on the construction of suitable solutions to the navier stokes equations and questions regarding the definition of large eddy simulation abstract this paper discusses the notion of large eddy simulation les as a mathematical concept and proposes to use the concept of suitable weak solutions as a building block for the development of a mathematical theory of les various techniques for constructing suitable weak solutions to the three dimensional navier stokes equations are reviewed and many features of these mathematical constructions are shown to comply with well accepted heuristic characterizations of les

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

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

stable parareal in time method for first and second order hyperbolic systems the parareal in time algorithm allows one to perform parallel simulations of time dependent problems this algorithm has been implemented on many types of time dependent problems with some success recent contributions have allowed one to extend the domain of application of the parareal in time algorithm so as to handle long time simulations of hamiltonian systems this improvement has managed to avoid the fatally large lack of accuracy of the plain parareal in time algorithm which does not conserve invariant quantities a somewhat similar difficulty occurs for problems where the solution lacks regularity either initially or during the evolution as is the case for hyperbolic systems of conservation laws in this paper we identify the reasons for instabilities of the parareal in time algorithm and propose a simple way to cure them we use the new method to solve a linear wave equation and a nonlinear burgers equation the results illustrate the stability of this variant of the parareal in time algorithm

chebyshev legendre super spectral viscosity method for nonlinear conservation laws in this paper a super spectral viscosity method using the chebyshev differential operator of high order d s sqrt 1 x 2 opx s is developed for nonlinear conservation laws the boundary conditions are treated by a penalty method compared with the second order spectral viscosity method the super one is much weaker while still guaranteeing the convergence of the bounded solution of the chebyshev galerkin chebyshev collocation or legendre galerkin approximations to nonlinear conservation laws which is proved by compensated compactness arguments

spectral simulation of supersonic reactive flows we present in this paper numerical simulations of reactive flows interacting with shock waves we argue that spectral methods are suitable for these problems and review the recent developments in spectral methods that have made them a powerful numerical tool appropriate for long term integrations of complicated flows even in the presence of shock waves a spectral code is described in detail and the theory that leads to each of its components is explained results of interactions of hydrogen jets with shock waves are presented and analyzed and comparisons with eno finite difference schemes are carried out

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

a generalized discrepancy and quadrature error bound an error bound for multidimensional quadrature is derived that includes the koksma hlawka inequality as a special case this error bound takes the form of a product of two terms one term which depends only on the integrand is defined as a generalized variation the other term which depends only on the quadrature rule is defined as a generalized discrepancy the generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the l p star discrepancy and p that arises in the study of lattice rules

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

stationarity tests for spatial point processes using discrepancies summary for testing stationarity of a given spatial point pattern guan 2008 proposed a model free statistic based on the deviations between observed and expected counts of points in expanding regions within the sampling window this article extends his method to a general class of statistics by incorporating also such information when points are projected to the axes and by allowing different ways to construct regions in which the deviations are considered the limiting distributions of the new statistics can be expressed in terms of integrals of a brownian sheet and hence asymptotic critical values can be approximated a simulation study shows that the new tests are always more powerful than that of guan when applied to the longleaf pine data where guan s test gave an inconclusive answer the new tests indicate a clear rejection of the stationarity hypothesis

local error estimates for discontinuous solutions of nonlinear hyperbolic equations let u x t be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data suppose u varepsilon x t is the solution of an approximate viscosity regularization where varepsilon 0 is the small viscosity amplitude it is shown that by post processing the small viscosity approximation u varepsilon pointwise values of u and its derivatives with an error as close to e as desired can be recovered the analysis relies on the adjoint problem of the forward error equation which in this case amounts to a backward linear transport equation with discontinuous coefficients the novelty of our approach is to use a generalized e condition of the forward problem in order to deduce a w 1 infty energy estimate for the discontinuous backward transport equation this in turn leads to e uniform estimate on moments of the error u varepsilon u the approach presented does not follow the characteristics and therefore applies mutatis mutandis to o

the convergence rate of approximate solutions for nonlinear scalar conservation laws let v varepsilon x t varepsilon 0 be a family of approximate solutions for the nonlinear scalar conservation law u t f u x 0 with c 0 1 initial data assume that v varepsilon x t are textit lip stable in the sense that they satisfy oleinik s e entropy condition it is shown that if these approximate solutions are textit lip consistent i e if v varepsilon cdot 0 u cdot 0 textit lip x v t varepsilon f v varepsilon x textit lip x t mathcal o varepsilon then they converge to the entropy solution and the convergence rate estimate v varepsilon cdot t u cdot t textit lip x mathcal o varepsilon holds consequently the familiar l p type and new pointwise error estimates are derived these convergence rate results are demonstrated in the context of entropy satisfying finite difference and glimm s schemes

variational multiscale turbulence modelling in a high order spectral element method in the variational multiscale vms approach to large eddy simulation les the governing equations are projected onto an a priori scale partitioning of the solution space this gives an alternative framework for designing and analyzing turbulence models we describe the implementation of the vms les methodology in a high order spectral element method with a nodal basis and discuss the properties of the proposed scale partitioning the spectral element code is first validated by doing a direct numerical simulation of fully developed plane channel flow the performance of the turbulence model is then assessed by several coarse grid simulations of channel flow at different reynolds numbers

boundary observer design for hyperbolic pde ode cascade systems the paper presents an observer design for a class of hyperbolic pde ode cascade systems with a boundary measurement the cascade systems consist of coupled pdes featuring one rightward and one leftward convecting first order transport pdes and a set of odes which enter the pdes through the left boundary of the systems the design which is based on the volterra integral transformation relies only on a single sensor at the right boundary of the system the observer consists of a copy of the plant plus output injection terms both in the pdes and the odes the observer is constructed in a collocated setup which means both sensing and actuation are located at the same boundary the observer gains are computed analytically by solving goursat type pdes in terms of bessel function of the first kind the observer design is tested against a field scale flow loop test experiment in stavanger by statoil oil company the results show that the observer converges to the actual values and that the design can be used as a process monitoring tool in oil well drilling

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

a finite element technique for solving first order pdes in lp an approximation technique for solving first order pdes in l p omega 1 le p infty is proposed the method is a generalization of the least squares method to non hilbertian settings a priori and a posteriori error estimates are proven numerical tests in l 1 omega show that this type of technique can handle discontinuities without resorting to limiting procedures

filtering in legendre spectral methods we discuss the impact of modal filtering in legendre spectral methods both on accuracy and stability for the former we derive sufficient conditions on the filter to recover high order accuracy away from points of discontinuity computational results confirm that less strict necessary conditions appear to be adequate we proceed to discuss a instability mechanism in polynomial spectral methods and prove that filtering suffices to ensure stability the results are illustrated by computational experiments

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

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

stability of switched linear hyperbolic systems by lyapunov techniques switched linear hyperbolic partial differential equation sa re considered in this technical note they model infinite dimensional systems of conservation laws and balance laws which are potentially affecte db y a distributed source or sink term the dynamics and the boundary condi tions are subject to abrupt changes given by a switching signal modeled as a piecewise constant function and possibly a dwell time by means of lyapunov techniques some sufficient conditions are obtained for the expo nential stability of the switching system uniformly for all switching signals different cases are considered with or without a dwell time assumption on the switching signals and on the number of positive characteristic veloci ties which may also depend on the switching signal some numerical sim ulations are also given to illustrate some main results and to motivate this study

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

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

adapting the spectral vanishing viscosity method for large eddy simulations in cylindrical configurations during the last decade the spectral vanishing viscosity svv method has been adopted successfully for large eddy type simulations les with high order discretizations in both cartesian and cylindrical coordinate systems for the latter case however previous studies were confined to annular domains in the present work we examine the applicability of svv in cylindrical coordinates to flows in which the axis region is included within the setting of an exponentially convergent spectral element fourier discretization in addition to the standard svv viscosity kernel two modified kernels with enhanced stabilization in the axis region are considered three fluid flow examples are considered including turbulent pipe flow the results on the one hand show a surprisingly small influence of the svv kernel while on the other they reveal the importance of spatial resolution in the axis region

switching rules for stabilization of linear systems of conservation laws in this paper the exponential convergence in l 2 norm is analyzed for a class of switched linear systems of conservation laws the boundary conditions are subject to switches we investigate the problem of synthesizing stabilizing switching controllers by means of lyapunov techniques three control strategies are developed based on steepest descent selection possibly combined with a hysteresis and a low pass filter for the first strategy we show the global exponential stabilizability but no result for the existence and uniqueness of trajectories can be stated for the other ones the problem is shown to be well posed and global exponential convergence can be obtained moreover we consider the robustness issues for these switching rules in presence of measurement noise some numerical examples illustrate our approach and show the merits of the proposed strategies particularly we have developed a model for a network of open channels with switching controllers in the gate operations

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

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

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

backstepping boundary control for first order hyperbolic pdes and application to systems with actuator and sensor delays abstract we consider a problem of boundary feedback stabilization of first order hyperbolic partial differential equations pdes these equations serve as a model for physical phenomena such as traffic flows chemical reactors and heat exchangers we design controllers using a backstepping method which has been recently developed for parabolic pdes with the integral transformation and boundary feedback the unstable pde is converted into a delay line system which converges to zero in finite time we then apply this procedure to finite dimensional systems with actuator and sensor delays to recover a well known infinite dimensional controller analog of the smith predictor for unstable plants we also show that the proposed method can be used for the boundary control of a korteweg de vries like third order pde the designs are illustrated with simulations

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

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

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

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

the exponential accuracy of fourier and chebyshev differencing methods it is shown that when differencing analytic functions using the pseudospectral fourier or chebyshev methods the error committed decays to zero at an exponential rate

stability analysis of finite difference pseudospectral and fourier galerkin approximations for time dependent problems we consider finite difference pseudospectral and fourier galerkin methods for the approximate solution of time dependent problems the paper provides a unified framework for the stability analysis of all three discrete methods in particular the problem of stability for highly accurate stencils is studied in some detail

statistical properties of generalized discrepancies when testing that a sample of n points in the unit hypercube 0 1 d comes from a uniform distribution the kolmogorov smirnov and the cramer von mises statistics are simple and well known procedures to encompass these measures of uniformity hickernell introduced the so called generalized p discrepancies these discrepancies can be used in numerical integration through monte carlo and quasi monte carlo methods design of experiments uniformity testing and goodness of fit tests the aim of this paper is to derive the statistical asymptotic properties of these statistics under monte carlo sampling in particular we show that under the hypothesis of uniformity of the sample of points the asymptotic distribution is a complex stochastic integral with respect to a pinned brownian sheet on the other hand if the points are not uniformly distributed then the asymptotic distribution is gaussian

sparse low energy decomposition for viscous conservation laws for viscous conservation laws solutions contain smooth but high contrast features which require the use of fine grids to properly resolve on coarse grids these high contrast jumps resemble shocks rather than their true viscous profiles which could lead to issues in the numerical approximation of their underlying dynamics in many cases the equations of motion emit traveling wave solutions which can be used to represent the viscous profiles analytically the traveling wave solutions can be thought of as a lower dimensional representation of the motion since they contain information from the evolution equation but are constant along certain time space curves using a parameterized basis involving the traveling waves along with the sparse low energy decompositions found in imaging sciences we propose an approximation to viscous conservation laws which separates the coarse smooth component from the sharp fine one our method provides an appropriate approximation to the solution on a coarse grid thereby accurately under resolving the viscous profile this is similar to the philosophy of shock capturing methods in the sense that we want to capture the viscous front without needing to resolve the profile theoretical results on the consistency of our method are shown in general we provide several computational examples for convex and non convex fluxes

testing multivariate uniformity and its applications some new statistics are proposed to test the uniformity of random samples in the multidimensional unit cube 0 1 d d 2 these statistics are derived from number theoretic or quasi monte carlo methods for measuring the discrepancy of points in 0 1 d under the null hypothesis that the samples are independent and identically distributed with a uniform distribution in 0 1 d we captain some asymptotic properties of the new statistics by monte carlo simulation it is found that the finite sample distributions of the new statistics are well approximated by the standard normal distribution n 0 1 or the chi squared distribution x 2 2 a power study is performed and possible applications of the new statistics to testing general multivariate goodness of fit problems are discussed

a 1d exact treatment of shock waves within spectral methods in plane geometry abstract we present a very exact numerical technique for solving 1d euler equations coupled with the transport equations for the entropy and the chemical abundances with or without shock formation two moving grids are used before and after the shock formation quantities are expanded on both sides of the matching point in chebychev polynomials series after the shock is formed rankine hugoniot conditions are used to determine the velocity of the shock and the matching conditions across the shock typical results are presented

spectral viscosity approximations to hamilton jacobi solutions the spectral viscosity approximate solution of convex hamilton jacobi equations with periodic boundary conditions is studied it is proved in this paper that the approximation and its gradient remain uniformly bounded formally spectral accurate and converge to the unique viscosity solution the l1 convergence rate of the order 1 varepsilon forall varepsilon 0 is obtained

accuracy of high order and spectral methods for hyperbolic conservation laws with discontinuous solutions higher order and spectral methods have been used with success for elliptic and parabolic initial and boundary value problems with smooth solutions on the other hand higher order methods have been applied to hyperbolic problems with less success as higher order approx imations of discontinuous solutions suffer from the gibbs phenomenon we extend past work and show that spectral methods yield spectral convergence of moments even when applied to problems with discontinuous solutions besides spectral fourier methods for periodic domains we also prove high order convergence for adjoint consistent non periodic numerical methods exemplified by the discontinuous galerkin finite element method

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

control of homodirectional and general heterodirectional linear coupled hyperbolic pdes research on stabilization of coupled hyperbolic pdes has been dominated by the focus on pairs of counter convecting heterodirectional transport pdes with distributed local coupling and with controls at one or both boundaries a recent extension allows stabilization using only one control for a system containing an arbitrary number of coupled transport pdes that convect at different speeds against the direction of the pde whose boundary is actuated in this paper we present a solution to the fully general case in which the number of pdes in either direction is arbitrary and where actuation is applied on only one boundary to all the pdes that convect downstream from that boundary to solve this general problem we solve as a special case the problem of control of coupled homodirectional hyperbolic linear pdes where multiple transport pdes convect in the same direction with arbitrary local coupling our approach is based on pde backstepping and yields solutions to stabilization by both full state and observer based output feedback and trajectory tracking problems

an interior penalty stabilised incompressible discontinuous galerkin fourier solver for implicit large eddy simulations abstract we present an implicit large eddy simulation iles h p high order 2 unstructured discontinuous galerkin fourier solver with sliding meshes the solver extends the laminar version of ferrer and willden 2012 34 to enable the simulation of turbulent flows at moderately high reynolds numbers in the incompressible regime this solver allows accurate flow solutions of the laminar and turbulent 3d incompressible navier stokes equations on moving and static regions coupled through a high order sliding interface the spatial discretisation is provided by the symmetric interior penalty discontinuous galerkin ip dg method in the x y plane coupled with a purely spectral method that uses fourier series and allows efficient computation of spanwise periodic three dimensional flows since high order methods e g discontinuous galerkin and fourier are unable to provide enough numerical dissipation to enable under resolved high reynolds computations i e as necessary in the iles approach we adapt the laminar version of the solver to increase controllably the dissipation and enhance the stability in under resolved simulations the novel stabilisation relies on increasing the penalty parameter included in the dg interior penalty ip formulation the latter penalty term is included when discretising the linear viscous terms in the incompressible navier stokes equations these viscous penalty fluxes substitute the stabilising effect of non linear fluxes which has been the main trend in implicit les discontinuous galerkin approaches the ip dg penalty term provides energy dissipation which is controlled by the numerical jumps at element interfaces e g large in under resolved regions such as to stabilise under resolved high reynolds number flows this dissipative term has minimal impact in well resolved regions and its implicit treatment does not restrict the use of large time steps thus providing an efficient stabilization mechanism for iles the ip dg stabilisation is complemented with a spectral vanishing viscosity svv method in the z direction to enhance stability in the continuous fourier space the coupling between the numerical viscosity in the dg plane and the svv damping provides an efficient approach to stabilise high order methods at moderately high reynolds numbers we validate the formulation for three turbulent flow cases a circular cylinder at re 3900 a static and pitch oscillating naca 0012 airfoil at re 10000 and finally a rotating vertical axis turbine at re 40000 with reynolds based on the circular diameter airfoil chord and turbine diameter respectively all our results compare favourably with published direct numerical simulations large eddy simulations or experimental data we conclude that the dg fourier high order solver with ip svv stabilisation proves to be a valuable tool to predict turbulent flows and associated statistics for both static and rotating machinery

reduced order models for parameter dependent geometries based on shape sensitivity analysis the proper orthogonal decomposition pod is widely used to derive low dimensional models of large and complex systems one of the main drawback of this method however is that it is based on reference data when they are obtained for one single set of parameter values the resulting model can reproduce the reference dynamics very accurately but generally lack of robustness away from the reference state it is therefore crucial to enlarge the validity range of these models beyond the parameter values for which they were derived this paper presents two strategies based on shape sensitivity analysis to partially address this limitation of the pod for parameters that define the geometry of the problem at hand design or shape parameters we first detail the methodology to compute both the pod modes and their lagrangian sensitivities with respect to shape parameters from them we derive improved reduced order bases to approximate a class of solutions over a range of parameter values secondly we demonstrate the efficiency and limitations of these approaches on two typical flow problems 1 the one dimensional burgers equation 2 the two dimensional flows past a square cylinder over a range of incidence angles

entropy functions for symmetric systems of conservation laws abstract using a simple symmetrizability criterion we show that symmetric systems of conservation laws are equipped with a one parameter family of entropy functions

stability analysis of discontinuous dynamical systems determined by semigroups we present lyapunov stability results for discontinuous dynamical systems dds determined by linear and nonlinear semigroups defined on banach space dds of the type considered herein arise in the modeling of a variety of finite and infinite dimensional systems including certain classes of hybrid systems discrete event systems switched systems systems subjected to impulse effects and the like we apply our results in the analysis of several important specific classes of dds

optimal numerical solution of pdes using the variational germano identity the variational germano identity vgi is a condition that must be satisfied if the variational approximation to a partial differential equation is optimal in a user defined metric the vgi is useful as it can be utilized to dynamically determine unknown parameters in the variational approximation without any a priori knowledge of the exact solution with these values it yields solutions that are close to the optimal solution in this manuscript we examine the relation between the definition of an optimal solution and the parameters computed using the vgi we demonstrate that this relation is made precise by a so called truncated restriction operator that appears in the final expression for the model parameters and is sensitive to the criterion used for selecting the optimal solution in order to facilitate the use of the vgi for determining model parameters we derive explicit expressions for this operator when three distinct approaches are used for defining the optimal solution through numerical examples that make use of these expressions we confirm that the vgi does indeed yield model parameters and numerical solutions that are consistent with the choice of the optimal solution we also observe that the numerical solutions obtained using the vgi are close to the best numerical solution that can be obtained from a given variational approximation indicating that it yields nearly optimal values for the parameters

resonance phenomenon for the galerkin truncated burgers and euler equations it is shown that the solutions of inviscid hydrodynamical equations with suppression of all spatial fourier modes having wave numbers in excess of a threshold k mathrm g exhibit unexpected features the study is carried out for both the one dimensional burgers equation and the two dimensional incompressible euler equation for large k mathrm g and smooth initial conditions the first symptom of truncation a localized short wavelength oscillation which we call a tyger is caused by a resonant interaction between fluid particle motion and truncation waves generated by small scale features shocks layers with strong vorticity gradients etc these tygers appear when complex space singularities come within one galerkin wavelength ensuremath lambda mathrm g 2 ensuremath pi k mathrm g from the real domain and typically arise far away from preexisting small scale structures at locations whose velocities match that of such structures tygers are weak and strongly localized at first in the burgers case at the time of appearance of the first shock their amplitudes and widths are proportional to k mathrm g ensuremath 2 3 and k mathrm g ensuremath 1 3 respectively but grow and eventually invade the whole flow they are thus the first manifestations of the thermalization predicted by t d lee q j appl math 10 69 1952 the sudden dissipative anomaly the presence of a finite dissipation in the limit of vanishing viscosity after a finite time t ensuremath star which is well known for the burgers equation and sometimes conjectured for the three dimensional euler equation has as counterpart in the truncated case the ability of tygers to store a finite amount of energy in the limit k mathrm g ensuremath rightarrow ensuremath infty this leads to reynolds stresses acting on scales larger than the galerkin wavelength and thus prevents the flow from converging to the inviscid limit solution there are indications that it may eventually be possible to purge the tygers and thereby to recover the correct inviscid limit behavior

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

hyperbolic phase transitions in traffic flow this paper provides a mathematical model of the phenomenon of phase transitions in traffic flow the model consists of a scalar conservation law coupled with a 2 2 system of conservation laws the coupling is achieved via a free boundary where the phase transition takes place for this model the riemann problem is stated and globally solved the cauchy problem is proved to admit a solution defined globally in time without any assumption about the smallness of the initial data or the number of phase boundaries qualitative properties of real traffic flow are shown to agree with properties of the solutions of the model

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

the behavior of flux difference splitting schemes near slowly moving shock waves abstract an investigation of the behavior of shock capturing schemes which compute the numerical flux from a solution of riemann s problem is performed the schemes of godunov roe and osher are examined for a one dimensional model problem consisting of a nearly stationary shock both scalar and systems of equations are examined it is found that for slow shocks there is a significant error generated when solving systems of equations while the scalar results are well behaved this error consists of a long wavelength noise in the downstream running wave families that is not effectively damped by the dissipation of the scheme the source of this error is shown and the implications for the performance of these schemes are considered this error may contribute to the slow convergence to steady state reported by many researchers

total variation and error estimates for spectral viscosity approximations we study the behavior of spectral viscosity approximations to nonlinear scalar conservation laws we show how the spectral viscosity method compromises between the total variation bounded viscosity approximations which are restricted to first order accuracy and the spectrally accurate yet unstable fourier method in particular we prove that the spectral viscosity method is l sup 1 stable and hence total variation bounded moreover the spectral viscosity solutions are shown to be lip sup stable in agreement with oleinik s e entropy condition this essentially nonoscillatory behavior of the spectral viscosity method implies convergence to the exact entropy solution and we provide convergence rate estimates of both global and local types 16 refs

asymptotic properties of the spectral test diaphony and related quantities this paper presents the limit laws of discrepancies defined via exponential sums and algorithms with error bounds to approximate the corresponding distribution functions the results cover the weighted and the nonweighted spectral test of hellekalek and various instances of the general discrepancies of hiekernell and hoogland and kleiss for the exponential function system as well as classical quantities like the spectral test diaphony and the zaremba figure of merit

generalised summation by parts operators and variable coefficients abstract high order methods for conservation laws can be highly efficient if their stability is ensured a suitable means mimicking estimates of the continuous level is provided by summation by parts sbp operators and the weak enforcement of boundary conditions recently there has been an increasing interest in generalised sbp operators both in the finite difference and the discontinuous galerkin spectral element framework however if generalised sbp operators are used the treatment of the boundaries becomes more difficult since some properties of the continuous level are no longer mimicked discretely interpolating the product of two functions will in general result in a value different from the product of the interpolations thus desired properties such as conservation and stability are more difficult to obtain here new formulations are proposed allowing the creation of discretisations using general sbp operators that are both conservative and stable thus several shortcomings that might be attributed to generalised sbp operators are overcome cf nordstrom and ruggiu 2017 38 and manzanero et al 2017 39

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

gas flow in fan shaped networks classical solutions and feedback stabilization we analyze the subcritical gas flow through fan shaped networks of pipes that is through tree shaped networks with exactly one node where more than two pipes meet the gas flow in pipe networks is modeled by the isothermal euler equations a hyperbolic pde system of balance laws for this system we analyze stationary states and classical nonstationary solutions locally around a stationary state on a finite time interval furthermore we present a lyapunov function and boundary feedback laws to stabilize a fan shaped network around a given stationary state

an adaptive wavelet viscosity method for systems of hyperbolic conservation laws for the burgers equation as an example of a hyperbolic conservation law we have considered in our previous paper castano diez et al 2008 8 a weak formulation with a stabilization for handling discontinuities commonly called a viscosity approach numerically this was realized by locally introducing degrees of freedom around the discontinuities by means of an adaptive wavelet method in an a posteriori fashion in the present paper we apply this method to systems of conservation laws specifically euler s equations for gas dynamics moreover as the viscosity stabilization produces some gibbs phenomena we discuss different post processing techniques known from data and image processing together with a number of numerical comparisons

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

stabilization of high dimensional harmonic balance solvers using time spectral viscosity we investigate the application of a temporal spectral viscosity operator to eliminate aliasing errors associated with the high dimensional harmonic balance technique which is an efficient method for modeling nonlinear time periodic problems previous studies have shown that aliasing errors resulting from the discrete fourier transformation may slow down convergence trigger a nonlinear instability or lead to nonphysical solutions a temporal spectral viscosity operator similar to that used for pseudospectral methods is introduced the temporal spectral viscosity is added to the high frequency modes of the solution to eliminate aliasing errors so as to ensure the convergence to the physical solution the implementation of the technique is straightforward and can be incorporated into the high dimensional harmonic balance solver as a matrix product operator the accuracy and effectiveness of the modified method is demonstrated for different test cases including a duffing oscillator and unsteady flow abou

disturbance rejection in 2 x 2 linear hyperbolic systems many interesting problems in the oil and gas industry face the challenge of responding to disturbances from afar typically the disturbance occurs at the inlet of a pipeline or the bottom of an oil well while sensing and actuation equipment is installed at the outlet only kilometers away from the disturbance the present paper develops an output feedback control law for such cases based on modelling the transport phenomenon as a 2 times 2 linear partial differential equation pde of hyperbolic type and the disturbance as a finite dimensional linear system affecting the left boundary of the pde sensing and actuation are co located at the right boundary of the pde the design provides a separation principle allowing a disturbance attenuating full state feedback control law to be combined with an observer the results are applied to a relevant problem from the oil and gas industry and demonstrated in simulations

a finite element filtered eddy viscosity method for the navier stokes equations with large reynolds number fluid turbulence is commonly modeled by the navier stokes equations with a large reynolds number the simulation of turbulence model is known to be very difficult we study artificial spectral viscosity models that render the simulation of turbulence tractable the models introduce several parameters we show that the models have solutions that converge in certain parameter limits to solutions of the navier stokes equations we also show using the mathematical analysis how effective choice for the parameter can be made the direct computational simulation of turbulence flow is a formidable task due to the disparate scales that have to be resolved turbulence modeling attempts to mitigate this situation by accounting for the effects of small scale behavior on that at large scales without explicitly esolving the small scales one such approach is to add viscosity to the problem the smagorin sky and ladyzhenskaya models and other eddy viscosity models are examples of this approach unfortunately this approach usually results in over dampening at the large scales i e large scale structures are unphysically smeared out to mitigate this fault of eddy viscosity modeling filtered eddy viscosity methods that add the artificial viscosity only to the high frequency modes were developed in the context of spectral methods we apply the filtered eddy viscosity idea to finite element methods with hierarchical basis functions we prove the existence and uniqueness of the finite element approximation and its convergence to a weak solution of the navier stokes system

analysis of the spectral vanishing viscosity method for periodic conservation laws the convergence of the spectral vanishing method for both the spectral and pseudospectral discretizations of the inviscid burgers equation is analyzed it is proved that this kind of vanishing viscosity is responsible for spectral decay of those fourier coefficients located toward the end of the computed spectrum consequently the discretization error is shown to be spectrally small independently of whether or not the underlying solution is smooth this in turn implies that the numerical solution remains uniformly bounded and convergence follows by compensated compactness arguments

exponential stability of switched linear hyperbolic initial boundary value problems we consider the initial boundary value problem governed by systems of linear hyperbolic partial differential equations in the canonical diagonal form and study conditions for exponential stability when the system discontinuously switches between a finite set of modes the switching system is fairly general in that the system matrix functions as well as the boundary conditions may switch in time we show how the stability mechanism developed for classical solutions of hyperbolic initial boundary value problems can be generalized to the case in which weaker solutions become necessary due to arbitrary switching we also provide an explicit dwell time bound for guaranteeing exponential stability of the switching system when for each mode the system is exponentially stable our stability conditions only depend on the system parameters and boundary data these conditions easily generalize to switching systems in the nondiagonal form under a simple commutativity assumption we present tutorial examples to illustrate the instabilities that can result from switching

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

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

boundary observer based control for hyperbolic pde ode cascade systems with stochastic jumps abstract this work is concerned with the boundary observer based control for a category of cascade systems with stochastic jumps the cascade system consists of ordinary differential equations odes entering the hyperbolic partial differential equations pdes through the boundary a continuous time discrete state homogeneous markov process models the jumping parameters of the system and its right continuous trajectory candidate values are taken from a finite set a boundary observer is designed which relies only on limited quantity of sensors at the boundary of the pdes based on the lyapunov functional approach the observer error system is firstly proved to be exponentially mean square stable then the controller is presented by utilizing the estimated states of the odes as feedback variables the evolution profiles of the pdes in the closed loop are proved to be exponentially mean square stable at the desired steady state the existence conditions of the boundary observer and the stability conditions of the pdes in the closed loop form are presented via a set of compact space varying linear matrix inequalities svlmis the svlmis are then approximated by a finite number of lmis which guarantee the svlmis and the computational cost is reduced greatly finally the obtained theoretical results are applied to a traffic flow control system which confirms that the proposed estimation and control methods are effective

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

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

convergence results for a class of spectrally hyperviscous models of 3 d turbulent flow abstract we consider the spectrally hyperviscous navier stokes equations shnse which add hyperviscosity to the nse but only to the higher frequencies past a cutoff wavenumber m 0 in guermond and prudhomme 2003 18 subsequence convergence of shnse galerkin solutions to dissipative solutions of the nse was achieved in a specific spectral vanishing viscosity setting our goal is to obtain similar results in a more general setting and to obtain convergence to the stronger class of leray solutions in particular we obtain subsequence convergence of shnse strong solutions to leray solutions of the nse by fixing the hyperviscosity coefficient while the spectral hyperviscosity cutoff m 0 goes to infinity this formulation presents new technical challenges and we discuss how its motivation can be derived from computational experiments e g those in borue and orszag 1996 1998 3 4 we also obtain weak subsequence convergence to leray weak solutions under the general assumption that the hyperviscous coefficient goes to zero with no constraints imposed on the spectral cutoff in both of our main results the aubin compactness theorem provides the underlying framework for the convergence to leray solutions