k p x omega we continue that investigation and describe a class of functions p x for which the set of smooth functions on omega is dense in w k p x omega as a corollary we obtain in terms of the distance function a condition on elements of w k p x omega sufficient to ensure that they belong to w 0 k p x omega

gradient estimates for the p x laplacean system we prove calderon and zygmund type estimates for a class of elliptic problems whose model is the non homogeneous p x laplacean system div jduj p x 2 du div jf j p x 2 f

existence of solutions for p x laplacian problems on a bounded domain in this paper we study the following p x laplacian problem div a x vd uv p x 2 d u b x v uv p x 2 u f x u x e u u 0 on p u where 1 p 1 l p x l p 2 n u s r n is a bounded domain and applying the mountain pass theorem we obtain the existence of solutions in w 0 1 p x for the p x laplacian problems in the superlinear and sublinear cases 2004 elsevier inc all rights reserved

differential equations of divergence form in musielak sobolev spaces and a sub supersolution method abstract we study some properties of differential operators of divergence form in musielak sobolev spaces and establish a sub supersolution method for differential equations of divergence form with homogeneous neumann or dirichlet boundary condition in musielak sobolev spaces

a 2d compact fourth order projection decomposition method a 2d fourth order compact direct scheme projection decomposition method for solving incompressible viscous flows in multi connected rectangular domains is devised in each subdomain the governing navier stokes equations are discretized by using fourth order compact schemes in space and second order scheme in time the coupling between subdomains is based on a direct non overlapping multidomain method it allows to solve each helmholtz poisson problem resulting of a projection method in complex geometries the major difficulty of the poisson neumann problem solvability is addressed and correctly treated the present numerical method is checked through some classical numerical experiments first the second order accuracy in time and the fourth order accuracy in space are shown by matching with the analytical solution of the taylor problem the method is also tested by simulating the flow in a 2d lid driven cavity the utility of the compact scheme projection decomposition method approach is further illustrated by two other benchmark problems viz the flow over a backward facing step and the laminar flow past a square prism the present results are in good agreement with the experimental data and other numerical solutions available in the literature