automatic mesh generation and adaptation by using contours a general and efficient remeshing algorithm is presented for the discretization of arbitrary planar domains into triangular elements in consistency with the given node spacing funstion the contour lines of the node spacing function at suitable calculated levels provide the natural lines of division of the problem domain into subregions where finite elements meshes of different element sizes zre generated using the available general purpose mesh generators

stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

integral equation preconditioning for the solution of poisson s equation on geometrically complex regions this paper is concerned with the implementation and investigation of integral equation based solvers as preconditioners for finite difference discretizations of poisson equations in geometrically complex domains the target discretizations are those associated with cut out grids we discuss such grids and also describe a software structure which enables their rapid construction computational results are presented

solution adaptive cartesian cell approach for viscous and inviscid flows a cartesian cell based approach for adaptively refined solutions of the euler and navier stokes equations in two dimensions is presented grids about geometrically complicated bodies are generated automatically by the recursive subdivision of a single cartesian cell encompassing the entire flow domain where the resulting cells intersect bodies polygonal cut cells are created using modified polygon clipping algorithms the grid is stored in a binary tree data structure that provides a natural means of obtaining cell to cell connectivity and of carrying out solution adaptive mesh refinement the euler and navier stokes equations are solved on the resulting grids using a finite volume formulation the convective terms are up winded a linear reconstruction of the primitive variables is performed providing input states to an approximate riemann solver for computing the fluxes between neighboring cells the results of a study comparing the accuracy and positivity of two classes of cell centered viscous gradient reconstruction procedures is briefly summarized adaptively refined solutions of the navier stokes equations are shown using the more robust of these gradient reconstruction procedures where the results computed by the cartesian approach are compared to theory experiment and other accepted computational results for a series of low and moderate reynolds number flows

multipeakons and the classical moment problem abstract classical results of stieltjes are used to obtain explicit formulas for the peakon antipeakon solutions of the camassa holm equation the closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem it is shown that collisions occur only in peakon antipeakon pairs and the details of the collisions are analyzed using results from the moment problem a sharp result on the steepening of the slope at the time of collision is given asymptotic formulas are given and the scattering shifts are calculated explicitly

persistence properties and unique continuation of solutions of the camassa holm equation it is shown that a strong solution of the camassa holm equation initially decaying exponentially together with its spacial derivative must be identically equal to zero if it also decays exponentially at a later time in particular a strong solution of the cauchy problem with compact initial profile can not be compactly supported at any later time unless it is the zero solution

traveling wave solutions of the camassa holm equation all weak traveling wave solutions of the camassa holm equation are classified we show that in addition to smooth solutions there are a multitude of traveling waves with singularities peakons cuspons stumpons and composite waves

new razumikhin type theorems on the stability for impulsive functional differential systems in this paper we investigate the stability of the trivial solution for impulsive functional differential systems using several lyapunov functions including partial components coupled with the razumikhin technique and obtained some new razumikhin type theorems which avoid using the auxiliary function p under less restrictive conditions our results improve some of the earlier findings and are suitable for many applications some examples are given to illustrate the advantages of the theorems obtained

implicit solution of the unsteady euler equations for high order accurate discontinuous galerkin discretizations efficient solution techniques for high order accurate time dependent problems are investigated for solving the two dimensional non linear euler equations in this work the spatial discretization consists of a high order accurate discontinuous galerkin dg approach implicit time integration techniques are considered exclusively in order to avoid the stability restrictions of explicit methods standard backwards differencing methods bdf1 and bdf2 as well as a second order crank nicholson cn2 and a fourth order implicit runge kutta irk4 scheme are considered in an attempt to balance the spatial and temporal accuracy of the overall approach the implicit system arising at each time step is solved using a p multigrid approach which is shown to produce h independent convergence rates while remaining relatively insensitive to the time step size the crank nicholson methodology although not l stable demonstrates superior performance compared to the bdf2 scheme for the problems chosen in this work however the fourth order accurate implicit runge kutta scheme is found to be the most efficient in terms of computational cost for a given accuracy level as compared to the lower order schemes in spite of the added cost per time step and the benefits of this scheme increase for tighter error tolerances

unstructured viscous grid generation by the advancing layers method a new method of generating unstructured triangular tetrahedral grids with high aspect ratio cells is proposed the method is based on new grid marching strategy referred to as advancing layers for construction of highly stretched cells in the boundary layer and the conventional advancing front technique for generation of regular equilateral cells in the inviscid flow region unlike the existing semi structured viscous grid generation techniques the new procedure relies on a totally unstructured advancing front grid strategy resulting in a substantially enhanced grid flexibility and efficiency the method is conceptually simple but powerful capable of producing high quality viscous grids for complex configurations with ease a number of two dimensional triangular grids are presented to demonstrate the methodology the basic elements of the method however have been primarily designed with three dimensional problems in mind making it extendible for tetrahedral viscous grid generation

the cauchy problem for the fokas olver rosenau qiao equation abstract it is shown that the initial value problem for the fokas olver rosenau qiao equation forq is well posed in sobolev spaces h s s 5 2 in the sense of hadamard furthermore it is proved that the dependence on initial data is sharp i e the data to solution map is continuous but not uniformly continuous also peakon travelling wave solutions are derived on both the circle and the line and are used to prove that the solution map is not uniformly continuous in h s for s 3 2

autonomous ovsyannikov theorem and applications to nonlocal evolution equations and systems abstract this work presents an ovsyannikov type theorem for an autonomous abstract cauchy problem in a scale of decreasing banach spaces which in addition to existence and uniqueness of solution provides an estimate about the analytic lifespan of the solution then using this theorem it studies the cauchy problem for camassa holm type equations and systems with initial data in spaces of analytic functions on both the circle and the line which is the main goal of this paper finally it studies the continuity of the data to solution map in spaces of analytic functions

unstructured mel modelling of nonlinear unsteady ship waves an unstructured mixed eulerian lagrangian model mel is described for the simulation of the nonlinear wave body interaction as opposed to the structured model used widely in this field in present model the free surface is interpolated accurately using a polynomial scheme coupled with the moving least squares method and is triangulated with a local interrogation method coupled with the advancing front method to provide a high quality mesh a new method is devised to update the free surface intersection with the body based on the desingularized boundary integral method double desingularized point sources are used for a control point at the intersection with one inside the body and the other above the free surface to satisfy both the body and free surface boundary conditions over there the desingularized point sources are positioned in terms of the element sizes as well as solid angles of the boundary surface so as to avoid possible overcrowding of the sources at the corners of the surface solid angles on the surface are computed analytically using the spherical triangle theory as an illustration the algorithm is used to simulate the wave generation for a wigley hull which impulsively starts to move at a constant speed on a calm water surface the anticipated kelvin ship wave pattern is well simulated the wave profiles along the hull agree well with the measured results of nobeless and mccarthy proceedings of the 2nd dtnsrdc workshop maryland 1983 pp 5 35 as compared to a structured approach the unstructured model reduces the cpu time and memory requirements and being robust in handling complex geometries is more versatile in practical applications

integrable peakon equations with cubic nonlinearity we present a new integrable partial differential equation found by vladimir novikov like the camassa holm and degasperis procesi equations this new equation admits peaked soliton peakon solutions but it has nonlinear terms that are cubic rather than quadratic we give a matrix lax pair for v novikov s equation and show how it is related by a reciprocal transformation to a negative flow in the sawada kotera hierarchy infinitely many conserved quantities are found as well as a bi hamiltonian structure the latter is used to obtain the hamiltonian form of the finite dimensional system for the interaction of n peakons and the two body dynamics n 2 is explicitly integrated finally all of this is compared with some analogous results for another cubic peakon equation derived by zhijun qiao

coupling a wall diffusion model with an immersed boundary technique immersed boundary methods showed a good compliance for a wide range of complex inviscid and viscous flows moreover the high reynolds number regime still remains an open issue the paper describes a wall layer approach to answer the need of a model that goes beyond the actual capability of classical wall functions a near wall region is established that gets information from the outer flowfield and returns back the wall stress the latter is obtained by integrating simplified thin boundary layer equations along a normal to the wall subgrid in the outer zone the compressible reynolds averaged navier stokes equations are solved by means of a finite volume method on two or three dimensional cartesian meshes the classic validation test dealing with a high reynolds number flow over a flat plate is carried out a benchmark two dimensional complex flow is numerically investigated to assess the performance of the wall model in case of a pressure induced separation the results are compared with both experime

non uniform dependence on initial data of solutions to the euler equations of hydrodynamics we show that continuous dependence on initial data of solutions to the euler equations of incompressible hydrodynamics is optimal more precisely we prove that the data to solution map is not uniformly continuous in sobolev h s topology for any s in mathbb r if the domain is the flat torus mathbb t n mathbb r n 2 pi mathbb z n and for any s 0 if the domain is the whole space mathbb r n

turbulent wall model for immersed boundary methods the paper describes the development of a wall model to extend the applicability of immersed boundary methods to high reynolds number flows a two layer approach based on a decomposition of the near wall region is adopted an outer region is governed by the compressible rans equations which are solved numerically by using a classical finite volume method in the proximity of the wall an inner zone is established and modelled by a simplified version of the thin boundary layer equations the simulation platform is based on cartesian meshes and an immersed boundary technique it is able to solve the steady euler rans equations in two and three dimensional coordinates the robustness and the accuracy of the methodology are discussed at present this work represents the last advance of a research activity whose final goal is a fast pre design tool for aeronautical industrial applications

tri hamiltonian duality between solitons and solitary wave solutions having compact support a simple scaling argument shows that most integrable evolutionary systems which are known to admit a bi hamiltonian structure are in fact governed by a compatible trio of hamiltonian structures we demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons solitary wave solutions having compact support or cusped and or peaked solitons a general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified korteweg de vries equation the nonlinear schr odinger equation the integrable boussinesq system used to model the two way propagation of shallow water waves and the ito system of coupled nonlinear wave equations these hierarchies include a remarkable variety of interesting integrable nonlinear differential equations textcopyright 1996 the american physical society

robust implicit direct discontinuous galerkin method for simulating the compressible turbulent flows this paper proposes a robust and accurate implicit direct discontinuous galerkin ddg method for simulating laminar and turbulent flows the reynolds averaged navier stokes rans equations couple

navier stokes calculations using cartesian grids i laminar flows a finite volume formulation for the navier stokes equations using cartesian grids is used to study flows past airfoils in addition to the solution of the complete equations solutions for two simplified versions of the governing equations were obtained and compared with those using body fitted grids results are presented for two airfoil sections naca 0012 and rae 2822 for a range of mach numbers angles of attack and reynolds numbers it is shown that the results are highly dependent on the smoothness of the surface grid without such smoothness the skin friction and pressure converge to nonuniform distributions on the other hand when surface cells with smoothly varying areas are used the results compare favorably with calculations employing body fitted grids urrently much of the work in computational fluid dynamics is focused on the solution of the navier stokes equations for increasingly complex airfoil and wing configura tions some of the issues being examined in these solutions are the efficiency of the algorithms the accuracy of the solutions and the generation of grids for complex geometries cartesian grids are being pursued as a simple alternative to complicated grid generation techniques for general configurations the successful implementation of cartesian grids for the solution of the euler equations1 demonstrated the simplicity and utility of these grids especially when multi element configurations were considered more recently use of cartesian grids was extended to the solution of the euler equations for flowfields around three dimensional geometries 2 the goal of the present work is to show the feasibility of using cartesian grids in the solution of the navier stokes equations for complex two dimensional configurations be cause of the many issues that were uncovered by the investi gation it was decided to limit the scope of the paper to laminar flows the algorithm employed is similar to that used in refs 1 and 2 and is based on the explicit runge kutta time stepping scheme of jameson et al 3 modifications pertaining to numerical damping were examined moreover the crucial influence of surface cells on the pressure and skin friction is investigated in detail a cell centered scheme is employed for such a scheme it is necessary to estimate the pressure and the various components of the stress tensor and the heat flux vector at the wall the wall pressure was estimated by extrapolation 1 2 two different methods for determining wall stresses are used and their effect on the solution is discussed in addition to the solution of the full equations two sim plified versions of the governing equations are considered

computation of solid liquid phase fronts in the sharp interface limit on fixed grids a finite difference formulation is applied to track solid liquid boundaries on a fixed underlying grid the interface is not of finite thickness but is treated as a discontinuity and is explicitly tracked the imposition of boundary conditions exactly on a sharp interface that passes through the cartesian grid is performed using simple stencil readjustments in the vicinity of the interface attention is paid to formulating difference schemes that are globally second order accurate in x and t error analysis and grid refinement studies are performed for test problems involving the diffusion and convection diffusion equations and for stable solidification problems issues concerned with stability and change of phase of grid points in the evolution of solid liquid phase fronts are also addressed it is demonstrated that the field calculation is second order accurate while the position of the phase front is calculated to first order accuracy furthermore the accuracy estimates hold for the cases where there is a property jump across the interface unstable solidification phenomena are simulated and an attempt is made to compare results with previously published work the results indicate the need to begin an effort to benchmark computations of instability phenomena

global dense multiscale reconstruction for a billion points we present a variational approach for surface reconstruction from a set of oriented points with scale information we focus particularly on scenarios with nonuniform point densities due to images taken from different distances in contrast to previous methods we integrate the scale information in the objective and globally optimize the signed distance function of the surface on a balanced octree grid we use a finite element discretization on the dual structure of the octree minimizing the number of variables the tetrahedral mesh is generated efficiently with a lookup table which allows to map octree cells to the nodes of the finite elements we optimize memory efficiency by data aggregation such that robust data terms can be used even on very large scenes the surface normals are explicitly optimized and used for surface extraction to improve the reconstruction at edges and corners

on a class of physically important integrable equations abstract a methodology introduced by fuchssteiner and the author is used to derive a class of physically important integrable evolution equations among these equations are integrable generalizations of the korteweg devries kdv of the modified kdv of the nonlinear schrodinger nls and of the sine gordon equations the modeling of water waves as well as general asymptotic considerations are used to illustrate the occurrence of the generalized modified kdv and nls equations respectively

a new comparison principle for impulsive differential systems with variable impulsive perturbations and stability theory in this paper the stability properties in terms of two measures for impulsive differential systems with variable impulses are investigated a new comparison principlem which allows trajectories strike the same hypersurface finite times is established and then be applied to obtain a stability criterion finally one example is worked out to show the advantage of our results

holder continuity of the solution map for the novikov equation the novikov equation ne has been discovered recently as a new integrable equation with cubic nonlinearities that is similar to the camassa holm and degasperis procesi equations which have quadratic nonlinearities ne is well posed in sobolev spaces hs on both the line and the circle for s 3 2 in the sense of hadamard and its data to solution map is continuous but not uniformly continuous this work studies the continuity properties of ne further for initial data in hs s 3 2 it is shown that the solution map for ne is holder continuous in hr topology for all 0 r s with exponent depending on s and r

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

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

a cartesian grid method with transient anisotropic adaptation a cartesian grid method with solution adaptive anisotropic refinement and coarsening is developed for simulating time dependent incompressible flows the cartesian grid cells and faces are managed using an unstructured data approach and algorithms are described for the time accurate transient anisotropic refinement and coarsening of the cells the governing equations are discretized using a collocated cell centered arrangement of velocity and pressure and advanced in time using the fractional step method significant savings in the memory requirement of the method can be realized by advancing the velocity field using a novel approximate factorization technique although an iterative technique is also presented the pressure poisson equation is solved using additive correction multigrid and an efficient coarse grid selection algorithm is presented finally the cartesian cell geometry allows the development of relatively simple analytic expressions for the optimal cell dimensions based on limiting the velocity interpolation error the overall method is validated by solving several benchmark flows including the 2d and 3d lid driven cavity flows and the 2d flow around a circular cylinder in this latter case an immersed boundary method is used to handle the embedded cylinder boundary

adjoint sensitivity formulation for discontinuous galerkin discretizations in unsteady inviscid flow problems this paper presents an unsteady discrete adjoint algorithm for high order implicit discontinuous galerkin discretizations in time dependent inviscid flow problems the major function of the adjoint approach is to obtain the sensitivity information in a time dependent functional output which in turn is used to drive an unsteady shape optimization process to deliver a minimum of the objective functional a gradient based optimization strategy is investigated in which the sensitivity derivatives of the objective functional with respect to input variables are formulated in the context of high order discontinuous galerkin discretizations while special emphasis is given to the variations and linearizations of curvilinear boundary elements implicit temporal discretizations consisting of a second order backward euler scheme and a fourth order implicit runge kutta scheme are considered exclusively in this work where the corresponding adjoint problem is required to be solved in a backward time integration manner due to the associated transpose operation two numerical examples for the unsteady shape design techniques are presented to verify the derived sensitivity formulations and to demonstrate the performance of the adjoint approach the first involves an inverse shape optimization case by matching a time dependent target pressure profile for a two dimensional periodic vortical gust impinging on a rae 2822 airfoil and the second considers minimization of the acoustic noise produced by subsonic flow over a naca0012 airfoil with a 0 03c thick blunt trailing edge

flow simulation using generalized static and dynamic grids a new approach is presented for a e ow simulation system using generalized grids in a generalized grid the physical domain of interest is decomposed into cells with an arbitrary number of edges or faces the grid can be of structured unstructured or hanging node type or an arbitrary combination of the types a cell face based data structure is used to storethegrid information a e ow simulation system is developed forgeneralized gridsthat can handle static and dynamic grids the full navier stokes equations in the integral form are taken as the relations that govern the e uid e ow a cell centered e nite volume scheme is developed for solving these governing equations the numerical e ux across the cell faces is evaluated by an upwind scheme based on roe s approximate riemann solver a higher order scheme is formulated by utilizing taylor s series expansion and green s theorem limiter functions are used to preserve monotonocity the skin friction coefe cient is used to evaluate the accuracy of the limiterfunctions thegeneralized minimalresidualmethod isutilized tosolvethesparselinearsystem ofequations resulting from thelinearization of the e ux vectors thespalart allmaras one equation turbulence model has been implemented for the generalized grids and is used to evaluate the turbulent viscosity for dynamically moving bodies the equations of classical mechanics are used to predict the trajectory based on the external aerodynamic and body forces a variety of computational examples are presented to demonstratethe wide rangeof applications and the results are compared with experimental data whenever available

3d delaunay mesh generation coupled with an advancing front approach abstract this paper describes a fully automatic mesh generation method suitable for domains of any shape in r 3 initially a so called boundary mesh is generated wherein internal points are created using advancing front point placement and inserted using a delaunay method the algorithm combines the advantages of efficiency and nice mathematical properties of a delaunay approach with advancing front high quality point placement strategy several three dimensional examples are presented to demonstrate the overall efficiency and the relevance of the combined procedure the present method can be extended to isotropic or anisotropic adaptive mesh generation

adjoint based h p adaptive discontinuous galerkin methods for the 2d compressible euler equations in this paper we investigate and present an adaptive discontinuous galerkin algorithm driven by an adjoint based error estimation technique for the inviscid compressible euler equations this approach requires the numerical approximations for the flow i e primal problem and the adjoint i e dual problem which corresponds to a particular simulation objective output of interest the convergence of these two problems is accelerated by an hp multigrid solver which makes use of an element gauss seidel smoother on each level of the multigrid sequence the error estimation of the output functional results in a spatial error distribution which is used to drive an adaptive refinement strategy which may include local mesh subdivision h refinement local modification of discretization orders p enrichment and the combination of both approaches known as hp refinement the selection between h and p refinement in the hp adaptation approach is made based on a smoothness indicator applied to the most recently available flow solution values numerical results for the inviscid compressible flow over an idealized four element airfoil geometry demonstrate that both pure h refinement and pure p enrichment algorithms achieve equivalent error reductions at each adaptation cycle compared to a uniform refinement approach but requiring fewer degrees of freedom the proposed hp adaptive refinement strategy is capable of obtaining exponential error convergence in terms of degrees of freedom and results in significant savings in computational cost a high speed flow test case is used to demonstrate the ability of the hp refinement approach for capturing strong shocks or discontinuities while improving functional accuracy

the power series method for nonlocal and nonlinear evolution equations abstract the initial value problem for a 4 parameter family of nonlocal and nonlinear evolution equations with data in a space of analytic functions is solved by using a power series method in abstract banach spaces in addition to determining the power series expansion of the solution this method also provides an estimate of the analytic lifespan expressed in terms of the norm of the initial data thus establishing an abstract cauchy kovalevsky type theorem for these equations

implicit method for the computation of unsteady flows on unstructured grids an implicit method for the computation of unsteady flows on unstructured grids is presented following a finite difference approximation for the time derivative the resulting nonlinear system of equations is solved at each time step by using an agglomeration multigrid procedure the method allows for arbitrarily large time steps and is efficient in terms of computational effort and storage inviscid and viscous unsteady flows are computed to validate the procedure the issue of the mass matrix which arises with vertex centered finite volume schemes is addressed the present formulation allows the mass matrix to be inverted indirectly a mesh point movement and reconnection procedure is described that allows the grids to evolve with the motion of bodies as an example of flow over bodies in relative motion flow over a multi element airfoil system undergoing deployment is computed

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

high order discontinuous galerkin method for computation of turbulent flows this paper presents a high order discontinuous galerkin method for computation of compressible turbulent flows in three space dimensions a modified spalart and allmaras turbulence model is implemented and discretized to the same order of accuracy as that for the reynolds averaged navier stokes equations the creation of curvilinear meshes for arbitrary three dimensional configurations is achieved through the aid of a computational analysis programming interface which enables direct communications with the computer aided design module to determine the true positions of surface quadrature points for high order geometric representations a modified linear elasticity approach is used sequentially and is of crucial importance for reshaping the interior mesh to allow high aspect ratio curved elements in turbulent boundary layers requirements of the mesh parameters including the wall spacing and viscous stretching factor are studied and it is concluded that for attached turbulent flows the conventional s

surface mesh generation by means of steiner triangulations a method is presented for the efficient and automatic generation of unstructured grids on surfaces the overall procedure uses a steiner triangulation where sites are added in arbitrary order and hence the tracking of fronts is not needed in the algorithm the connectivity of the new points is initially generated by direct subdivision and further improved by using local reconnection the lawson s method is used to obtain a delaunay tessellation but other reconnection criteria may be used with minor modifications e g minimise the maximum angle the mesh generation procedure uses an approximate physical space grid to define the surface during the grid generation process at the end the mapped space coordinates are transformed back to the actual surface the field point distribution is controlled by an element size function obtained from the solution of a laplace equation whose boundary conditions are derived from the boundary node information additional control is possible by prescribing dummy boundaries the combination of grid smoothing and the elimination of nodes of degree 3 4 pairs of nodes of degree 5 and nodes of degree higher than 7 as a postprocessing step has proved to be an efficient tool to improve the grid quality results are presented which demonstrate that high quality unstructured grids can be efficiently generated on surfaces using the present method

decay properties of solutions to a 4 parameter family of wave equations in this paper persistence properties of solutions are investigated for a 4 parameter family k abc equation of evolution equations having k 1 degree nonlinearities and containing as its integrable members the camassa holm the degasperis procesi novikov and fokas olver rosenau qiao equations these properties will imply that strong solutions of the k abc equation will decay at infinity in the spatial variable provided that the initial data does furthermore it is shown that the equation exhibits unique continuation for appropriate values of the parameters k a b and c

generalizations of the camassa holm equation we classify generalized camassa holm type equations which possess infinite hierarchies of higher symmetries we show that the obtained equations can be treated as negative flows of integrable quasi linear scalar evolution equations of orders 2 3 and 5 we present the corresponding lax representations or linearization transformations for these equations some of the obtained equations seem to be new

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

finite element euler computations in three dimensions an explicit finite element solution procedure for the three dimensional euler equations is presented the solution domain is automatically meshed using a tetrahedral mesh generator which is an extension of our previous two dimensional work several examples are included to illustrate the performance of the generator and solver an adaptive mesh regeneration procedure is used for the first time in three dimensions

octree advancing front method for generation of unstructured surface and volume meshes amethodisdescribedforthegeneration ofunstructuredgridsforcomplexthree dimensionalgeometriesincluding multibodydomains acombinedoctree advancing frontmethod ispresentedforgenerationoftheunstructured surface mesh as well as the tetrahedral volume mesh the unique feature of this generator is that a special octree provides the local length scales of the mesh which can vary signi cantly this octree is automatically created and eliminates the need for a user constructed background mesh to determine the grid spacing and stretching parameters the grid generator isrobust isgeometry independent and requires minimal userinteraction a techniqueis also presented for generation of anisotropic surface meshes that result in a signi cant reduction in the number of triangular faces automatic local remeshing is also employed to improve the quality of the tetrahedral mesh the geometry independence of the octree advancing front method is demonstrated through both surface and volume

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

an optimal order interior penalty discontinuous galerkin discretization of the compressible navier stokes equations in this article we propose a new symmetric version of the interior penalty discontinuous galerkin finite element method for the numerical approximation of the compressible navier stokes equations here particular emphasis is devoted to the construction of an optimal numerical method for the evaluation of certain target functionals of practical interest such as the lift and drag coefficients of a body immersed in a viscous fluid with this in mind the key ingredients in the construction of the method include i an adjoint consistent imposition of the boundary conditions ii an adjoint consistent reformulation of the underlying target functional of practical interest iii design of appropriate interior penalty stabilization terms numerical experiments presented within this article clearly indicate the optimality of the proposed method when the error is measured in terms of both the l 2 norm as well as for certain target functionals computational comparisons with other discontinuous galerkin schemes proposed in the literature including the second scheme of bassi and rebay cf f bassi s rebay gmres discontinuous galerkin solution of the compressible navier stokes equations in b cockburn g karniadakis c w shu eds discontinuous galerkin methods lecture notes in comput sci engrg vol 11 springer berlin 2000 pp 197 208 f bassi s rebay numerical evaluation of two discontinuous galerkin methods for the compressible navier stokes equations int j numer methods fluids 40 2002 197 207 the standard sipg method outlined in r hartmann p houston symmetric interior penalty dg methods for the compressible navier stokes equations i method formulation int j numer anal model 3 1 2006 1 20 and an nipg variant of the new scheme will be undertaken

wave breaking and peakons for a modified camassa holm equation in this paper we investigate the formation of singularities and the existence of peaked traveling wave solutions for a modified camassa holm equation with cubic nonlinearity the equation is known to be integrable and is shown to admit a single peaked soliton and multi peakon solutions of a different character than those of the camassa holm equation singularities of the solutions can occur only in the form of wave breaking and a new wave breaking mechanism for solutions with certain initial profiles is described in detail

periodic boundary value problems for first order functional differential equations with impulse by applying the well known fixed point theorem of cone expansion and compression this paper investigates the existence of multiple positive solutions of periodic boundary value problems for first order functional differential equations with impulse meanwhile four examples are worked out to demonstrate the main results

a new flux splitting scheme a new flux splitting scheme is proposed the scheme is remarkably simple and yet its accuracy rivals and in some cases surpasses that of roe s solver in the euler and navier stokes solutions carried out in this study the scheme is robust and converges as fast as the roe splitting we propose an appropriately defined cell face advection mach number using values from the two straddling cells via associated characteristic speeds this interface mach number is then used to determine the upwind extrapolation for the convective quantities accordingly the name of the scheme is coined as the advection upstream splitting method ausm we also introduce a new pressure splitting which is shown to behave successfully yielding much smoother results than other existing pressure splittings of particular interest is the supersonic blunt body problem in which the roe scheme gives anomalous solutions the ausm produces correct solutions without difficulty for a wide range of flow conditions as well as grids

a versatile sharp interface immersed boundary method for incompressible flows with complex boundaries a sharp interface immersed boundary method for simulating incompressible viscous flow past three dimensional immersed bodies is described the method employs a multi dimensional ghost cell methodology to satisfy the boundary conditions on the immersed boundary and the method is designed to handle highly complex three dimensional stationary moving and or deforming bodies the complex immersed surfaces are represented by grids consisting of unstructured triangular elements while the flow is computed on non uniform cartesian grids the paper describes the salient features of the methodology with special emphasis on the immersed boundary treatment for stationary and moving boundaries simulations of a number of canonical two and three dimensional flows are used to verify the accuracy and fidelity of the solver over a range of reynolds numbers flow past suddenly accelerated bodies are used to validate the solver for moving boundary problems finally two cases inspired from biology with highly complex three dimensional bodies are simulated in order to demonstrate the versatility of the method

an advancing front delaunay triangulation algorithm designed for robustness abstract a new algorithm is described for generating an unstructured mesh about an arbitrary two dimensional configuration mesh points are generated automatically by the algorithm in a manner which ensures a smooth variation of elements and the resulting triangulation constitutes the delaunay triangulation of these points the algorithm combines the mathematical elegance and efficiency of delaunay triangulation algorithms with the desirable point placement features boundary integrity and robustness that are traditionally associated with advancing front type mesh generation strategies the method offers increased robustness over previous algorithms in that it cannot fail regardless of the initial boundary point distribution and the prescribed cell size distribution throughout the flow field

some tricks from the symmetry toolbox for nonlinear equations generalizations of the camassa holm equation abstract the main subject of the paper is to give a survey and to present new methods on how integrability results i e results for symmetry groups inverse scattering formulations action angle transformations and the like can be transferred from one equation to others in case the equations are not related by backlund transformations as a main example the so called camassa holm equation is chosen for which the relevant results are obtained by having a look on the korteweg de vries kdv equation the camassa holm equation turns out to be a different factorization equation of the kdv it describes shallow water waves and reconciles the properties which were known for different orders of shallow water wave approximations we follow here an old method already marginally mentioned in fuchssteiner and fokas 1981 and fuchssteiner 1983 and recently applied by others olver and rosenau 1995 the method allows an immediate recovery of the recursion operator for the camassa holm equation from the invariance structure of the kdv although both equations are not related by backlund transformations however in addition and different from other approaches from there by use of the squared eigenfunction relation for the kdv equation the lax pair formulation for the different factorization equation is derived for the example under consideration it is of course the one obtained in camassa and holm 1993 since the methods proposed here can be transferred to any compatible factorization of recursion operators its application even in the special case which was chosen for illustration leads to a large class of integrable equations among which the camassa holm equation can be found as well as a three parameter family of generalizations of the equation the advantage of the general approach to the lax pair presentation is that direct transformations between action and angle variables are obtained so using this lax pair formulation as a novel result a direct transformation between action and angle variables for the camassa holm equation is derived further novel results in the paper are a hodograph link back from a backlund transformation of the camassa holm equation to a particular member of the kdv hierarchy additional symmetries and the construction of the conformal algebra for the hierarchy of the camassa holm equation the methods involved include hereditary operators bi hamiltonian formulations nilpotent flows and scaling symmetries

a simple approach to discretize the viscous term with spatially varying eddy viscosity a simple approach to discretize the viscous dissipation term in the incompressible navier stokes equations with spatially varying viscosity is presented unlike the case where the viscosity remains constant its discretization may be quite cumbersome especially for high order staggered formulations to circumvent this problem we propose an alternative form of the viscous term whose discretization is straightforward notice that this approach is also suitable for eddy viscosity models for large eddy simulation moreover since it is based on already available operators it can be easily implemented on any structured or unstructured code the supra convergence of the method is numerically shown on both a fourth order cartesian staggered and an unstructured collocated formulation

a high order semi explicit discontinuous galerkin solver for 3d incompressible flow with application to dns and les of turbulent channel flow abstract we present an efficient discontinuous galerkin scheme for simulation of the incompressible navier stokes equations including laminar and turbulent flow we consider a semi explicit high order velocity correction method for time integration as well as nodal equal order discretizations for velocity and pressure the non linear convective term is treated explicitly while a linear system is solved for the pressure poisson equation and the viscous term the key feature of our solver is a consistent penalty term reducing the local divergence error in order to overcome recently reported instabilities in spatially under resolved high reynolds number flows as well as small time steps this penalty method is similar to the grad div stabilization widely used in continuous finite elements we further review and compare our method to several other techniques recently proposed in literature to stabilize the method for such flow configurations the solver is specifically designed for large scale computations through matrix free linear solvers including efficient preconditioning strategies and tensor product elements which have allowed us to scale this code up to 34 4 billion degrees of freedom and 147 456 cpu cores we validate our code and demonstrate optimal convergence rates with laminar flows present in a vortex problem and flow past a cylinder and show applicability of our solver to direct numerical simulation as well as implicit large eddy simulation of turbulent channel flow at r e 180 as well as 590

perturbative symmetry approach the aim of our paper is to formulate a perturbative version of the symmetry approach in the symbolic representation and to generalize it in order to make it suitable for the study of nonlocal and non evolution equations our formalism is the development and incorporation of the perturbative approach of zakharov and schulman the symbolic method of sanders and wang and the standard symmetry approach of shabat et al we apply our theory to describe integrable generalizations of the benjamin ono type equations and to isolate integrable cases of the camassa holm type equations

lax integrability and the peakon problem for the modified camassa holm equation peakons are special weak solutions of a class of nonlinear partial differential equations modelling non linear phenomena such as the breakdown of regularity and the onset of shocks we show that the natural concept of weak solutions in the case of the modified camassa holm equation studied in this paper is dictated by the distributional compatibility of its lax pair and as a result it differs from the one proposed and used in the literature based on the concept of weak solutions used for equations of the burgers type subsequently we give a complete construction of peakon solutions satisfying the modified camassa holm equation in the sense of distributions our approach is based on solving certain inverse boundary value problem the solution of which hinges on a combination of classical techniques of analysis involving stieltjes continued fractions and multi point pade approximations we propose sufficient conditions needed to ensure the global existence of peakon solutions and analyze the large time asymptotic behaviour whose special features include a formation of pairs of peakons that share asymptotic speeds as well as toda like sorting property

a new integrable equation with cuspons and w m shape peaks solitons in this paper we propose a new completely integrable wave equation mt mx u2 ux2 2m2ux 0 m u uxx the equation is derived from the two dimensional euler equation and is proven to have lax pair and bi hamiltonian structures this equation possesses new cusp solitons cuspons instead of regular peakons ce x ct with speed c through investigating the equation we develop a new kind of soliton solutions w m shape peaks solitons there exist no smooth solitons for this integrable water wave equation

the cauchy problem for a 4 parameter family of equations with peakon traveling waves abstract the initial value problem for a novel 4 parameter family of evolution equations which are nonlinear and nonlocal and possess peakon traveling wave solutions is studied on both the line and the circle it is proved that this family of equations is well posed in the sense of hadamard when the initial data belong to the sobolev spaces h s with s 5 2 also it is shown that the data to solution map is not uniformly continuous however if h s s 5 2 is equipped with a weaker h r norm 0 r s then the solution map becomes holder continuous