quantum algebras as quantizations of dual poisson lie groups a systematic computational approach for the explicit construction of any quantum hopf algebra uz g z starting from the lie bialgebra g that gives the first order deformation of the coproduct map z is presented the procedure is based on the well known quantum duality principle namely the fact that any quantum algebra can be viewed as the quantization of the unique poisson lie structure g g on the dual group g which is obtained by exponentiating the lie algebra g defined by the dual map from this perspective the coproduct for uz g is just the pull back of the group law for g and the poisson analogues of the quantum commutation rules for uz g are given by the unique poisson lie structure g on g whose linearization is the poisson analogue of the initial lie algebra g this approach is shown to be a very useful technical tool in order to solve the lie bialgebra quantization problem explicitly since once a lie bialgebra g is given the full dual poisson lie group g can be obtained either by applying standard poisson lie group techniques or by implementing the algorithm presented here with the aid of symbolic manipulation programs as a consequence the quantization of g will give rise to the full uz g quantum algebra provided that ordering problems are appropriately fixed through the choice of certain local coordinates on g whose coproduct fulfils a precise quantum symmetry property the applicability of this approach is explicitly demonstrated by reviewing the construction of several instances of quantum deformations of physically relevant lie algebras such as the 2 1 anti de sitter algebra so 2 2 and the poincare algebra in 3 1 dimensions

relativistic and newtonian κ space times the deformations of the galilei algebra and their associated noncommutative newtonian space times are investigated this is done by analyzing the possible nonrelativistic limits of an eleven generator pseudo extended poincare algebra p and their implications for the existence of a first order differential calculus the additional one form needed to achieve a consistent calculus on minkowski space is shown to be related to the additional central generator entering in the p hopf algebra in the process deformations of the extended galilei and galilei algebras are introduced which have respectively a cocycle and a bicrossproduct structure

solution of the implicitly discretised reacting flow equations by operator splitting abstract a non iterative method for handling the coupling of the implicitly discretised time dependent fluid flow equations is described the method is based on the use of pressure and velocity as dependent variables and is hence applicable to both the compressible and incompressible versions of the transport equations the main feature of the technique is the splitting of the solution process into a series of steps whereby operations on pressure are decoupled from those on velocity at each step with the split sets of equations being amenable to solution by standard techniques at each time step the procedure yields solutions which approximate the exact solution of the difference equations the accuracy of this splitting procedure is assessed for a linearised form of the discretised equations and the analysis indicates that the solution yielded by it differs from the exact solution of the difference equations by terms proportional to the powers of the time step size by virtue of this it is possible to dispense with iteration thus resulting in an efficient implicit scheme while retaining simplicity of implementation relative to contemporary block simultaneous methods this is verified in a companion paper which presents results of computations carried out using the method

accuracy analysis of explicit runge kutta methods applied to the incompressible navier stokes equations this paper investigates the temporal accuracy of the velocity and pressure when explicit runge kutta methods are applied to the incompressible navier stokes equations it is shown that at least up to and including fourth order the velocity attains the classical order of accuracy without further constraints however in case of a time dependent gradient operator which can appear in case of time varying meshes additional order conditions need to be satisfied to ensure the correct order of accuracy furthermore the pressure is only first order accurate unless additional order conditions are satisfied two new methods that lead to a second order accurate pressure are proposed which are applicable to a certain class of three and four stage methods a special case appears when the boundary conditions for the continuity equation are independent of time since in that case the pressure can be computed to the same accuracy as the velocity field without additional cost relevant computations of decaying vortices and of an actuator disk in a time dependent inflow support the analysis and the proposed methods

dealiasing techniques for high order spectral element methods on regular and irregular grids high order methods are becoming increasingly attractive in both academia and industry especially in the context of computational fluid dynamics however before they can be more widely adopted issues such as lack of robustness in terms of numerical stability need to be addressed particularly when treating industrial type problems where challenging geometries and a wide range of physical scales typically due to high reynolds numbers need to be taken into account one source of instability is aliasing effects which arise from the nonlinearity of the underlying problem in this work we detail two dealiasing strategies based on the concept of consistent integration the first uses a localised approach which is useful when the nonlinearities only arise in parts of the problem the second is based on the more traditional approach of using a higher quadrature the main goal of both dealiasing techniques is to improve the robustness of high order spectral element methods thereby reducing aliasing driven instabilities we demonstrate how these two strategies can be effectively applied to both continuous and discontinuous discretisations where in the latter both volumetric and interface approximations must be considered we show the key features of each dealiasing technique applied to the scalar conservation law with numerical examples and we highlight the main differences in terms of implementation between continuous and discontinuous spatial discretisations

quantum 1 1 extended galilei algebras from lie bialgebras to quantum r matrices and integrable systems the lie bialgebras of the 1 1 extended galilei algebra are obtained and classified into four multiparametric families their quantum deformations are obtained together with the corresponding deformed casimir operators for the coboundary cases quantum universal r matrices are also given applications of the quantum extended galilei algebras to classical integrable systems are explicitly developed

1 1 schrodinger lie bialgebras and their poisson lie groups all lie bialgebra structures for the 1 1 dimensional centrally extended schrodinger algebra are explicitly derived and proved to be of coboundary type therefore since all of them come from a classical r matrix the complete family of schrodinger poisson lie groups can be deduced by means of the sklyanin bracket all possible embeddings of the harmonic oscillator extended galilei and gl 2 lie bialgebras within the schrodinger classification are studied as an application new quantum hopf algebra deformations of the schrodinger algebra including their corresponding quantum universal r matrices are constructed

quantum e 2 groups and lie bialgebra structures lie bialgebra structures on e 2 are classified for two lie bialgebra structures which are not coboundaries i e those which are not determined by a classical r matrix we solve the cocycle condition find the lie poisson brackets and obtain quantum group relations there is one to one correspondence between lie bialgebra structures on e 2 and possible quantum deformations of u e 2 and e 2

classification of real three dimensional poisson lie groups all real three dimensional poisson lie pl groups are explicitly constructed and fully classified under group automorphisms by making use of their one to one correspondence with the complete classification of real three dimensional lie bialgebras given in gomez 2000 j math phys 41 4939 many of these 3d pl groups are non coboundary structures whose poisson brackets are given here for the first time casimir functions for all three dimensional pl groups are given and some features of several pl structures are commented

classification of low dimensional lie super bialgebras a thorough analysis of lie super bialgebra structures on lie super algebras osp 1 2 and super e 2 is presented a combined technique of computer algebraic computations and subsequent identification of equivalent structures is applied in all the cases poisson lie brackets on supergroups are found the possibility of quantizing them in order to obtain quantum groups is discussed it turns out to be straightforward for all but one structure for the super e 2 group

lie bi algebra structures for centrally extended two dimensional galilei algebra and their lie poisson counterparts all bi algebra structures for centrally extended galilei algebra are classified the corresponding lie poisson structures on centrally extended galilei group are found

constraint consistent runge kutta methods for one dimensional incompressible multiphase flow new time integration methods are proposed for simulating incompressible multiphase flow in pipelines described by the one dimensional two fluid model the methodology is based on halfexplicit runge kutta methods being explicit for the mass and momentum equations and implicit for the volume constraint these half explicit methods are constraint consistent i e they satisfy the hidden constraints of the two fluid model namely the volumetric flow incompressibility constraint and the poisson equation for the pressure a novel analysis shows that these hidden constraints are present in the continuous semi discrete and fully discrete equations next to constraint consistency the new methods are conservative the original mass and momentum equations are solved and the proper shock conditions are satisfied efficient the implicit constraint is rewritten into a pressure poisson equation and the time step for the explicit part is restricted by a cfl condition based on the convective wave speeds and accurate achieving high order temporal accuracy for all solution components masses velocities and pressure highorder accuracy is obtained by constructing a new third order runge kutta method that satisfies the additional order conditions arising from the presence of the constraint in combination with time dependent boundary conditions two test cases kelvin helmholtz instabilities in a pipeline and liquid sloshing in a cylindrical tank show that for time independent boundary conditions the half explicit formulation with a classic fourth order runge kutta method accurately integrates the two fluid model equations in time while preserving all constraints a third test case ramp up of gas production in a multiphase pipeline shows that our new third order method is preferred for cases featuring time dependent boundary conditions

boundary treatment for fourth order staggered mesh discretizations of the incompressible navier stokes equations harlow and welch phys fluids 8 1965 2182 2189 introduced a discretization method for the incompressible navier stokes equations conserving the secondary quantities kinetic energy and vorticity besides the primary quantities mass and momentum this method was extended to fourth order accuracy by several researchers 25 14 21 in this paper we propose a new consistent boundary treatment for this method which is such that continuous integration by parts identities including boundary contributions are mimicked in a discrete sense in this way kinetic energy is exactly conserved even in case of non zero tangential boundary conditions we show that requiring energy conservation at the boundary conflicts with order of accuracy conditions and that the global accuracy of the fourth order method is limited to second order in the presence of boundaries we indicate how non uniform grids can be employed to obtain full fourth order accuracy

half explicit runge kutta methods for differential algebraic systems of index 2 half explicit runge kutta methods for differential algebraic problems of index 2 are investigated it is shown how the arising order conditions can be solved and a particular method of order 4 is constructed in addition this paper simplifies the known convergence theory for such methods and demonstrates by numerical experiments their excellent properties when applied to constrained multibody systems

why many theories of shock waves are necessary convergence error in formally path consistent schemes we are interested in nonlinear hyperbolic systems in nonconservative form arising in fluid dynamics and for solutions containing shock waves we investigate the convergence of finite difference schemes applied to such systems according to dal maso lefloch and murat s theory a shock wave theory for a given nonconservative system requires prescribing a priori a family of paths in the phase space in the present paper we consider schemes that are formally consistent with a given family of paths and we investigate their limiting behavior as the mesh is refined we first generalize to systems a property established earlier by hou and lefloch for scalar conservation laws and we prove that nonconservative schemes generate at the level of the limiting hyperbolic system an convergence error source term which provided the total variation of the approximations remains uniformly bounded is a locally bounded measure this convergence error measure is supported on the shock trajectories and as we demonstrate here is usually small in the special case that the scheme converges in the sense of graphs a rather strong convergence property often violated in practice then this measure source term vanishes we also discuss the role of the equivalent equation associated with a difference scheme here the distinction between scalar equations and systems appears most clearly since for systems the equivalent equation of a scheme that is formally path consistent depends upon the prescribed family of paths the core of this paper is devoted to investigate numerically the approximation of several simplified or full hyperbolic models arising in fluid dynamics this leads us to the conclusion that for systems having nonconservative products associated with linearly degenerate characteristic fields the convergence error vanishes for more general models this measure is evaluated very accurately especially by plotting the shock curves associated with each scheme under consideration as we demonstrate plotting the shock curves provide a convenient approach for evaluating the range of validity of a given scheme

a nonlinear model of heat conduction a nonlinear model of heat propagation is presented from which a new heat conduction equation is derived an exact similarity solution in closed form of this equation is obtained which reveals the travelling wave characteristics for the transient temperature distribution it is shown that the temperature disturbances propagate with finite velocity which is a monotonically decreasing function of time

heisenberg xxz model and quantum galilei group the 1d heisenberg spin model with an anisotropy of the xxz type is analysed in terms of the symmetry given by the quantum galilei group gamma q 1 for a chain with an infinite number of sites the authors show that the magnon excitations and the s 1 2 n magnon bound states are determined by the algebra in this case the gamma q 1 symmetry provides a description naturally compatible with the bethe ansatz the recurrence relations determined by gamma q 1 permit one to express the energy of the n magnon bound states in a closed form in terms of tchebischeff polynomials

an approximate linearized riemann solver for a two fluid model an approximate linearized riemann solver is presented for the numerical simulation of two phase flows this new solver is based on a linearization of nonconservative products and uses an extension of roe s approximate riemann solver the scheme is applied to shock tube problem and to a standard test for two fluid codes

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

quantum 2 1 kinematical algebras a global approach in this paper we give an approach to quantum deformations of the 2 1 kinematical lie algebras within a scheme that simultaneously describes all groups of motions of classical geometries in n 3 dimensions we cover at once all the kinematical geometries including the quantum versions of inonu wigner contractions which are defined in a natural way and relate q deformations as expected we thus obtain some q deformations previously known for the three dimensional euclidean and 2 1 poincare algebras and also some new q deformations for these and other kinematical algebras such as the 2 1 de sitter galilei and newton hooke algebras

on the wave structure of two phase flow models we explore the relationship between two common two phase flow models usually denoted as the two fluid and drift flux models they differ in their mathematical description of momentum transfer between the phases in this paper we provide a framework in which these two model formulations are unified the drift flux model employs a mixture momentum equation and treats interphasic momentum exchange indirectly through the slip relation which gives the relative velocity between the phases as a function of the flow parameters this closure law is in general highly complex which makes it difficult to analyze the model algebraically to facilitate the analysis we express the quasi linear formulation of the drift flux model as a function of three parameters the derivatives of the slip with respect to the vector of unknown variables the wave structure of this model is investigated using a perturbation technique then we rewrite the drift flux model with a general slip relation such that it is expressed in term

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