immersed finite element methods for 4th order differential equations we propose three new finite element methods for solving boundary value problems of 4th order differential equations with discontinuous coefficients typical differential equations modeling the small transverse displacement of a beam and a thin plate formed by multiple uniform materials are considered one important feature of these finite element methods is that their meshes can be independent of the interface between different materials finite element spaces based on both the conforming and mixed formulations are presented numerical examples are given to illustrate capabilities of these methods

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

reduced aliasing formulations of the convective terms within the navier stokes equations for a compressible fluid the effect on aliasing errors of different formulations describing the cubically nonlinear convective terms within the discretized navier stokes equations is examined in the presence of a non trivial density spectrum fourier analysis shows that the existing skew symmetric forms of the convective term result in reduced aliasing errors relative to the conservation form several formulations of the convective term including a new formulation proposed for cubically nonlinear terms are tested in direct numerical simulation dns of decaying compressible isotropic turbulence both in chemically inert small density fluctuations and reactive cases large density fluctuations and for different degrees of resolution in the dns of reactive turbulent flow the new cubic skew symmetric form gives the most accurate results consistent with the spectral error analysis and at the lowest cost in marginally resolved dns and les poorly resolved by definition the new cubic skew symmetric form represents a robust convective formulation which minimizes both aliasing and computational cost while also allowing a reduction in the use of computationally expensive high order dissipative filters

analysis of numerical errors in large eddy simulation using statistical closure theory this paper develops a dynamic error analysis procedure for the numerical errors arising from spatial discretization in large eddy simulation the analysis is based on edqnm closure theory and is applied to the les of decaying isotropic turbulence first the effects of finite differencing truncation error aliasing error and the dynamic smagorinsky model are independently considered the time evolution of kinetic energy and spectra predicted by the analysis are compared to actual les using the navier stokes equations and good agreement is obtained the analysis is then extended to simultaneously consider all sources of error in a second order discretely energy conserving central difference les solver good agreement between the analysis and actual les is obtained the analysis is used to compare the contribution of the subgrid model to that of numerical errors and it is shown that the contribution of the subgrid scale model is much higher than the numerical errors the proposed one dimensional edqnm les model shows potential as a more general tool for the analysis of numerical error and sgs model in simulations of turbulent flow

finite volume treatment of dispersion relation preserving and optimized prefactored compact schemes for wave propagation in developing suitable numerical techniques for computational aero acoustics the dispersion relation preserving drp scheme by tam and co workers and the optimized prefactored compact opc scheme by ashcroft and zhang have shown desirable properties of reducing both dissipative and dispersive errors these schemes originally based on the finite difference attempt to optimize the coefficients for better resolution of short waves with respect to the computational grid while maintaining pre determined formal orders of accuracy in the present study finite volume formulations of both schemes are presented to better handle the nonlinearity and complex geometry encountered in many engineering applications linear and nonlinear wave equations with and without viscous dissipation have been adopted as the test problems highlighting the principal characteristics of the schemes and utilizing linear and nonlinear wave equations with different wavelengths as the test cases the performance of these approaches is documented for the linear wave equation there is no major difference between the drp and opc schemes for the nonlinear wave equations the finite volume version of both drp and opc schemes offers substantially better solutions in regions of high gradient or discontinuity

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

comparison of some lie symmetry based integrators lie symmetry based integrators are constructed in order to preserve the local invariance properties of the equations the geometrical methods leading to discretized equations for numerical computations involve many different concepts therefore they give rise to numerical schemes that vary in the accuracy in the computational cost and in the implementation in this paper a comparison is made between some alternative lie symmetry based methods illustrated on the example of the burgers equation the importance of the symmetry preservation is numerically highlighted

verification of a fluid dynamics solver using correlations with linear stability results a novel method is described for verification of fluid dynamics solvers based on correlations with solutions from linear stability analysis a difficulty with the linear stability analysis solutions for spatially developing flows is that flow fields typically exhibit exponentially growing features compromising the performance of classical error metrics this motivates the construction of a projection based metric that only assumes the shape of the solution and not the growth rate of the perturbations thus also allowing the latter to be determined the proposed correlation metric complements classical error metrics such as p norms and can also be used for time dependent problems with realistic boundary conditions we demonstrate how the present method can be applied in the verification of an euler solver for the instability behavior of laminar compressible free and confined shear layers

energy preserving turbulent simulations at a reduced computational cost energy conserving discretizations are widely regarded as a fundamental requirement for high fidelity simulations of turbulent flows the skew symmetric splitting of the nonlinear term is a well known approach to obtain semi discrete conservation of energy in the inviscid limit however its computation is roughly twice as expensive as that of the divergence or advective forms alone a novel time advancement strategy that retains the conservation properties of skew symmetric based schemes at a reduced computational cost has been developed this method is based on properly constructed runge kutta schemes in which a different form advective or divergence for the convective term is adopted at each stage a general framework is presented to derive schemes with prescribed accuracy on both solution and energy conservation simulations of homogeneous isotropic turbulence show that the new procedure is effective and can be considerably faster than skew symmetric based techniques

an adaptive implicit explicit scheme for the dns and les of compressible flows on unstructured grids an adaptive implicit explicit scheme for direct numerical simulation dns and large eddy simulation les of compressible turbulent flows on unstructured grids is developed the method uses a node based finite volume discretization with summation by parts sbp property which in conjunction with simultaneous approximation terms sat for imposing boundary conditions leads to a linearly stable semi discrete scheme the solution is marched in time using an implicit explicit runge kutta imex rk time advancement scheme a novel adaptive algorithm for splitting the system into implicit and explicit sets is developed the method is validated using several canonical laminar and turbulent flows load balance for the new scheme is achieved by a dual constraint domain decomposition algorithm the scalability and computational efficiency of the method is investigated and memory savings compared with a fully implicit method is demonstrated a notable reduction of computational costs compared to both fully implicit and fully explicit schemes is observed

higher order numerical discretizations for exterior and biharmonic type pdes a higher order numerical discretization technique based on minimum sobolev norm msn interpolation was introduced in our previous work in this article the discretization technique is presented as a tool to solve two hard classes of pdes namely the exterior laplace problem and the biharmonic problem the exterior laplace problem is compactified and the resultant near singular pde is solved using this technique this finite difference type method is then used to discretize and solve biharmonic type pdes a simple book keeping trick of using ghost points is used to obtain a perfectly constrained discrete system numerical results such as discretization error condition number estimate and solution error are presented for both classes of pdes variable coefficient examples on complicated geometries and irregular grids are considered the method is seen to have high order of convergence in all these cases through numerical evidence perhaps for the first time such a systematic higher order procedure for irregular grids and variable coefficient cases is now available though not discussed in the paper the idea seems to be easily generalizable to finite element type techniques as well

isotropic finite volume discretization finite volume methods traditionally employ dimension by dimension extension of the one dimensional reconstruction and averaging procedures to achieve spatial discretization of the governing partial differential equations on a structured cartesian mesh in multiple dimensions this simple approach based on tensor product stencils introduces an undesirable grid orientation dependence in the computed solution the resulting anisotropic errors lead to a disparity in the calculations that is most prominent between directions parallel and diagonal to the grid lines in this work we develop isotropic finite volume discretization schemes which minimize such grid orientation effects in multidimensional calculations by eliminating the directional bias in the lowest order term in the truncation error explicit isotropic expressions that relate the cell face averaged line and surface integrals of a function and its derivatives to the given cell area and volume averages are derived in two and three dimensions respectively it is found that a family of isotropic approximations with a free parameter can be derived by combining isotropic schemes based on next nearest and next next nearest neighbors in three dimensions use of these isotropic expressions alone in a standard finite volume framework however is found to be insufficient in enforcing rotational invariance when the flux vector is nonlinear and or spatially non uniform the rotationally invariant terms which lead to a loss of isotropy in such cases are explicitly identified and recast in a differential form various forms of flux correction terms which allow for a full recovery of rotational invariance in the lowest order truncation error terms while preserving the formal order of accuracy and discrete conservation of the original finite volume method are developed numerical tests in two and three dimensions attest the superior directional attributes of the proposed isotropic finite volume method prominent anisotropic errors such as spurious asymmetric distortions on a circular reaction diffusion wave that feature in the conventional finite volume implementation are effectively suppressed through isotropic finite volume discretization furthermore for a given spatial resolution a striking improvement in the prediction of kinetic energy decay rate corresponding to a general two dimensional incompressible flow field is observed with the use of an isotropic finite volume method instead of the conventional discretization c 2014 elsevier inc all rights reserved

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

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

application of the entropically damped artificial compressibility model to direct numerical simulation of turbulent channel flow abstract the feasibility of the entropically damped artificial compressibility edac model for simulation of wall bounded turbulence is examined a favourable feature of edac is the purely parabolic type of governing equations resulting from the introduction of a damping term to the pressure evolution equation this significantly reduces noise in the velocity divergence field when central schemes are applied for the spatial discretisation the method of lines is used for the solution of the edac equations a conservative finite difference method of purely 2nd and mixed 4th 2nd order is applied the explicit 4th order six stage low storage runge kutta method is used for time advancement as expected the mixed scheme is superior to the 2nd order one in terms of both accuracy and computational efficiency for the first time the edac model is assessed for the direct numerical simulation of wall bounded turbulent flow requiring a non uniform mesh as a particular test case we have chosen the channel flow at the friction reynolds numbers r e 180 and 395 a very good agreement with the reference data is obtained as documented by the chosen one point velocity statistics the mean and r m s profiles and the budgets of the reynolds stresses the use of the explicit time discretisation and local rather than compact spatial schemes results in easily and efficiently parallelisable solution algorithm we achieved very high computational performance on a desktop computer when solving the edac equations using a code dedicated for the graphics processing unit gpu the aspects of the gpu implementation are discussed

l2 well posedness of planar div curl systems criteria for the existence and uniqueness of solutions of div curl boundary value problems on bounded planar regions with nice boundaries are developed the boundary conditions to be treated include prescribed normal component of the field tangential component of the field and disjoint combinations of these conditions under natural assumptions on the data when either tangential or normal components are given on the whole boundary weak finite energy solutions exist if and only if a compatibility condition holds if the region is simply connected this solution is unique when the region is multiply connected there is a finite dimensional family of solutions the dimension of the solution space is the betti number of the region the problem is well posed with a unique solution when certain line integrals are further prescribed l 2 estimates of the solutions are given if mixed tangential and normal components of the field are specified on different parts of the boundary no compatibility condition is required for solvability in general though there is considerable non uniqueness of solutions well posedness is recovered by specifying certain line integrals l 2 estimates of the solutions are given the dimensionality of the solution space depends on the topology of the boundary data these results depend on certain weighted orthogonal decompositions of l 2 vector fields on the region which are related to classical hodge weyl decomposition results

on the symmetric form of systems of conservation laws with entropy abstract this paper reviews the symmetrizability of systems of conservation laws which possess entropy functions symmetric formulations in conservation form for the equations of gas dynamics are presented

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

on the accuracy of the rotation form in simulations of the navier stokes equations the rotation form of the navier stokes equations nonlinearity is commonly used in high reynolds number flow simulations it was pointed out by a few authors and not widely known apparently that it can also lead to a less accurate approximate solution than the usual email protected u form we give a different explanation of this effect related to i resolution of the bernoulli pressure and ii the scaling of the coupling between velocity and pressure error with respect to the reynolds number we show analytically that i the difference between the two nonlinearities is governed by the difference in the resolution of the bernoulli and kinematic pressures and ii a simple linear grad div stabilization ameliorates much of the bad scaling of the velocity error with respect to re the rotation form does have superior conservation properties to the alternatives and it appears to be amenable to more efficient preconditioners thus the rotational form with grad div stabilization is a promising method we also give experiments that show bad velocity approximation is tied to poor pressure resolution in either form

fully conservative higher order finite difference schemes for incompressible flow conservation properties of the mass momentum and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discrete equations existing finite difference schemes in regular and staggered grid systems are checked for violations of the conservation requirements and a few important discrepancies are pointed out in particular it is found that none of the existing higher order schemes for a staggered mesh system simultaneously conserve mass momentum and kinetic energy this deficiency is corrected through the derivation of a general family of fully conservative higher order accurate finite difference schemes for staggered grid systems finite difference schemes in a collocated grid system are also analyzed and a violation of kinetic energy conservation is revealed the predicted conservation properties are demonstrated numerically in simulations of inviscid white noise performed in a two dimensional periodic domain the proposed fourth order schemes in a staggered grid system are generalized for the case of a non uniform mesh and the resulting scheme is used to perform large eddy simulations of turbulent channel flow

an efficient semi implicit compressible solver for large eddy simulations in this paper a pressure based semi implicit algorithm for compressible flows is described this type of solver is of great interest in combustion devices where the flow speed is small but where the unsteady heat release can be coupled to the acoustic modes and lead to instabilities the fractional step method used in this algorithm is based on a characteristic splitting which clearly separates the acoustics from the advection this splitting allows to use adapted numerics for these different time scales the algorithm is second order in space and time for linear acoustics and for low mach advection without iterating the time step moreover when kinetic energy conserving schemes are used for the predictor step the algorithm discretely conserves the kinetic energy in the low mach limit all these properties are illustrated on simple test cases and the algorithm is finally validated by performing the les of the cold flow of an industrial burner

meromorphic continuation of l functions of p adic representations x det i td x b xqd x l b xq b x where x is the teichmiiller lifting of zt and d x denotes the degree of x over fq this infinite product is a well defined power series with coefficients in rp because there are only finitely many closed points of a given degree if v fq is an affine variety contained in an then one defines l b v t to be the product in equation 1 1 except that we now restrict x to run only over the closed points of v fq using a standard reduction argument and dwork s

higher entropy conservation and numerical stability of compressible turbulence simulations we present a numerical formulation for the treatment of nonlinear instabilities in shock free compressible turbulence simulations the formulation is high order and contains no artificial dissipation numerical stability is enhanced through semi discrete satisfaction of global conservation properties stemming from the second law of thermodynamics and the entropy equation the numerical implementation is achieved using a conservative skew symmetric splitting of the nonlinear terms the robustness of the method is demonstrated by performing unresolved numerical simulations and large eddy simulations of compressible isotropic turbulence at a very high reynolds number results show the scheme is capable of capturing the statistical equilibrium of low mach number compressible turbulent fluctuations at infinite reynolds number comparisons with the entropy splitting technique j comput phys 162 2000 33 j comput phys 178 2002 307 staggered method j comput phys 191 2 2003 392 and skew symmetric like schemes j comput phys 161 2000 114 confirm the superiority of the current approach we also discuss a flaw in the skew symmetric splitting implemented in the literature very good results are obtained based on the proper splitting

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

multigrid solution of automatically generated high order discretizations for the biharmonic equation in this work we use a symbolic algebra package to derive a family of finite difference approximations for the biharmonic equation on a 9 point compact stencil the solution and its first derivatives are carried as unknowns at the grid points dirichlet boundary conditions are thus incorporated naturally since the approximations use the 9 point compact stencil no special formulas are needed near the boundaries both second order and fourth order discretizations are derived the fourth order approximation produce more accurate results than the 13 point classical stencil or the commonly used system of two second order equations coupled with the boundary condition the method suffers from slow convergence when classical iteration methods such as gauss seidel or sor are employed in order to alleviate this problem we propose several multigrid techniques that exhibit grid independent convergence and solve the biharmonic equation in a small amount of computer time test results from three different problems including stokes flow in a driven cavity are reported

a minimum sobolev norm technique for the numerical discretization of pdes partial differential equations pdes are discretized into an under determined system of equations and a minimum sobolev norm solution is shown to be efficient to compute and converge under very generic conditions numerical results of a single code that can handle pdes in first order form on complicated polygonal geometries are shown for a variety of pdes variable coefficient div curl scalar elliptic pdes elasticity equation stationary linearized navier stokes scalar fourth order elliptic pdes telegrapher s equations singular pdes etc

driven cavity flows by efficient numerical techniques abstract efficient and reliable numerical techniques of high order accuracy are presented for solving problems of steady viscous incompressible flow in the plane and are used to obtain accurate solutions for the driven cavity a solution is obtained at reynolds number 10 000 on a 180 180 grid the numerical methods combine an efficient linear system solver an adaptive newton like method for nonlinear systems and a continuation procedure for following a branch of solutions over a range of reynolds numbers

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

generalized conservative approximations of split convective derivative operators we propose a strategy to design locally conservative finite difference approximations of convective derivatives for shock free compressible flows with arbitrary order of accuracy that generalizes the approach of ducros et al 2000 1 and that can be applied as a building block of low dissipative hybrid shock capturing methods the approximations stem from application of standard central difference formulas to split forms of the convective terms in the compressible euler equations which guarantee strong numerical stability and near energy preservation in the inviscid limit a convenient implementation of the high order fluxes is suggested which guarantees improved computational efficiency over existing methods numerical tests performed for isotropic turbulence at zero viscosity show stability of schemes with order of accuracy up to ten and effectiveness of convective splitting of kennedy and gruber 2008 2 in providing extra stability in the presence of strong density variations numerical simulations of compressible turbulent boundary layer flow indicate suitability of the method for non uniform grids and overall support superior computational efficiency of high order schemes

designing an efficient solution strategy for fluid flows 1 a stable high order finite difference scheme and sharp shock resolution for the euler equations abstract we derive high order finite difference schemes for the compressible euler and navier stokes equations that satisfy a semidiscrete energy estimate and present an efficient strategy for the treatment of discontinuities that leads to sharp shock resolution the formulation of the semidiscrete energy estimate is based on symmetrization of the equations a canonical splitting of the flux derivative vector and the use of difference operators that satisfy a discrete analogue to the integration by parts procedure used in the continuous energy estimate for the euler equations the symmetrization is designed such as to preserve the homogeneity of the flux vectors around discontinuities or sharp gradients refined grids are created on which the discrete equations are solved after adding artificial viscosity the positioning of the subgrids and computation of the viscosity are aided by a detection algorithm which is based on a multiscale wavelet analysis of the pressure grid function the wavelet theory provides easy to implement mathematical criteria to detect discontinuities sharp gradients and spurious oscillations quickly and efficiently as the detection algorithm does not depend on the numerical method used it is of general interest the numerical method described and the detection algorithm are part of a general solution strategy for fluid flows which is currently being developed by the authors and collaborators

split form nodal discontinuous galerkin schemes with summation by parts property for the compressible euler equations fisher and carpenter 2013 12 found a remarkable equivalence of general diagonal norm high order summation by parts operators to a subcell based high order finite volume formulation this equivalence enables the construction of provably entropy stable schemes by a specific choice of the subcell finite volume flux we show that besides the construction of entropy stable high order schemes a careful choice of subcell finite volume fluxes generates split formulations of quadratic or cubic terms thus by changing the subcell finite volume flux to a specific choice we are able to generate in a systematic way all common split forms of the compressible euler advection terms such as the ducros splitting and the kennedy and gruber splitting although these split forms are not entropy stable we present a systematic way to prove which of those split forms are at least kinetic energy preserving with this we construct a unified high order split form dg framework we investigate with three dimensional numerical simulations of the inviscid taylorgreen vortex and show that the new split forms enhance the robustness of high order simulations in comparison to the standard scheme when solving turbulent vortex dominated flows in fact we show that for certain test cases the novel split form discontinuous galerkin schemes are more robust than the discontinuous galerkin scheme with over integration

short note straightforward high order numerical dissipation via the viscous term for direct and large eddy simulation in this short note we show how to use a highly accurate finite difference scheme to compute second derivatives in the navier stokes equations while ensuring targeted numerical dissipation this approach essentially non conservative is shown to be close to an upwind method and is straightforward to implement with a negligible computational extra cost the benefit offered by the resulting discrete operator is illustrated for the direct computation of sound in aeroacoustics and in the more general context of large eddy simulation through connections with hyperviscosity and spectral vanishing viscosity

short note new connections between finite element formulations of the navier stokes equations we show the velocity solutions to the convective skew symmetric and rotational galerkin finite element formulations of the navier stokes equations are identical if scott vogelius elements are used and thus all three formulations will be the same pointwise divergence free solution velocity a connection is then established between the formulations for grad div stabilized taylor hood elements under mild restrictions the formulations velocity solutions converge to each other and to the scott vogelius solution as the stabilization parameter tends to infinity thus the benefits of using scott vogelius elements can be obtained with the less expensive taylor hood elements and moreover the benefits of all the formulations can be retained if the rotational formulation is used numerical examples are provided that confirm the theory

computational design for long term numerical integration of the equations of fluid motion the integral constraints on quadratic quantities of physical importance such as conservation of mean kinetic energy and mean square vorticity will not be maintained in finite difference analogues of the equation of motion for two dimensional incompressible flow unless the finite difference jacobian expression for the advection term is restricted to a form which properly represents the interaction between grid points as derived in this paper it is shown that the derived form of the finite difference jacobian prevents nonlinear computational instability and thereby permits long term numerical integrations

on coordinate transformation and grid stretching for sparse grid pricing of basket options we evaluate two coordinate transformation techniques in combination with grid stretching for pricing basket options in a sparse grid setting the sparse grid technique is a basic technique for solving a high dimensional partial differential equation by creating a small hypercube sub grid in the composite sparse grid we can also determine hedge parameters accurately we evaluate these techniques for multi asset examples with up to five underlying assets in the basket

the main conjecture of iwasawa theory for totally real fields let p be an odd prime let mathcal g be a compact p adic lie group with a quotient isomorphic to p we give an explicit description of k 1 of the iwasawa algebra of mathcal g in terms of iwasawa algebras of abelian subquotients of mathcal g we also prove a result about k 1 of a certain canonical localisation of the iwasawa algebra of mathcal g which occurs in the formulation of the main conjectures of noncommutative iwasawa theory these results predict new congruences between special values of artin l functions which we then prove using the q expansion principle of deligne ribet as a consequence we prove the noncommutative main conjecture for totally real fields assuming a suitable version of iwasawa s conjecture about vanishing of the cyclotomic invariant in particular we get an unconditional result for totally real pro p p adic lie extension of abelian extensions of

boundary consistent b spline filtering schemes and application to high fidelity simulations of turbulence abstract a filtering operation based on b spline discretizations is introduced to target weakly growing mesh scale oscillations that can arise in high fidelity turbulence simulations this is a spectral regularization that can be described using the singular values of a banded matrix operator with the filtering strength set by a scalar or vector valued penalty parameter the penalty parameter can be specified though it can also be advantageously selected to minimize the generalized cross validation gcv measure of distance between the pre and post filtered solutions efficient algorithms are developed to compute both the scalar and vector penalty parameters the b spline filter has a sharper localization to high wavenumber than compact or explicit filters of the same stencil width and is demonstrated for solutions of the burgers equation decaying burgers turbulence and compressible navier stokes turbulent channel flow these simulations confirm the scheme s numerical stability and ability to narrowly target the high wavenumber components of numerical solutions an advantage over finite difference filters is that these are stable on bounded domains and even preserve formal order of accuracy

the compressible hybrid rans les formulation using an additive operator in this work the compressible governing equations for the hybrid reynolds averaged large eddy simulations are formally derived by applying a hybrid filter to the navier stokes equations this filter is constructed by linearly combining the reynolds average rans and large eddy simulation les operators with a continuous blending function the derived hybrid equations include additional terms that represent the interactions between rans and les formulations the relevance of these terms is investigated in flat plate turbulent boundary layer simulations and indicate that these additional terms play a fundamental role in compensating for the turbulence that is neither modeled nor resolved in the transition region between rans and les results also show that when the additional terms are included the calculations are not very sensitive to the blending function implemented in the hybrid filter in the contrary when these terms are neglected and a step function is implemented in the hybrid filter nonphysical discontinuities are predicted in the flow statistics

a discontinuous galerkin method with lagrange multiplier for hyperbolic conservation laws with boundary conditions we introduce a discontinuous galerkin method with lagrange multiplier dglm to approximate the solution to the hyperbolic conservation laws with boundary conditions lagrange multipliers are introduced on the edge face of the element via weak divergence wang and ye 2014 the final global system has reduced numbers of unknowns of the standard dg methods numerical fluxes from finite volume difference method are not considered for the time discretization backward euler difference method is used stability of the approximate solution is proved in energy norm discontinuity of the solution is allowed in the error analysis local error estimates of o hr 12 t with pr e elements rd 12 are derived where h and t are the maximum diameter of the elements and time steps respectively and d is the dimension of the spatial domain the high order approximation is obtained under an appropriate condition on the stabilizing parameter it is shown that the method preserves the property of the local mass conservation an explanation on algorithmic aspects is given

discretely conservative finite difference formulations for nonlinear conservation laws in split form theory and boundary conditions the lax wendroff theorem stipulates that a discretely conservative operator is necessary to accurately capture discontinuities the discrete operator however need not be derived from the divergence form of the continuous equations indeed conservation law equations that are split into linear combinations of the divergence and product rule form and then discretized using any diagonal norm skew symmetric summation by parts sbp spatial operator yield discrete operators that are conservative furthermore split form discretely conservation operators can be derived for periodic or finite domain sbp spatial operators of any order examples are presented of a fourth order sbp finite difference operator with second order boundary closures sixth and eighth order constructions are derived and are supplied in an accompanying text file

symmetry preserving discretization of turbulent flow we propose to perform turbulent flow simulations in such manner that the difference operators do have the same symmetry properties as the underlying differential operators i e the convective operator is represented by a skew symmetric coefficient matrix and the diffusive operator is approximated by a symmetric positive definite matrix mimicking crucial properties of differential operators forms in itself a motivation for discretizing them in a certain manner we give it a concrete form by noting that a symmetry preserving discretization of the navier stokes equations is stable on any grid and conserves the total mass momentum and kinetic energy for the latter the physical dissipation is to be turned off of coarse being stable on any grid the choice of the grid may be based on the required accuracy solely and the main question becomes how accurate is a symmetry preserving discretization its accuracy is tested for a turbulent flow in a channel by comparing the results to those of physical experiments and previous numerical studies the comparison is carried out for a reynolds number of 5600 which is based on the channel width and the mean bulk velocity based on the channel half width and wall shear velocity the reynolds number becomes 180 the comparison shows that with a fourth order symmetry preserving method a 64 64 32 grid suffices to perform an accurate numerical simulation

a weighted meshfree collocation method for incompressible flows using radial basis functions abstract a weighted strong form collocation method using radial basis functions and explicit time integration is proposed to solve the incompressible navier stokes equations the velocities and pressure are solved directly at the same time step and the continuity equation is satisfied at each time step which improve the solution accuracy and stability no artificial compressibility coefficient needs to be introduced for modeling the incompressible flows and no pressure oscillation arises in the numerical solutions optimal convergence can be achieved by imposing the derived proper weights on the boundaries and the continuity equation radial basis collocation method in a lagrangian form is quite easy to capture the moving boundary or free surface in flow problems moreover solid boundary conditions can be enforced directly and no special treatments are required further critical time step for the explicit time integration is predicted in the stability analysis and the influences on the stability are evaluated numerical studies validate the high accuracy as well as good stability of the presented method

immersed boundary methods for general finite difference and finite volume navier stokes solvers we present an immersed boundary algorithm for incompressible flows with complex boundaries suitable for cartesian or curvilinear grid system the key stages of any immersed boundary technique are the interpolation of a velocity field given on a mesh onto a general boundary a line in 2d a surface in 3d and the spreading of a force field from the immersed boundary to the neighboring mesh points to enforce the desired boundary conditions on the immersed boundary points we propose a technique that uses the reproducing kernel particle method w k liu s jun y f zhang reproducing kernel particle methods int j numer methods fluids 20 8 1995 1081 1106 for the interpolation and spreading unlike other methods presented in the literature the one proposed here has the property that the integrals of the force field and of its moment on the grid are conserved independent of the grid topology uniform or non uniform cartesian or curvilinear the technique is easy to implement and is able to maintain the order of the original underlying spatial discretization applications to two and three dimensional flows in cartesian and non cartesian grid system with uniform and non uniform meshes are presented

global l functions over function fields let c be a smooth projective curve over a finite field k with a marked point and let a be the ring of regular functions on c r furthermore let c be the completion of an algebraic closure of the function field k of c at a place above in this situation goss attaches a global l function to any family of drinfeld a modules via an infinite euler product on the domain s c zp cf 11 as in the classical situation it converges on some half plane of s a similar procedure yields for any closed point v of c and any family m of a motives a global lfunction l v m s which converges on a half plane of a suitably defined domain sv 12 8 the theme of the current article is to derive some consequences for such global l functions from the theory of crystals over function fields introduced by r pink and the current author in 2 this theory encompasses a motives as defined by anderson 1 indeed any family of a motives on a scheme x is represented by an a crystal on x furthermore for any compactifiable morphism f y x there is a functor direct image with compact support from a crystals on y to a crystals on x at least in the derived context for such f a trace formula is given in 2 for l functions of crystals it is this trace formula and the relations between goss global l functions and l functions of a crystals which we will mainly exploit slightly generalizing goss definition we obtain for any a scheme x of finite type over k any a crystal f on x and any closed point v of c a vadic l function l v f s as an euler product which converges on some half plane of sv cf definition 2 8 our principal goal is to prove that any such function has a meromorphic essentially algebraic continuation to all of sv in the sense of 12 8 along the way we will prove some interesting

kinetic energy and entropy preserving schemes for compressible flows by split convective forms abstract this study proposes a kinetic energy and entropy preserving scheme to achieve stable and non dissipative compressible flow simulations the proposed scheme is built in such a way that the numerical formulations satisfy the analytical relations among the governing equations at the discrete level the mass and momentum convective terms are recast into split convective forms once the mass and momentum equations are discretized the constraints given by the analytical relations subsequently determine the formulations of the numerical fluxes solved in the total energy equation an analysis reveals that satisfying the analytical relations at the discrete level is essential for solving the energy exchange between the kinetic energy and internal energy correctly in the total energy equation and therefore important for entropy conservation taylor green vortex and euler isotropic turbulence simulations at infinite reynolds number are conducted to verify the analysis given in this paper the proposed scheme achieves both kinetic energy conservation in the incompressible limit and entropy conservation as well as convergences of thermodynamic variable fluctuations whereas existing kinetic energy preserving schemes fail entropy conservation and the thermodynamic variable fluctuations diverge

skew selfadjoint form for systems of conservation laws abstract hyperbolic systems of conservation laws augumented with an entropy inequality are studied it is shown that such systems can be written in a quasilinear skew selfadjoint form centered differencing of such a form under the smooth regime ends up with a systematic recipe for constructing quasiconservative schemes where the global entropy conservation is recovered employing the above formulation in bounded regions under the nonsmooth regime as well a local entropy decay estimate is also concluded examples of the shallow water and the full gasdynamics equations are explicitly treated

an efficient time advancing strategy for energy preserving simulations energy conserving numerical methods are widely employed within the broad area of convection dominated systems semi discrete conservation of energy is usually obtained by adopting the so called skew symmetric splitting of the non linear convective term defined as a suitable average of the divergence and advective forms although generally allowing global conservation of kinetic energy it has the drawback of being roughly twice as expensive as standard divergence or advective forms alone in this paper a general theoretical framework has been developed to derive an efficient time advancement strategy in the context of explicit runge kutta schemes the novel technique retains the conservation properties of skew symmetric based discretizations at a reduced computational cost it is found that optimal energy conservation can be achieved by properly constructed runge kutta methods in which only divergence and advective forms for the convective term are used as a consequence a considerable improvement in computational efficiency over existing practices is achieved the overall procedure has proved to be able to produce new schemes with a specified order of accuracy on both solution and energy the effectiveness of the method as well as the asymptotic behavior of the schemes is demonstrated by numerical simulation of burgers equation

skew symmetric form of convective terms and fully conservative finite difference schemes for variable density low mach number flows the form of convective terms for compressible flow equations is discussed in the same way as for an incompressible one by morinishi et al y morinishi t s lund o v vasilyev p moin fully conservative higher order finite difference schemes for incompressible flow j comput phys 124 1998 90 and fully conservative finite difference schemes suitable for shock free unsteady compressible flow simulations are proposed commutable divergence advective and skew symmetric forms of convective terms are defined by including the temporal derivative term for compressible flow these forms are analytically equivalent if the continuity is satisfied and the skew symmetric form is secondary conservative without the aid of the continuity while the divergence form is primary conservative the relations between the present and existing quasi skew symmetric forms are also revealed commutable fully discrete finite difference schemes of convection are then derived in a staggered grid system and they are fully conservative provided that the corresponding discrete continuity is satisfied in addition a semi discrete convection scheme suitable for compact finite difference is presented based on the skew symmetric form the conservation properties of the present schemes are demonstrated numerically in a three dimensional periodic inviscid flow the proposed fully discrete fully conservative second order accurate scheme is also used to perform the dns of compressible isotropic turbulence and the simulation of open cavity flow

b spline method and zonal grids for simulations of complex turbulent flows a numerical technique for computations of turbulent flows is described the technique is based on b splines and allows grid embedding in physically significant flow regions numerical tests which include solutions of nonlinear advection diffusion equations and computations of flow over a circular cylinder at reynolds numbers up to 300 indicate that the method is accurate and efficient in computations of flow over a cylinder the lift drag and base suction coefficients agree well with existing experimental data and previous numerical simulations

a low numerical dissipation immersed interface method for the compressible navier stokes equations a numerical method to solve the compressible navier stokes equations around objects of arbitrary shape using cartesian grids is described the approach considered here uses an embedded geometry representation of the objects and approximate the governing equations with a low numerical dissipation centered finite difference discretization the method is suitable for compressible flows without shocks and can be classified as an immersed interface method the objects are sharply captured by the cartesian mesh by appropriately adapting the discretization stencils around the irregular grid nodes located around the boundary in contrast with available methods no jump conditions are used or explicitly derived from the boundary conditions although a number of elements are adopted from previous immersed interface approaches a new element in the present approach is the use of the summation by parts formalism to develop stable non stiff first order derivative approximations at the irregular grid points second order derivative approximations as those appearing in the transport terms can be stiff when irregular grid points are located too close to the boundary this is addressed using a semi implicit time integration method moreover it is shown that the resulting implicit equations can be solved explicitly in the case of constant transport properties convergence studies are performed for a rotating cylinder and vortex shedding behind objects of varying shapes at different mach and reynolds numbers

on main conjectures in non commutative iwasawa theory and related conjectures the main conjecture of non commutative iwasawa theory of coates fukaya kato sujatha and venjakob is shown to be equivalent modulo an appropriate torsion hypothesis to a family of main conjectures over cyclotomic zp extensions together with suitable compatibility relations between the associated abelian p adic l functions this result combines with techniques of ritter and weiss and of kakde to give a concrete strategy for deducing main conjectures for non abelian compact p adic lie extensions of arbitrarily large rank from main conjectures in the classical sense as an explicit example of this approach we combine it with a recent result of ritter and weiss to prove modulo an appropriate vanishing hypothesis on invariants the main conjecture of non commutative iwasawa theory for tate motives over compact p adic lie extensions of totally real fields of arbitrarily large rank we then deduce several interesting consequences of this result concerning a range of related conjectures including the equivariant tamagawa number conjecture 1991 mathematics subject classification primary 11s40 secondary 11r29 20c05

mass momentum and energy conserving mamec discretizations on general grids for the compressible euler and shallow water equations the paper explains a method by which discretizations of the continuity and momentum equations can be designed such that they can be combined with an equation of state into a discrete energy equation the resulting mamec discretizations conserve mass momentum as well as energy although no explicit conservation law for the total energy is present essential ingredients are i discrete convection that leaves the discrete energy invariant and ii discrete consistency between the thermodynamic terms of particular relevance is the way in which finite volume fluxes are related to nodal values the method is an extension of existing methods based on skew symmetry of discrete operators because it allows arbitrary equations of state and a larger class of grids than earlier methods the method is first illustrated with a one dimensional example on a highly stretched staggered grid in which the mamec method calculates qualitatively correct results and a non skew symmetric finite volume method becomes unstable a further example is a two dimensional shallow water calculation on a rectilinear grid as well as on an unstructured grid the conservation of mass momentum and energy is checked and losses are found negligible up to machine accuracy

a finite difference scheme for three dimensional incompressible flows in cylindrical coordinates a finite difference scheme for the direct simulation of the incompressible time dependent three dimensional navier stokes equations in cylindrical coordinates is presented the equations in primitive variables vr v vzandp are solved by a fractional step method together with an approximate factorization technique cylindrical coordinates are singular at the axis the introduction of the radial fluxqr r vron a staggered grid simplifies the treatment of the region atr 0 the method is tested by comparing the evolution of a free vortex ring and its collision with a wall with the theory experiments and other numerical results the formation of a tripolar vortex where the highest vorticity is atr 0 is also considered finally to emphasize the accurate treatment near the axis the motion of a lamb dipole crossing the origin is simulated

improving the speed and accuracy of projection type incompressible flow solvers abstract superseding so called first generation incompressible flow solvers of the projection type based on taylor galerkin advection second order pressure damping and element based data structures the current second generation solvers based on high order upwind advection fourth order pressure damping and edge based data structures have now been in use for half a decade and have proven remarkably robust and efficient for many large scale problems in order to achieve higher accuracy and speed these solvers have recently been enhanced in a variety of ways a substepping for advection b implicit treatment of advective terms via sgs and gmres lu sgs iterative solvers c fully implicit time accurate advancement of pressure and velocities and d linelet preconditioning for the pressure poisson equation the combined effect of these third generation improvements leads to speedups of the order of o 1 5 1 10 with similar or even better temporal accuracy as demonstrated on a variety of academic and industrial problems

numerically stable formulations of convective terms for turbulent compressible flows abstract a systematic analysis of the discrete conservation properties of non dissipative central difference approximations of the compressible navier stokes equations is reported a generalized splitting of the nonlinear convective terms is considered and energy preserving formulations are fully characterized by deriving a two parameter family of split forms previously developed formulations reported in literature are shown to be particular members of this family novel splittings are introduced and discussed as well furthermore the conservation properties yielded by different choices for the energy equation i e total and internal energy entropy are analyzed thoroughly it is shown that additional preserved quantities can be obtained through a suitable adaptive selection of the split form within the derived family local conservation of primary invariants which is a fundamental property to build high fidelity shock capturing methods is also discussed in the paper numerical tests performed for the taylor green vortex at zero viscosity fully confirm the theoretical findings and show that a careful choice of both the splitting and the energy formulation can provide remarkably robust and accurate results

compressible large eddy simulation of separation control on a wall mounted hump compressible large eddy simulations of turbulent flow over a wall mounted hump with active flow control are performed and compared with previous experiments the flow is characterized by the unsteady separation before the steep trailing edge which naturally reattaches downstream of the hump to form an unsteady turbulent separation bubble the low mach number large eddy simulation demonstrated a good prediction of surface pressure coefficient separation bubble length and velocity profiles compared with experiments the effect of compressibility on the baseline flow is documented and analyzed and is found to increase the separation bubble size due to a reduced growth rate control is applied just before the natural separation point via steady suction and zero net mass flux oscillatory forcing and steady suction is shown to be more effective in decreasing the size of the separation bubble and pressure drag for the control parameters investigated controlled flow at a compressible subsonic mach number is applied and found to be slightly less effective than the same control parameters at low mach numbers

locally corrected spectral methods and overdetermined elliptic systems we present fast locally corrected spectral methods for linear constant coefficient elliptic systems of partial differential equations in d dimensional periodic geometry first arbitrary second order elliptic systems are converted to overdetermined first order systems overdetermination preserves ellipticity while first order systems eliminate mixed derivatives resolve convection diffusion conflicts and simplify derivative computations second a periodic fundamental solution is derived by fourier analysis and mollified for rapid convergence independent of the regularity of the elliptic problem third a new ewald summation technique for first order elliptic systems locally corrects the mollified solution to achieve high order accuracy we also discuss second kind boundary integral equations based on single layer potentials formed with the mollified and corrected fundamental solution which form a useful toolkit for solving general elliptic boundary value problems in general domains the resulting spectral methods provide highly accurate solutions and derivatives for periodic problems

stabilized non dissipative approximations of euler equations in generalized curvilinear coordinates we discuss stabilization strategies for finite difference approximations of the compressible euler equations in generalized curvilinear coordinates that do not rely on explicit upwinding or filtering of the physical variables our approach rather relies on a skew symmetric like splitting of the convective derivatives that guarantees preservation of kinetic energy in the semi discrete low mach number limit a locally conservative formulation allows efficient implementation and easy incorporation into existing compressible flow solvers the validity of the approach is tested for benchmark flow cases including the propagation of a cylindrical vortex and the head on collision of two vortex dipoles the tests support high accuracy and superior stability over conventional central discretization of the convective derivatives the potential use for dns les of turbulent compressible flows in complex geometries is discussed

a novel concept for the design of immersed interface methods the objective of this paper is to present a novel robust high order accurate immersed interface method iim for advection diffusion type equations in contrast to other immersed methods that were designed for consistency and accuracy with a posteriori check of the numerical stability we combine local taylor series expansion at irregular grid points with a local stability constraint as part of the design process stability investigations of the iim are employed to demonstrate that the local stability constraint is sufficient for obtaining a globally stable method as long as the neumann number is less than its limiting value one of the key aspects of this iim is that the irregular finite difference stencils can be isolated from the rest of the computational domain to validate our novel immersed interface approach two dimensional and three dimensional test cases for model equations are presented in addition this method is applied to the incompressible navier stokes equations to conduct stability investigations of a boundary layer flow over a rough surface and for investigations of pulsatile stenotic flows stability investigations of wall bounded flows are challenging for immersed methods because the near wall accuracy is important for correctly capturing the characteristics of the hydrodynamic instability mechanisms in particular regarding the wave relation between the wave velocity components close to the wall

truncation error analysis of the rotational form for the convective terms in the navier stokes equation the truncation error of the rotational form for the convective terms in the navier stokes equation is examined in the direct numerical simulation dns of the fully developed turbulent channel flow in which the low order finite difference method was used for the partial derivatives in the wall normal direction an estimate of the truncation error using the taylor expansion revealed that this truncation error term is comparable to the rotational stress generation term represented by the nonlineark model in the reynolds averaged turbulence models in the governing equations for the reynolds shear stress and the normal stress due to the coriolis force term acting in a channel flow rotating about the spanwise axis the effective angular velocity due to the truncation error term was dependent on the distance from the wall and the turbulence was reduced on both sides of the walls analogously to the laminarization of turbulence on the suction side of the conventional rotating channel flow this analogy was further assessed in the dns of the channel flow rotating with this effective angular velocity the time development of the wall friction velocity was similar to that obtained using the rotational form

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

an implicit forcing immersed boundary method for simulating viscous flows in irregular domains abstract we present a method for solving the incompressible navier stokes equations in irregular domains these equations are discretized using finite difference method in a uniform cartesian grid stationary rigid boundaries are embedded in the cartesian grid and singular forces are applied at the rigid boundaries to impose the no slip conditions the singular forces are then distributed to the nearby cartesian grid points using the immersed boundary method in the present work the singular forces are computed implicitly by solving a small system of equations at each time step this system of equations is derived from a second order projection method the main advantage of this method is that it imposes the no slip boundary condition exactly and avoids the need for small time step to maintain stability the ability of the method to simulate viscous flows in irregular domains is demonstrated by applying to 2 dimensional flows past a circular cylinder multiple rigid obstacles and 3 dimensional flow past a sphere

augmented lagrangian preconditioner for large scale hydrodynamic stability analysis abstract hydrodynamic linear stability analysis of large scale three dimensional configurations is usually performed with a time stepping approach based on the adaptation of existing solvers for the unsteady incompressible navier stokes equations we propose instead to solve the nonlinear steady equations with the newton method and to determine the largest growth rate eigenmodes of the linearized equations using a shift and invert spectral transformation and a krylov schur algorithm the solution of the shifted linearized navier stokes problem which is the bottleneck of this approach is computed via an iterative krylov subspace solver preconditioned by the modified augmented lagrangian mal preconditioner benzi et al 2011 the well known efficiency of this preconditioned iterative strategy for solving the real linearized steady state equations is assessed here for the complex shifted linearized equations the effect of various numerical and physical parameters is investigated numerically on a two dimensional flow configuration confirming the reduced number of iterations over state of the art steady state and time stepping based preconditioners a parallel implementation of the steady navier stokes and eigenvalue solvers developed in the freefem language suitably interfaced with the petsc slepc libraries is described and made openly available to tackle three dimensional flow configurations its application on a small scale three dimensional problem shows the good performance of this iterative approach over a direct l u factorization strategy in regards of memory and computational time on a large scale three dimensional problem with 75 million unknowns a 80 parallel efficiency on 256 up to 2048 processes is obtained

entropy splitting for high order numerical simulation of compressible turbulence a stable high order numerical scheme for direct numerical simulation dns of shock free compressible turbulence is presented the method is applicable to general geometries it contains no upwinding artificial dissipation or filtering instead the method relies on the stabilizing mechanisms of an appropriate conditioning of the governing equations and the use of compatible spatial difference operators for the interior points interior scheme as well as the boundary points boundary scheme an entropy splitting approach splits the inviscid flux derivatives into conservative and nonconservative portions the spatial difference operators satisfy a summation by parts condition leading to a stable scheme combined interior and boundary schemes for the initial boundary value problem using a generalized energy estimate a laplacian formulation of the viscous and heat conduction terms on the right hand side of the navier stokes equations is used to ensure that any tendency to odd even decoupling associated with central schemes can be countered by the fluid viscosity the resulting methods are able to minimize the spurious high frequency oscillations associated with pure central schemes especially for long time integration applications such as dns for validation purposes the methods are tested in a dns of compressible turbulent plane channel flow at low values of friction mach number where reference turbulence data bases exist it is demonstrated that the methods are robust in terms of grid resolution and in good agreement with published channel data accurate turbulence statistics can be obtained with moderate grid sizes stability limits on the range of the splitting parameter are determined from numerical tests

stable and accurate schemes for the compressible navier stokes equations minimal stencil width discretizations of combined mixed and non mixed second order derivatives are analyzed with respect to accuracy and stability we show that these discretizations lead to stability for cauchy problems with a careful boundary treatment we also show that the stability holds for initial boundary value problems the analysis is verified by numerical simulations of burgers and navier stokes equations in two and three space dimensions

l functions of symmetric powers of kloosterman sums unit root l functions and p adic estimates the l function of symmetric powers of classical kloosterman sums is a polynomial whose degree is now known as well as the complex absolute values of the roots in this paper we provide estimates for the p adic absolute values of these roots our method is indirect we first develop a dwork type p adic cohomology theory for the two variable infinite symmetric power l function associated to the kloosterman family and then study p adic estimates of the eigenvalues of frobenius a continuity argument then provides the desired p adic estimates

high order fluxes for conservative skew symmetric like schemes in structured meshes developing high order non dissipative schemes is an important research task for both steady and unsteady flow computations we take as a starting point the built in de aliasing property of the discretized skew symmetric form for the non linear terms of the navier stokes equations recalled in kravchenko and moin 1 two families of high order locally conservative schemes matching this discretized skew symmetric form are considered and rewritten in terms of telescopic fluxes for both finite difference and finite volume approximations in the context of compressible flows the jameson s scheme 2 is shown to be the second order member of larger families of skew symmetric like centered schemes the fourth order finite volume and finite difference and the sixth order finite difference schemes which belong to this family are provided the proposed schemes are extended to shock capturing schemes either by modifying the jameson s artificial viscosity or by hybriding the centered flux with weno 3 fluxes an adapted interpolation is proposed to extend the use of the proposed schemes to non regular grids several tests are provided showing that the conjectured order is properly recovered even with irregular meshes and that the shock capturing properties allow us to improve the second order results for standard test cases the improvement due to fourth order is then confirmed for the estimation of the growth of two ts waves and three crow instability dimensional unstable modes for both confined and free shear flows the last application concerns the steady computation using the spalart allmaras model of a separated boundary layer it confirms that the use of a high order scheme improves the results even in this type of steady applications

an entropy stable nodal discontinuous galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry we design an arbitrary high order accurate nodal discontinuous galerkin spectral element approximation for the non linear two dimensional shallow water equations with non constant possibly discontinuous bathymetry on unstructured possibly curved quadrilateral meshes the scheme is derived from an equivalent flux differencing formulation of the split form of the equations we prove that this discretization exactly preserves the local mass and momentum furthermore combined with a special numerical interface flux function the method exactly preserves the mathematical entropy which is the total energy for the shallow water equations by adding a specific form of interface dissipation to the baseline entropy conserving scheme we create a provably entropy stable scheme that is the numerical scheme discretely satisfies the second law of thermodynamics finally with a particular discretization of the bathymetry source term we prove that the numerical approximation is well balanced we provide numerical examples that verify the theoretical findings and furthermore provide an application of the scheme for a partial break of a curved dam test problem

comparison of conservative and rotational forms in large eddy simulation of turbulent channel flow abstract fully developed turbulent channel flow is studied numerically using large eddy simulation les with a fourier finite difference method a prominent feature of this paper lies in the use of a conservative form of the arakawa type for the convective terms in the navier stokes equations the results are compared with earlier ones of moin and kim 1 based on the rotational form of the convective terms noticeable differences are found in quantities such as turbulence intensities two point correlations etc the difference is most prominent in the grid scale portion of the turbulent shear stress the present results agree well with the recent direct simulation of a mildly curved channel flow by moser and moin 2 using the fourier chebyshev polynomial expansions an estimate is made of the error inherent in the rotational form combined with the second order central finite difference method the relationship between the energy production mechanism and helicity is pointed out

statistical condition estimation for linear systems the standard approach to measuring the condition of a linear system compresses all sensitivity information into one number thus a loss of information can occur in situations in which the standard condition number with respect to inversion does not accurately reflect the actual sensitivity of a solution or particular entries of a solution it is shown that a new method for estimating the sensitivity of linear systems addresses these difficulties the new procedure measures the effects on the solution of small random changes in the input data and by properly scaling the results obtains reliable condition estimates for each entry of the computed solution moreover this approach which is referred to as small sample statistical condition estimation is no more costly than the standard 1 norm or power method 2 norm condition estimates and it has the advantage of considerable flexibility for example it easily accommodates restrictions on or structure associated with allowable perturbations the method also has a rigorous statistical theory available for the probability of accuracy of the condition estimates however it gives no estimate of an approximate null vector for nearly singular systems the theory of this approach is discussed along with several illustrative examples