symmetry and perturbation of the vector nonlinear schrodinger equation abstract we show that the symmetry algebra of the n component vector nonlinear schrodinger nls equation is given by the loop algebra of su n u 1 the recursion operator and the virasoro type algebra of the master symmetries suitable for any n or even for an operator nls equation are also constructed we have shown that a general su n invariant third order correction of the vector nls equation can be moved in the next order of perturbation theory i e the corresponding normal form of the equation is shown to be integrable

on some third order nonlinear boundary value problems existence location and multiplicity results abstract we prove an ambrosetti prodi type result for the third order fully nonlinear equation u t f t u t u t u t s p t with f 0 1 r 3 r and p 0 1 r continuous functions s r under several two point separated boundary conditions from a nagumo type growth condition an a priori estimate on u is obtained an existence and location result will be proved by degree theory for s r such that there exist lower and upper solutions the location part can be used to prove the existence of positive solutions if a non negative lower solution is considered the existence nonexistence and multiplicity of solutions will be discussed as s varies

fully nonlinear fourth order equations with functional boundary conditions abstract the aim of this paper consists in to give sufficient conditions to ensure the existence and location of the solutions of a nonlinear fully fourth order equation with functional boundary conditions the arguments make use of the upper and lower solutions method a laplacian operator and a fixed point theorem an application of the beam theory to a nonlinear continuous model of the human spine allows to estimate its deformation under some loading forces

soliton solutions of the kadomtsev petviashvili ii equation we study a general class of line soliton solutions of the kadomtsev petviashvili ii kpii equation by investigating the wronskian form of its tau function we show that in addition to the previously known line soliton solutions of kpii this class also contains a large variety of multisoliton solutions many of which exhibit nontrivial spatial interaction patterns we also show that in general such solutions consist of unequal numbers of incoming and outgoing line solitons from the asymptotic analysis of the tau function we explicitly characterize the incoming and outgoing line solitons of this class of solutions we illustrate these results by discussing several examples

bright n soliton solutions in terms of the triple wronskian for the coupled nonlinear schrodinger equations in optical fibers the coupled nonlinear schrodinger nls equations are usually used to describe the dynamics of two component solitons in optical fibers via the darboux transformation the coupled nls equations of the manakov type are found to have triple wronskian solutions proof is finished by virtue of some new triple wronskian identities by solving the zero potential lax pair the triple wronskian solutions give the bright n soliton solutions which are characterized by 3n complex parameters to obtain an understanding of the asymptotic behavior of the bright n soliton solutions with arbitrary n some algebraic properties of the triple wronskian are analyzed and an algebraic procedure is presented to derive the explicit expressions of the asymptotic solitons as t

fourth order problems with nonlinear boundary conditions we develop a new method of lower and upper solutions for a fourth order nonlinear boundary value problem where the differential equation has dependence on all lower order derivatives our boundary conditions are nonlinear we will assume the functions that define the nonlinear boundary conditions are either monotone or nonmonotone as a result we obtain existence principles which improve recent results in the literature

space time discontinuous galerkin finite element method with dynamic grid motion for inviscid compressible flows ii efficient flux quadrature a new and efficient quadrature rule for the flux integrals arising in the space time discontinuous galerkin discretization of the euler equations in a moving and deforming space time domain is presented and analyzed the quadrature rule is a factor three more efficient than the commonly applied quadrature rule and does not affect the local truncation error and stability of the numerical scheme the local truncation error of the resulting numerical discretization is determined and is shown to be the same as when product gauss quadrature rules are used details of the approximation of the dissipation in the numerical flux are presented which render the scheme consistent and stable the method is successfully applied to the simulation of a three dimensional transonic flow over a deforming wing

positive solutions of multi point boundary value problems at resonance abstract we establish new results on the existence of positive solutions for some multi point boundary value problems at resonance our results are based on a recent leggett williams norm type theorem due to o regan and zima we also derive a new result for a three point problem previously studied by several authors

quasi ale finite element method for nonlinear water waves this paper presents a newly developed quasi arbitrary lagrangian eulerian finite element method qale fem for simulating water waves based on fully nonlinear potential theory the main difference of this method from the conventional finite element method developed by one of authors of this paper and others see e g q w ma g x wu r eatock taylor finite element simulation of fully non linear interaction between vertical cylinders and steep waves part 1 methodology and numerical procedure and part 2 numerical results and validation int j numer methods fluids 36 2001 265 308 and g x wu z z hu simulation of nonlinear interactions between waves and floating bodies through a finite element based numerical tank proc r soc a 460 2004 2050 3037 3058 is that the complex mesh is generated only once at the beginning and is moved at all other time steps in order to conform to the motion of the free surface and structures this feature allows one to use an unstructured mesh with any degree of complexity without the need of regenerating it every time step which is generally inevitable and very costly due to this feature the qale fem has high potential in enhancing computational efficiency when applied to problems associated with the complex interaction between large steep waves and structures since the use of an unstructured mesh in such a case is likely to be necessary to achieve overall high efficiency the numerical techniques involved in the qale fem are developed including the method to move interior nodes technique to re distribute the nodes on the free surface scheme to calculate velocities and so on the model is validated by water waves generated by a wavemaker in a tank and the interaction between water waves and periodic bars on the bed of tank satisfactory agreement is achieved with analytical solutions experimental data and numerical results from other methods

solvability for a third order discontinuous fully equation with nonlinear functional boundary conditions abstract we prove an existence and location result for the third order functional nonlinear boundary value problem u t f t u u t u t for t a b 0 l 0 u u u t 0 0 l 1 u u u a u a 0 l 2 u u u b u b with t 0 a b given f i c i r 2 r is a l 1 caratheodory function and l 0 l 1 l 2 are continuous functions depending functionally on u and u the arguments make use of an a priori estimate on u lower and upper solutions method and degree theory applications to a multipoint problem and to a beam equation will be presented

periodic n dimensional first order coupled systems and periodic schizophrenia phenomena abstract this paper introduces some existence and location results for periodic solutions of first order coupled nonlinear systems of n ordinary differential equations the arguments used are based on the arzela ascoli theorem and schauder s fixed point theorem combined with the upper and lower solution technique to illustrate the applicability of these problems an application to a mathematical model of the thyroid pituitary interaction and their homeostatic mechanism is included for specific cases of periodic events such as relapsing catatonia

arbitrary lagrangian eulerian finite element analysis of free surface flow an arbitrary lagrangian eulerian ale description of newtonian fluid flow is presented it allows the simulation of free surfaces with the finite element method fem we develop a robust node repositioning algorithm which consists of two parts an energy minimization principle controls the positions of the internal nodes a good approximation of the boundaries is ensured by explicitly enforcing the free surface conditions numerical examples are done employing the mini element

upper and lower solution methods for fully nonlinear boundary value problems abstract sufficient conditions are given for the existence of a solution of a fourth order nonlinear boundary value problem with nonlinear boundary conditions the conditions assume the existence of a strong upper solution lower solution pair a concept that is defined in the paper the differential equation has nonlinear dependence on all lower order derivatives of the unknown in particular appropriate nagumo conditions are obtained and employed

arbitrary lagrangian eulerian and free surface methods in fluid mechanics abstract a computational procedure is developed to solve problems of viscous incompressible flows under large free surface motions the arbitrary lagrangian eulerian ale method is used to move the free surface nodes as well as the internal nodes the coupling of the mesh motion equations and the fluid equations is essentially done through the free surface boundary conditions an elastic model with free surface boundary conditions is used to model the mesh motion the boundary condition at the free surface is deduced from the fact that no fluid particle can cross the free surface the fluid problem is solved using a galerkin least square gls finite element formulation

higher order boundary value problems with φ laplacian and functional boundary conditions we study the existence of solutions of the boundary value problem f u n 1 t f t u t u t u n 1 t 0 email protected 0 1 g i u u u n 1 u i 0 0 i 0 n 2 g n 1 u u u n 1 u n 2 1 0 where n 2 f and g i i 0 n 1 are continuous and f is a caratheodory function we obtain an existence criterion based on the existence of a pair of coupled lower and upper solutions we also apply our existence theorem to derive some explicit conditions for the existence of a solution of a special case of the above problem in our problem both the differential equation and the boundary conditions may have dependence on all lower order derivatives of the unknown function and many boundary value problems with various boundary conditions studied extensively in the literature are special cases of our problem consequently our results improve and cover a number of known results in the literature examples are given to illustrate the applicability of our theorems

a robust elliptic grid generator a variational principle is proposed that results in a robust elliptic grid generator having many of the strengths of original winslow or homogeneous thompson thames mastin method httm the new grid generator places grid lines more uniformly over the domain than does httm without loss of orthogonality numerically generated examples are given to demonstrate these effects grid quality measures are introduced to quantify differences between discrete grids both the httm and the new grid generator can generate folded grids on certain pathological regions but overall they are very robust grid weighting for solution adaptive calculations is briefly considered generalization of the new method to surface and volume grid generation is straightforward

regular article free surface flow simulation using hp spectral elements computational instabilities inherent in the solution of inviscid incompressible free surface flow have been apparent for some time and different techniques have been used to overcome them in this paper we attempt to explain why these instabilities occur and in doing so a method is developed to study the spatially semi discrete eigenvalues and eigenvectors which govern the stability of the system it is found that the asymmetric spatial discretisation of the fluid domain causes the instabilities after mechanisms for stability are recognised and implemented unsteady inviscid incompressible linear and non linear free surface flow is simulated using a hp spectral element code ensuring fast convergence which incorporates arbitrary lagrangian eulerian ale techniques to decrease deformation of the computational mesh

on the one dimensional p laplacian with a singular nonlinearity abstract in this paper we are concerned with the existence and nonexistence of positive solutions of boundary value problems for the one dimensional p laplacian with a singular nonlinearity in the case of the model equation we give the necessary and sufficient conditions of the existence of positive solutions for both the dirichlet problem and the periodic problem

space time discontinuous galerkin finite element method with dynamic grid motion for inviscid compressible flows i general formulation a new space time discontinuous galerkin finite element method for the solution of the euler equations of gas dynamics in time dependent flow domains is presented the discontinuous galerkin discretization results in an efficient elementwise conservative upwind finite element method which is particularly well suited for local mesh refinement the upwind scheme uses a formulation of the hllc flux applicable to moving meshes and several formulations for the stabilization operator to ensure that monotone solutions around discontinuities are investigated the non linear equations of the space time discretization are solved using a multigrid accelerated pseudo time integration technique with an optimized runge kutta method the linear stability of the pseudo time integration method is investigated for the linear advection equation the numerical scheme is demonstrated with simulations of the flow field in a shock tube a channel with a bump and an oscillating naca 0012 airfoil these simulations show that using the data at the superconvergence points the accuracy of the numerical discretization is o h5 2 in space for smooth subsonic flows both on structured and on locally refined meshes and that the space time adaptation can significantly improve the accuracy and efficiency of the numerical method

space time discontinuous galerkin finite element method for shallow water flows a space time discontinuous galerkin dg finite element method is presented for the shallow water equations over varying bottom topography the method results in nonlinear equations per element which are solved locally by establishing the element communication with a numerical hllc flux to deal with spurious oscillations around discontinuities we employ a dissipation operator only around discontinuities using krivodonova s discontinuity detector the numerical scheme is verified by comparing numerical and exact solutions and validated against a laboratory experiment involving flow through a contraction we conclude that the method is second order accurate in both space and time for linear polynomials

an arbitrary lagrangian eulerian finite element method for solving three dimensional free surface flows abstract this paper discusses numerical solution of unsteady three dimensional free surface flows the governing equilibrium equations are written in the framework of the arbitrary lagrangian eulerian kinematic description the corresponding variational formulation is established afterwards since the variational problems are nonlinear with respect to the moving coordinates a second order approximate variational problem is derived after a consistent linearization of the referential motion stability of the discrete formulations is ensured with the help of a new stabilization method a robust preconditioned gmres algorithm is then used to solve the resulting set of nonlinear equations finally the computational algorithms are assessed through numerical studies of various problems a large sloshing flow in a three dimensional reservoir a discharge flow from a reservoir simulation of a liquid vortex produced inside a cylindrical container with a disk rotating at the bottom and a three dimensional practical hydraulic problem

a high order discontinuous galerkin fourier incompressible 3d navier stokes solver with rotating sliding meshes we present the development of a sliding mesh capability for an unsteady high order order 3 h p discontinuous galerkin solver for the three dimensional incompressible navier stokes equations a high order sliding mesh method is developed and implemented for flow simulation with relative rotational motion of an inner mesh with respect to an outer static mesh through the use of curved boundary elements and mixed triangular quadrilateral meshes a second order stiffly stable method is used to discretise in time the arbitrary lagrangian eulerian form of the incompressible navier stokes equations spatial discretisation is provided by the symmetric interior penalty galerkin formulation with modal basis functions in the x y plane allowing hanging nodes and sliding meshes without the requirement to use mortar type techniques spatial discretisation in the z direction is provided by a purely spectral method that uses fourier series and allows computation of spanwise periodic three dimensional flows the developed solver is shown to provide high order solutions second order in time convergence rates and spectral convergence when solving the incompressible navier stokes equations on meshes where fixed and rotating elements coexist in addition an exact implementation of the no slip boundary condition is included for curved edges circular arcs and naca 4 digit airfoils where analytic expressions for the geometry are used to compute the required metrics the solver capabilities are tested for a number of two dimensional problems governed by the incompressible navier stokes equations on static and rotating meshes the taylor vortex problem a static and rotating symmetric naca0015 airfoil and flows through three bladed cross flow turbines in addition three dimensional flow solutions are demonstrated for a three bladed cross flow turbine and a circular cylinder shadowed by a pitching naca0012 airfoil

hp multigrid as smoother algorithm for higher order discontinuous galerkin discretizations of advection dominated flows part i multilevel analysis the hp multigrid as smoother algorithm hp mgs for the solution of higher order accurate space time discontinuous galerkin discretizations of advection dominated flows is presented this algorithm combines p multigrid with h multigrid at all p levels where the h multigrid acts as smoother in the p multigrid the performance of the hp mgs algorithm is further improved using semi coarsening in combination with a new semi implicit runge kutta method as smoother a detailed multilevel analysis of the hp mgs algorithm is presented to obtain more insight into the theoretical performance of the algorithm as model problem a fourth order accurate space time discontinuous galerkin discretization of the advection diffusion equation is considered the multilevel analysis shows that the hp mgs algorithm has excellent convergence rates both for steady state and time dependent problems and low and high cell reynolds numbers including highly stretched meshes

state transformations of colliding optical solitons and possible application to computation in bulk media using explicit bright soliton solutions for the coupled manakov system recently described by radhakrishnan lakshmanan and hietarinta we show that collisions of these solitons can be completely described by explicit linear fractional transformations of a complex valued polarization state we design sequences of solitons operating on other sequences of solitons that effect logic operations including controlled not gates both data and logic operators have the self restoring and reusability features of digital logic circuits this suggests a method for implementing computation in a bulk nonlinear medium without interconnecting discrete components

arbitrary lagrangian eulerian method for navier stokes equations with moving boundaries this article deals with the numerical approximation of navier stokes equations in a domain with moving boundaries using the arbitrary lagrangian eulerian ale method we transform the problem from a moving domain to a fixed reference domain through an artificial domain velocity then we apply the characteristic method to solve the navier stokes equation in the fixed domain suitable boundary conditions are used for the interfaces three examples are provided to show the efficiency of the present method

interactions of vector solitons in this paper we study the interaction of two widely separated vector solitons in the nonintegrable coupled nonlinear schrodinger nls equations using a modification of karpman solov ev perturbation method we derive dynamical equations for the evolution of both solitons internal parameters we show that these dynamical equations allow fixed points that correspond to stationary two vector soliton bound states if these solitons have the same phase in one component same sign and pi phase difference in the other component opposite sign however linear stability analysis indicates that these bound states are always unstable due to a phase related unstable eigenvalue we also investigate vector soliton interactions and show that in contrast to soliton interactions in the single nls equation vector solitons repel or attract each other depending not only on their relative phases but also on their initial position separation lastly interaction of an arbitrary number of vector solitons is also studied in brief all our analytical results are supported by direct numerical simulations

existence of positive solutions for an n 2 order nonlinear bvp under some suitable assumptions we shall show that the n 2 order nonlinear boundary value problem bvp e f p u n t f t u t u 1 t u n 1 t bc u i 0 0 i 0 1 2 n 3 u n 2 0 email protected n 2 1 u n 1 1 email protected n 1 0 u n 0 email protected n d u n 1 email protected n d has at least three solutions in c n 0 1 n 2

the vector nonlinear schrodinger hierarchy abstract an article published recently by one of the authors jne presents closed form solutions for zero curvature representations of the vector nonlinear schrodinger hierarchy several of these results are confirmed computationally but left unproven in this article we begin by providing strict algebraic proofs of these results the forms of the hierarchy s associated spectral curves are investigated and proven to have only odd genus without the introduction of integration constants we admit specific non zero constants of integration to the hierarchy and show that even genus curves may be introduced in a very straightforward manner thereby accessing the full family of finite gap solutions to the vector nonlinear schrodinger equation finally an original construction of the infinite set of conserved densities of the hierarchy is given

exact soliton solutions of coupled nonlinear schrodinger equations shape changing collisions logic gates and partially coherent solitons the different dynamical features underlying soliton interactions in coupled nonlinear schrodinger equations which model multimode wave propagation under varied physical situations in nonlinear optics are studied in this paper by explicitly constructing multisoliton solutions up to four soliton solutions for two coupled and arbitrary n coupled nonlinear schrodinger equations using the hirota bilinearization method we bring out clearly the various features underlying the fascinating shape changing intensity redistribution collisions of solitons including changes in amplitudes phases and relative separation distances and the very many possibilities of energy redistributions among the modes of solitons however in this multisoliton collision process the pairwise collision nature is shown to be preserved in spite of the changes in the amplitudes and phases of the solitons detailed asymptotic analysis also shows that when solitons undergo multiple collisions there exists the exciting possibility of shape restoration of at least one soliton during interactions of more than two solitons represented by three and higher order soliton solutions from an application point of view we have shown from the asymptotic expressions how the amplitude intensity redistribution can be written as a generalized linear fractional transformation for the n component case also we indicate how the multisolitons can be reinterpreted as various logic gates for suitable choices of the soliton parameters leading to possible multistate logic in addition we point out that the various recently studied partially coherent solitons are just special cases of the bright soliton solutions exhibiting shape changing collisions thereby explaining their variable profile and shape variation in collision process

a finite element method for fully nonlinear water waves we introduce a numerical method for fully nonlinear three dimensional water surface waves described by standard potential theory the method is based on a transformation of the dynamic water volume onto a fixed domain regridding at each time step is thereby avoided the transformation introduces an elliptic boundary value problem which is solved by a preconditioned conjugate gradient method moreover a simple domain imbedding precedure is introduced to solve problems with an obstacle in the water volume numerical experiments are presented and they show nice convergence properties of the iterative solver as well as convergence of the entire solution towards a reference solution computed by another scheme

space time discontinuous galerkin method for nonlinear water waves a space time discontinuous galerkin dg finite element method for nonlinear water waves in an inviscid and incompressible fluid is presented the space time dg method results in a conservative numerical discretization on time dependent deforming meshes which follow the free surface evolution the algorithm is higher order accurate both in space and time and closely related to an arbitrary lagrangian eulerian ale approach a detailed derivation of the numerical algorithm is given including an efficient procedure to solve the nonlinear algebraic equations resulting from the space time discretization numerical examples are shown on a series of model problems to demonstrate the accuracy and capabilities of the method

on higher order fully periodic boundary value problems abstract in this paper we present sufficient conditions for the existence of periodic solutions of the higher order fully differential equation u n x f x u x u x u n 1 x with n 3 x a b and f a b r n r a continuous function verifying a nagumo type growth condition a new type of lower and upper solutions eventually non ordered allows us to obtain not only the existence but also some qualitative properties on the solution the last section contains two examples to stress the application to both cases of n odd and n even

the use of backlund transformations in obtaining n soliton solutions in wronskian form the wronskian formulation of the n soliton solutions of various nonlinear evolution equations modified korteweg de vries with both zero and non zero asymptotic conditions sine gordon korteweg de vries kadomtsev petviashvili and boussinesq equations are obtained by inductive use of backlund transformations in hirota s 1980 bilinear notation

variational space time dis continuous galerkin method for nonlinear free surface water waves a new variational finite element method is developed for nonlinear free surface gravity water waves using the potential flow approximation this method also handles waves generated by a wave maker its formulation stems from miles variational principle for water waves together with a finite element discretization that is continuous in space and discontinuous in time one novel feature of this variational finite element approach is that the free surface evolution is variationally dependent on the mesh deformation vis a vis the mesh deformation being geometrically dependent on free surface evolution another key feature is the use of a variational dis continuous galerkin finite element discretization in time moreover in the absence of a wave maker it is shown to be equivalent to the second order symplectic stormer verlet time stepping scheme for the free surface degrees of freedom these key features add to the stability of the numerical method finally the resulting numerical scheme is verified against nonlinear analytical solutions with long time simulations and validated against experimental measurements of driven wave solutions in a wave basin of the maritime research institute netherlands

painleve analysis and integrability of coupled non linear schrodinger equations considering a system of coupled non linear schrodinger nls equations the authors discuss the integrability properties through painleve p analysis for the two coupled nls equations they show that there exists a pair of parametric values possessing the p property for which the associated backlund transformation bt and the hirota bilinearisation are constructed these parametric choices are identical to those of zakharov and schulman 1982 who established the integrability in terms of motion invariants finally they extend the p analysis to the n coupled nls system and identify two parametric choices possessing the p property which are natural generalisations of the two coupled nls cases

space time discontinuous galerkin finite element method for two fluid flows a novel numerical method for two fluid flow computations is presented which combines the space time discontinuous galerkin finite element discretization with the level set method and cut cell based interface tracking the space time discontinuous galerkin stdg finite element method offers high accuracy an inherent ability to handle discontinuities and a very local stencil making it relatively easy to combine with local hp refinement the front tracking is incorporated via cut cell mesh refinement to ensure a sharp interface between the fluids to compute the interface dynamics the level set method lsm is used because of its ability to deal with merging and breakup also the lsm is easy to extend to higher dimensions small cells arising from the cut cell refinement are merged to improve the stability and performance the interface conditions are incorporated in the numerical flux at the interface and the stdg discretization ensures that the scheme is conservative as long as the numerical fluxes are conservative the numerical method is applied to one and two dimensional two fluid test problems using the euler equations

space time discontinuous galerkin method for the compressible navier stokes equations a space time discontinuous galerkin finite element method for the compressible navier stokes equations is presented we explain the space time setting derive the weak formulation and discuss our choices for the numerical fluxes the resulting numerical method allows local grid adaptation as well as moving and deforming boundaries which we illustrate by computing the flow around a 3d delta wing on an adapted mesh and by simulating the dynamic stall phenomenon of a 2d airfoil in rapid pitch up maneuver

traveling water waves we present a numerical continuation method for traveling wave solutions to the full water wave problem using a spectral collocation discretization the water wave problem is reformulated in terms of surface variables giving rise to the zakharov craig sulem formulation and traveling waves are studied by introducing a phase velocity vector as a parameter we follow non trivial solution branches bifurcating from the trivial solution branch via numerical continuation methods techniques such as projections and filtering allow the computation to proceed for greater distances up the branch and parallelism allows the computation of larger problems we conclude with results including the formation of hexagonal patterns for the three dimensional problem

on the backlund gauge transformation and homoclinic orbits of a coupled nonlinear schrodinger system abstract the backlund gauge transformation for a system of coupled nls nonlinear schrodinger equations with a degenerate associated spectral operator is derived from an algebraic perspective extending aspects of other results m boiti tu guizhang il nuovo cimento 71b 1982 253 264 d h sattinger v d zurkowski physica d 26 1 3 1987 225 250 that apply in the context of non degenerate spectral operators moreover we demonstrate how the backlund gauge transformation can be used to explicitly construct the entire unstable manifold via superpositions of homoclinic orbits of a plane wave solution with both self phase instabilities and coupling instabilities this work builds on the results of ercolani et al n ercolani m g forest d w mclaughlin physica d 18 1986 472 474 n ercolani m g forest d w mclaughlin physica d 43 2 3 1990 349 384 for the sine gordon equation and forest et al m g forest d w mclaughlin d j muraki o c wright j nonlinear sci in press m g forest s p sheu o c wright phys lett a in press for the integrable coupled nls system

a hybrid spectral dg method for solving the phase averaged ocean wave equation algorithm and validation we develop a new high order hybrid discretization of the phased averaged action balance equation to simulate ocean waves we employ discontinuous galerkin dg discretization on an unstructured grid in geophysical space and fourier collocation along the directional and frequency coordinates the original action balance equation is modified to facilitate absorbing boundary conditions along the frequency direction this modification enforces periodicity at the frequency boundaries so that the fast convergence of fourier collocation holds in addition a mapping along the directional coordinate is introduced to cluster the collocation points around steep directional spectra time discretization is accomplished by the tvd runge kutta scheme the overall convergence of the scheme is exponential spectral we successfully verified and validated the method against several analytical solutions observational data and experimental results

a multi dimensional high order dg ale method based on gas kinetic theory with application to oscillating bodies this paper presents a multi dimensional high order discontinuous galerkin dg method in an arbitrary lagrangian eulerian ale formulation to simulate flows over variable domains with moving and deforming meshes it is an extension of the gas kinetic dg method proposed by the authors for static domains x ren et al 2015 22 a moving mesh gas kinetic dg method is proposed for both inviscid and viscous flow computations a flux integration method across a translating and deforming cell interface has been constructed differently from the previous ale type gas kinetic method with piecewise constant mesh velocity at each cell interface within each time step the mesh velocity variation inside a cell and the mesh moving and rotating at a cell interface have been accounted for in the finite element framework as a result the current scheme is applicable for any kind of mesh movement such as translation rotation and deformation the accuracy and robustness of the scheme have been improved significantly in the oscillating airfoil calculations all computations are conducted in a physical domain rather than in a reference domain and the basis functions move with the grid movement therefore the numerical scheme can preserve the uniform flow automatically and satisfy the geometric conservation law gcl the numerical accuracy can be maintained even for a largely moving and deforming mesh several test cases are presented to demonstrate the performance of the gas kinetic dg ale method

numerical simulation of gravity waves abstract we present a new spectral method to simulate numerically the waterwave problem in a channel for a fluid of finite or infinite depth it is formulated in terms of the free surface elevation and the velocity potential the numerical method is based on the reduction of this problem to a lower dimensional computation involving surface variables alone to accomplish this we describe the taylor expansion of the dirichlet neumann operator in homogeneous powers of the surface elevation each term is a concatenation of fourier multipliers with powers of and its derivatives and is valid uniformly in wavenumber these are easily calculated using the fast fourier transform the method is illustrated by computing the long time evolution of modulated wave packets and of approximations to the stokes steady wave train by imposing a surface pressure we observe surface steepening in large amplitude evolution and wake and bow wave development for flows with a close to critical froude number finally we give an example of nonlinear evolution of the distribution of energy among normal modes

a space time galerkin least squares finite element formulation of the navier stokes equations for moving domain problems a space time galerkin least squares finite element formulation of the navier stokes equations is presented for the analysis of free surface flows moving spatial configurations and deforming fluid structure interfaces the variational equation is based on the time discontinuous galerkin method employing the physical entropy variables the space time elements are oriented in time to accommodate the spatial deformations if the elements are oriented along the particle paths the formulation is lagrangian and if they are fixed in time it is eulerian consequently this formulation is analogous to the arbitrary lagrangian eulerian ale technique a novel mesh rezoning strategy is presented to orient the elements in time and adapt the fluid mesh to the changing spatial configuration numerical results are presented to show the performance of the method