uncertainty quantification methodology for hyperbolic systems with application to blood flow in arteries abstract we present a stochastic finite volume ader sfv ader methodology for uncertainty quantification uq in the general framework of systems of hyperbolic balance laws with uncertainty in parameters the resulting method is weakly intrusive meaning that deterministic solvers need minor modifications to include random contributions and has no theoretical accuracy barrier an illustration of the second order version for the viscous burgers equation with uncertain viscosity coefficient is first given in detail under different deterministic initial conditions the attainment of the theoretically expected convergence rate is demonstrated empirically results and features of our scheme are discussed we then extend the sfv ader method to a non linear hyperbolic system with source terms that models one dimensional blood flow in arteries assuming uncertainties in a set of parameters of the problem results are then compared with measurements in a well defined 1 1 experimental replica of the network of largest arteries in the human systemic circulation for a 99 confidence level severe influence of parameter fluctuations is seen in pressure profiles while secondary effects on flow rate profiles become visible particularly in terminal branches results suggest that the proposed methodology could successfully be applied to other physical problems involving hyperbolic balance laws

a stochastic projection method for fluid flow ii random process an uncertainty quantification scheme is developed for the simulation of stochastic thermofluid processes the scheme relies on spectral representation of uncertainty using the polynomial chaos pc system the solver combines a galerkin procedure for the determination of pc coefficients with a projection method for efficiently simulating the resulting system of coupled transport equations implementation of the numerical scheme is illustrated through simulations of natural convection in a 2d square cavity with stochastic temperature distribution at the cold wall the properties of the uncertainty representation scheme are analyzed and the predictions are contrasted with results obtained using a monte carlo approach

multi resolution analysis scheme for uncertainty quantification in chemical systems this paper presents a multi resolution approach for the propagation of parametric uncertainty in chemical systems it is motivated by previous studies where galerkin formulations of wiener hermite expansions were found to fail in the presence of steep dependences of the species concentrations with regard to the reaction rates the multi resolution scheme is based on representation of the uncertain concentration in terms of compact polynomial multi wavelets allowing for the control of the convergence in terms of polynomial order and resolution level the resulting representation is shown to greatly improve the robustness of the galerkin procedure in presence of steep dependences however this improvement comes with a higher computational cost which drastically increases with the number of uncertain reaction rates to overcome this drawback an adaptive strategy is proposed to control locally in the parameter space and in time the resolution level the efficiency of the method is demonstrated for an uncertain chemical system having eight random parameters

roe solver with entropy corrector for uncertain hyperbolic systems this paper deals with intrusive galerkin projection methods with a roe type solver for treating uncertain hyperbolic systems using a finite volume discretization in physical space and a piecewise continuous representation at the stochastic level the aim of this paper is to design a cost effective adaptation of the deterministic dubois and mehlman corrector to avoid entropy violating shocks in the presence of sonic points the adaptation relies on an estimate of the eigenvalues and eigenvectors of the galerkin jacobian matrix of the deterministic system of the stochastic modes of the solution and on a correspondence between these approximate eigenvalues and eigenvectors for the intermediate states considered at the interface we derive some indicators that can be used to decide where a correction is needed thereby reducing the computational costs considerably the effectiveness of the proposed corrector is assessed on the burgers and euler equations including sonic points

robust and efficient discontinuous galerkin methods for under resolved turbulent incompressible flows abstract we present a robust and accurate discretization approach for incompressible turbulent flows based on high order discontinuous galerkin methods the dg discretization of the incompressible navier stokes equations uses the local lax friedrichs flux for the convective term the symmetric interior penalty method for the viscous term and central fluxes for the velocity pressure coupling terms stability of the discretization approach for under resolved turbulent flow problems is realized by a purely numerical stabilization approach consistent penalty terms that enforce the incompressibility constraint as well as inter element continuity of the velocity field in a weak sense render the numerical method a robust discretization scheme in the under resolved regime the penalty parameters are derived by means of dimensional analysis using penalty factors of order 1 applying these penalty terms in a postprocessing step leads to an efficient solution algorithm for turbulent flows the proposed approach is applicable independently of the solution strategy used to solve the incompressible navier stokes equations i e it can be used for both projection type solution methods as well as monolithic solution approaches since our approach is based on consistent penalty terms it is by definition generic and provides optimal rates of convergence when applied to laminar flow problems robustness and accuracy are verified for the orr sommerfeld stability problem the taylor green vortex problem and turbulent channel flow moreover the accuracy of high order discretizations as compared to low order discretizations is investigated for these flow problems a comparison to state of the art computational approaches for large eddy simulation indicates that the proposed methods are highly attractive components for turbulent flow solvers

stochastic galerkin framework with locally reduced bases for nonlinear two phase transport in heterogeneous formations abstract the generalized polynomial chaos method with multiwavelet basis functions is applied to the buckley leverett equation we consider a spatially homogeneous domain modeled as a random field the problem is projected onto stochastic basis functions which yields an extended system of partial differential equations analysis and numerical methods leading to reduced computational cost are presented for the extended system of equations the accurate representation of the evolution of a discontinuous stochastic solution over time requires a large number of stochastic basis functions adaptivity of the stochastic basis to reduce computational cost is challenging in the stochastic galerkin setting since the change of basis affects the system matrix itself to achieve adaptivity without adding overhead by rewriting the entire system of equations for every grid cell we devise a basis reduction method that distinguishes between locally significant and insignificant modes without changing the actual system matrices results are presented for problems in one and two spatial dimensions with varying number of stochastic dimensions we show how to obtain stochastic velocity fields from realistic permeability fields and demonstrate the performance of the stochastic galerkin method with local basis reduction the system of conservation laws is discretized with a finite volume method and we demonstrate numerical convergence to the reference solution obtained through monte carlo sampling

attractors for the semilinear reaction diffusion equation with distribution derivatives we investigate the asymptotic behavior of solutions to a nonlinear reaction diffusion equation with distribution derivatives in the inhomogeneous term because the solutions of this equation are not very regular i e u only belongs to lp rn h1 rn and ut is only in h 1 rn for the forcing term in h 1 rn the standard method does not directly work in our case we demonstrate the asymptotic regularity of the solution to obtain the l2 rn h1 rn asymptotic compactness of the semigroup and therefore the existence of a l2 rn h1 rn global attractor in particular cases our results enable us to improve on some previously known results

regularity analysis of metamaterial maxwell s equations with random coefficients and initial conditions abstract in this paper we develop and analyze the stochastic collocation method for solving the time dependent metamaterial maxwell s equations subject to random coefficients and random initial conditions we provide a rigorous regularity analysis of the solution with respect to the random variables to our best knowledge this is the first theoretical results derived for the stochastic metamaterial maxwell equations the rate of convergence is proved depending on the regularity of the solution numerical results are presented to confirm the theoretical analysis we also demonstrate that the backward wave propagation phenomenon still exists in random metamaterial

filtered stochastic galerkin methods for hyperbolic equations abstract uncertainty quantification for nonlinear hyperbolic problems becomes a challenging task in the vicinity of shocks standard intrusive methods such as stochastic galerkin sg lead to oscillatory solutions and can result in non hyperbolic moment systems the intrusive polynomial moment ipm method guarantees hyperbolicity but comes at higher numerical costs in this paper we filter the generalized polynomial chaos gpc coefficients of the sg approximation which allows a numerically cheap reduction of oscillations the derived filter is based on lasso regression which sets small gpc coefficients of high order to zero we adaptively and automatically choose the filter strength to obtain a zero valued highest order moment the filtered sg method is tested for burgers and the euler equations results show a reduction of oscillations at shocks which leads to an improved approximation of expectation values and the variance compared to sg and ipm

the attractor for a nonlinear reaction diffusion system in an unbounded domain in this paper the quasi linear second order parabolic systems of reaction diffusion type in an unbounded domain are considered our aim is to study the long time behavior of parabolic systems for which the nonlinearity depends explicitly on the gradient of the unknown functions to this end we give a systematic study of given parabolic systems and their attractors in weighted sobolev spaces dependence of the hausdorff dimension of attractors on the weight of the sobolev spaces is considered 2001 john wiley sons inc

multi element generalized polynomial chaos for arbitrary probability measures we develop a multi element generalized polynomial chaos me gpc method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs given a stochastic input with an arbitrary probability measure its random space is decomposed into smaller elements subsequently in each element a new random variable with respect to a conditional probability density function pdf is defined and a set of orthogonal polynomials in terms of this random variable is constructed numerically then the generalized polynomial chaos gpc method is implemented element by element numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost efficiency and convergence of me gpc are studied numerically by considering some commonly used random variables me gpc provides an efficient and flexible approach to solving differential equations with random inputs especially for problems related to long term integration large perturbation and stochastic discontinuities

a stochastic galerkin method for first order quasilinear hyperbolic systems with uncertainty this paper is concerned with generalized polynomial chaos gpc approximation for first order quasilinear hyperbolic systems with uncertainty the one dimensional 1d hyperbolic system is first symmetrized with the aid of left eigenvector matrix of the jacobian matrix then the gpc stochastic galerkin method is applied to derive a provably symmetrically hyperbolic equations for the gpc expansion coefficients the resulting deterministic gpc galerkin system is discretized by a path conservative finite volume weno scheme in space and a third order total variation diminishing runge kutta method in time the method is further extended to two dimensional 2d quasilinear hyperbolic system with uncertainty where the symmetric hyperbolicity of the one dimensional gpc galerkin system is carried over via an operator splitting technique several numerical experiments are conducted to demonstrate the accuracy and effectiveness of the proposed gpc stochastic galerkin method

interior penalty variational multiscale method for the incompressible navier stokes equation monitoring artificial dissipation in this paper we apply the interior penalty method analyzed in e burman and m a fernndez numer math 107 2007 no 1 39 77 mr2317827 2008c 65317 to flows at high reynolds number as a possible measure of solution quality we propose to monitor the ratio between the artificial dissipation induced by the numerical method and the computed physical dissipation for smooth flows we prove that for our method the artificial dissipation serves as an a posteriori error estimator and also that it vanishes at an optimal rate in the case of flows with multiscale features we discuss a heuristic approach relating the decay of the artificial dissipation to the decay rate of the power spectrum some numerical results in two space dimensions are presented examining the relation between the numerical dissipation and solution quality

simplified csp analysis of a stiff stochastic ode system abstract we develop a simplified computational singular perturbation csp analysis of a stochastic dynamical system we focus on the case of parametric uncertainty and rely on polynomial chaos pc representations to quantify its impact we restrict our attention to a system that exhibits distinct timescales and that tends to a deterministic steady state irrespective of the random inputs a detailed analysis of eigenvalues and eigenvectors of the stochastic system jacobian is conducted which provides a relationship between the pc representation of the stochastic jacobian and the jacobian of the galerkin form of the stochastic system the analysis is then used to guide the application of a simplified csp formalism that is based on relating the slow and fast manifolds of the uncertain system to those of a nominal deterministic system two approaches are specifically developed with the resulting simplified csp framework the first uses the stochastic eigenvectors of the uncertain system as csp vectors whereas the second uses the eigenvectors of the nominal system as csp vectors numerical experiments are conducted to demonstrate the results of the stochastic eigenvalue and eigenvector analysis and illustrate the effectiveness of the simplified csp algorithms in addressing the stiffness of the system dynamics

simplex stochastic collocation with eno type stencil selection for robust uncertainty quantification multi element uncertainty quantification approaches can robustly resolve the high sensitivities caused by discontinuities in parametric space by reducing the polynomial degree locally to a piecewise linear approximation it is important to extend the higher degree interpolation in the smooth regions up to a thin layer of linear elements that contain the discontinuity to maintain a highly accurate solution this is achieved here by introducing essentially non oscillatory eno type stencil selection into the simplex stochastic collocation ssc method for each simplex in the discretization of the parametric space the stencil with the highest polynomial degree is selected from the set of candidate stencils to construct the local response surface approximation the application of the resulting ssc eno method to a discontinuous test function shows a sharper resolution of the jumps and a higher order approximation of the percentiles near the singularity ssc eno is also applied to a chemical model problem and a shock tube problem to study the impact of uncertainty both on the formation of discontinuities in time and on the location of discontinuities in space

uncertainty propagation using wiener haar expansions an uncertainty quantification scheme is constructed based on generalized polynomial chaos pc representations two such representations are considered based on the orthogonal projection of uncertain data and solution variables using either a haar or a legendre basis governing equations for the unknown coefficients in the resulting representations are derived using a galerkin procedure and then integrated in order to determine the behavior of the stochastic process the schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of rayleigh benard instability for situations involving random parameters close to a critical point the computational implementations show that the wiener haar wha representation provides more robust predictions that those based on a wiener legendre wle decomposition however when the solution depends smoothly on the random data the wle scheme exhibits superior convergence suggestions regarding future extensions are finally drawn based on these experiences

attractors for nonlinear reaction diffusion systems in unbounded domains via the method of short trajectories article history we consider a nonlinear reaction diffusion equation on the whole space r d we prove the well posedness of the corresponding cauchy problem in a general functional setting namely when the initial datum is uniformly locally bounded in l 2 only then we adapt the short trajectory method to establish the existence of the global attractor and if d 3 we find an upper bound of its kolmogorov s e entropy

multi resolution analysis of wiener type uncertainty propagation schemes a multi resolution analysis mra is applied to an uncertainty propagation scheme based on a generalized polynomial chaos pc representation the mra relies on an orthogonal projection of uncertain data and solution variables onto a multi wavelet basis consisting of compact piecewise smooth polynomial functions the coefficients of the expansion are computed through a galerkin procedure the mra scheme is applied to the simulation of the lorenz system having a single random parameter the convergence of the solution with respect to the resolution level and expansion order is investigated in particular results are compared to two monte carlo sampling strategies demonstrating the superiority of the mra for more complex problems however the mra approach may require excessive cpu times adaptive methods are consequently developed in order to overcome this drawback two approaches are explored the first is based on adaptive refinement of the multi wavelet basis while the second is based on adaptive block partitioning of the space of random variables computational tests indicate that the latter approach is better suited for large problems leading to a more efficient flexible and parallelizable scheme

intrusive galerkin methods with upwinding for uncertain nonlinear hyperbolic systems this paper deals with stochastic spectral methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws we consider problems with parametric uncertainty in initial conditions and model coefficients whose solutions exhibit discontinuities in the spatial as well as in the stochastic variables the stochastic spectral method relies on multi resolution schemes where the stochastic domain is discretized using tensor product stochastic elements supporting local polynomial bases a galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution hyperbolicity of the resulting galerkin system is analyzed a finite volume scheme with a roe type solver is used for discretization of the spatial and time variables an original technique is introduced for the fast evaluation of approximate upwind matrices which is particularly well adapted to local polynomial bases efficiency and robustness of the overall method are assessed on the burgers and euler equations with shocks

a fast and stable well balanced scheme with hydrostatic reconstruction for shallow water flows we consider the saint venant system for shallow water flows with nonflat bottom it is a hyperbolic system of conservation laws that approximately describes various geophysical flows such as rivers coastal areas and oceans when completed with a coriolis term or granular flows when completed with friction numerical approximate solutions to this system may be generated using conservative finite volume methods which are known to properly handle shocks and contact discontinuities however in general these schemes are known to be quite inaccurate for near steady states as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions this difficulty can be overcome by using the so called well balanced schemes we describe a general strategy based on a local hydrostatic reconstruction that allows us to derive a well balanced scheme from any given numerical flux for the homogeneous problem whenever the initial solver satisfies some classical stability properties it yields a simple and fast well balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality

existence of a global attractor for the parabolic equation with nonlinear laplacian principal part in an unbounded domain abstract in this paper we study the long time behavior of solutions for the parabolic equation with non linear laplacian principal part in r n we prove the existence of a global l 2 r n l r n attractor when n p and the existence of a global l 2 r n l n p n p r n attractor when n p

stochastic discrete equation method sdem for two phase flows a new scheme for the numerical approximation of a five equation model taking into account uncertainty quantification uq is presented in particular the discrete equation method dem for the discretization of the five equation model is modified for including a formulation based on the adaptive semi intrusive asi scheme thus yielding a new intrusive scheme sdem for simulating stochastic two phase flows some reference test cases are performed in order to demonstrate the convergence properties and the efficiency of the overall scheme the propagation of initial conditions uncertainties is evaluated in terms of mean and variance of several thermodynamic properties of the two phases

existence of a global attractor for a p laplacian equation in rn abstract in this paper we prove the existence of the l 2 r n w 1 p r n l q r n global attractor for the p laplacian equation u t div u p 2 u u p 2 u f u g with a more general nonlinear term f f 1 a x f 2 in r n where a l 1 r n l r n and f 1 f 2 satisfy the arbitrary q order polynomial growth condition without any restriction on q p q 2 p 2

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

multilevel monte carlo finite volume methods for shallow water equations with uncertain topography in multi dimensions the initial data and bottom topography used as inputs in shallow water models are prone to uncertainty due to measurement errors we model this uncertainty statistically in terms of random shallow water equations we extend the multilevel monte carlo mlmc algorithm to numerically approximate the random shallow water equations efficiently the mlmc algorithm is suitably modified to deal with uncertain and possibly uncorrelated data on each node of the underlying topography grid by the use of a hierarchical topography representation numerical experiments in one and two space dimensions are presented to demonstrate the efficiency of the mlmc algorithm

on the divergence constraint in mixed finite element methods for incompressible flows the divergence constraint of the incompressible navier stokes equations is revisited in the mixed finite element framework while many stable and convergent mixed elements have been developed throughout the past four decades most classical methods relax the divergence constraint and only enforce the condition discretely as a result these methods introduce a pressure dependent consistency error which can potentially pollute the computed velocity these methods are not robust in the sense that a contribution from the right hand side which influences only the pressure in the continuous equations impacts both velocity and pressure in the discrete equations this article reviews the theory and practical implications of relaxing the divergence constraint several approaches for improving the discrete mass balance or even for computing divergence free solutions will be discussed grad div stabilization higher order mixed methods derived on the basis of an exact de rham complex h div conforming finite

uncertainty quantification in kinematic wave models we develop a probabilistic approach to quantify parametric uncertainty in first order hyperbolic conservation laws kinematic wave equations the approach relies on the derivation of a deterministic equation for the cumulative density function cdf of a system state in which probabilistic descriptions probability density functions or pdfs of system parameters and or initial and boundary conditions serve as inputs in contrast to pdf equations which are often used in other contexts cdf equations allow for straightforward and unambiguous determination of boundary conditions with respect to sample variables the accuracy and robustness of solutions of the cdf equation for one such system the saint venant equations of river flows are investigated via comparison with monte carlo simulations

attractors for reaction diffusion equations in unbounded domains abstract in this paper we study the asymptotic behaviour of solutions for parabolic non linear evolution equations in r n we prove the asymptotic compactness of the solutions and then establish the existence of the global attractor in l 2 r n

hyperbolicity preserving and well balanced stochastic galerkin method for shallow water equations a stochastic galerkin formulation for a stochastic system of balanced or conservation laws may fail to preserve the hyperbolicity of the original system in this work we develop a hyperbolicity pr

model order reduction for parametrized nonlinear hyperbolic problems as an application to uncertainty quantification abstract in this work we present a model order reduction mor technique for hyperbolic conservation laws with applications in uncertainty quantification uq the problem consists of a parametrized time dependent hyperbolic system of equations where the parameters affect the initial conditions and the fluxes in a non linear way the procedure utilized to reduce the order is a combination of a greedy algorithm in the parameter space a proper orthogonal decomposition pod in time and empirical interpolation method eim to deal with non linearities drohmann 2012 we provide under some hypothesis an error bound for the reduced solution with respect to the high order one the algorithm shows small errors and savings of the computational time up to 90 in the uq simulations which are performed to validate the algorithm

large time behavior via the method of l trajectories abstract the method of l trajectories is presented in a general setting as an alternative approach to the study of the large time behavior of nonlinear evolutionary systems it can be successfully applied to the problems where solutions suffer from lack of regularity or when the leading elliptic operator is nonlinear here we concentrate on systems of a parabolic type and apply the method to an abstract nonlinear dissipative equation of the first order and to a class of equations pertinent to nonlinear fluid mechanics in both cases we prove the existence of a finite dimensional global attractor and the existence of an exponential attractor

uncertainty quantification in chemical systems summary we demonstrate the use of multiwavelet spectral polynomial chaos techniques for uncertainty quantification in non isothermal ignition of a methane air system we employ bayesian inference for identifying the probabilistic representation of the uncertain parameters and propagate this uncertainty through the ignition process we analyze the time evolution of moments and probability density functions of the solution we also examine the role and significance of dependence among the uncertain parameters we finish with a discussion of the role of non linearity and the performance of the algorithm copyright q 2009 john wiley sons ltd

a stochastic collocation method for the second order wave equation with a discontinuous random speed in this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions the speed is piecewise smooth in the physical space and depends on a finite number of random variables the numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space this approach leads to the solution of uncoupled deterministic problems as in the monte carlo method we consider both full and sparse tensor product spaces of orthogonal polynomials we provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data in particular we show that unlike in elliptic and parabolic problems the solution to hyperbolic problems is not in general analytic with respect to the random variables therefore the rate of convergence may only be algebraic an exponential fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data we present numerical examples which confirm the analysis and show that the collocation method is a valid alternative to the more traditional monte carlo method for this class of problems

attractors of reaction diffusion systems in unbounded domains and their spatial complexity the nonlinear reaction diffusion system in an unbounded domain is studied it is proven that under some natural assumptions on the nonlinear term and on the diffusion matrix this system possesses a global attractor in the corresponding phase space since the dimension of the attractor happens to be infinite we study its kolmogorov s entropy upper and lower bounds of this entropy are obtained moreover we give a more detailed study of the attractor for the spatially homogeneous rde in n in this case a group of spatial shifts acts on the attractor in order to study the spatial complexity of the attractor we interpret this group as a dynamical system with multidimensional time if n 1 acting on a phase space it is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy in order to clarify the nature of this chaos we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic and even lipschitz continuous embedding of this system into the spatial shifts on the attractor finally we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution 2003 wiley periodicals inc

solution of stochastic partial differential equations using galerkin finite element techniques this paper presents a framework for the construction of galerkin approximations of elliptic boundary value problems with stochastic input data a variational formulation is developed which allows among others numerical treatment by the finite element method a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized the paper develops a foundation for treating stochastic partial differential equations pdes which can be further developed in many directions

on the existence of the compact global attractor for semilinear reaction diffusion systems on rn we show that a class of reaction diffusion systems on rngenerates an asymptotically compact semiflow on the banach space of bounded uniformly continuous functions if such a semiflow is dissipative then a unique non empty compact minimal attractor is known to exist we apply this abstract result to obtain the existence of the compact minimal attractor for reaction diffusion systems on rnthat contain appropriate weight functions we also state conditions which guarantee that the attractor has finite hausdorff dimension

uncertainty quantification for systems of conservation laws uncertainty quantification through stochastic spectral methods has been recently applied to several kinds of non linear stochastic pdes in this paper we introduce a formalism based on kinetic theory to tackle uncertain hyperbolic systems of conservation laws with polynomial chaos pc methods the idea is to introduce a new variable the entropic variable in bijection with our vector of unknowns which we develop on the polynomial basis by performing a galerkin projection we obtain a deterministic system of conservation laws we state several properties of this deterministic system in the case of a general uncertain system of conservation laws we then apply the method to the case of the inviscid burgers equation with random initial conditions and we present some preliminary results for the euler system we systematically compare results from our new approach to results from the stochastic galerkin method in the vicinity of discontinuities the new method bounds the oscillations due to gibbs phenomenon to a certain range through the entropy of the system without the use of any adaptative random space discretizations it is found to be more precise than the stochastic galerkin method for smooth cases but above all for discontinuous cases

the multi element probabilistic collocation method me pcm error analysis and applications stochastic spectral methods are numerical techniques for approximating solutions to partial differential equations with random parameters in this work we present and examine the multi element probabilistic collocation method me pcm which is a generalized form of the probabilistic collocation method in the me pcm the parametric space is discretized and a collocation cubature grid is prescribed on each element both full and sparse tensor product grids based on gauss and clenshaw curtis quadrature rules are considered we prove analytically and observe in numerical tests that as the parameter space mesh is refined the convergence rate of the solution depends on the quadrature rule of each element only through its degree of exactness in addition the l 2 error of the tensor product interpolant is examined and an adaptivity algorithm is provided numerical examples demonstrating adaptive me pcm are shown including low regularity problems and long time integration we test the me pcm on two dimensional navier stokes examples and a stochastic diffusion problem with various random input distributions and up to 50 dimensions while the convergence rate of me pcm deteriorates in 50 dimensions the error in the mean and variance is two orders of magnitude lower than the error obtained with the monte carlo method using only a small number of samples e g 100 the computational cost of me pcm is found to be favorable when compared to the cost of other methods including stochastic galerkin monte carlo and quasi random sequence methods

adaptive stochastic galerkin fem abstract a framework for residual based a posteriori error estimation and adaptive mesh refinement and polynomial chaos expansion for general second order linear elliptic pdes with random coefficients is presented a parametric deterministic elliptic boundary value problem on an infinite dimensional parameter space is discretized by means of a galerkin projection onto finite generalized polynomial chaos gpc expansions and by discretizing each gpc coefficient by a fem in the physical domain an anisotropic residual based a posteriori error estimator is developed it contains bounds for both contributions to the overall error the error due to gpc discretization and the error due to finite element discretization of the gpc coefficients in the expansion the reliability of the residual estimator is established based on the explicit form of the residual estimator an adaptive refinement strategy is presented which allows to steer the polynomial degree adaptation and the dimension adaptation in the stochastic galerkin discretization and embedded in the gpc adaptation loop also the finite element mesh refinement of the gpc coefficients in the physical domain asynchronous mesh adaptation for different gpc coefficients is permitted subject to a minimal compatibility requirement on meshes used for different gpc coefficients details on the implementation with the open source software framework alea are presented it is generic and is based on available stiffness and mass matrices of a fem for the deterministic nonparametric nominal problem evaluated in the fenics environment preconditioning of the resulting matrix equation and iterative solution are discussed numerical experiments in two spatial dimensions for membrane and plane stress boundary value problems on polygons are presented they indicate substantial savings in total computational complexity due to fe mesh coarsening in high gpc coefficients

a stochastic galerkin method for the boltzmann equation with uncertainty we develop a stochastic galerkin method for the boltzmann equation with uncertainty the method is based on the generalized polynomial chaos gpc approximation in the stochastic galerkin framework and can handle random inputs from collision kernel initial data or boundary data we show that a simple singular value decomposition of gpc related coefficients combined with the fast fourier spectral method in velocity space allows one to compute the high dimensional collision operator very efficiently in the spatially homogeneous case we first prove that the analytical solution preserves the regularity of the initial data in the random space and then use it to establish the spectral accuracy of the proposed stochastic galerkin method several numerical examples are presented to illustrate the validity of the proposed scheme

longer time accuracy for incompressible navier stokes simulations with the emac formulation abstract in this paper we consider the recently introduced emac formulation for the incompressible navier stokes ns equations which is the only known ns formulation that conserves energy momentum and angular momentum when the divergence constraint is only weakly enforced since its introduction the emac formulation has been successfully used for a wide variety of fluid dynamics problems we prove that discretizations using the emac formulation are potentially better than those built on the commonly used skew symmetric formulation by deriving a better longer time error estimate for emac while the classical results for schemes using the skew symmetric formulation have gronwall constants dependent on exp c r e t with r e the reynolds number it turns out that the emac error estimate is free from this explicit exponential dependence on the reynolds number additionally it is demonstrated how emac admits smaller lower bounds on its velocity error since incorrect treatment of linear momentum angular momentum and energy induces lower bounds for l 2 velocity error and emac treats these quantities more accurately results of numerical tests for channel flow past a cylinder and 2d kelvin helmholtz instability are also given both of which show that the advantages of emac over the skew symmetric formulation increase as the reynolds number gets larger and for longer simulation times

subcell resolution in simplex stochastic collocation for spatial discontinuities subcell resolution has been used in the finite volume method fvm to obtain accurate approximations of discontinuities in the physical space stochastic methods are usually based on local adaptivity for resolving discontinuities in the stochastic dimensions however the adaptive refinement in the probability space is ineffective in the non intrusive uncertainty quantification framework if the stochastic discontinuity is caused by a discontinuity in the physical space with a random location the dependence of the discontinuity location in the probability space on the spatial coordinates then results in a staircase approximation of the statistics which leads to first order error convergence and an underprediction of the maximum standard deviation to avoid these problems we introduce subcell resolution into the simplex stochastic collocation ssc method for obtaining a truly discontinuous representation of random spatial discontinuities in the interior of the cells discretizing the probability space the presented ssc sr method is based on resolving the discontinuity location in the probability space explicitly as function of the spatial coordinates and extending the stochastic response surface approximations up to the predicted discontinuity location the applications to a linear advection problem the inviscid burgers equation a shock tube problem and the transonic flow over the rae 2822 airfoil show that ssc sr resolves random spatial discontinuities with multiple stochastic and spatial dimensions accurately using a minimal number of samples

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

high order exactly divergence free hybrid discontinuous galerkin methods for unsteady incompressible flows abstract in this paper we present an efficient discretization method for the solution of the unsteady incompressible navier stokes equations based on a high order hybrid discontinuous galerkin formulation the crucial component for the efficiency of the discretization method is the distinction between stiff linear parts and less stiff non linear parts with respect to their temporal and spatial treatment exploiting the flexibility of operator splitting time integration schemes we combine two spatial discretizations which are tailored for two simpler sub problems a corresponding hyperbolic transport problem and an unsteady stokes problem for the hyperbolic transport problem a spatial discretization with an upwind discontinuous galerkin method and an explicit treatment in the time integration scheme is rather natural and allows for an efficient implementation the treatment of the stokes part involves the solution of linear systems in this case a discretization with hybrid discontinuous galerkin methods is better suited we consider such a discretization for the stokes part with two important features h div conforming finite elements to guarantee exactly divergence free velocity solutions and a projection operator which reduces the number of globally coupled unknowns we present the method discuss implementational aspects and demonstrate the performance on two and three dimensional benchmark problems

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

time asymptotics and the self organization hypothesis for 2d navier stokes equations in this paper we study the long time behaviour of solutions with a two dimensional structure of the navier stokes equations on a periodic grid from a rigorous investigation of the decrease of the energy e and the enstrophy w it follows that the rayleigh quotient image is monotonically decreasing on solutions this is shown to imply that the spatial structure of any solution tends to a critical point of q which is the structure of some planar vortex and that all these structures are unstable except for the fundamental one which has the longest wavelength the time dependence of the approach towards this self organized state is investigated in some detail in the spectral plane the mean squared wave numbers of both the spectral energy and enstrophy density are shown to decrease as a function of time for the enstrophy this implies that the normal cascade does not match the discriminating effect of the viscous dissipation

an assessment of two models for the subgrid scale tensor in the rational les model les models seek to approximate the large scales of a flow which are defined by a space average u p of the velocity u and the pressure p of the flow a natural question which arises is given reliable data for u p how accurate is the approximation of u p by the solution computed with a les model this paper presents numerical studies of this question at a 2d and 3d mixing layer problem for the rational les model with two types of models for the subgrid scale tensor the smagorinsky model and a model proposed by iliescu and layton whereas in the 2d mixing layer problem the model by iliescu and layton showed better results the behaviour of both models was similar in the 3d mixing layer problem

dynamically orthogonal numerical schemes for efficient stochastic advection and lagrangian transport quantifying the uncertainty of lagrangian motion can be performed by solving a large number of ordinary differential equations with random velocities or equivalently a stochastic transport partial differential equation pde for the ensemble of flow maps the dynamically orthogonal do decomposition is applied as an efficient dynamical model order reduction to solve for such stochastic advection and lagrangian transport its interpretation as the method that applies instantaneously the truncated svd on the matrix discretization of the original stochastic pde is used to obtain new numerical schemes fully linear explicit central advection schemes stabilized with numerical filters are selected to ensure efficiency accuracy stability and direct consistency between the original deterministic and stochastic do advections and flow maps various strategies are presented for selecting a time stepping that accounts for the curvature of the fixed rank manifold and the error related to closely singular coefficient matrices efficient schemes are developed to dynamically evolve the rank of the reduced solution and to ensure the orthogonality of the basis matrix while preserving its smooth evolution over time finally the new schemes are applied to quantify the uncertain lagrangian motions of a 2d double gyre flow with random frequency and of a stochastic flow past a cylinder

convergence rates of multilevel and sparse tensor approximations for a random elliptic pde partial differential equations with random coefficients can be cast as parametric problems with a potentially infinite dimensional parameter domain for a class of elliptic equations we derive convergence rates of approximations based on a tensorized polynomial basis on the parameter domain and a sequence of spatial discretizations we prove that a sparse tensor product construction achieves essentially the same convergence rate as a multilevel approximation in which the spatial discretization level is chosen separately for each coefficient of the solution with respect to the polynomial basis in some cases the same rate is attained also by a single level approximation using just one spatial discretization for all coefficients we suggest an adaptive algorithm that reaches the optimal convergence rate with respect to the total number of degrees of freedom in a multilevel setting without prior knowledge of this rate numerical computations confirm theoretical results

attractors of partial differential evolution equations in an unbounded domain there is a large number of papers in which attractors of parabolic reaction diffusion equations in bounded domains are investigated in this paper these equations are considered in the whole unbounded space and a theory of attractors of such equations is built while investigating these equations specific difficulties arise connected with the noncompactness of operators with the continuity of their spectra etc therefore some new conditions on nonlinear terms arise in this paper weighted spaces are widely applied an important feature of this problem is worth mentioning namely properties of semigroups corresponding to equations with solutions in spaces of growing and of decreasing functions essentially differ

a sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes a smolyak type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms input data of the model to compute solution statistics the sparse grid stochastic collocation method uses approximate solutions produced here by finite elements corresponding to a deterministic set of points in the random input space this naturally requires solving uncoupled deterministic problems as in the monte carlo method if the number of random variables needed to describe the input data is moderately large full tensor product spaces are computationally expensive to use due to the curse of dimensionality in this case the sparse grid approach is still expected to be competitive with the classical monte carlo method therefore it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than monte carlo this work provides error estimates for the fully discrete solution using l q norms and analyzes the computational efficiency of the proposed method in particular it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem number of input random variables in the final estimates the derived estimates are then used to compare the method with monte carlo indicating for which problems the former is more efficient than the latter computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and monte carlo approaches

sparse grid collocation schemes for stochastic natural convection problems in recent years there has been an interest in analyzing and quantifying the effects of random inputs in the solution of partial differential equations that describe thermal and fluid flow problems spectral stochastic methods and monte carlo based sampling methods are two approaches that have been used to analyze these problems as the complexity of the problem or the number of random variables involved in describing the input uncertainties increases these approaches become highly impractical from implementation and convergence points of view this is especially true in the context of realistic thermal flow problems where uncertainties in the topology of the boundary domain boundary flux conditions and heterogeneous physical properties usually require high dimensional random descriptors the sparse grid collocation method based on the smolyak algorithm offers a viable alternate method for solving high dimensional stochastic partial differential equations an extension of the collocation approach to include adaptive refinement in important stochastic dimensions is utilized to further reduce the numerical effort necessary for simulation we show case the collocation based approach to efficiently solve natural convection problems involving large stochastic dimensions equilibrium jumps occurring due to surface roughness and heterogeneous porosity are captured comparison of the present method with the generalized polynomial chaos expansion and monte carlo methods are made

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

asymptotic behavior and attractors for reaction diffusion equations in unbounded domains abstract in this paper we give general and flexible conditions for a reaction diffusion equation to be dissipative in an unbounded domain the functional setting is based on standard lebesgue and sobolev lebesgue spaces we show how the reaction and diffusion mechanisms have to work together to obtain the asymptotic compactness of solutions and therefore the existence of the compact attractor in particular cases our results allow us to improve some previous known results

efficient stochastic galerkin methods for random diffusion equations we discuss in this paper efficient solvers for stochastic diffusion equations in random media we employ generalized polynomial chaos gpc expansion to express the solution in a convergent series and obtain a set of deterministic equations for the expansion coefficients by galerkin projection although the resulting system of diffusion equations are coupled we show that one can construct fast numerical methods to solve them in a decoupled fashion the methods are based on separation of the diagonal terms and off diagonal terms in the matrix of the galerkin system we examine properties of this matrix and show that the proposed method is unconditionally stable for unsteady problems and convergent for steady problems with a convergent rate independent of discretization parameters numerical examples are provided for both steady and unsteady random diffusions to support the analysis

a hyperbolicity preserving discontinuous stochastic galerkin scheme for uncertain hyperbolic systems of equations abstract intrusive uncertainty quantification methods such as stochastic galerkin are gaining popularity whereas the classical stochastic galerkin approach is not ensured to preserve hyperbolicity of the underlying hyperbolic system we apply a modification of this method that uses a slope limiter to retain admissible solutions of the system while providing high order approximations in the physical and stochastic space this is done using a spatial discontinuous galerkin scheme and a multi element stochastic galerkin ansatz in the random space we analyze the convergence of the resulting scheme and apply it to the compressible euler equations with various uncertain initial states in one and two spatial domains with up to three uncertainties the performance in multiple stochastic dimensions is compared to the non intrusive stochastic collocation method the numerical results underline the strength of our method especially if discontinuities are present in the uncertainty of the solution

adaptive anisotropic spectral stochastic methods for uncertain scalar conservation laws this paper deals with the design of adaptive anisotropic discretization schemes for conservation laws with stochastic parameters a finite volume scheme is used for the deterministic discretization while a piecewise polynomial representation is used at the stochastic level the methodology is designed in the context of intrusive galerkin projection methods with a roe type solver the adaptation aims at selecting the stochastic resolution level based on the local smoothness of the solution in the stochastic domain in addition the stochastic features of the solution greatly vary in space and time so that the constructed stochastic approximation space depends on space and time the dynamically evolving stochastic discretization uses a tree structure representation that allows for the efficient implementation of the various operators needed to perform anisotropic multiresolution analysis efficiency of the overall adaptive scheme is assessed on a stochastic nonlinear conservation law with uncertain initial

probabilistic collocation method for strongly nonlinear problems 1 transform by location 1 in this work we propose a new collocation method for uncertainty quantification in strongly nonlinear problems based on polynomial construction the traditional probabilistic collocation method pcm approximates the model output response which is a function of the random input parameter from the eulerian point of view in specific locations in some cases especially when the advection dominates the model response has a strongly nonlinear profile with a discontinuous shock or large gradient this nonlinearity in the space domain is then translated to nonlinearity in the random parametric domain which causes nonphysical oscillation and inaccurate estimation using the traditional pcm to address this issue a new location based transformed probabilistic collocation method xtpcm is developed in this study inspired by the lagrangian point of view in which model response is represented by an alternative variable i e the location of a particular response value which is relatively linear to the random parameter with a smooth profile the location is then approximated by polynomial construction from which a sufficient number of location samples are randomly generated and transformed back to obtain the response samples and to estimate the statistical properties the advantage of the xtpcm is demonstrated through applications to multiphase flow and solute transport in porous media which shows that the xtpcm achieves higher statistical accuracy than does the pcm and produces more reasonable realizations without oscillation while computational effort is greatly reduced compared to the direct sampling monte carlo method

high order methods as an alternative to using sparse tensor products for stochastic galerkin fem solutions of random elliptic boundary value problems admit efficient approximations by polynomials on the parameter domain each coefficient in such an expansion is a spatially dependent function and can be approximated within a hierarchy of finite element spaces if the finite elements are of sufficiently high order using just a single spatial mesh is predicted to achieve the same convergence rate with respect to the total number of degrees of freedom as sparse tensor product constructions and other multilevel stochastic galerkin approximations numerical computations for an elliptic two point boundary value problem confirm this and indicate no loss of accuracy for a single level method compared to using a sparse tensor product with the same total number of degrees of freedom

asymptotic preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings in this paper we develop a set of stochastic numerical schemes for hyperbolic and transport equations with diffusive scalings and subject to random inputs the schemes are asymptotic preserving ap in the sense that they preserve the diffusive limits of the equations in discrete setting without requiring excessive refinement of the discretization our stochastic ap schemes are extensions of the well developed deterministic ap schemes to handle the random inputs we employ generalized polynomial chaos gpc expansion and combine it with stochastic galerkin procedure we apply the gpc galerkin scheme to a set of representative hyperbolic and transport equations and establish the ap property in the stochastic setting we then provide several numerical examples to illustrate the accuracy and effectiveness of the stochastic ap schemes

asymptotic behavior of a class of non autonomous degenerate parabolic equations abstract we investigate the asymptotic behavior of solutions of a class of non autonomous degenerate parabolic equations on r n with unbounded external forcing terms the existence of a pullback global attractor is proved in l 2 r n l p r n w 1 r r n for any p r 2

numerical approximation of poroelasticity with random coefficients using polynomial chaos and hybrid high order methods abstract in this work we consider the biot problem with uncertain poroelastic coefficients the uncertainty is modeled using a finite set of parameters with prescribed probability distribution we present the variational formulation of the stochastic partial differential system and establish its well posedness we then discuss the approximation of the parameter dependent problem by non intrusive techniques based on polynomial chaos decompositions we specifically focus on sparse spectral projection methods which essentially amount to performing an ensemble of deterministic model simulations to estimate the expansion coefficients the deterministic solver is based on a hybrid high order discretization supporting general polyhedral meshes and arbitrary approximation orders we numerically investigate the convergence of the probability error of the polynomial chaos approximation with respect to the level of the sparse grid finally we assess the propagation of the input uncertainty onto the solution considering an injection extraction problem

on reference solutions and the sensitivity of the 2d kelvin helmholtz instability problem abstract two dimensional kelvin helmholtz instability problems are popular examples for assessing discretizations for incompressible flows at high reynolds number unfortunately the results in the literature differ considerably this paper presents computational studies of a kelvin helmholtz instability problem with high order divergence free finite element methods reference results in several quantities of interest are obtained for three different reynolds numbers up to the beginning of the final vortex pairing a mesh independent prediction of the final pairing is not achieved due to the sensitivity of the considered problem with respect to small perturbations a theoretical explanation of this sensitivity to small perturbations is provided based on the theory of self organization of 2d turbulence possible sources of perturbations that arise in almost any numerical simulation are discussed

a hyperbolicity preserving stochastic galerkin approximation for uncertain hyperbolic systems of equations abstract uncertainty quantification through stochastic spectral methods is rising in popularity we derive a modification of the classical stochastic galerkin method that ensures the hyperbolicity of the underlying hyperbolic system of partial differential equations the modification is done using a suitable slope limiter based on similar ideas in the context of kinetic moment models we apply the resulting modified stochastic galerkin method to the compressible euler equations and the m 1 model of radiative transfer our numerical results show that it can compete with other uq methods like the intrusive polynomial moment method while being computationally inexpensive and easy to implement

uncertainty quantification of geochemical and mechanical compaction in layered sedimentary basins abstract in this work we propose an uncertainty quantification methodology for sedimentary basins evolution under mechanical and geochemical compaction processes which we model as a coupled time dependent non linear monodimensional depth only system of pdes with uncertain parameters while in previous works formaggia et al 2013 porta et al 2014 we assumed a simplified depositional history with only one material in this work we consider multi layered basins in which each layer is characterized by a different material and hence by different properties this setting requires several improvements with respect to our earlier works both concerning the deterministic solver and the stochastic discretization on the deterministic side we replace the previous fixed point iterative solver with a more efficient newton solver at each step of the time discretization on the stochastic side the multi layered structure gives rise to discontinuities in the dependence of the state variables on the uncertain parameters that need an appropriate treatment for surrogate modeling techniques such as sparse grids to be effective we propose an innovative methodology to this end which relies on a change of coordinate system to align the discontinuities of the target function within the random parameter space the reference coordinate system is built upon exploiting physical features of the problem at hand we employ the fact that the locations of the material interfaces display a smooth dependence on the random parameters and are therefore amenable to sparse grid polynomial approximations we showcase the capabilities of our numerical methodologies through two synthetic test cases in particular we show that our methodology reproduces with high accuracy multi modal probability density functions displayed by target state variables e g porosity

a stochastic projection method for fluid flow i basic formulation we describe the construction and implementation of a stochastic navier stokes solver the solver combines a spectral stochastic uncertainty representation scheme with a finite difference projection method for flow simulation the uncertainty quantification scheme is adapted from the spectral stochastic finite element method ssfem which is based on regarding uncertainty as generating a new dimension and the solution as being dependent on this dimension in the ssfem formalism the stochastic dependence is represented in terms of the polynomial chaos system and the coefficients in the corresponding spectral representation are obtained using a galerkin approach it is shown that incorporation of the spectral uncertainty representation scheme into the projection method results in a coupled system of advection diffusion equations for the various uncertainty fields and in a decoupled system of pressure projection steps this leads to a very efficient stochastic solver whose advantages are illustrated using steady and transient simulations of transport and mixing in a microchannel

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

an explicit divergence free dg method for incompressible flow abstract we present an explicit divergence free dg method for incompressible flow based on velocity formulation only an h div conforming and globally divergence free finite element space is used for the velocity field and the pressure field is eliminated from the equations by design the resulting ode system can be discretized using any explicit time stepping methods we use the third order strong stability preserving runge kutta method in our numerical experiments our spatial discretization produces the identical velocity field as the divergence conforming dg method of cockburn et al 2007 based on a velocity pressure formulation when the same dg operators are used for the convective and viscous parts due to the global nature of the divergence free constraint and its interplay with the boundary conditions it is very hard to construct local bases for our finite element space here we present a key result on the efficient implementation of the scheme by identifying the equivalence of the mass matrix inversion of the globally divergence free finite element space to a standard hybrid mixed poisson solver hence in each time step a hybrid mixed poisson solver is used which reflects the global nature of the incompressibility condition in the actual implementation of this fully discrete scheme the pressure field is also computed via the hybrid mixed poisson solver hence the scheme can be interpreted as a velocity pressure formulation that treats the incompressibility constraint and pressure forces implicitly but the viscous and convective part explicitly since we treat viscosity explicitly for the navier stokes equation our method shall be best suited for unsteady high reynolds number flows so that the cfl constraint is not too restrictive

uncertainty quantification in hybrid dynamical systems uncertainty quantification uq techniques are frequently used to ascertain output variability in systems with parametric uncertainty traditional algorithms for uq are either system agnostic and slow such as monte carlo or fast with stringent assumptions on smoothness such as polynomial chaos and quasi monte carlo in this work we develop a fast uq approach for hybrid dynamical systems by extending the polynomial chaos methodology to these systems to capture discontinuities we use a wavelet based wiener haar expansion we develop a boundary layer approach to propagate uncertainty through separable reset conditions we also introduce a transport theory based approach for propagating uncertainty through hybrid dynamical systems here the expansion yields a set of hyperbolic equations that are solved by integrating along characteristics the solution of the partial differential equation along the characteristics allows one to quantify uncertainty in hybrid or switching dynamical systems the above methods are demonstrated on example problems

existence and finite dimensionality of the global attractor for evolution equations on unbounded domains abstract this paper is devoted to the asymptotic behavior for some evolution equations when the underlying euclidean domain is unbounded we present a method that is able to overcome the lack of compactness of the trajectories in such situations and that allows us to obtain some results on the global attractor similar to those on bounded domains this method is based on the utilization of an appropriate weight integration function that depends on x and t and becomes effective in x for large t s