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

entropy stable high order discontinuous galerkin methods for ideal compressible mhd on structured meshes abstract we present a discontinuous galerkin dg scheme with suitable quadrature rules 15 for ideal compressible magnetohydrodynamic mhd equations on structural meshes the semi discrete scheme is analyzed to be entropy stable by using the symmetrizable version of the equations as introduced by godunov 32 the entropy stable dg framework with suitable quadrature rules 15 the entropy conservative flux in 14 inside each cell and the entropy dissipative approximate godunov type numerical flux at cell interfaces to make the scheme entropy stable the main difficulty in the generalization of the results in 15 is the appearance of the non conservative source terms added in the modified mhd model introduced by godunov 32 which do not exist in the general hyperbolic system studied in 15 special care must be taken to discretize these source terms adequately so that the resulting dg scheme satisfies entropy stability total variation diminishing bounded tvd tvb limiters and bound preserving limiters are applied to control spurious oscillations we demonstrate the accuracy and robustness of this new scheme on standard mhd examples

grid refinement for lattice bgk models lattice boltzmann models proposed at the end of the 1980s as the noise free version of lattice gas models are based on gas kinetic representation of fluid flow their recent modifications the lattice bgk models provide especially simple effective and stable algorithms for the solution of hydrodynamical problems a local second order grid refinement scheme for the lattice bgk model is proposed in this work the refinement scheme and a boundary fitting scheme for complicated geometries are applied to simulate a benchmark problem of flow past a cylinder in a channel with small and moderate reynolds numbers

why nonconservative schemes converge to wrong solutions error analysis this paper attempts to give a qualitative and quailtitative description of the numerical error introduced by using finite difference schemes in nonconservative form for scalar conservation laws we show that these schemes converge strongly in li norm to the solution of an inhomogeneous conservation law containing a borel measure source term moreover we analyze the properties of this borel measure and derive a sharp estimate for the l1 error between the limit function given by the scheme and the correct solution in general the measure source term is of the order of the entropy dissipation measure associated with the scheme in certain cases the error can be small for short times which makes it difficult to detect numerically but generically such an error will grow in time and this would lead to a large error for large time calculations finally we show that a local correction of any high order accurate scheme in nonconservative form is sufficient to ensure its convergence to the correct solution

on designing observers for time delay systems with non linear disturbances in this paper the observer design problem is studied for a class of time delay non linear systems the system under consideration is subject to delayed state and non linear disturbances the time delay is allowed to be time varying and the non linearities are assumed to satisfy global lipschitz conditions the problem addressed is the design of state observers such that for the admissible time delay as well as non linear disturbances the dynamics of the observation error is globally exponentially stable an effective algebraic matrix inequality approach is developed to solve the non linear observer design problem specifically some conditions for the existence of the desired observers are derived and an explicit expression of desired observers is given in terms of some free parameters a simulation example is included to illustrate the practical applicability of the proposed theory

lattice boltzmann method for the convection diffusion equation in curvilinear coordinate systems a lattice boltzmann method for simulating convection and diffusion using a curvilinear grid system is presented the proposed method does not require an interpolation or coarse graining procedure and thus maintains the algorithmic simplicity of the original lattice boltzmann scheme the lattice boltzmann scheme is based on uniformly distributed lattice points in a transformed coordinate system and the apparent anisotropy of diffusion that arises due to the coordinate transformation is properly handled using the multiple relaxation time collision operator an asymptotic analysis of the lattice boltzmann equation shows that the proposed method appropriately reproduces the transformed convection diffusion equation several specific problems are numerically analyzed in order to validate the proposed method including an axially symmetric two dimensional problem in which the diffusion flux at an oblate hemispheroid is simulated using a body fitted orthogonal curvilinear grid system

fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains a time dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered by applying the energy method well posed boundary conditions for the continuous problem are derived summation by parts sbp operators for the space and time discretization together with a weak imposition of boundary and initial conditions using simultaneously approximation terms sats lead to a provable fully discrete energy stable conservative finite difference scheme we show how to construct a time dependent sat formulation that automatically imposes boundary conditions when and where they are required we also prove that a uniform flow field is preserved i e the numerical geometric conservation law ngcl holds automatically by using sbp sat in time and space the developed technique is illustrated by considering an application using the linearized euler equations the sound generated by moving boundaries numerical calculations corroborate the stability and accuracy of the new fully discrete approximations

robust approaches to handling complex geometries with galerkin difference methods abstract the galerkin difference gd basis is a set of continuous piecewise polynomials defined using a finite difference like grid of degrees of freedom the one dimensional gd basis functions are naturally extended to multiple dimensions using a tensor product construction on quadrilateral elements the gd basis can be used to define the solution space for a discontinuous galerkin finite element discretization of partial differential equations in this work we propose two approaches to handling complex geometries within this setting 1 using nonconforming curvilinear gd elements and 2 coupling affine gd elements with curvilinear simplicial elements in both cases the semidiscrete discontinuous galerkin method is provably energy stable even when variational crimes are committed additionally for both element types a weight adjusted mass matrix is used which ensures that only the reference mass matrix must be inverted we also present sufficient conditions on the treatment of metric terms for the curvilinear nonconforming gd elements to ensure that the scheme is both constant preserving and conservative numerical experiments confirm the stability results and demonstrate the accuracy of the coupled schemes

free surface neumann boundary condition for the advection diffusion lattice boltzmann method the main objective of this paper is the derivation and validation of a free surface neumann boundary condition for the advection diffusion lattice boltzmann method most literature boundary conditions are applied on straight walls and sometimes on curved geometries or fixed free surfaces but dynamic free surfaces especially with fluid motion in normal direction are hardly addressed a chapman enskog expansion is the basis for the derivation of the advection diffusion equation using the advection diffusion lattice boltzmann method and the bgk collision operator for this numerical scheme a free surface neumann boundary condition with no flux in normal direction to the free surface is derived finally the boundary condition is validated in different static and dynamic test scenarios including a detailed view on the conservation of the diffusive scalar the normal and tangential flux components to the free surface and the accuracy the validation scenarios reveal the superiority of the new approach to the compared literature schemes especially for arbitrary fluid motion

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

boundary conditions of the lattice boltzmann method for convection diffusion equations in this paper we employ an asymptotic analysis technique and construct two boundary schemes accompanying the lattice boltzmann method for convection diffusion equations with general robin boundary conditions one scheme is for straight boundaries with the boundary points locating at any distance from the lattice nodes and has second order accuracy the other is for curved boundaries has only first order accuracy and is much simpler than the existing schemes unlike those in the literature our schemes involve only the current lattice node such a single node boundary schemes are highly desirable for problems with complex geometries the two schemes are validated numerically with a number of examples the numerical results show the utility of the constructed schemes and very well support our theoretical predications

asymptotic analysis of multiple relaxation time lattice boltzmann schemes for mixture modeling a new lattice boltzmann model for simulating ideal mixtures has been developed by means of the multiple relaxation time mrt approach when compared with the previous single relaxation time srt formulation of the same model based on the continuous kinetic theory the new model offers the possibility to independently tune the mutual diffusivity and the effects of cross collisions on the effective stress tensor the additional degrees of freedom due to the increased set of relaxation time constants used for modeling the cross collisions allow us to match the experimental data on macroscopic transport coefficients two different integration rules i e the forward euler and the modified mid point integration rule were used in order to numerically integrate the developed model unfortunately the simpler forward euler integration rule violates the mass conservation and there is no way to fix the problem by changing the definition of the macroscopic velocity on the other hand a small correction has been purposely designed for compensating this error by means of the mid point integration rule some numerical simulations are reported for proving the effectiveness of the proposed corrective factor for the considered application the asymptotic analysis recently suggested as an effective tool for analyzing the macroscopic equations corresponding to the lattice boltzmann schemes offers a remarkable advantage in comparison with the classical chapman enskog technique because it easily deals with leading terms in the distribution functions which are no more maxwellian

theory of the lattice boltzmann method two fluid model for binary mixtures a two fluid lattice boltzmann model for binary mixtures is developed the model is derived formally from kinetic theory by discretizing two fluid boltzmann equations in which mutual collisions and self collisions are treated independently in the resulting lattice boltzmann model viscosity and diffusion coefficients can be varied independently by a suitable choice of mutual and self collision relaxation time scales further the proposed model can simulate miscible and immiscible fluids by changing the sign of the mutual collision term this is in contrast to most existing single fluid lattice boltzmann models that employ a single relaxation time scale and hence are restricted to unity prandtl and schmidt numbers the extension of binary mixing model to multiscalar mixing is quite straightforward

low and high order accurate boundary conditions the present contribution focuses on the accuracy of reflection type boundary conditions in the stokesbrinkmandarcy modeling of porous flows solved with the lattice boltzmann method lbm which we operate with the two relaxation time trt collision and the brinkman force based scheme bf called bf trt scheme in parallel we compare it with the stokesbrinkmandarcy linear finite element method fem where the dirichlet boundary conditions are enforced on grid vertices in bulk both bf trt and fem share the same defect in their discretization a correction to the modeled brinkman equation appears given by the discrete laplacian of the velocity proportional resistance force this correction modifies the effective brinkman viscosity playing a crucial role in the triggering of spurious oscillations in the bulk solution while the exact form of this defect is available in lattice aligned straight or diagonal flows in arbitrary flow lattice orientations its approximation is constructed at boundaries we verify that such a brinkman viscosity correction has an even more harmful impact already at the first order it shifts the location of the no slip wall condition supported by traditional lbm boundary schemes such as the bounce back rule for that reason this work develops a new class of boundary schemes to prescribe the dirichlet velocity condition at an arbitrary wall boundary node distance and that supports a higher order accuracy in the accommodation of the trt brinkman solutions for their modeling we consider the standard bf scheme and its improved version called ibf this latter is generalized in this work to suppress or to reduce the viscosity correction in arbitrarily oriented flows our framework extends the one and two point families of linear and parabolic link wise boundary schemes respectively called b li and b mli which avoid the interference of the brinkman viscosity correction in their closure relations the performance of lbm and fem is thoroughly evaluated in three benchmark tests which are run throughout three distinctive permeability regimes the first configuration is a horizontal porous channel studied with a symbolic approach where we construct the exact solutions of fem and bf ibf with different boundary schemes the second problem refers to an inclined porous channel flow which brings in as new challenge the formation of spurious boundary layers in lbm that is numerical artefacts that arise due to a deficient accommodation of the bulk solution by the low accurate boundary scheme the third problem considers a porous flow past a periodic square array of solid cylinders which intensifies the previous two tests with the simulation of a more complex flow pattern the ensemble of numerical tests provides guidelines on the effect of grid resolution and the trt free collision parameter over the accuracy and the quality of the velocity field spanning from stokes to darcy permeability regimes it is shown that with the use of the high order accurate boundary schemes the simple uniform mesh based trt lbm formulation can even surpass the accuracy of fem employing hardworking body fitted meshes

consistent second order boundary implementations for convection diffusion lattice boltzmann method in this study an alternative second order boundary scheme is proposed under the framework of the convection diffusion lattice boltzmann lb method for both straight and curved geometries with the proposed scheme boundary implementations are developed for the dirichlet neumann and linear robin conditions in a consistent way the chapman enskog analysis and the hermite polynomial expansion technique are first applied to derive the explicit expression for the general distribution function with second order accuracy then the macroscopic variables involved in the expression for the distribution function is determined by the prescribed macroscopic constraints and the known distribution functions after streaming see the paragraph after eq 29 for the discussions of the streaming step in lb method after that the unknown distribution functions are obtained from the derived macroscopic information at the boundary nodes for straight boundaries boundary nodes are directly placed at the physical boundary surface and the present scheme is applied directly when extending the present scheme to curved geometries a local curvilinear coordinate system and first order taylor expansion are introduced to relate the macroscopic variables at the boundary nodes to the physical constraints at the curved boundary surface in essence the unknown distribution functions at the boundary node are derived from the known distribution functions at the same node in accordance with the macroscopic boundary conditions at the surface therefore the advantages of the present boundary implementations are i the locality i e no information from neighboring fluid nodes is required ii the consistency i e the physical boundary constraints are directly applied when determining the macroscopic variables at the boundary nodes thus the three kinds of conditions are realized in a consistent way it should be noted that the present focus is on two dimensional cases and theoretical derivations as well as the numerical validations are performed in the framework of the two dimensional five velocity lattice model

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

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

lattice boltzmann models based on half range gauss hermite quadratures we discuss general features of thermal lattice boltzmann models based on half range gauss quadratures specialising to the half range gauss hermite and gauss laguerre cases the main focus of the paper is on the construction of high order half range hermite lattice boltzmann hhlb models the performance of the hhlb models is compared with that of laguerre lattice boltzmann llb and full range hermite lattice boltzmann hlb models by conducting convergence tests with respect to the quadrature order on stationary profiles of the particle number density macroscopic velocity temperature and heat fluxes in the two dimensional couette flow the bhatnagar gross krook bgk collision term is used throughout the paper to reduce the computational costs of the numerical simulations we use mixed lattice boltzmann models constructed using different quadrature methods on each cartesian axis for kn 0 01 the hlb models require the least number of velocities to satisfy our convergence test when kn 0 05 the hlb models are outperformed in terms of number of velocities employed by both the llb and the hhlb models moreover we find that the hhlb models require less quadrature points than the llb models at all tested values of kn which we attribute to the maxwellian form of the weight function for the half range hermite polynomials

nonlinear observers with linearizable error dynamics we present a new method for designing asymptotic observers for a class of nonlinear systems the error between the state of the system and the state of the observer in appropriate coordinates evolves linearly and can be made to decay arbitrarily exponentially fast

a well balanced and entropy conservative discontinuous galerkin spectral element method for the shallow water equations in this work we design an arbitrary high order accurate nodal discontinuous galerkin spectral element type method for the one dimensional shallow water equations the novel method uses a skew symmetric formulation of the continuous problem we prove that this discretisation exactly preserves the local mass and momentum furthermore we show that combined with a special numerical interface flux function the method exactly preserves the entropy which is also the total energy for the shallow water equations finally we prove that the surface fluxes the skew symmetric volume integrals and the source term are well balanced numerical tests are performed to demonstrate the theoretical findings

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

analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice boltzmann bgk model in this paper we analytically solve the velocity of the lattice boltzmann bgk equation lbgk for several simple flows the analysis provides a framework to theoretically analyze various boundary conditions in particular the analysis is used to derive the slip velocities generated by various schemes for the nonslip boundary condition we find that the slip velocity is zero as long as f e 0 at boundaries no matter what combination of distributions is chosen the schemes proposed by nobleet al and by inamuroet al yield the correct zeroslip velocity while some other schemes such as the bounce back scheme and the equilibrium distribution scheme would inevitably generate a nonzero slip velocity the bounce back scheme with the wall located halfway between a flow node and a bounce back node is also studied for the simple flows considered and is shown to produce results of second order accuracy the momentum exchange at boundaries seems to be highly related to the slip velocity at boundaries to be specific the slip velocity is zero only when the momentum dissipated by boundaries is equal to the stress provided by fluids

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

linearization by output injection and nonlinear observers abstract observers can easily be constructed for those nonlinear systems which can be transformed into a linear system by change of state variables and output injection necessary and sufficient conditions for the existence of such a transformation are given

consistent lattice boltzmann modeling of low speed isothermal flows at finite knudsen numbers in slip flow regime ii application to curved boundaries gaseous flows inside microfluidic devices often fall in the slip flow regime according to this theoretical description the navier stokes model remains applicable in bulk while at solid walls a slip velocity boundary model shall be considered physically it is well established that to properly account for the wall curvature the wall slip velocity must be determined by the shear stress rather than the normal component of the velocity derivative alone as commonly applied to planar surfaces it follows that the numerical transcription of this type of boundary condition is generally a challenging task for standard computational fluid dynamics cfd techniques this paper aims to show that the realization of the slip velocity condition on arbitrarily shaped boundaries can be accomplished in a natural way with the lattice boltzmann method lbm to substantiate this conclusion this work undertakes the following three studies first we examine the conditions under which the generic reflection type boundary rules used by lbm become consistent models for the slip velocity boundary condition this effort makes use of the second order chapman enskog expansion method where we address both planar and curved boundaries the analysis also clarifies the capabilities and limitations behind the considered reflection type slip schemes second we revisit the family of parabolic accurate lbm slip boundary schemes originally formulated in phys rev e 96 013311 2017 2470 004510 1103 physreve 96 013311 on the basis of the multireflection framework and discuss their characteristics when operating on curved boundaries as well as the limitations of other less accurate lbm slip boundary formulations such as the linearly accurate slip schemes and the widely popular kinetic based boundary schemes in addition we also discuss the numerical stability of the parabolic slip schemes previously developed providing an heuristic strategy to improve their stable range of operation third we evaluate the performance of the several slip boundary schemes debated in this paper the numerical tests correspond to two classical 2d benchmark flow problems of slip over non planar solid surfaces namely i the velocity profile of the cylindrical couette flow and ii the permeability of a slow rarefied gas over a periodic array of circular cylindrical obstacles the obtained numerical results confirm the competitiveness of the lbm when equipped with slip boundary schemes of parabolic accuracy as cfd tool to simulate slippage phenomena over arbitrarily non planar surfaces indeed although operating on a simple uniform mesh discretization the lbm yields a similar or even superior level of accuracy compared to state of the art fem simulations conducted on hardworking body fitted meshes this conclusion establishes the lbm as a very appealing cfd technique for simulating microfluidic flows in the slip flow regime a result that deserves further exploration in future studies

on discretely entropy stable weight adjusted discontinuous galerkin methods curvilinear meshes abstract we construct entropy conservative and entropy stable discontinuous galerkin dg discretizations for time dependent nonlinear hyperbolic conservation laws on curvilinear meshes the resulting schemes preserve a semi discrete quadrature approximation of a continuous global entropy inequality the proof requires the satisfaction of a discrete geometric conservation law which we enforce through an appropriate polynomial approximation we extend the construction of entropy conservative and entropy stable dg schemes to the case when high order curvilinear mass matrices are approximated using low storage weight adjusted approximations and describe how to retain global conservation properties under such an approximation for certain types of curvilinear meshes these weight adjusted approximations deliver optimal rates of convergence finally the high order accuracy local conservation and discrete conservation or dissipation of entropy for under resolved solutions are verified through numerical experiments for the compressible euler equations on triangular and tetrahedral meshes

a new multigrid formulation for high order finite difference methods on summation by parts form multigrid schemes for high order finite difference methods on summation by parts form are studied by comparing the effect of different interpolation operators by using the standard linear prolonga

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

nonlinear observers for two classesof perturbed systems in this paper we study two important classes of nonlinear systems with perturbation namely perturbation with output injection and perturbation of pure feedback systems with sufficiently good initial error estimates and sufficiently restricted external inputs and perturbation parameter nonlinear observers for each class of perturbed systems are designed so that the error between the exact states and the estimated states goes to zero our simulation results illustrate the applicability of our technique in building nonlinear observers for these two classes of perturbed systems

truncation effect on taylor aris dispersion in lattice boltzmann schemes the taylor dispersion in parabolic velocity field provides a well known benchmark for advection diffusion ade schemes and serves as a first step towards accurate modeling of the high order non gaussian effects in heterogeneous flow while applying the lattice boltzmann ade two relaxation times trt scheme for a transport with given peclet number pe one should select six free tunable parameters namely i molecular diffusion scale equilibrium parameter ii three families of equilibrium weights assigned to the terms of mass velocity and numerical diffusion correction and iii two relaxation rates we analytically and numerically investigate the respective roles of all these degrees of freedom in the accuracy and stability in the evolution of a gaussian plume for this purpose the third and fourth order transient multi dimensional analysis of the recurrence equations of the trt ade scheme is extended for a spatially variable velocity field the key point is in the coupling of the truncation and taylor dispersion analysis which allows us to identify the second order numerical correction k t to taylor dispersivity coefficient k t the procedure is exemplified for a straight poiseuille flow where k t is given in a closed analytical form in equilibrium and relaxation parameter spaces the predicted longitudinal dispersivity is in excellent agreement with the numerical experiments over a wide parameter range in relatively small pe range the relative dispersion error increases with peclet number this deficiency reduces in the intermediate and high pe range where it becomes pe independent and velocity amplitude independent eliminating k t by a proper parameter choice and employing specular reflection for zero flux condition on solid boundaries the d2q9 trt ade scheme may reproduce the taylor aris result quasi exactly from very coarse to fine grids and from very small to arbitrarily high peclet numbers since free tunable product of two eigenfunctions also controls stability of the model the validity of the analytically established von neumann stability diagram is examined in poiseuille profile the simplest coordinate stencil subclass which is the d2q5 trt bounce back scheme demonstrates the best performance and achieves the maximum accuracy for most stable relaxation parameters closed analytical form is derived for truncation dispersivity of the d2q9 advection diffusion lbm scheme in higher peclet range its relative contribution to taylor result becomes pe and velocity independent numerical dispersivity vanishes for equilibrium velocity weight dependent eigenfunction product in trt collision analytically predicted longitudinal dispersivity is in excellent agreement with the numerical experiments the d2q5 trt bounce back scheme achieves maximum accuracy for most stable relaxation parameters

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

canonical form observer design for non linear time variable systems an observer of canonical phase variable form for non linear time variable systems is introduced the development of this non linear time variable form requires regularity of the non linear time variable observability matrix of the system from the relationships derived during the development it follows that a non linear time variable observer can be dimensioned by an eigenvalue assignment with respect to the canonical state coordinates if a linearization of system non linearities about the reconstructed state trajectories is permissible this is an assumption similar to that of the extended kalman filter based on a linearization about the current estimate

lattice boltzmann model for nonlinear convection diffusion equations a lattice boltzmann model for convection diffusion equation with nonlinear convection and isotropic diffusion terms is proposed through selecting equilibrium distribution function properly the model can be applied to the common real and complex valued nonlinear evolutionary equations such as the nonlinear schrodinger equation complex ginzburg landau equation burgers fisher equation nonlinear heat conduction equation and sine gordon equation by using a real and complex valued distribution function and relaxation time detailed simulations of these equations are performed and it is found that the numerical results agree well with the analytical solutions and the numerical solutions reported in previous studies

multiple relaxation time lattice boltzmann model for the convection and anisotropic diffusion equation a lattice boltzmann model with a multiple relaxation time mrt collision operator for the convection diffusion equation is presented the model uses seven discrete velocities in three dimensions d3q7 model the off diagonal components of the relaxation time matrix which originate from the rotation of the principal axes enable us to take into account full anisotropy of diffusion an asymptotic analysis of the model equation with boundary rules for the dirichlet and neumann type specified flux conditions is carried out to show that the model is first and second order accurate in time and space respectively the results of the analysis are verified by several numerical examples it is also shown numerically that the error of the mrt model is less sensitive to the variation of the relaxation time coefficients than that of the classical bgk model in addition an alternative treatment for the neumann type boundary condition that improves the accuracy on a curved boundary is presented along with a numerical example of a spherical boundary

second order curved boundary treatments of the lattice boltzmann method for convection diffusion equations in this paper we present a kind of second order curved boundary treatments for the lattice boltzmann method solving two dimensional convection diffusion equations with general nonlinear robin boundary conditions the key idea is to derive approximate boundary values or normal derivatives on computational boundaries with second order accuracy by using the prescribed boundary condition once the approximate information is known the second order bounce back schemes can be perfectly adopted our boundary treatments are validated with a number of numerical examples the results show the utility of our boundary treatments and very well support our theoretical predications on the second order accuracy thereof the idea is quite universal it can be directly generalized to 3 dimensional problems multiple relaxation time models and the navier stokes equations

lattice boltzmann method based on dual mrt model for three dimensional natural convection and entropy generation in cuo water nanofluid filled cuboid enclosure included with discrete active walls abstract in the present study the three dimensional natural convection and entropy generation in a cuboid enclosure included with various discrete active walls is analyzed using lattice boltzmann method the enclosure is filled with cuo water nanofluid to predict thermo physical properties dynamic viscosity and thermal conductivity of cuo water nanofluid the kkl model is applied to consider the effect of brownian motion on nanofluid properties in lattice boltzmann simulation two different mrt models are used to solve the problem the d3q7 mrt model is used to solve the temperature filed and the d3q19 is employed to solve the fluid flow of natural convection within the enclosure the influences of different rayleigh numbers 1 0 3 r a 1 0 6 and solid volume fractions 0 0 04 and four different arrangements of discrete active walls on the fluid flow heat transfer total entropy generation local heat transfer irreversibility and local fluid friction irreversibility are presented comprehensively

right coprime factorizations and stabilization for nonlinear systems the problem of constructing right coprime factorizations which are based on the graph of an input output map instead of the bezout identity of nonlinear input output maps is considered the map is assumed to arise from a state variable realization with a fixed initial state the main result is that the existence of a stabilizing state feedback implies the existence of a right coprime factorization for the map the technique is illustrated by application to nonlinear systems which are affine in the control and have a controllable linear past and nonlinear systems which are feedback linearizable a notion of input output stability that requires a bound on the magnitude of the input signals is introduced methods of constructing such bounds are developed for a locally feedback linearizable system the problem of input bounds is transferred to the equivalent linear system this leads to a technique that allows state and input constraints for a feedback linearizable system to be mapped to the equivalent linear system

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

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

observers for nonlinear systems in steady state after constructing a nonlinear observer and applying state feedback depending on the estimated states to asymptotically stabilize the system we introduce new inputs of bounded norm to the system we then show that the parameters for the observer with sufficiently good initial error estimates can be chosen so that the error between states and estimates goes to zero particular remarks are appropriate for the class of global feedback linearizable systems where the bounds on the norm of the initial states and the amplitude of the new inputs can be as large as desired

local second order boundary methods for lattice boltzmann models a new way to implement solid obstacles in lattice boltzmann models is presented the unknown populations at the boundary nodes are derived from the locally known populations with the help of a second order chapman enskog expansion and dirichlet boundary conditions with a given momentum steady flows near a flat wall arbitrarily inclined with respect to the lattice links are then obtained with a third order error in particular couette and poiseuille flows are exactly recovered without the knudsen layers produced for inclined walls by the bounce back condition

three dimensional multi relaxation time mrt lattice boltzmann models for multiphase flow in this paper three dimensional 3d multi relaxation time mrt lattice boltzmann lb models for multiphase flow are presented in contrast to the bhatnagar gross krook bgk model a widely employed kinetic model in mrt models the rates of relaxation processes owing to collisions of particle populations may be independently adjusted as a result the mrt models offer a significant improvement in numerical stability of the lb method for simulating fluids with lower viscosities we show through the chapman enskog multiscale analysis that the continuum limit behavior of 3d mrt lb models corresponds to that of the macroscopic dynamical equations for multiphase flow we extend the 3d mrt lb models developed to represent multiphase flow with reduced compressibility effects the multiphase models are evaluated by verifying the laplace young relation for static drops and the frequency of oscillations of drops the results show satisfactory agreement with available data and significant gains in numerical stability

regular article an accurate curved boundary treatment in the lattice boltzmann method the lattice boltzmann equation lbe is an alternative kinetic method capable of solving hydrodynamics for various systems major advantages of the method are due to the fact that the solution for the particle distribution functions is explicit easy to implement and natural to parallelize because the method often uses uniform regular cartesian lattices in space curved boundaries are often approximated by a series of stairs that leads to reduction in computational accuracy in this work a second order accurate treatment of the boundary condition in the lbe method is developed for a curved boundary the proposed treatment of the curved boundaries is an improvement of a scheme due to o filippova and d hanel 1998 j comput phys 147 219 the proposed treatment for curved boundaries is tested against several flow problems 2 d channel flows with constant and oscillating pressure gradients for which analytic solutions are known flow due to an impulsively started wall lid driven square cavity flow and uniform flow over a column of circular cylinders the second order accuracy is observed with a solid boundary arbitrarily placed between lattice nodes the proposed boundary condition has well behaved stability characteristics when the relaxation time is close to 1 2 the zero limit of viscosity the improvement can make a substantial contribution toward simulating practical fluid flow problems using the lattice boltzmann method

multireflection boundary conditions for lattice boltzmann models we present a general framework for several previously introduced boundary conditions for lattice boltzmann models such as the bounce back rule and the linear and quadratic interpolations the objectives are twofold first to give theoretical tools to study the existing link type boundary conditions and their corresponding accuracy second to design boundary conditions for general flows which are third order kinetic accurate using these new boundary conditions couette and poiseuille flows are exact solutions of the lattice boltzmann models for a reynolds number mathrm re 0 stokes limit for arbitrary inclination with the lattice directions numerical comparisons are given for stokes flows in periodic arrays of spheres and cylinders linear periodic array of cylinders between moving plates and for navier stokes flows in periodic arrays of cylinders for mathrm re l200 these results show a significant improvement of the overall accuracy when using the linear interpolations instead of the bounce back reflection up to an order of magnitude on the hydrodynamics fields further improvement is achieved with the new multireflection boundary conditions reaching a level of accuracy close to the quasianalytical reference solutions even for rather modest grid resolutions and few points in the narrowest channels more important the pressure and velocity fields in the vicinity of the obstacles are much smoother with multireflection than with the other boundary conditions finally the good stability of these schemes is highlighted by some simulations of moving obstacles a cylinder between flat walls and a sphere in a cylinder

lattice boltzmann method for 3 d flows with curved boundary in this work we investigate two issues that are important to computational efficiency and reliability in fluid dynamic applications of the lattice boltzmann equation lbe 1 computational stability and accuracy of different lattice boltzmann models and 2 the treatment of the boundary conditions on curved solid boundaries and their 3 d implementations three athermal 3 d lbe models q15d3 q19d3 and q27d3 are studied and compared in terms of efficiency accuracy and robustness the boundary treatment recently developed by filippova and hanel 1998 j comp phys 147 219 and mei et al 1999 j comp phys 155 307 in 2 d is extended to and implemented for 3 d the convergence stability and computational efficiency of the 3 d lbe models with the boundary treatment for curved boundaries were tested in simulations of four 3 d flows 1 fully developed flows in a square duct 2 flow in a 3 d lid driven cavity 3 fully developed flows in a circular pipe and 4 a uniform flow over a sphere we found that while the 15 velocity 3 d q15d3 model is more prone to numerical instability and the q27d3 is more computationally intensive the q19d3 model provides a balance between computational reliability and efficiency through numerical simulations we demonstrated that the boundary treatment for 3 d arbitrary curved geometry has second order accuracy and possesses satisfactory stability characteristics

thermal boundary condition for the thermal lattice boltzmann equation a thermal boundary condition for a double population thermal lattice boltzmann equation tlbe is introduced and numerically demonstrated the unknown distribution population at the boundary node is decomposed into its equilibrium part and nonequilibrium parts and then the nonequilibrium part is approximated with a first order extrapolation of the nonequilibrium part of the populations at the neighboring fluid nodes numerical tests with dirichlet and neumann boundary constraints show that the numerical results of the tlbe together with the present boundary schemes agree well with the analytical solutions and those of the finite volume method

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

boundary conditions for thermal lattice boltzmann equation method we propose a thermal boundary condition treatment based on the bounce back idea and interpolation of the distribution functions for both the dirichlet and neumann normal derivative conditions in the thermal lattice boltzmann equation tlbe method the coefficients for the distribution functions involved are determined to satisfy the dirichlet or neumann condition with second order accuracy for the dirichlet condition there is an adjustable parameter in the treatment and three particular schemes are selected for demonstration while for the neumann condition the second order accurate scheme is unique when applied to inclined or curved boundaries the dirichlet condition treatment can be directly used while the neumann condition given in the normal direction of the boundary should be converted into derivative conditions in the discrete velocity directions of the tlbe model a spatially coupled formula relating the boundary temperature boundary normal heat flux and the distribution functions near the boundary is thus derived for the neumann problems on curved boundaries the applicability and accuracy of the present boundary treatments are examined with several numerical tests for which analytical solutions are available including the 2 dimensional 2 d steady state channel flow the 1 d transient heat conduction in an inclined semi infinite solid the 2 d steady state and transient heat conduction inside a circle and the 3 d steady state circular pipe flow while the dirichlet condition treatment leads to second order accuracy for the temperature field it only gives a first order accurate boundary heat flux because of the irregularity of the cuts by the curved boundary with the lattices with the neumann condition on the curved boundary the accuracy for the temperature field obtained is first order when the tangential temperature gradient on the boundary is decoupled second order convergence of the temperature field can be obtained with neumann conditions

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

a lattice boltzmann model for multiphase flows with large density ratio a lattice boltzmann model for simulating multiphase flows with large density ratios is described in this paper the method is easily implemented it does not require solving the poisson equation and does not involve the complex treatments of derivative terms the interface capturing equation is recovered without any additional terms as compared to other methods m r swift w r osborn j m yeomans lattice boltzmann simulation of liquid gas and binary fluid systems phys rev e 54 1996 5041 5052 t inamuro t ogata s tajima n konishi a lattice boltzmann method for incompressible two phase flows with large density differences j comput phys 198 2004 628 644 t lee c l lin a stable discretization of the lattice boltzmann equation for simulation of incompressible two phase flows at high density ratio j comput phys 206 2005 16 47 besides it requires less discrete velocities as a result its efficiency could be greatly improved especially in 3d applications it is validated by several cases a bubble in a stationary flow and the capillary wave the numerical surface tension obtained from the laplace law and the interface profile agrees very well with the respective analytical solution the method is further verified by its application to capillary wave and the bubble rising under buoyancy with comparison to other methods all the numerical experiments show that the present approach can be used to model multiphase flows with large density ratios

boundary scheme for linear heterogeneous surface reactions in the lattice boltzmann method a boundary scheme is proposed to treat the linear heterogeneous surface reaction i e general robin boundary condition for the lattice boltzmann method lbm in this study the basic idea of the present scheme is to compute the scalar variable gradient in the boundary condition with the moment of the nonequilibrium distribution functions the unknown distribution functions on the boundary can then be obtained on the basis of the given boundary condition for a straight wall where the lattice nodes are located on the boundaries both the scalar variable and its gradient can be expressed in terms of the distribution functions at the boundary node and the scheme is purely local the scheme is also extended to problems with a curved wall in which a linear extrapolation is employed to realize the exact boundary position a common feature of the two schemes lies in the easy treatment of the heterogenous reactions in comparison with existing methods a number of simulations are performed to test the accuracy of the schemes the results show that for flat walls the scheme can achieve second order accuracy while for curved walls the order of the accuracy is between 1 0 and 1 5

an entropy stable nodal discontinuous galerkin method for the resistive mhd equations part i theory and numerical verification abstract the first paper of this series presents a discretely entropy stable discontinuous galerkin dg method for the resistive magnetohydrodynamics mhd equations on three dimensional curvilinear unstructured hexahedral meshes compared to other fluid dynamics systems such as the shallow water equations or the compressible navier stokes equations the resistive mhd equations need special considerations because of the divergence free constraint on the magnetic field for instance it is well known that for the symmetrization of the ideal mhd system as well as the continuous entropy analysis a non conservative term proportional to the divergence of the magnetic field typically referred to as the powell term must be included as a consequence the mimicry of the continuous entropy analysis in the discrete sense demands a suitable dg approximation of the non conservative terms in addition to the ideal mhd terms this paper focuses on the resistive mhd equations our first contribution is a proof that the resistive terms are symmetric and positive definite when formulated in entropy space as gradients of the entropy variables which enables us to show that the entropy inequality holds for the resistive mhd equations this continuous analysis is the key for our dg discretization and guides the path for the construction of an approximation that discretely mimics the entropy inequality typically termed entropy stability our second contribution is a detailed derivation and analysis of the discretization on three dimensional curvilinear meshes the discrete analysis relies on the summation by parts property which is satisfied by the dg spectral element method dgsem with legendre gauss lobatto lgl nodes although the divergence free constraint is included in the non conservative terms the resulting method has no particular treatment of the magnetic field divergence errors which might pollute the solution quality our final contribution is the extension of the standard resistive mhd equations and our dg approximation with a divergence cleaning mechanism that is based on a generalized lagrange multiplier glm as a conclusion to the first part of this series we provide detailed numerical validations of our dgsem method that underline our theoretical derivations in addition we show a numerical example where the entropy stable dgsem demonstrates increased robustness compared to the standard dgsem