rings and modules which are stable under automorphisms of their injective hulls it is proved among other results that a prime right nonsingular ring in particular a simple ring r is right self injective if rr is invariant under automorphisms of its injective hull this answers two questions raised by singh and srivastava and clark and huynh an example is given to show that this conclusion no longer holds when prime ring is replaced by semiprime ring in the above assumption also shown is that automorphism invariant modules are precisely pseudo injective modules answering a recent question of lee and zhou furthermore rings whose cyclic modules are automorphism invariant are investigated

dynamics for the energy critical nonlinear wave equation in high dimensions t duyckaerts and f merle 2008 studied the variational structure near the ground state solution w of the energy critical wave equation and classified the solutions with the threshold energy e w 0 in dimensions d 3 4 5 in this paper we extend the results to all dimensions d 6 the main issue in high dimensions is the non lipschitz continuity of the nonlinearity which we get around by making full use of the decay property of w

on the effect of numerical errors in large eddy simulations of turbulent flows aliased and dealiased numerical simulations of a turbulent channel flow are performed using spectral and finite difference methods analytical and numerical studies show that aliasing errors are more destructive for spectral and high order finite difference calculations than for low order finite difference simulations numerical errors have different effects for different forms of the nonlinear terms in the navier stokes equations for divergence and convective forms spectral methods are energy conserving only if dealiasing is performed for skew symmetric and rotational forms both spectral and finite difference methods are energy conserving even in the presence of aliasing errors it is shown that discrepancies between the results of dealiased spectral and standard nondialiased finite difference methods are due to both aliasing and truncation errors with the latter being the leading source of differences the relative importance of aliasing and truncation errors as compared to subgrid scale model terms in large eddy simulations is analyzed and discussed for low order finite difference simulations truncation errors can exceed the magnitude of the subgrid scale term

stability analysis for linear discretisations of the advection equation with runge kutta time integration for the 1 dim linear advection problem stability limits of runge kutta rk methods from 1st to 7th order in combination with upwind or centered difference schemes from 1st to 6th order are presented the analysis can be carried out in a rather general way by introduction of a broad class of runge kutta methods here called linear case runge kutta lc rk methods which behave completely similar for linear time independent and homogeneous ode systems and contain the classical order stage rk methods the set of conditions for the coefficients of these lc rk schemes could be derived explicitly for arbitrary order n from an efficiency viewpoint the lc rk 3rd order methods in combination with upwind 3rd or 5th order or the lc rk 4th order scheme with 4th order centered difference advection are a good choice the analysis can be extended easily to multidimensional splited advection for which a necessary stability condition is presented

one dimensional characteristic boundary conditions using nonlinear invariants a new treatment of boundary conditions using the flow decomposition in characteristics is derived for inviscid one dimensional flows using nonlinear invariants the new set of equations is equivalent to the relations of thompson 1987 5 but it has the advantage to provide a physical interpretation of the characteristics for nonlinear perturbations this interpretation is a major advantage to deal with the nonlinear injection of waves in the domain in particular the limitation of the standard relations to the injection of waves without nonlinear interactions on the boundary is addressed associated errors are evaluated analytically and numerically and a clear improvement of the results is demonstrated with the new expressions to avoid a drift of the mean values relaxation terms are usually added in the relations and the boundary conditions become almost non reflecting the consequences of these relaxation terms on outgoing and ingoing waves are widely investigated in the present paper and a nonlinear correction is proposed to recover perfectly non reflecting conditions without drift to end simulations are performed on the generation of indirect combustion noise through a nozzle to illustrate the advantages of the new formulation

application of dispersion relation preserving theory to develop a two dimensional convection diffusion scheme in this paper a finite difference scheme is developed within the nine point semi discretization framework for the convection diffusion equation the employed pade approximation renders a fourth order temporal accuracy and the spatial approximation of convection terms accommodates the dispersion relation the artificial viscosity introduced in the two dimensional convection diffusion reaction cdr equation for stability reasons is analytically derived constraints on the mesh size and time interval for rendering a monotonic matrix are also rigorously derived to validate the proposed method we investigate several problems that are amenable to the exact solutions the results with good rates of convergence are obtained for the investigated scalar and navier stokes problems

modules which are coinvariant under automorphisms of their projective covers abstract in this paper we study modules coinvariant under automorphisms of their projective covers we first provide an alternative and in fact a more succinct and conceptual proof for the result that a module m is invariant under automorphisms of its injective envelope if and only if given any submodule n of m any monomorphism f n m can be extended to an endomorphism of m and then as a dual of it we show that over a right perfect ring a module m is coinvariant under automorphisms of its projective cover if and only if for every submodule n of m any epimorphism m m n can be lifted to an endomorphism of m

sound radiation by a moving line source above an impedance plane with frequency dependent properties abstract an analytic solution for the problem of sound radiation by a harmonic line source moving at a uniform subsonic speed parallel to an impedance plane is proposed the main originality of this work is that the variation of the impedance with the frequency is taken into account compared to the case of a constant impedance the reflection coefficient and the location of its poles in the complex plane are modified a uniform asymptotic expression is then developed for moderate mach numbers and a closed form expression corresponding to a weyl van der pol formula is proposed for a grazing incidence for hard grounds and for low mach numbers unlike previous analytical solutions derived in the literature for a point source the impedance is evaluated at the doppler frequency instead of at the source frequency the analytical solution and asymptotic expressions are then compared satisfactorily to a numerical solution obtained from a time domain solver of the linearized euler equations finally a parametric study is carried out showing that the assumption of a constant impedance is valid if the source mach number remains small typically less than 0 2 and if the source is not too close to the ground

automorphism invariant modules satisfy the exchange property abstract warfield proved that every injective module has the exchange property this was generalized by fuchs who showed that quasi injective modules satisfy the exchange property we extend this further and prove that a module invariant under automorphisms of its injective hull satisfies the exchange property we also show that automorphism invariant modules are clean and that directly finite automorphism invariant modules satisfy the internal cancellation and hence the cancellation property

an analysis of numerical errors in large eddy simulations of turbulence the reliability of numerical simulations of turbulence depend on our ability to quantify and control discretization errors in the classical literature on error analysis typically simple linear equations are studied estimates of errors derived from such analyses depend on the assumption that each dependent variable can be characterized by a unique amplitude and scale of spatial variation that can be normalized to unity this assumption is not valid for strongly nonlinear problems such as turbulence where nonlinear interactions rapidly redistribute energy resulting in the appearance of a broad continuous spectrum of amplitudes in such situations the numerical error as well as the subgrid model can change with grid spacing in a complicated manner that cannot be inferred from the results of classical error analysis in this paper a formalism for analyzing errors in such nonlinear problems is developed in the context of finite difference approximations for the navier stokes equations when the flow is fully turbulent analytical expressions for the power spectra of these errors are derived by exploiting the joint normal approximation for turbulent velocity fields these results are applied to large eddy simulation of turbulence to obtain quantitative bounds on the magnitude of numerical errors an assessment of the significance of these errors in made by comparing their magnitudes with that of the nonlinear and subgrid terms one method of controlling the errors is suggested and its effectiveness evaluated through quantitative measures although explicit evaluations are presented only for large eddy simulation the expressions derived for the power spectra of errors are applicable to direct numerical simulation as well

on the blow up phenomenon of the critical nonlinear schrodinger equation abstract in this paper we consider the blow up phenomenon of critical nonlinear schrodinger equations in dimension 1d and 2d we define the minimal mass as the l 2 norm necessary to ignite a wave collapse and we stress its role in the blow up mechanism asymptotic compactness properties and l 2 concentration are proved the proof relies on linear and nonlinear profile decompositions

compact accurately boundary adjusting high resolution technique for fluid dynamics a novel high resolution numerical method is presented for one dimensional hyperbolic problems based on the extension of the original upwind leapfrog scheme to quasi linear conservation laws the method is second order accurate on non uniform grids in space and time has a very small dispersion error and computational stencil defined within one space time cell for shock capturing the scheme is equipped with a conservative non linear correction procedure which is directly based on the maximum principle plentiful numerical examples are provided for linear advection quasi linear scalar hyperbolic conservation laws and gas dynamics and comparisons with other computational methods in the literature are discussed

analysis of the dissipation and dispersion properties of the multi domain chebyshev pseudospectral method the accuracy of the multi domain chebyshev pseudospectral method is investigated for wave propagation problems by examining the properties of the method in the wavenumber space theoretically in terms of dispersion and dissipation errors for a number of n 1 points in the subdomains used in the literature with n typically between 8 to 32 significant errors can be obtained for waves discretized by more than points per wavelength the dispersion and dissipation errors determined from the analysis in the wavenumber space are found to be in good agreement with those obtained in test cases accuracy limits based on arbitrary criteria are proposed yielding minimum resolutions of 7 7 5 2 and 4 0 points per wavelength for n 8 16 and 32 respectively the numerical efficiency of the method is estimated showing that it is preferable to choose n between 16 and 32 in practice the stability of the method is also assessed using the standard fourth order runge kutta algorithm finally 1 d and 2 d problems involving long range wave propagation are solved to illustrate the dissipation and dispersion errors for short waves the error anisotropy is studied in the 2 d case in particular for a hybrid fourier chebyshev configuration

the focusing energy critical nonlinear schrodinger equation in dimensions five and higher we consider the focusing energy critical nonlinear schr odinger equation iu t delta u u 4 over d 2 u in dimensions d geq 5 we prove that if a maximal lifespan solution u colon i times bbb r d to bbb c obeys sup t in i nabla u t 2 nabla w 2 then it is global and scatters both forward and backward in time here w denotes the ground state which is a stationary solution of the equation in particular if a solution has both energy and kinetic energy less than those of the ground state w at some point in time then the solution is global and scatters we also show that any solution that blows up with bounded kinetic energy must concentrate at least the kinetic energy of the ground state similar results were obtained by kenig and merle for spherically symmetric initial data and dimensions d 3 4 5

short note on the application of explicit spatial filtering to the variables or fluxes of linear equations the need for filtering high frequency waves is a recurrent issue in numerical simulations these waves might indeed lead to instability and they are in general not calculated accurately by the discretization algorithms selective filters have therefore been designed in order to damp high frequency waves without affecting significantly low frequency disturbances 1 6 these filters are particularly used in computational aeroacoustics but they appear also suitable for large eddy simulations les in which only the scales larger than the grid size are computed and whose equations are derived formally by applying a filter operator to the navier stokes equations 7 moreover les based specially on explicit filtering have been also developed 8 10 in practice the flow variables are usually filtered explicitly after each time step consider for example the time integration of the following differential equation

numerical simulation of sound generated by vortex pairing in a mixing layer a numerical code solving the filtered navier stokes equations is developed using special techniques of computational aeroacoustics this approach allows a direct determination of the compressible field on a computational domain including the acoustic far field a two dimensional mixing layer between two flows at m 1 0 12 and m 2 0 48 is simulated the reynolds number built up from the initial vorticity thickness and the velocity difference across the mixing layer is re 1 28 x 10 4 an appropriate forcing of the mixing layer is defined to have only one pairing in the computational domain at a fixed location the sound generation pattern for a single pairing displays a double spiral structure corresponding to the rotating quadrupole associated to two corotative vortices successive pairings produce an acoustic radiation at the frequency of this mechanism the directly computed far field sound is then compared to the prediction of lighthill s acoustic analogy lighthill m j on sound generated aerodynamically i general theory proceedings of the royal society of london vol 211 series a 1107 1952 pp 564 587 based on the aerodynamic fluctuations provided by the large eddy simulation code two integral formulations of the analogy based on spatial derivatives and time derivatives respectively are tested results are in good qualitative agreement with the results of direct simulation the accuracy is however greater with the formulation using time derivatives instead of spatial derivatives

runge kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics in this paper a new runge kutta method with minimal dispersion and dissipation error is developed the chebyshev pseudospectral method is utilized using spatial discretization and a new fourth order six stage runge kutta scheme is used for time advancing the proposed scheme is more efficient than the existing ones for acoustic computations

global wellposedness of defocusing critical nonlinear schrodinger equation in the radial case is globally wellposed in time more precisely we obtain a unique solution u u ch1 0 such that for all time u t depends continuously on the data in fact the dependence is even real analytic here moreover there is scattering for t the same statement holds for radial data h s 1 and proves in particular global existence of classical solutions in the radially symmetric case also this issue was open thus this is the analogue for nls of the result for the wave equation with quintic nonlinearity obtained by m struwe str in the radial case and by m grillakis gr s s in general in the case of the wave equation the proof is based on the following two different facts

global well posedness and scattering for the energy critical nonlinear schrodinger equation in r 3 we obtain global well posedness scattering and global l 10 spacetime bounds for energy class solutions to the quintic defocusing schrodinger equa tion in r 1 3 which is energy critical in particular this establishes global existence of classical solutions our work extends the results of bourgain 4 and grillakis 20 which handled the radial case the method is similar in spirit to the induction on energy strategy of bourgain 4 but we perform the induction analysis in both frequency space and physical space simultaneously and replace the morawetz inequality by an interaction variant first used in 12 13 the principal advantage of the interaction morawetz estimate is that it is not localized to the spatial origin and so is better able to handle nonradial solutions in particular this interaction estimate together with an almost conservation argument controlling the movement of l 2 mass in fre quency space rules out the possibility of energy concentration

a divergence free condition compensated method for incompressible navier stokes equations the present study aims to develop a new formulation to effectively calculate the incompressible navier stokes solutions in non staggered grids the distinguished feature of the proposed method which avoids directly solving the divergence free equation is to add a rigorously derived source term to the momentum equation to ensure satisfaction of the fluid incompressibility for the sake of numerical accuracy dispersion relation preserving upwind scheme developed within the two dimensional context was employed to approximate the convection terms the validity of the proposed mass preserving navier stokes method is justified by solving two benchmark problems at high reynolds and rayleigh numbers based on the simulated navier stokes solutions the proposed formulation is shown to outperform the conventional segregated method in terms of the reduction of cpu time 2007 elsevier b v all rights reserved

comparison between lattice boltzmann method and navier stokes high order schemes for computational aeroacoustics computational aeroacoustic caa simulation requires accurate schemes to capture the dynamics of acoustic fluctuations which are weak compared with aerodynamic ones in this paper two kinds of schemes are studied and compared the classical approach based on high order schemes for navier stokes like equations and the lattice boltzmann method the reference macroscopic equations are the 3d isothermal and compressible navier stokes equations a von neumann analysis of these linearized equations is carried out to obtain exact plane wave solutions three physical modes are recovered and the corresponding theoretical dispersion relations are obtained then the same analysis is made on the space and time discretization of the navier stokes equations with the classical high order schemes to quantify the influence of both space and time discretization on the exact solutions different orders of discretization are considered with and without a uniform mean flow three different lattice boltzmann models are then presented and studied with the von neumann analysis the theoretical dispersion relations of these models are obtained and the error terms of the model are identified and studied it is shown that the dispersion error in the lattice boltzmann models is only due to the space and time discretization and that the continuous discrete velocity boltzmann equation yield the same exact dispersion as the navier stokes equations finally dispersion and dissipation errors of the different kind of schemes are quantitatively compared it is found that the lattice boltzmann method is less dissipative than high order schemes and less dispersive than a second order scheme in space with a 3 step runge kutta scheme in time the number of floating point operations at a given error level associated with these two kinds of schemes are then compared

a computational study of three numerical methods for some advection diffusion problems three numerical methods have been used to solve two problems described by advection diffusion equations with specified initial and boundary conditions the methods used are the third order upwind scheme 5 fourth order upwind scheme 5 and non standard finite difference scheme nsfd 10 we considered two test problems the first test problem we considered has steep boundary layers near x 1 and this is challenging problem as many schemes are plagued by non physical oscillation near steep boundaries 16 many methods suffer from computational noise when modeling the second test problem we compute some errors namely l2 and l errors dissipation and dispersion errors total variation and the total mean square error for both problems we then use an optimization technique 1 to find the optimal value of the time step at a given value of the spatial step which minimizes the dispersion error and this is validated by numerical experiments

on richardson extrapolation for low dissipation low dispersion diagonally implicit runge kutta schemes abstract in the modeling of wave propagation phenomena it is necessary to use time integration methods which are not only sufficiently accurate but also properly describe the amplitude and phase of the propagating waves it is not clear if amending the developed schemes by extrapolation methods to obtain a high order of accuracy preserves the qualitative properties of these schemes in the perspective of dissipation dispersion and stability analysis it is illustrated that the combination of various optimized schemes with richardson extrapolation is not optimal for minimal dissipation and dispersion errors optimized third order and fourth order methods are obtained and it is shown that the proposed methods combined with richardson extrapolation result in fourth and fifth orders of accuracy correspondingly while preserving optimality and stability the numerical applications include the linear wave equation a stiff system of reaction diffusion equations and the nonlinear euler equations with oscillatory initial conditions it is demonstrated that the extrapolated third order scheme outperforms the recently developed fourth order diagonally implicit runge kutta scheme in terms of accuracy and stability

optimization of high order diagonally implicit runge kutta methods abstract this article presents constrained numerical optimization of high order linearly and algebraically stable diagonally implicit runge kutta methods after satisfying the desired order conditions undetermined coefficients are optimized with respect to objective functions which consider accuracy stability and computational cost constraints are applied during the optimization to enforce stability properties to ensure a well conditioned method and to limit the domain of the abscissa two promising third order methods are derived using this approach labelled sdirk 3 1 2 2 3 l 14 and sdirk 3 1 4 l sa 5 both optimized schemes have a good balance of properties the relative error norm of the latter the l 2 norm scaled by a function of the number of implicit stages is a factor of two smaller than comparable methods found in the literature variations on these methods are discussed relative to trade offs in their accuracy and stability properties a novel fifth order scheme sdirk 5 1 5 l 02 is derived with a significantly lower relative error norm than the comparable fifth order a stable reference method in addition the optimized scheme is l stable the accuracy and relative efficiency of the runge kutta methods are verified through numerical simulation of van der pol s equation as well as numerical simulation of vortex shedding in the laminar wake of a circular cylinder and in the turbulent wake of a naca 0012 airfoil these results demonstrate the value of numerical optimization for selecting undetermined coefficients in the construction of high order runge kutta methods with a balance between competing objectives

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

a controllable canonical form implementation of time domain impedance boundary conditions for broadband aeroacoustic computation a new method which can be effectively and efficiently applied in the simulations of broadband noise problems is proposed for time domain impedance boundary condition implementations by using the so called controllable canonical form that is well known in linear system usually the impedance boundary condition can be defined in frequency domain as a rational polynomial function with poles in the negative half of the complex plane to guarantee stability otherwise causality might be violated in the corresponding time domain implementation to address this issue various methodologies have been proposed previously that usually lead to complicated polynomials whose numerical implementations are often indirect and intricate the proposed method with a controllable canonical form on the other hand directly transforms the frequency domain transfer function a quotient of rational polynomials to an equivalent state space model which consists of a series of first order ordinary differential equations that can be numerically implemented in a straightforward way the proposed method is demonstrated by using two benchmark problems a two dimensional gaussian pulse propagating in a uniform flow with a lined wall and the test cases from the nasa langley grazing incidence tube experiments good agreements demonstrate the potential of the proposed computational method

spatial eigenanalysis of spectral hp continuous galerkin schemes and their stabilisation via dg mimicking spectral vanishing viscosity for high reynolds number flows abstract this study considers the spatial eigensolution analysis of spectral hp continuous galerkin cg schemes complementing a recent work by moura et al 2016 15 which addressed cg s temporal analysis while the latter assumes periodic boundary conditions the spatial approach presumes inflow outflow type conditions and therefore provides insights for a different class of problems the linear advection diffusion problem is considered for a wide range of peclet numbers allowing for viscous effects at different intensities the inviscid linear advection case receives particular attention owing to the manifestation of peculiar characteristics previously observed for discontinuous galerkin dg schemes in the limit of strong over upwinding these effects are discussed in detail due to their potential to negatively affect solution quality and numerical stability of under resolved simulations at high reynolds numbers the spectral vanishing viscosity svv technique is subsequently considered as a natural stabilization strategy in the context of linear advection an optimization procedure is employed to match svv diffusion levels to those of dg at appropriate polynomial orders the resulting cg svv discretisations are tested against under resolved computations of spatially developing vortex dominated flows and display excellent robustness at high reynolds numbers along with superior eddy resolving characteristics at higher polynomial orders this highlights the importance of appropriate stabilization techniques to improve the potential of spectral hp cg methods for high fidelity simulations of transitional and turbulent flows including implicit les under resolved dns approaches

on the effective accuracy of spectral like optimized finite difference schemes for computational aeroacoustics the present article aims at highlighting the strengths and weaknesses of the so called spectral like optimized explicit central finite difference schemes when the latter are used for numerically approximating spatial derivatives in aeroacoustics evolution problems with that view we first remind how differential operators can be approximated using explicit central finite difference schemes the possible spectral like optimization of the latter is then discussed the advantages and drawbacks of such an optimization being theoretically studied before they are numerically quantified for doing so two popular spectral like optimized schemes are assessed via a direct comparison against their standard counterparts such a comparative exercise being conducted for several academic test cases at the end general conclusions are drawn which allows us discussing the way spectral like optimized schemes shall be preferred or not to standard ones when it comes to simulate real life aeroacoustics problems

an effective explicit pressure gradient scheme implemented in the two level non staggered grids for incompressible navier stokes equations in this paper an improved two level method is presented for effectively solving the incompressible navier stokes equations this proposed method solves a smaller system of nonlinear navier stokes equations on the coarse mesh and needs to solve the oseen type linearized equations of motion only once on the fine mesh level within the proposed two level framework a prolongation operator which is required to linearize the convective terms at the fine mesh level using the convergent navier stokes solutions computed at the coarse mesh level is rigorously derived to increase the prediction accuracy this indispensable prolongation operator can properly communicate the flow velocities between the two mesh levels because it is locally analytic solution convergence can therefore be accelerated for the sake of numerical accuracy momentum equations are discretized by employing the general solution for the two dimensional convection diffusion reaction model equation the convective instability problem can be simultaneously eliminated thanks to the proper treatment of convective terms the converged solution is thus very high in accuracy as well as in yielding a quadratic spatial rate of convergence for the sake of programming simplicity and computational efficiency pressure gradient terms are rigorously discretized within the explicit framework in the non staggered grid system the proposed analytical prolongation operator for the mapping of solutions from the coarse to fine meshes and the explicit pressure gradient discretization scheme which accommodates the dispersion relation preserving property have been both rigorously justified from the predicted navier stokes solutions

semi implicit runge kutta schemes development and application to compressible channel flow time stepconstraintdueto thefine meshnear thewall the explicitsubschemeis asix stage fourth order low storage runge kutta scheme based on analysis in fourier space and results obtained for propagation test cases the semiimplicitschemesareshowntobeoforder3and forwavesdiscretizedbyanumberofpointsperperiodbetween4and 16 tobeasaccurateas ormoreaccuratethan thestandardexplicitfourth orderrunge kuttaalgorithmintermsof dissipationanddispersion thelarge eddysimulationofacompressibleturbulentchannelflowatafrictionreynolds numberof360andamachnumberof0 1isthencarriedoutwithoneoftheproposedalgorithms thecomputational time is reduced by a factor 1 33 with respect to a large eddy simulation using the explicit subscheme in all directions wall pressure and velocity spectra from the large eddy simulation are presented to give insights into the flow turbulent structures in particular wave number frequency spectra are calculated acoustic components appear to be identified in these spectra

frequency optimized rbf fd for wave equations abstract we present a method to obtain optimal rbf fd formulas which maximize their frequency range of validity the optimization is based on the idea of keeping an error of interest dispersion phase or group velocity errors below a given threshold for a wavenumber interval as large as possible to find the weights of these optimal finite difference formulas we solve an optimization problem in a previous work we developed a method to optimize the frequency range of validity for finite difference weights that method required to solve a system of nonlinear equations with as many unknowns as half of the number of weights which is a very hard task when the number of nodes gets large the current method requires solving an optimization problem with only one parameter which makes finding a global minimum easier and thus can be used for bigger stencils we also study which of the standard rbf are more appropriate for this problem and introduce a new rbf that depends on two parameters this new rbf improves the resulting frequency response of the rbf fd methods while keeping the cost of the optimization problem low

a family of high order finite difference schemes with good spectral resolution this paper presents a family of finite difference schemes for the first and second derivatives of smooth functions the schemes are hermitian and symmetric and may be considered a more general version of the standard compact pade schemes discussed by lele they are different from the standard pade schemes in that the first and second derivatives are evaluated simultaneously for the same stencil width the proposed schemes are two orders higher in accuracy and have significantly better spectral representation eigenvalue analysis and numerical solutions of the one dimensional advection equation are used to demonstrate the numerical stability of the schemes the computational cost of computing both derivatives is assessed and shown to be essentially the same as the standard pade schemes the proposed schemes appear to be attractive alternatives to the standard pade schemes for computations of the navier stokes equations

finite difference schemes for long time integration a general method for constructing finite difference schemes for longtime integration problems is presented it is demonstrated for discretizations of first and second space derivatives however the approach is not limited to these cases the schemes are constructed so as to minimize the global truncation error taking into account the initial data the resulting second order compact schemes can be used for integration times fourfold or more longer than previously studied schemes with similar computational complexity a similar approach was used to obtain improved integration schemes

aeroacoustics of t junction merging flow this paper reports a numerical study of the aeroacoustics of merging flow at t junction the primary focus is to elucidate the acoustic generation by the flow unsteadiness the study is conducted by performing direct aeroacoustic simulation approach which solves the unsteady compressible navier stokes equations and the perfect gas equation of state simultaneously using the conservation element and solution element method for practical flows the reynolds number based on duct width is usually quite high 105 in order to properly account for the effects of flow turbulence a large eddy simulation methodology together with a wall modeling derived from the classical logarithm wall law is adopted the numerical simulations are performed in two dimensions and the acoustic generation physics at different ratios of side branch to main duct flow velocities vr 0 5 0 67 1 0 2 0 are studied both the levels of unsteady interactions of merging flow structures and the efficiency of acoustic generation are obser

the extended fourier pseudospectral time domain method for atmospheric sound propagation an extended fourier pseudospectral time domain pstd method is presented to model atmospheric sound propagation by solving the linearized euler equations in this method evaluation of spatial derivatives is based on an eigenfunction expansion evaluation on a spatial grid requires only two spatial points per wavelength time iteration is done using a low storage optimized six stage runge kutta method this method is applied to two dimensional non moving media models one with screens and one for an urban canyon with generally high accuracy in both amplitude and phase for a moving atmosphere accurate results have been obtained in models with both a uniform and a logarithmic wind velocity profile over a rigid ground surface and in the presence of a screen the method has also been validated for three dimensional sound propagation over a screen for that application the developed method is in the order of 100 times faster than the second order accurate fdtd solution to the linearized euler equations the method is found to be well suited for atmospheric sound propagation simulations where effects of complex meteorology and straight rigid boundary surfaces are to be investigated

the dynamics of the 3d radial nls with the combined terms in this paper we show the scattering and blow up result of the radial solution with the energy below the threshold for the nonlinear schrodinger equation nls with the combined terms iu t delta u u 4 u u 2 u qquad qquad qquad qquad rm cnls in the energy space h 1 mathbb r 3 the threshold is given by the ground state w for the energy critical nls iu t u u 4 u this problem was proposed by tao visan and zhang in comm pdes 32 1281 1343 2007 the main difficulty is lack of the scaling invariance illuminated by ibrahim et al in analysis pde 4 3 405 460 2011 we need to give the new radial profile decomposition with the scaling parameter then apply it to the scattering theory our result shows that the defocusing dot h 1 subcritical perturbation u 2 u does not affect the determination of the threshold of the scattering solution of cnls in the energy space

finite difference approximations of first derivatives for three dimensional grid singularities explicit finite difference approximations of first derivatives are developed for two dimensional three way and five way grid singularities the schemes for three way singularities also apply for six way singularities the development objectives are such that the finite difference schemes are non dissipative and compatible with regular difference approximations for convection dominated phenomena special emphasis is given to the directional dependency of numerical phase speeds and stability limits for explicit time integration for that purpose free coefficients are used to minimize the isotropy error of the numerical phase velocity it is shown that numerical phase speeds and stability limits are almost isotropic hence employment of the proposed schemes is nearly independent of the local orientation of the differencing stencil the time step limits are greater than those of regular finite differences indicating that the overall time step of a computation is not reduced by employing the proposed schemes the developed schemes for three way singularities are of second fourth and sixth order of accuracy whereas the five way difference approximations are second and fourth order accurate respectively the spectral characteristics the formal order of accuracy and the stability limits of the proposed schemes imply that they can favorably be used in combination with regular difference approximations of the same order of accuracy on block structured grids

development of arbitrary order multioperators based schemes for parallel calculations 1 higher than fifth order approximations to convection terms further results concerning arbitrary order approximations to grid functionals via linear combinations of basis operators obtained by fixing sets of free parameters multioperators are presented a parallel algorithm for their calculations is described as basis operators a version of one parametric families of the fifth order compact upwind differencing operators cud as well as the fourth order non centered approximations to first derivatives are considered the resulting conservative schemes for fluid dynamics type of equations or other equations with convection terms are outlined the existence and uniqueness of the corresponding multioperators are discussed it is shown that for properly chosen parameters multioperators preserve the upwind downwind properties of the basis operators that is their positivity negativity in appropriate hilbert spaces of grid functions as examples the seventh and ninth order multioperators based schemes with very good dispersion and dissipation properties are described their possible optimization being discussed numerical examples illustrating their extremely high accuracy are presented

compact implicit maccormack type schemes with high accuracy in this work the maccormack methodology is extended to implicit compact differencing schemes a prefactorization method is developed which splits the implicit matrix into two independent upper and lower matrices which are easier to invert using this method a new class of high order accurate compact maccormack type schemes is derived two fourth order schemes are described and results are shown for three linear and nonlinear caa benchmark problems

high order low dissipation low dispersion diagonally implicit runge kutta schemes high order low dissipation low dispersion diagonally implicit runge kutta schemes are analyzed and introduced based on the optimization of amplification and phase errors for wave propagation various optimized schemes can be obtained the new scheme shows negligible dissipation it is illustrated mathematically and numerically that the new scheme preserves fourth order accuracy while the recently developed diagonally implicit runge kutta scheme does not the numerical applications contain the advection equation with and without a stiff nonlinear source term and an oscillatory test the new scheme is a stable as desired for the solution of stiff problems

a multi domain fourier pseudospectral time domain method for the linearized euler equations the fourier pseudospectral time domain f pstd method is computationally one of the most cost efficient methods for solving the linearized euler equations for wave propagation through a medium with smoothly varying spatial inhomogeneities in the presence of rigid boundaries as the method utilizes an equidistant discretization local fine scale effects of geometry or medium inhomogeneities require a refinement of the whole grid which significantly reduces the computational efficiency for this reason a multi domain f pstd methodology is presented with a coarse grid covering the complete domain and fine grids acting as a subgrid resolution of the coarse grid near local fine scale effects data transfer between coarse and fine grids takes place utilizing spectral interpolation with super gaussian window functions to impose spatial periodicity local time stepping is employed without intermediate interpolation the errors introduced by the window functions and the multi domain implementation are quantified and compared to errors related to the initial conditions and from the time iteration scheme it is concluded that the multi domain methodology does not introduce significant errors compared to the single domain method examples of scattering from small scale density scatters sound reflecting from a slitted rigid object and sound propagation through a jet are accurately modelled by the proposed methodology for problems that can be solved by f pstd the presented methodology can lead to a significant gain in computational efficiency

smoothing properties and retarded estimates for some dispersive evolution equations smoothing properties in the form of space time integrability properties play an important role in the study of dispersive evolution equations a number of them follow from a combination of general arguments and specific estimates we present a general formulation which makes the separation between the two types of ingredients as clear as possible and we illustrate it with the examples of the schrodinger equation of the wave equation and of a class of 1 1 dimensional equations related to the benjamin ono equation of special interest for the cauchy problem are retarded estimates expressed in terms of those properties we derive a number of such estimates associated with the last example and we mention briefly an application of those estimates to the cauchy problem for the generalized benjamin ono equation

continuous modules are clean ar ingr is said to be clean if every element of r is a sum of an idempotent and a unit the class of clean rings is quite large and includes for instance semiperfect rings and thus finite rings and rings of linear transformations of vector spaces we prove that the endomorphism ring of every continuous or discrete module is clean

uniformly best wavenumber approximations by spatial central difference operators we construct accurate central difference stencils for problems involving high frequency waves or multi frequency solutions over long time intervals with a relatively coarse spatial mesh and with an easily obtained bound on the dispersion error this is done by demonstrating that the problem of constructing central difference stencils that have minimal dispersion error in the infinity norm can be recast into a problem of approximating a continuous function from a finite dimensional subspace with a basis forming a chebyshev set in this new formulation characterising and numerically obtaining optimised schemes can be done using established theory

dynamics for the energy critical nonlinear schrodinger equation in high dimensions in t duyckaerts f merle dynamic of threshold solutions for energy critical nls preprint arxiv 0710 5915 math ap t duyckaerts and f merle studied the variational structure near the ground state solution w of the energy critical nls and classified the solutions with the threshold energy e w in dimensions d 3 4 5 under the radial assumption in this paper we extend the results to all dimensions d 6 the main issue in high dimensions is the non lipschitz continuity of the nonlinearity which we get around by making full use of the decay property of w

on the inadvisability of using single parameter impedance models for representing the acoustical properties of ground surfaces although semi empirical one parameter models are used widely for representing outdoor ground impedance they are not physically admissible even when corrected to satisfy a passivity condition in respect of surface impedance they do not satisfy the condition that the real part of complex density must be greater than zero comparison of predictions with frequency domain data for short range propagation have indicated that physically admissible models provide superior overall agreement a two parameter variable porosity model yields better agreement for many grassland surfaces and a two parameter version of the slit pore microstructural impedance model yields better agreement with data obtained over low flow resistivity surfaces such as forest floors and gravel impedance models and conditions for physical admissibility are summarised in addition to those examined previously the slit pore model is shown to be physically admissible after providing further examples of the better agreement with short range data that can be achieved using two parameter models it is shown that differences between frequency domain predictions at longer ranges using physically admissible models rather than one parameter models are significantly greater than those resulting from short range spatial variability and comparable with seasonal variability over grassland

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

a general strategy for the optimization of runge kutta schemes for wave propagation phenomena we analyze optimized explicit runge kutta schemes rk for computational aeroacoustics and wave propagation phenomena in general exploiting the analysis developed in s pirozzoli performance analysis and optimization of finite difference schemes for wave propagation problems j comput phys 222 2007 809 831 we rigorously evaluate the performance of several time integration schemes in terms of appropriate error and cost metrics and provide a general strategy to design runge kutta methods tailored for specific applications we present families of optimized second and third order runge kutta schemes with up to seven stages and describe their implementation in the framework of williamson s 2n storage formulation j h williamson low storage runge kutta schemes j comput phys 35 1980 48 56 numerical simulations of the 1d linear advection equation and of the 2d linearized euler equations are performed to demonstrate the validity of the theory and to quantify the improvement provided by optimized schemes

summation by parts operators with minimal dispersion error for coarse grid flow calculations we present a procedure for constructing summation by parts operators with minimal dispersion error both near and far from numerical interfaces examples of such operators are constructed and compar

finite differences for coarse azimuthal discretization and for reduction of effective resolution near origin of cylindrical flow equations in this paper the errors generated by the computation of derivatives in the azimuthal direction when flow equations are solved in cylindrical coordinates using finite differences are investigated they might be large for coarse discretizations even using high order schemes which led us to design explicit finite differences specially for 8 16 32 and 64 points per circle these schemes are shown to improve accuracy with respect to standard finite differences and to provide solutions for a two dimensional propagation problem similar to those obtained using fourier spectral methods in the direction a method is then presented to alleviate the time step limitation resulting from explicit time integration near cylindrical origin when finite differences are used it consists in calculating azimuthal derivatives at coarser resolutions than permitted by the grid in the same way as usually done using spectral methods in practice a series of doublings of the effective discretization in is implemented thus simulations can for instance be performed on a grid containing n 256 points with a time step 32 times larger with an accuracy comparable to that achieved in corresponding simulations involving fourier spectral methods in the direction

a general class of commutative filters for les in complex geometries a class of filters for large eddy simulations of turbulent inhomogeneous flows is presented a general set of rules for constructing discrete filters in complex geometry is given and examples of such filters are presented with these filters the commutation error between numerical differentiation and filtering can be made arbitrarily small allowing for derivation of a consistent set of equations for the large scale field the application of such filters for explicit filtering in large eddy simulations and the issue of boundary conditions for the filtered field are also discussed

high accuracy algorithms for computational aeroacoustics this paper presents an analysis of high bandwidth operators developed for use with an essentially nonoscillatory eno method the spatial operators of a sixth order eno code are modified to resolve waves with as few as 7 points per wavelength ppw by decreasing the formal order of the algorithm numerical and analytical solutions are compared for the model problems of plane wave propagation and sound generation by an oscillating sphere these problems involve linear propagation wave steepening and shock formation an analysis of the ppw required for sufficient accuracy shows that low order algorithms need an excessive number of grid points to produce acceptable solutions in contrast high order codes provide good predictions on relatively coarse grids the high bandwidth operators produce only modest improvements over the original sixth order operators for nonlinear problems in which wave steepening is significant however they clearly outperform the original operators for long distance linear propagation because the high bandwidth operators have the same stencil as the original sixth order operators these gains are achieved with no additional computational work

grid stabilization of high order one sided differencing i first order hyperbolic systems we construct stable maximal order boundary closures for high order central difference methods the stability is achieved by adding a small number of additional subcell nodes near the boundaries at experimentally determined locations we find that methods up through 8th order can be stabilized by the addition of a single node up through 16th order by the addition of two nodes and up through 22nd order with three extra nodes we also consider the application of the technique to dispersion relation preserving methods and we construct and test artificial dissipation operators

global well posedness and scattering for the defocusing energy critical nonlinear schrodinger equation in r1 4 we obtain global well posedness scattering uniform regularity and global l6t x spacetime bounds for energy space solutions to the defocusing energy critical nonlinear schrodinger equation in a 4 our arguments closely follow those of colliander keel et al though our derivation of the frequency localized interaction morawetz estimate is somewhat simpler as a consequence our method yields a better bound on the l6t x norm

dual automorphism invariant modules a module m is called an automorphism invariant module if every isomorphism between two essential submodules of m extends to an automor phism of m this paper introduces the notion of dual of such modules we call a module m to be a dual automorphism invariant module if whenever k1 and k2 are small submodules of m then any epimorphism m k1 m k2 with small kernel lifts to an endomorphism of m in this paper we give vari ous examples of dual automorphism invariant module and study its properties in particular we study abelian groups and prove that dual automorphism invariant abelian groups must be reduced it is shown that over a right perfect ring r a lifting right r module m is dual automorphism invariant if and only if m is quasi projective all our rings have identity element and modules are right unital a right r module m is called an automorphism invariant module if every isomorphism between two essential submodules of m extends to an automorphism of m equivalently m is an automorphism invariant module if for any automorphismofe m m m where e m is the injective hull of m see 6 and 10

time space domain dispersion relation based finite difference method with arbitrary even order accuracy for the 2d acoustic wave equation high order finite difference fd methods have been widely used for numerical solution of acoustic wave equations it has been reported that the modeling accuracy is of 2nd order when the conventional 2m th order space domain fd and the 2nd order time domain fd stencils are directly used to solve the acoustic wave equation under the same discretization the present version of time space domain dispersion relation based fd method can improve the accuracy from 2nd order to 2m th order along eight directions for the 2d acoustic wave equation to increase the accuracy further we propose a new fd stencil for 2d acoustic wave equation modeling this new time space domain dispersion relation based fd stencil can reach the same arbitrary even order accuracy along all directions and is more accurate and more stable than the conventional one for the same m dispersion analysis and modeling examples demonstrate its advantages

development of a wavenumber preserving scheme for solving maxwell s equations in curvilinear non staggered grids abstract in this paper a compact finite difference solver for solving the maxwell s equations in curvilinear coordinates is presented the scheme formulated in time domain can theoretically preserve zero divergence condition and scaled wavenumber characteristics in non staggered grids the inherent local conservation laws are also retained discretely all the time the space and time derivative terms are approximated to yield an equal fourth order spatial and temporal accuracy in irregular physical domain maxwell s equations are recast in terms of the contravariant and covariant field variables so that the developed dual preserving solver can be directly implemented in addition in curvilinear coordinates the four components in the metric tensor have been calculated under the guideline that the determinant of the transformation matrix is computed exactly through the computational exercises it is demonstrated that the newly proposed solver with a fairly small numerical scaled wavenumber error in curvilinear coordinates is computationally efficient for use to get the long time accurate maxwell s solutions in irregular physical domain

three time integration methods for incompressible flows with discontinuous galerkin boltzmann method abstract this paper presents three time integration methods for incompressible flows with finite element method in solving the lattice bgk boltzmann equation the space discretization is performed using nodal discontinuous galerkin method which employs unstructured meshes with triangular elements and high order approximation degrees the time discretization is performed using three different kinds of time integration methods namely direct decoupling and splitting from the storage cost temporal accuracy numerical stability and time consumption we systematically compare three time integration methods then benchmark fluid flow simulations are performed to highlight efficient time integration methods numerical results are in good agreement with others or exact solutions

development of a dispersively accurate conservative level set scheme for capturing interface in two phase flows a two step interface capturing scheme implemented within the framework of conservative level set method is developed in this study to simulate the gas water two phase fluid flow in addition to solving the pure advection equation which is used to advect the level set function for tracking interface both nonlinear and stabilized features are taken into account for the level set function so that a sharply varying interface can be stably predicted to preserve the conservative property the mapping given by x x u t is performed between two coordinates x and x for the advection diffusion equation in a flow field with velocity u to capture the interface the flux term capable of compressing the level set contours is also adopted in the construction of linear inviscid burgers equation which is indispensable in the level set method to resolve the physically sharp interface without incurring contact discontinuity oscillations a damping term which is nonlinear in terms of the level set function and can render an adequate artificial diffusion to stabilize the contact surface is added into the reinitialization step in the modified level set method for accurately predicting the level set function the advection scheme for solving the linear inviscid burgers equation in the advection step of the modified level set method is developed to accommodate the true dispersion relation the solution computed from the resulting two dimensional dispersion relation preserving advection scheme can minimize the phase error less artificial damping is needed to damp the oscillations in the vicinity of contact surface and the interface can be less numerically smeared for the sake of programming simplicity the incompressible two phase flow will be discretized in non staggered grids without incurring checkerboard oscillations by the developed explicit compact scheme for the approximation of pressure gradient terms for the verification of the proposed two dimensional dispersion relation preserving scheme and the non staggered incompressible flow solver three benchmark problems have been chosen in this study the proposed conservative level set method for capturing the interface in incompressible fluid flows is also verified by solving the dam break bubble rising in water droplet falling in water and rayleigh taylor instability problems good agreements with the referenced solutions are demonstrated for all the four investigated problems

optimised prefactored compact schemes for linear wave propagation phenomena a family of space and time optimised prefactored compact schemes are developed that minimise the computational cost for given levels of numerical error in wave propagation phenomena with special reference to aerodynamic sound this work extends the approach of pirozzoli 1 to the maccormack type prefactored compact high order schemes developed by hixon 2 in which their shorter pade stencil from the prefactorisation leads to a simpler enforcement of numerical boundary conditions an explicit low storage multi step runge kutta integration advances the states in time theoretical predictions for spatial and temporal error bounds are derived for the cost optimised schemes and compared against benchmark schemes of current use in computational aeroacoustic applications in terms of computational cost for a given relative numerical error value one and two dimensional test cases are presented to examine the effectiveness of the cost optimised schemes for practical flow computations an effectiveness up to about 50 higher than the standard schemes is verified for the linear one dimensional advection solver which is a popular baseline solver kernel for computational physics problems a substantial error reduction for a given cost is also obtained in the more complex case of a two dimensional acoustic pulse propagation provided the optimised schemes are made to operate close to their nominal design points

a family of low dispersive and low dissipative explicit schemes for flow and noise computations explicit numerical methods for spatial derivation filtering and time integration are proposed they are developed with the aim of computing flow and noise with high accuracy and fidelity all the methods are constructed in the same way by minimizing the dispersion and the dissipation errors in the wavenumber space up to k x 2 corresponding to four points per wavelength they are shown to be more accurate and also more efficient numerically than most of the standard explicit high order methods for uniform and slowly non uniform grids two problems involving long range sound propagation are resolved to illustrate their respective precisions remarks about their practical applications are then made especially about the connection with the boundary conditions finally their relevance for the simulation of turbulent flows is emphasized

prefactored small stencil compact schemes abstract a new type of prefactored compact schemes is presented that obtains high order accuracy while using a very small stencil size these schemes require fewer boundary stencils and offer simpler boundary condition implementation than the existing compact schemes results are shown for linear and nonlinear 1 d benchmark problems and a 2 d linear benchmark problem boundary stencils for both exterior and interior multiblock boundaries are given and boundary condition specification is described and demonstrated

adaptive filtering for the lattice boltzmann method in this study a new selective filtering technique is proposed for the lattice boltzmann method this technique is based on an adaptive implementation of the selective filter coefficient the proposed model makes the latter coefficient dependent on the shear stress in order to restrict the use of the spatial filtering technique in sheared stress region where numerical instabilities may occur different parameters are tested on 2d test cases sensitive to numerical stability and on a 3d decaying taylorgreen vortex the results are compared to the classical static filtering technique and to the use of a standard subgrid scale model and give significant improvements in particular for low order filter consistent with the lbm stencil