an entropy stable discontinuous galerkin approximation for the incompressible navier stokes equations with variable density and artificial compressibility abstract we present a provably stable discontinuous galerkin spectral element method for the incompressible navier stokes equations with artificial compressibility and variable density stability proofs which include boundary conditions that follow a continuous entropy analysis are provided we define a mathematical entropy function that combines the traditional kinetic energy and an additional energy term for the artificial compressibility and derive its associated entropy conservation law the latter allows us to construct a provably stable split form nodal discontinuous galerkin dg approximation that satisfies the summation by parts simultaneous approximation term sbp sat property the scheme and the stability proof are presented for general curvilinear three dimensional hexahedral meshes we use the exact riemann solver and the bassi rebay 1 br1 scheme at the inter element boundaries for inviscid and viscous fluxes respectively and an explicit low storage runge kutta rk3 scheme to integrate in time we assess the accuracy and robustness of the method by solving a manufactured solution the kovasznay flow a lid driven cavity the inviscid taylor green vortex and the rayleigh taylor instability

on the singularity analysis of ordinary differential equations invariant under time translation and rescaling the pt is applied to the general second and third order ordinary differential equations invariant under the two symmetries associated with time translation and rescaling in order to investigate their solvability and global integrability the effect of the two symmetries on the compatibility conditions is determined and we show that generally these conditions are automatically a consequence of the resonance condition use is made of truncated laurent series both in ascending and descending powers as an example the case of the generalized chazy equation is presented

on the jacobi last multiplier integrating factors and the lagrangian formulation of differential equations of the painleve gambier classification abstract we use a formula derived almost seventy years ago by madhav rao connecting the jacobi last multiplier of a second order ordinary differential equation and its lagrangian and determine the lagrangians of the painleve equations indeed this method yields the lagrangians of many of the equations of the painleve gambier classification using the standard legendre transformation we deduce the corresponding hamiltonian functions while such hamiltonians are generally of non standard form they are found to be constants of motion on the other hand for second order equations of the lienard class we employ a novel transformation to deduce their corresponding lagrangians we illustrate some particular cases and determine the conserved quantity first integral resulting from the associated noetherian symmetry finally we consider a few systems of second order ordinary differential equations and deduce their lagrangians by exploiting again the relation between the jacobi last multiplier and the lagrangian

comment on on the general solution for the modified emden type equation ddot x alpha x pdot x beta x 3 0 it is shown that the modified emden type equation in the title belongs to a class of lienard s equations that may be mapped into separable abel s equations and are therefore integrable first integrals for equations of this class are easily derived there are three of them corresponding to three distinct dynamical regimes and integrated to yield parametric representations of the corresponding solutions

real gromov witten theory in all genera and real enumerative geometry construction the first part of this work constructs positive genus real gromov witten invariants of realorientable symplectic manifolds of odd complex dimensions the second part studies the orientations on the moduli spaces of real maps used in constructing these invariants the present paper applies the results of the latter to obtain quantitative and qualitative conclusions about the invariants defined in the former after describing large collections of real orientable symplectic manifolds we show that the genus 1 real gromov witten invariants of sufficiently positive almost kahler threefolds are signed counts of real genus 1 curves only and thus provide direct lower bounds for the counts of these curves in such targets we specify real orientations on the real orientable complete intersections in projective spaces the real gromov witten invariants they determine are in a sense canonically determined by the complete intersection itself at least in most cases we also obtain equivariant localization data that computes the real invariants of projective spaces and determines the contributions from many torus fixed loci for other complete intersections our results confirm walcher s predictions for the vanishing of these invariants in certain cases and for the localization data in other cases

a splitting method for incompressible flows with variable density based on a pressure poisson equation a new fractional time stepping technique for solving incompressible flows with variable density is proposed the main feature of this method is that as opposed to other known algorithms the pressure is determined by just solving one poisson equation per time step which greatly reduces the computational cost the stability of the method is proved and the performance of the method is numerically illustrated

hypergeometric first integrals of the duffing and van der pol oscillators abstract the autonomous duffing oscillator and its van der pol modification are known to admit time dependent first integrals for specific values of parameters this corresponds to the existence of darboux polynomials and in fact more can be shown that there exist liouvillian first integrals which do not depend on time they can be expressed in terms of the gauss and kummer hypergeometric functions and are neither analytic algebraic nor meromorphic a criterion for this to happen in a general dynamical system is formulated as well

nonstandard conserved hamiltonian structures in dissipative damped systems nonlinear generalizations of damped harmonic oscillator in this paper we point out the existence of a remarkable nonlocal transformation between the damped harmonic oscillator and a modified emden type nonlinear oscillator equation with linear forcing x xx x3 x 0 which preserves the form of the time independent integral conservative hamiltonian and the equation of motion generalizing this transformation we prove the existence of nonstandard conservative hamiltonian structure for a general class of damped nonlinear oscillators including lienard type systems further using the above hamiltonian structure for a specific example namely the generalized modified emden equation x xqx x2q 1 0 where and q are arbitrary parameters the general solution is obtained through appropriate canonical transformations we also present the conservative hamiltonian structure of the damped mathews lakshmanan oscillator equation the associated lagrangian description for all the above systems is also briefly discussed

analysis and solution of a nonlinear second order differential equation through rescaling and through a dynamical point of view the solutions of the equation y yy y3 0 where is a free parameter are investigated for 1 9 the equation is linearizable through an eight parameter symmetry group and is completely integrable for 1 9 only two symmetries subsist but through a dynamical description the analytical asymptotic solutions and their behavior are given according to the value of and according to the initial conditions

traveling wave solutions for some factorized nonlinear pdes in this work some new special traveling wave solutions of the convective fisher equation the time delayed burgers fisher equation the burgers fisher equation and a nonlinear dispersive dissipative equation kakutani and kawahara 1970 j phys soc japan 29 1068 are obtained through the factorization technique all of them share the same type of factorization scheme which reduces the original equation to a riccati equation of the same kind whose general solution is given in terms of bessel and neumann functions in addition some novel particular solutions of the nonlinear dispersive dissipative equation are provided

energy conserving runge kutta methods for the incompressible navier stokes equations energy conserving methods have recently gained popularity for the spatial discretization of the incompressible navier stokes equations in this paper implicit runge kutta methods are investigated which keep this property when integrating in time firstly a number of energy conserving runge kutta methods based on gauss radau and lobatto quadrature are constructed these methods are suitable for convection dominated problems such as turbulent flows because they do not introduce artificial diffusion and are stable for any time step secondly to obtain robust time integration methods that work also for stiff problems the energy conserving methods are extended to a new class of additive runge kutta methods which combine energy conservation with l stability in this class the radau iia b method has the best properties results for a number of test cases on two stage methods indicate that for pure convection problems the additive radau iia b method is competitive with the gauss methods however for stiff problems such as convection dominated flows with thin boundary layers both the higher order gauss and radau iia b method suffer from order reduction overall the gauss methods are the preferred method for energy conserving time integration of the incompressible navier stokes equations

on error estimates of the projection methods for the navier stokes equations second order schemes we present in this paper a rigorous error analysis of several projection schemes for the approximation of the unsteady incompressible navier stokes equations the error analysis is accomplished by interpreting the respective projection schemes as second order time discretizations of a perturbed system which approximates the navier stokes equations numerical results in agreement with the error analysis are also presented

a method to tackle first order ordinary differential equations with liouvillian functions in the solution the usual prelle singer ps approach misses many first order ordinary differential equations presenting liouvillian functions in the solution lfoodes we point out why and propose a method extending the ps method to solve a class of these previously unsolved lfoodes although our method does not cover all the lfoodes it maintains the semi decision nature of the usual ps method

a nonlocal connection between certain linear and nonlinear ordinary differential equations oscillators we explore a nonlocal connection between certain linear and nonlinear ordinary differential equations odes representing physically important oscillator systems and identify a class of integrable nonlinear odes of any order we also devise a method to derive explicit general solutions of the nonlinear odes interestingly many well known integrable models can be accommodated into our scheme and our procedure thereby provides further understanding of these models

mass conservative and energy stable finite difference methods for the quasi incompressible navier stokes cahn hilliard system primitive variable and projection type schemes abstract in this paper we describe two fully mass conservative energy stable finite difference methods on a staggered grid for the quasi incompressible navier stokes cahn hilliard q nsch system governing a binary incompressible fluid flow with variable density and viscosity both methods namely the primitive method finite difference method in the primitive variable formulation and the projection method finite difference method in a projection type formulation are so designed that the mass of the binary fluid is preserved and the energy of the system equations is always non increasing in time at the fully discrete level we also present an efficient practical nonlinear multigrid method comprised of a standard fas method for the cahn hilliard equation and a method based on the vanka type smoothing strategy for the navier stokes equation for solving these equations we test the scheme in the context of capillary waves rising droplets and rayleigh taylor instability quantitative comparisons are made with existing analytical solutions or previous numerical results that validate the accuracy of our numerical schemes moreover in all cases mass of the single component and the binary fluid was conserved up to 1 0 8 and energy decreases in time

a group theoretical identification of integrable cases of the lienard type equation x f x x g x 0 i equations having nonmaximal number of lie point symmetries we carry out a detailed lie point symmetry group classification of the lienard type equation x f x x g x 0 where f x and g x are arbitrary smooth functions of x we divide our analysis into two parts in the present first part we isolate equations that admit lesser parameter lie point symmetries namely one two and three parameter symmetries and in the second part we identify equations that admit maximal eight parameter lie point symmetries in the former case the invariant equations form a family of integrable equations and in the latter case they form a class of linearizable equations under point transformations further we prove the integrability of all the equations obtained in the present paper through equivalence transformations either by providing the general solution or by constructing time independent hamiltonians several of these equations are being identified for the first time from the group theoretical analysis

on the non integrability of a family of duffing van der pol oscillators we investigate the non integrability of a family of duffing van der pol oscillators x alpha x x2 1 x beta x3 gamma cos omega t by studying the analytic properties of the dynamics in complex time we find that the solutions of have no worse than algebraic singularities at t with only t t 1 2 terms present in their series expansions unlike for example the alpha 0 duffing case where typically log t t terms arise still when integrating around long enough contours a remarkably intricate pattern of square root singularities emerges on different sheets which appears to prevent solutions from ever returning to the original sheet such evidence of infinitely sheeted solutions termed the iss property has also been observed in a number of hamiltonian systems and is illustrated here on a simple example of a single first order differential equation we suggest that the iss property is a necessary condition for non integrability i e non existence of a complete set of analytic single valued constants of the motion which would permit the complete integration of a dynamical system in terms of quadratures

on conservation laws of navier stokes galerkin discretizations abstract we study conservation properties of galerkin methods for the incompressible navier stokes equations without the divergence constraint strongly enforced in typical discretizations such as the mixed finite element method the conservation of mass is enforced only weakly and this leads to discrete solutions which may not conserve energy momentum angular momentum helicity or vorticity even though the physics of the navier stokes equations dictate that they should we aim in this work to construct discrete formulations that conserve as many physical laws as possible without utilizing a strong enforcement of the divergence constraint and doing so leads us to a new formulation that conserves each of energy momentum angular momentum enstrophy in 2d helicity and vorticity for reference the usual convective formulation does not conserve most of these quantities several numerical experiments are performed which verify the theory and test the new formulation

solving second order ordinary differential equations by extending the prelle singer method we propose a method for solving second order ordinary differential equations odes which is based on the ideas behind the prelle singer ps procedure for first order odes while the ps procedure treats differential equations des of the form y p x y q x y with p and q polynomials whose coefficients lie in the field of complex numbers our method is applicable to des of the form y p x y y q x y y the key to our approach is to focus not on the final solution but on the first order invariants of the equation our method is an attempt to address algorithmically the solution of second order odes with solutions in terms of elementary functions

an hybrid finite volume finite element method for variable density incompressible flows this paper is devoted to the numerical simulation of variable density incompressible flows modeled by the navier stokes system we introduce an hybrid scheme which combines a finite volume approach for treating the mass conservation equation and a finite element method to deal with the momentum equation and the divergence free constraint the breakthrough relies on the definition of a suitable footbridge between the two methods through the design of compatibility condition in turn the method is very flexible and allows to deal with unstructured meshes several numerical tests are performed to show the scheme capabilities in particular the viscous rayleigh taylor instability evolution is carefully investigated

a conservative finite element method for the incompressible euler equations with variable density abstract we construct a finite element discretization and time stepping scheme for the incompressible euler equations with variable density that exactly preserves total mass total squared density total energy and pointwise incompressibility the method uses raviart thomas or brezzi douglas marini finite elements to approximate the velocity and discontinuous polynomials to approximate the density and pressure to achieve exact preservation of the aforementioned conserved quantities we exploit a seldom used weak formulation of the momentum equation and a second order time stepping scheme that is similar but not identical to the midpoint rule we also describe and prove stability of an upwinded version of the method we present numerical examples that demonstrate the order of convergence of the method

mixed mimetic spectral element method for stokes flow a pointwise divergence free solution in this paper we apply the recently developed mimetic discretization method to the mixed formulation of the stokes problem in terms of vorticity velocity and pressure the mimetic discretization presented in this paper and in kreeft et al 51 is a higher order method for curvilinear quadrilaterals and hexahedrals fundamental is the underlying structure of oriented geometric objects the relation between these objects through the boundary operator and how this defines the exterior derivative representing the grad curl and div through the generalized stokes theorem the mimetic method presented here uses the language of differential k forms with k cochains as their discrete counterpart and the relations between them in terms of the mimetic operators reduction reconstruction and projection the reconstruction consists of the recently developed mimetic spectral interpolation functions the most important result of the mimetic framework is the commutation between differentiation at the continuous level with that on the finite dimensional and discrete level as a result operators like gradient curl and divergence are discretized exactly for stokes flow this implies a pointwise divergence free solution this is confirmed using a set of test cases on both cartesian and curvilinear meshes it will be shown that the method converges optimally for all admissible boundary conditions

on the general solution for the modified emden type equation in this paper we demonstrate that the modified equation of emden type mee is integrable either explicitly or by quadrature for any value of and we also prove that the mee possesses appropriate time independent hamiltonian functions for the full range of parameters and in addition we show that the mee is intimately connected with two well known nonlinear models namely the equation of force free duffing type oscillator and the two dimensional lotka volterra equation and thus the complete integrability of the latter two models can also be understood in terms of the mee

reply to comment on on the general solution for the modified emden type equation explicit implicit solutions for general parametric values of the modified emden equation and its generalizations were obtained recently by us using a modified prelle singer approach coupled with a hamiltonian formalism in his comment iacono has suggested an interesting alternate approach here we point out the limitations and incompleteness of the approach as well as some additional elementary approaches to the above equations

isogeometric analysis of the navier stokes cahn hilliard equations with application to incompressible two phase flows abstract in this work we provide a unified and comparative description of the most prominent phase field based two phase flow models and present our numerical results of the application of galerkin based isogeometric analysis iga to incompressible navier stokes cahn hilliard nsch equations in velocity pressure phase field chemical potential formulation for the approximation of the velocity and pressure fields lbb compatible non uniform rational b spline spaces are used which can be regarded as smooth generalizations of taylor hood pairs of finite element spaces the one step scheme is used for the discretization in time the static and rising bubble in addition to the nonlinear rayleigh taylor instability flow problems are considered in two dimensions as model problems in order to investigate the numerical properties of the scheme

a projection fem for variable density incompressible flows abstract this work describes a new finite element projection method for the computation of incompressible viscous flows of nonuniform density one original idea of the proposed method consists in factorizing the density variable partly outside and partly inside the time evolution operator in the momentum equation to prevent spatial discretization errors in the mass conservation to affect the kinetic energy balance of the fluid it is shown that unconditional stability in the incremental version of the projection method is possible provided two projections are performed per time step in particular a second order accurate bdf projection method is presented and its numerical performance is illustrated by test computations and comparisons

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

an efficient phase field based multiple relaxation time lattice boltzmann model for three dimensional multiphase flows in this paper an efficient phase field based lattice boltzmann lb model with multiple relaxation time mrt collision operator is developed for the simulation of three dimensional multiphase flows this model is an extension of our previous two dimensional model liang etal 2014 to the three dimensions using the d3q7 seven discrete velocities in three dimensions lattice for the cahnhilliard equation and the d3q15 lattice for the navierstokes equations due to the less lattice velocity directions used the computational efficiency can be significantly improved in the study of three dimensional multiphase flows and simultaneously the present model can recover the cahnhilliard equation and the navierstokes equations correctly through the chapmanenskog procedure we compare the present mrt model with its single relaxation time version and the previous lb model by using two benchmark interface tracking problems and numerical results show that the present mrt model can achieve a significant improvement in the accuracy and stability of the interface capturing the developed model can also be able to deal with multiphase fluids with high viscosity ratio which is demonstrated by the simulation of the layered poiseuille flow and rayleightaylor instability at various viscosity ratios the numerical results are found to be in good agreement with the analytical solutions or some available results in addition it is also found that the instability induces a more complex structure of the interface at a low viscosity

symmetries and reductions of order for certain nonlinear third and second order differential equations with arbitrary nonlinearity we examine the reductions of the order of certain third and second order nonlinear equations with arbitrary nonlinearity through their symmetries and some appropriate transformations we use the folding transformation which enables one to change from a nonlinearity with an arbitrary exponent to a nonlinearity with a specific numerical exponent

the painleve test hidden symmetries and the equation y yy ky3 0 for general values of the parameter k the equation y yy ky3 0 can be reduced to quadrature via a lie algebraic approach either direct or through hidden symmetries for specific values of k mostly in 1 9 1 8 the solution can be expressed in parametric form for these values of k the equation passes the weak painleve test for some other values of k the equation passes the painleve test but the solution cannot in general be expressed parametrically

an energy and helicity conserving finite element scheme for the navier stokes equations we present a new finite element scheme for solving the navier stokes equations that exactly conserves both energy int omega u 2 and helicity int omega u cdot nabla times u in the absence of viscosity and external force we prove stability exact conservation and convergence for the scheme energy and helicity are exactly conserved by using a combination of the usual convective form with the rotational form of the nonlinearity and solving for both velocity and a projected vorticity in a trapezoidal time discretization numerical results are presented that compare the scheme to the usual trapezoidal schemes

enumeration of genus two curves with a fixed complex structure in ℙ 2 and ℙ 3 the main concrete result of this paper is enumeration of genus two curves with complex structure fixed in p and p along the way rational curves with certain simple singularities are counted as well while the methods described can be used to count positive genus curves in some other cases the most powerful direct applications of the machinery developed are to enumeration of rational curves with a very large class of singularities in projective spaces

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

gauge uzawa methods for incompressible flows with variable density two new gauge uzawa schemes are constructed for incompressible flows with variable density one is in the conserved form while the other is in the convective form it is shown that the first order versions of both schemes in their semi discretized form are unconditionally stable numerical experiments indicate that the first order resp second order versions of the two schemes lead to first order resp second order convergence rate for all variables and that these schemes are suitable for handling problems with large density ratios such as in the situation of air bubble rising in water

nonparametric identification of asymmetric nonlinear vibration systems with the hilbert transform abstract the objective of the paper is presenting a simple and more accurate technique for precise identification of nonlinear elastic force functions acting in asymmetric vibration systems the identification procedure based on the hilbert transform is a nonparametric one it does not require a priori information about the system structure or its parameters the examples of the identification of asymmetric classic vibration nonlinear models the helmholtz and the double well duffing oscillators are investigated

lower bounds for enumerative counts of positive genus real curves abstract we transform the positive genus real gromov witten invariants of many real orientable symplectic threefolds into signed counts of curves these integer invariants provide lower bounds for counts of real curves of a given genus that pass through conjugate pairs of constraints we conclude with some implications and related conjectures for hodge integrals

on mass conserving least squares methods least squares variational methods have several practical and theoretical advantages for solving elliptic partial differential equations including symmetric positive definite discrete operators and a sharp error measure one of the potential drawbacks especially in three dimensions is that mass conservation is achieved only in a least squares sense and underresolved solutions are especially vulnerable to poor conservation for the stationary navier stokes equations which are typically rewritten as a larger system of first order equations the loss of mass in the approximate solution is strongly dependent upon the boundary conditions used a new first order system reformulation of the navier stokes equations is presented that admits a wider range of mass conserving boundary conditions this new formulation is shown to provide both excellent mass conservation and excellent algebraic multigrid performance for three different problems a square channel backward facing step and branching tubes with two generations of bifurcations

a conservative phase field method for solving incompressible two phase flows in this paper a conservative phase field method based on the work of sun and beckermann y sun c beckermann sharp interface tracking using the phase field equation j comput phys 220 2007 626 653 for solving the two and three dimensional two phase incompressible navier stokes equations is proposed the present method can preserve the total mass as the cahn hilliard equation but the calculation and implementation are much simpler than that the dispersion relation preserving schemes are utilized for the advection terms while the helmholtz smoother is applied to compute the surface tension force term to verify the proposed method several benchmarks are examined and shown to have good agreements with previous results it also shows that the satisfactions of mass conservations are guaranteed

reduced genus one gromov witten invariants in a previous paper we described a natural closed subset m 0 1 k x a j of the moduli space m1 k x a j of stable genus one j holomorphic maps into a symplectic manifold x in this paper we generalize the definition of the main component to moduli spaces of perturbed in a restricted way j holomorphic maps this generalization implies that m 0 1 k x a j just like m1 k x a j carries a virtual fundamental class and can be used to define symplectic invariants these truly genus one invariants constitute part of the standard genus one gromovwitten invariants which arise from the entire moduli space m1 k x a j the new invariants are more geometric and can be used to compute the genus one gw invariants of complete intersections as shown in a separate paper

energy stable and momentum conserving hybrid finite element method for the incompressible navier stokes equations a hybrid method for the incompressible navier stokes equations is presented the method inherits the attractive stabilizing mechanism of upwinded discontinuous galerkin methods when momentum advection becomes significant equal order interpolations can be used for the velocity and pressure fields and mass can be conserved locally using continuous lagrange multiplier spaces to enforce flux continuity across cell facets the number of global degrees of freedom is the same as for a continuous galerkin method on the same mesh different from our earlier investigations on the approach for the navier stokes equations the pressure field in this work is discontinuous across cell boundaries it is shown that this leads to very good local mass conservation and for an appropriate choice of finite element spaces momentum conservation also a new form of the momentum transport terms for the method is constructed such that global energy stability is guaranteed even in the absence of a pointwise solenoidal velocity field mass conservation momentum conservation and global energy stability are proved for the time continuous case and for a fully discrete scheme the presented analysis results are supported by a range of numerical simulations

fully iterative ilu preconditioning of the unsteady navier stokes equations for gpgpu abstract in this work we investigate the numerical difficulties that arise in optimizing the efficiency of newtonian fluids simulations on a massively parallel computing hardware like a gpu in particular we will concentrate on the resulting algebraic problem we will present an approximate fully iterative ilu preconditioner and we will show that this solution is very efficient on a gpu if compared with an intrinsic massively parallel preconditioner like the diagonal preconditioner which indeed goes faster than more robust techniques like ilu despite their strong decrease in the number of iterations we refer to gmres as the iterative scheme used to solve the linear system in particular we will deal with the solution of incompressible flows with variable density and we will investigate the interplay between reynolds and atwood numbers we will show that the numerical simulation at medium high reynolds numbers produces linear systems whose matrices can be reasonably preconditioned with the diagonal preconditioner while at low reynolds numbers the higher viscosity of the fluid makes the diagonal preconditioner ineffective in the solution time requested from gmres and decreasing the reynolds number unable to let gmres converge at all in this situation we will show how an adequate iterative approach to the parallel solution of the triangular systems that result from the ilu preconditioning turns out to be robust and efficient we will show numerical results for variable density fluids discretized with the scheme described in calgaro et al 2008 in classical benchmarks and in particular in the well known rayleigh taylor instability

a nonlocal connection between certain linear and nonlinear ordinary differential equations part ii complex nonlinear oscillators in this paper we present a method to identify integrable complex nonlinear oscillator systems and construct their solutions for this purpose we introduce two types of nonlocal transformations which relate specific classes of nonlinear complex ordinary differential equations odes with complex linear odes thereby proving the integrability of the former we also show how to construct the solutions using the two types of nonlocal transformations with several physically interesting cases as examples

on the approximation of the unsteady navier stokes equations by finite element projection methods this paper provides an analysis of a fractional step projection method to compute incompressible viscous flows by means of finite element approximations the analysis is based on the idea that the appropriate functional setting for projection methods must accommodate two different spaces for representing the velocity fields calculated respectively in the viscous and the incompressible half steps of the method such a theoretical distinction leads to a finite element projection method with a poisson equation for the incremental pressure unknown and to a very practical implementation of the method with only the intermediate velocity appearing in the numerical algorithm error estimates in finite time are given an extension of the method to a problem with unconventional boundary conditions is also considered to illustrate the flexibility of the proposed method

new aspects of integrability of force free duffing van der pol oscillator and related nonlinear systems in this paper we show that the force free duffing van der pol oscillator is completely integrable for a specific parametric choice we derive a general solution for this parametric choice further we describe a procedure to construct the transformation which removes the time dependent part from the first integral and provide the general solution by quadrature the procedure is shown to have a wider applicability through additional examples we also show that through our method one can deduce linearizing transformations in a simple and straightforward way and illustrate it with a specific example

on the complete integrability and linearization of certain second order nonlinear ordinary differential equations a method for finding general solutions of second order nonlinear ordinary differential equations by extending the prelle singer ps method is briefly discussed we explore integrating factors integrals of motion and the general solution associated with several dynamical systems discussed in the current literature by employing our modifications and extensions of the ps method we also introduce a novel way of deriving linearizing transformations from the first integrals to linearize the second order nonlinear ordinary differential equations to free particle equations we illustrate the theory with several potentially important examples and show that our procedure is widely applicable

analysis of velocity components decoupled projection method for the incompressible navier stokes equations we study the temporal accuracy and stability of the velocity components decoupled projection method vdpm for fully discrete incompressible navier stokes equations in particular we investigate the effect of three formulations of the nonlinear convection term which include the advective skew symmetric and divergence forms on the temporal accuracy and stability second order temporal accuracy of the vdpm for both velocity and pressure is verified by establishing global error estimates in terms of a discrete l 2 norm considering the energy evolution we demonstrate that the vdpm is stable when the time step is less than or equal to a constant stability diagrams which display the distributions of the maximum magnitude of the eigenvalues of the corresponding amplification matrices are obtained using von neumann analysis these diagrams indicate that the advective form is more stable than the other formulations of the nonlinear convection term numerical tests are performed in order to support the mathematical findings involving temporal accuracy and stability and the effects of the formulations of the nonlinear convection term are analyzed overall our results indicate that the vdpm along with an advective discrete convection operator is almost unconditionally stable second order accurate in time and computationally efficient because of the non iterative solution procedure in solving the decoupled momentum equations

structure preserving discretization of incompressible fluids the geometric nature of euler fluids has been clearly identified and extensively studied over the years culminating with lagrangian and hamiltonian descriptions of fluid dynamics where the configuration space is defined as the volume preserving diffeomorphisms and kelvin s circulation theorem is viewed as a consequence of noether s theorem associated with the particle relabeling symmetry of fluid mechanics however computational approaches to fluid mechanics have been largely derived from a numerical analytic point of view and are rarely designed with structure preservation in mind and often suffer from spurious numerical artifacts such as energy and circulation drift in contrast this paper geometrically derives discrete equations of motion for fluid dynamics from first principles in a purely eulerian form our approach approximates the group of volume preserving diffeomorphisms using a finite dimensional lie group and associated discrete euler equations are derived from a variational principle with non holonomic constraints the resulting discrete equations of motion yield a structure preserving time integrator with good long term energy behavior and for which an exact discrete kelvin s circulation theorem holds

a second order projection method for variable density flows this paper describes a second order projection method for variabledensity incompressible flows the method is suitable for both finite amplitude density variations and for fluids that are modeled using a boussinesq approximation it is based on a second order fractional step scheme in which diffusion convection terms are advanced without enforcing the incompressibility condition and the resulting intermediate velocity field is then projected onto the space of discretely divergencefree vector fields the nonlinear convection terms are treated using a godunov type procedure that is second order for smooth flow and remain stable and non oscillatory for nonsmooth flows with low fluid viscosities the method is described for finite amplitude density variation and the simplifications for a boussinesq approximation are sketched numerical results are presented that validate the convergence properties of the method and demonstrate its performance on more realistic problems

a new fractional time stepping method for variable density incompressible flows this paper describes a new method for solving variable density incompressible viscous flows we have dealt with the momentum equation and the divergence free constraint in a new manner by rewriting the original equations the originality of the proposed approach is that we have used different numerical methods to evaluate the evolution of the velocity and pressure compared with some established methods the proposed approach is parameter free more flexible and can avoid the difficulties caused by the original equations the stability analysis of the method is performed to show that our method is stable finally numerical experiments are given to show the accuracy efficiency and validness of this method for variable density incompressible flows

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 mass conservation and solvability of the discretized variable density zero mach navier stokes equations abstract this paper presents a study of a variable density zero mach navier stokes discretization and subtle difficulties associated with the pressure a consistent compatibility condition between the temperature and velocity is first derived it is shown that one can accurately and efficiently solve all primitive variables with the uniform order of accuracy of the specified discretization and time marching schemes it is shown that higher density ratios than those commonly reported are achievable and it is in part traced to the solvability of the pressure poisson equation numerical properties are assessed with the method of manufactured solutions and several classical testing cases are carried out to validate the code ranging from variable density mixing problems to premixed flames with forcing

error estimate of gauge uzawa methods for incompressible flows with variable density abstract in this paper we construct a positivity preserving gauge uzawa method for the semi discrete in time scheme of incompressible viscous flows with variable density and establish its stability and error estimates we also construct and implement a fully discrete scheme with finite elements in space and derive its positivity preserving and stability result

on the properties of a variant of the riccati system of equations a variant of the generalized riccati system of equations is considered it is shown that for 2n 3 the system admits a bilagrangian description and the dynamics has a node at the origin whereas for much smaller than a critical value the dynamics is periodic the origin being a centre it is found that the solution changes from being periodic to aperiodic at a critical point which is independent of the initial conditions this behaviour is explained by finding a scaling argument via which the phase trajectories corresponding to different initial conditions collapse onto a single universal orbit numerical evidence for the transition is shown further using a perturbative renormalization group argument it is conjectured that the oscillator exhibits isochronous oscillations the correctness of the conjecture is established numerically

multi fluid flows with the particle finite element method abstract particle methods are those in which the problem is represented by a discrete number of particles each particle moves accordingly with its own mass and the external internal forces applied on it in this paper the particle finite element method based on finite element shape functions is used to solve the continuous fluid mechanics equations in the case of heterogeneous density to evaluate the external applied forces to each particle the incompressible navier stokes equations are solved at each time step using a lagrangian formulation all the information in the fluid is transmitted via the particles all kinds of density heterogeneous fluids and multiphase flows with internal interfaces including or not free surfaces breaking waves and fluid separations may be easily solved with this methodology

unusual lienard type nonlinear oscillator a lienard type nonlinear oscillator of the form x kxx k2 9 x3 lambda1x 0 which may also be considered as a generalized emden type equation is shown to possess unusual nonlinear dynamical properties it is shown to admit explicit nonisolated periodic orbits of conservative hamiltonian type for lambda1 0 these periodic orbits exhibit the unexpected property that the frequency of oscillations is completely independent of amplitude and continues to remain as that of the linear harmonic oscillator this is completely contrary to the standard characteristic property of nonlinear oscillators interestingly the system though appears deceptively a dissipative type for lambda1 or 0 does admit a conserved hamiltonian description where the characteristic decay time is also independent of the amplitude the results also show that the criterion for conservative hamiltonian system in terms of divergence of flow function needs to be generalized

standard and non standard lagrangians for dissipative dynamical systems with variable coefficients dynamical systems described by equations of motion with the first order time derivative dissipative terms of even and odd powers and coefficients varying either in time or in space are considered methods to obtain standard and non standard lagrangians are presented and used to identify classes of equations of motion that admit a lagrangian description it is shown that there are two general classes of equations that have standard lagrangians and one special class of equations that can only be derived from non standard lagrangians in addition each general class has a subset of equations with non standard lagrangians conditions required for the existence of standard and non standard lagrangians are derived and a relationship between these two types of lagrangians is introduced by obtaining lagrangians for several dynamical systems and some basic equations of mathematical physics it is demonstrated that the presented methods can be applied to a broad range of physical problems

geometric variational discretization of continuum theories this study derives geometric variational discretization of continuum theories arising in fluid dynamics magnetohydrodynamics mhd and the dynamics of complex fluids a central role in these discretizations is played by the geometric formulation of fluid dynamics which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume preserving diffeomorphisms of the fluid domain inspired by this framework we construct a finite dimensional approximation to the diffeomorphism group and its lie algebra thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group the extension to mhd and complex fluid flow is then made through an appeal to the theory of euler poincare systems with advection which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters upon deriving a family of structured integrators for these systems we test their performance via a numerical implementation of the update schemes on a cartesian grid among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries automatic satisfaction of solenoidal constraints on vector fields good long term energy behavior robustness with respect to the spatial and temporal resolution of the discretization and applicability to irregular meshes

stokes complexes and the construction of stable finite elements with pointwise mass conservation two families of conforming finite elements for the two dimensional stokes problem are developed guided by two discrete smoothed de rham complexes which we coin stokes complexes we show that the finite element pairs are inf sup stable and also provide pointwise mass conservation on very general triangular meshes

an extended validation of the last generation of particle finite element method for free surface flows in this paper a new generation of the particle method known as particle finite element method pfem which combines convective particle movement and a fixed mesh resolution is applied to free surface flows this interesting variant previously described in the literature as pfem 2 is able to use larger time steps when compared to other similar numerical tools which implies shorter computational times while maintaining the accuracy of the computation pfem 2 has already been extended to free surface problems being the main topic of this paper a deep validation of this methodology for a wider range of flows to accomplish this task different improved versions of discontinuous and continuous enriched basis functions for the pressure field have been developed to capture the free surface dynamics without artificial diffusion or undesired numerical effects when different density ratios are involved a collection of problems has been carefully selected such that a wide variety of froude numbers density ratios and dominant dissipative cases are reported with the intention of presenting a general methodology not restricted to a particular range of parameters and capable of using large time steps the results of the different free surface problems solved which include rayleigh taylor instability sloshing problems viscous standing waves and the dam break problem are compared to well validated numerical alternatives or experimental measurements obtaining accurate approximations for such complex flows

calculation of incompressible viscous flows by an unconditionally stable projection fem this paper investigates the numerical performance of a finite element implementation of a new incremental fractional step method to compute steady and unsteady incompressible viscous flows under general boundary conditions and using unstructured meshes a variational framework is adopted which accommodates two different spaces for representing and approximating the velocity fields calculated respectively in the viscous and inviscid phases of the method but which leads to a very simple numerical scheme in terms of only one discrete velocity field an unconditionally stable semi implicit approximation of the nonlinear term is used to eliminate any time step restriction as far as the numerical stability is concerned numerical results for five test problems in two dimensions are reported to illustrate the flexibility of the proposed method

on the genus one gromov witten invariants of complete intersections we state and prove a long elusive relation between genus one gromov witten of a complete intersection and twisted gromov witten invariants of the ambient projective space as shown in a previous paper certain naturally arising cones of holomorphic vector bundle sections over the main component m 0 1 k p d of the moduli space of stable genus one holomorphic maps into p have a well defined euler class in this paper we extend this result to moduli spaces of perturbed in a restricted way j holomorphic maps this extension is used to show that these cones are the correct genus one analogues of the vector bundles relating genus zero gromov witten invariants of a complete intersection to those of the ambient projective space a relationship for highergenus invariants is conjectured as well

multiphase lattice boltzmann flux solver for incompressible multiphase flows with large density ratio a multiphase lattice boltzmann flux solver mlbfs is proposed in this paper for incompressible multiphase flows with low and large density ratios in the solver the flow variables at cell centers are given from the solution of macroscopic governing differential equations navier stokes equations recovered by multiphase lattice boltzmann lb model by the finite volume method at each cell interface the viscous and inviscid fluxes are evaluated simultaneously by local reconstruction of solution for the standard lattice boltzmann equation lbe the forcing terms in the governing equations are directly treated by the finite volume discretization the phase interfaces are captured by solving the phase field cahn hilliard equation with a fifth order upwind scheme unlike the conventional multiphase lb models which restrict their applications on uniform grids with fixed time step the mlbfs has the capability and advantage to simulate multiphase flows on non uniform grids the proposed solver is validated by several benchmark problems such as two phase co current flow taylor couette flow in an annulus rayleigh taylor instability and droplet splashing on a thin film at density ratio of 1000 with reynolds numbers ranging from 20 to 1000 numerical results show the reliability of the proposed solver for multiphase flows with high density ratio and high reynolds number

on adjoint symmetry equations integrating factors and solutions of nonlinear odes we consider the role of the adjoint equation in determining explicit integrating factors and first integrals of nonlinear odes in chandrasekar et al 2006 j math phys 47 023508 the authors have used an extended version of the prelle singer method for a class of nonlinear odes of the oscillator type in particular we show that their method actually involves finding a solution of the adjoint symmetry equation next we consider a coupled second order nonlinear ode system and derive the corresponding coupled adjoint equations we illustrate how the coupled adjoint equations can be solved to arrive at a first integral

a conservative adaptive projection method for the variable density incompressible navier stokes equations in this paper we present a method for solving the equations governing time dependent variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids the method is based on a projection formulation in which we first solve advection diffusion equations to predict intermediate velocities and then project these velocities onto a space of approximately divergence free vector fields our treatment of the first step uses a specialized second order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high reynolds number flow density and other scalars are advected in such a way as to maintain conservation if appropriate and free stream preservation our approach to adaptive refinement uses a nested hierarchy of logically rectangular girds with simultaneous refinement of the girds in both space and time the integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized the single grid algorithm is described briefly but the emphasis here is on the time stepping procedure for the adaptive hierarchy numerical examples are presented to demonstrate the algorithms s accuracy and convergence properties and illustrate the behavior of the method an additional example demonstrates the performance of the method on a more realistic problem namely a three dimensional variable density shear layer

on the local analytic integrability for a generalized damped nonlinear oscillator equation we consider the general damped nonlinear oscillator x y y a1x a2xn 1 a3x2n 1 y a4 a5xn where ai are real parameters for i 1 2 3 4 5 and n is an integer we note that these class of lienard differential systems include a large number of physically important nonlinear oscillators we provide a complete and rigorous characterization of these systems that admit a local analytic first integral in a neighborhood of the origin

integrability and symmetries for the helmholtz oscillator with friction this paper deals with the helmholtz oscillator which is a simple nonlinear oscillator whose equation presents a quadratic nonlinearity and the possibility of escape when a periodic external force is introduced the width of the stochastic layer which is a region around the separatrix where orbits may exhibit transient chaos is calculated in the absence of friction and external force it is well known that analytical solutions exist since it is completely integrable when only friction is included there is no analytical solution for all parameter values however by means of the lie theory for differential equations we find a relation between parameters for which the oscillator is integrable this is related to the fact that the system possesses a symmetry group and the corresponding symmetries are computed finally the analytical explicit solutions are shown and related to the basins of attraction

a simple and unified approach to identify integrable nonlinear oscillators and systems in this paper we consider a generalized second order nonlinear ordinary differential equation ode of the form x k1xq k2 x k3x2q 1 k4xq 1 1x 0 where ki s i 1 2 3 4 1 and q are arbitrary parameters which includes several physically important nonlinear oscillators such as the simple harmonic oscillator anharmonic oscillator force free helmholtz oscillator force free duffing and duffing van der pol oscillators modified emden type equation and its hierarchy generalized duffing van der pol oscillator equation hierarchy and so on and investigate the integrability properties of this rather general equation we identify several new integrable cases for arbitrary value of the exponent q q r the q 1 and q 2 cases are analyzed in detail and the results are generalized to arbitrary q our results show that many classical integrable nonlinear oscillators can be derived as subcases of our results and significantly enlarge the list of integrable equations that exists in the contemporary literature t

an adaptive pressure correction method without spurious velocities for diffuse interface models of incompressible flows in this article we propose to study two issues associated with the use of the incremental projection method for solving the incompressible navier stokes equation the first one is the combination of this time splitting algorithm with an adaptive local refinement method the second one is the reduction of spurious velocities due to the right hand side of the momentum balance we propose a new variant of the incremental projection method for solving the navier stokes equations with variable density and illustrate its behavior with the example of two phase flows simulations using a cahn hilliard navier stokes model

solution and asymptotic blow up behaviour of a class of nonlinear dissipative systems in this paper we consider a three parameter class of lienard type nonlinear dissipative systems of the form x b 3kx x k2x3 bkx2 x 0 since such dissipative systems admit an eight parameter lie group of point transformations it follows that there exists a complex point transformation mapping such a system into the free particle system x 0 normally such an explicit point transformation cannot be found here we find such an explicit point transformation through exploiting the group properties of the determining equations that lead to it consequently we obtain the explicit general solution of such dissipative systems moreover we completely characterize the asymptotic and or finite time blow up behaviour of such systems in terms of their three parameters and initial data

on the structure of certain natural cones over moduli spaces of genus one holomorphic maps we show that certain naturally arising cones over the main component of a moduli space of j0 holomorphic maps into pn have a well defined euler class we also prove that this is the case if the standard complex structure j0 on pn is replaced by a nearby almost complex structure j the genus zero analogue of the cone considered in this paper is a vector bundle the genus zero gromov witten invariant of a projective complete intersection can be viewed as the euler class of such a vector bundle as shown in a separate paper this is also the case for the genus one part of the genus one gw invariant the remaining part is a multiple of the genus zero gw invariant

isogeometric divergence conforming b splines for the unsteady navier stokes equations divergence conforming b splines are developed for application to the incompressible navier stokes equations on geometrically mapped domains these enable smooth pointwise divergence free solutions and thus satisfy mass conservation in the strongest possible sense semi discrete methods based on divergence conforming b splines are shown to conserve linear and angular momentum and satisfy balance laws for energy vorticity enstrophy and helicity these are geometric structure preserving quantities and numerical simulations that are sensitive to them are shown to be qualitatively correct and quantitatively accurate the methods developed are anticipated to open new doors to the practical calculation of complex flows and to studies of their physical behavior

on certain new integrable second order nonlinear differential equations and their connection with two dimensional lotka volterra system in this paper we consider a second order nonlinear ordinary differential equation of the form x k1 x2 x k2 k3x x k4x3 k5x2 k6x 0 where ki s i 1 2 6 are arbitrary parameters by using the modified prelle singer procedure we identify five new integrable cases in this equation besides two known integrable cases namely i k2 0 k3 0 and ii k1 0 k2 0 k5 0 among these five four equations admit time dependent first integrals and the remaining one admits time independent first integral from the time independent first integral nonstandard hamiltonian structure is deduced thereby proving the liouville sense of integrability in the case of time dependent integrals we either explicitly integrate the system or transform to a time independent case and deduce the underlying hamiltonian structure we also demonstrate that the above second order ordinary differential equation is intimately related to the two dimensional lotka volterra system from the integrable parametric choices of the above non

first integrals for the modified emden equation q α t q qn 0 it is shown that the modified emden equation q t q qn 0 possesses first integrals for functions t other than kt 1 the function t is obtained explicitly in the case n 3 and parametrically for other n 2 the case n 2 is seen to be particularly difficult to solve

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100