solution of the dispersionless hirota equations the dispersionless differential pay identity is shown to be equivalent to a kernel expansion providing a universal algebraic characterization and solution of the dispersionless hirota equations some calculations based on d bar data of the action are also indicated

a smooth transition model between kinetic and diffusion equations this paper presents a model which provides a smooth transition between a kinetic and a diffusion domain the idea is to use a buffer zone in which both diffusion and kinetic equations will be solved the solution of the original kinetic equation will be recovered as the sum of the solutions of these two equations we use an artificial connecting function which makes the equation on each domain degenerate at the end of the buffer zone thus no boundary condition is needed at the transition point this model avoids the delicate issue of finding the interface condition or iteration in a typical domain decomposition method that couples a kinetic equation with hydrodynamic equations a new asymptotic preserving method for this model is introduced and numerical examples are used to validate this new model and the new numerical method

simple finite element method in vorticity formulation for incompressible flows a very simple and efficient finite element method is introduced for two and three dimensional viscous incompressible flows using the vorticity formulation this method relies on recasting the traditional finite element method in the spirit of the high order accurate finite difference methods introduced by the authors in another work optimal accuracy of arbitrary order can be achieved using standard finite element or spectral elements the method is convectively stable and is particularly suited for moderate to high reynolds number flows

a fully semi lagrangian discretization for the 2d incompressible navier stokes equations in the vorticity streamfunction formulation abstract a numerical method for the two dimensional incompressible navier stokes equations in vorticity streamfunction form is proposed which employs semi lagrangian discretizations for both the advection and diffusion terms thus achieving unconditional stability without the need to solve linear systems beyond that required by the poisson solver for the reconstruction of the streamfunction a description of the discretization of dirichlet boundary conditions for the semi lagrangian approach to diffusion terms is also presented numerical experiments on classical benchmarks for incompressible flow in simple geometries validate the proposed method

dispersionless toda hierarchy and two dimensional string theory the dispersionless toda hierarchy turns out to lie in the heart of a recently proposed landau ginzburg formulation of two dimensional string theory at self dual compactification radius the dynamics of massless tachyons with discrete momenta is shown to be encoded into the structure of a special solution of this integrable hierarchy this solution is obtained by solving a riemann hilbert problem equivalence to the tachyon dynamics is proven by deriving recursion relations of tachyon correlation functions in the machinery of the dispersionless toda hierarchy fundamental ingredients of the landau ginzburg formulation such as landau ginzburg potentials and tachyon landau ginzburg fields are translated into the language of the lax formalism furthermore a wedge algebra is pointed out to exist behind the riemann hilbert problem and speculations on its possible role as generators of extra states and fields are presented

finite band solutions of the coupled dispersionless hierarchy the coupled dispersionless hierarchy is derived with the help of the zero curvature equation based on the lax matrix we introduce an algebraic curve of arithmetic genus n from which we establish the corresponding meromorphic function the baker akhiezer function and dubrovin type equations the straightening out of all the flows is given under the abel jacobi coordinates using the asymptotic properties of and we obtain the explicit theta function representations of the meromorphic function the baker akhiezer function and of solutions for the whole hierarchy

the immersed interface method for simulating prescribed motion of rigid objects in an incompressible viscous flow in the immersed interface method a boundary immersed in a fluid is represented as a singular force in the navier stokes equations this paper presents an explicit approach for computing the singular force to enforce prescribed motion of a rigid boundary in an incompressible viscous flow the tangential component of the singular force is related to the surface vorticity and is calculated from the normal derivative of the velocity the normal component of the singular force is determined from a predictor and a corrector the predictor uses the normal derivative of the vorticity the corrector superposes a homogeneous solution to the pressure poisson equation to achieve the desired normal derivative of the pressure in the current immersed interface method the velocity and the pressure are solved using the mac scheme with the incorporation of jump conditions induced by the singular force and a discontinuous finite body force the body force is applied to obtain the rigid motion of the fluid enclosed by the boundary circular couette flow flow past a cylinder and flow around flappers are simulated to test the accuracy stability and efficiency of the method as well as the effect of the corrector with no stiff springs to model rigid boundaries the method is stable at relatively high reynolds numbers

an efficient compact difference scheme for solving the streamfunction formulation of the incompressible navier stokes equations recently a new paradigm for solving the steady navier stokes equations using a streamfunction velocity formulation was proposed by gupta and kalita m m gupta j c kalita a new paradigm for solving navier stokes equations streamfunction velocity formulation j comput phys 207 2005 52 68 which avoids difficulties inherent in the conventional streamfunction vorticity and primitive variable formulations it is discovered that this formulation can reached second order accurate and obtained accuracy solutions with little additional cost for a couple of fluid flow problems in this paper an efficient compact finite difference approximation named as five point constant coefficient second order compact 5pcc soc scheme is proposed for the streamfunction formulation of the steady incompressible navier stokes equations in which the grid values of the streamfunction and the values of its first derivatives velocities are carried as the unknown variables the derivation approach is simple and can be easily used to derive compact difference schemes for other similar high order elliptic differential equations numerical examples including the lid driven cavity flow problem and a problem of flow in a rectangular cavity with the hight width ratio of 2 are solved numerically to demonstrate the accuracy and efficiency of the newly proposed scheme the results obtained are compared with ones by different available numerical methods in the literature the present scheme not only shows second order accurate but also proves more effective than the existing second order compact scheme of the streamfunction formulation in the aspect of computational cost

vorticity vector potential method for 3d viscous incompressible flows in time dependent curvilinear coordinates e and liu j comput phys 138 1997 57 82 put forward a finite difference method for 3d viscous incompressible flows in the vorticity vector potential formulation on non staggered grids in this paper we will extend this method to the case of flows in the presence of a deformable surface by use of two kinds of surface differential operators the implementation of boundary conditions on a plane is generalized to a curved smooth surface with given velocity distribution whether this be an inflow outflow interface or a curved wall to deal with the irregular and varying physical domain time dependent curvilinear coordinates are constructed and the corresponding tensor analysis is adopted in deriving the component form of the governing equations therefore the equations can be discretized and solved in a regular and fixed parametric domain numerical results are presented for a 3d lid driven cavity with a deforming surface and a 3d duct flow with a deforming boundary a new way to validate numerical simulations is proposed based on an expression for the rate of strain tensor on a deformable surface

a hybrid ψ v hoc approach for surface tension driven flows in level set framework abstract in this study a higher order compact hoc finite difference scheme is proposed for the simulation of multiphase flows with surface tension effects by coupling the pure streamfunction formulation of navier stokes n s equations with the level set method unlike most of the existing methods for multiphase flows that solve the incompressible navier stokes equations in the velocity pressure form or the stream function vorticity form we recast the governing equations into the biharmonic form of the n s equations which carries stream function as the only variable to be computed the coupling of this formulation with level set ls enables the simulation of two phase flows by the core computation of only the single variable free from discontinuities in the whole physical domain in the process we also propose a novel hybrid hoc algorithm for post processing the pressure on either side of the interface including the jump across it the robustness of this method is established through a series of benchmark problems such as dispersion of capillary waves static bubble and oscillating bubble at equilibrium in a fluid at rest and deforming circular drops in shear flows we compare our numerical results with available benchmark results and excellent match is obtained in all the cases along with accurate representation of overall trends our approach is proved to be very efficient in terms of its easy implementation relatively low computational time and high order of accuracy

fast direct solver for the biharmonic equation on a disk and its application to incompressible flows we develop a simple and efficient fft based fast direct solver for the biharmonic equation on a disk the biharmonic equation is split into a coupled system of harmonic problems we first use the truncated fourier series expansion to derive a set of coupled singular odes then we solve those singular equations by second order finite difference discretizations using a radial grid with shifting a half mesh away from the origin we can handle the coordinate singularity easily without pole conditions the sherman morrison formula is then applied to solve the resultant linear system in a cost efficient way the computational complexity of the method consists of o mnlog 2n arithmetic operations for mxn grid points the numerical accuracy check and some applications to the incompressible navier stokes flows inside a disk are conducted

conformal maps and integrable hierarchies we show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2d toda integrable hierarchy the maps are determined by a particular solution to the hierarchy singled out by the conditions known as string equations the same hierarchy locally solves the 2d inverse potential problem i e reconstruction of the domain out of a set of its harmonic moments this is the same solution which is known to describe 2d gravity coupled to c matter we also introduce a concept of the function for analytic curves

meshless schemes for unsteady navier stokes equations in vorticity formulation using radial basis functions in this paper we provide a new scheme for unsteady incompressible flows in vorticity stream function formulation combined with the radial basis functions method it is an efficient meshless method optimal accuracy can be achieved using this method the efficiency and accuracy are demonstrated by numerical examples

a pure compact scheme for the streamfunction formulation of navier stokes equations a pure streamfunction formulation is introduced for the numerical simulation of the two dimensional incompressible navier stokes equations the idea is to replace the vorticity in the vorticity streamfunction evolution equation by the laplacian of the streamfunction the resulting formulation includes the streamfunction only thus no inter function relations need to be invoked a compact numerical scheme which interpolates streamfunction values as well as its first order derivatives is presented and analyzed a number of numerical experiments are presented including driven and double driven cavities where the reynolds numbers are sufficiently large leading to symmetry breaking of asymptotic solutions

a high order discontinuous galerkin method for 2d incompressible flows in this paper we introduce a high order discontinuous galerkin method for two dimensional incompressible flow in the vorticity stream function formulation the momentum equation is treated explicitly utilizing the efficiency of the discontinuous galerkin method the stream function is obtained by a standard poisson solver using continuous finite elements there is a natural matching between these two finite element spaces since the normal component of the velocity field is continuous across element boundaries this allows for a correct upwinding gluing in the discontinuous galerkin framework while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability the method is efficient for inviscid or high reynolds number flows optimal error estimates are proved and verified by numerical experiments

on a direct approach to quasideterminant solutions of a noncommutative kp equation a noncommutative version of the modified kp equation and a family of its solutions expressed as quasideterminants are discussed the origin of these solutions is explained by means of darboux transformations and the solutions are verified directly we also verify directly an explicit connection between quasideterminant solutions of the noncommutative mkp equation and the noncommutative kp equation arising from the miura transformation

a mixed spectral wavelet method for the solution of the stokes problem the paper presents a mixed wavelet spectral chebychev method for solving the unsteady 2d stokes equations in the vorticity stream function formulation with periodicity condition in one direction after an appropriate time discretisation of the equations one has to solve at each time step a stationary stokes like problem a capacitance matrix method is used to eliminate the problem of boundary conditions this leads to solving a series of helmholtz problems the spatial discretisation makes use of the wavelet method in the periodic direction and the spectral collocation chebychev method in the non periodic direction the resolution of the discrete helmholtz problem is done by means of the diagonalisation technique in the non periodic direction the system then splits into a sequence of one dimensionnal periodic helmholtz problems which are efficiently inverted using ffts numerical tests show both the stability and the accuracy of the method

energy and helicity preserving schemes for hydro and magnetohydro dynamics flows with symmetry we propose a class of simple and efficient numerical scheme for incompressible fluid equations with coordinate symmetry by introducing a generalized vorticity stream formulation the divergence free constraints are automatically satisfied in addition with explicit treatment of the nonlinear terms and local vorticity boundary condition the navier stokes mhd respectively equation essentially decouples into 2 4 respectively scalar equation and thus the scheme is very efficient moreover with proper discretization of the nonlinear terms the scheme preserves both energy and helicity identities numerically this is achieved by recasting the nonlinear terms convection vorticity stretching geometric source lorentz force and electro motive force in terms of jacobians this conservative property is valid even in the presence of the pole singularity for axisymmetric flows the exact conservation of energy and helicity has effectively eliminated excessive numerical viscosity numerical examples have demonstrated both accuracy and efficiency of the scheme finally local mesh refinement near the boundary can also be easily incorporated into the scheme without extra cost

weakly compressible high order i stable central difference schemes for incompressible viscous flows abstract in this paper we propose a weakly compressible model for numerical simulation of the incompressible viscous flows this model asymptotically approximates the incompressible navier stokes equations when the mach number tends to zero the main advantage of this model is that its numerical discretization avoids any poisson solver thus is very attractive for problems with complicated geometries this model is discretized by high order center differences in space and the so called i stable method for time the linear stability region of an i stable method contains part of the imaginary axis when solving a system of convection diffusion equations with a small viscosity the i stable method allows a very large cell reynolds number thus is particularly suitable for the simulation of fluid flows with large reynolds numbers numerical experiments illustrate the efficiency and robustness of this approach

a high order adaptive numerical method for recirculating flows at large reynolds number abstract a numerical method is constructed for two dimensional navier stokes flows in a circular domain adaptive computational grids are employed to resolve the boundary layer and the boundary vorticity values are obtained in high order of accuracy numerical results show a quite good agreement with the asymptotic expansion calculation kim siam j appl math 58 5 1998 1394 1413 extension to general shape domains is briefly addressed

a cartesian grid method for modeling multiple moving objects in 2d incompressible viscous flow we present an efficient method for solving 2d incompressible viscous flows around multiple moving objects our method employs an underlying regular cartesian grid to solve the system using a streamfunction vorticity formulation and with discontinuities representing the embedded objects the no penentration condition for the moving geometry is satisfied by superposing a homogenous solution to the poisson s equation for the streamfunction the no slip condition is satisfied by generating vorticity on the surfaces of the objects both the initial poisson solution and the evaluation of the homogenous solution require embedding irregular discontinuities in a fast poisson solver computation time is dictated by the time required to do a fast poisson solution plus solve an integral form of laplace s equation there is no significant increase in computational cost if the geometry of the embedded objects is variable and moving relative to the underlying grid we test the method against the canonical example of flow past a cylinder and obtained new results on the flow and forces of two cylinders moving relative to each other

the dispersionless lax equations and topological minimal models it is shown that perturbed rings of the primary chiral fields of the topological minimal models coincide with some particular solutions of the dispersionless lax equations the exact formulae for the tree level partition functions of an topological minimal models are found the virasoro constraints for the analogue of the function of the dispersionless lax equation correspond ing to these models are proved

quasideterminant solutions of the generalized heisenberg magnet model in this paper we present the darboux transformation for the generalized heisenberg magnet ghm model based on the general linear lie group gl n and construct multi soliton solutions in terms of quasideterminants further we relate the quasideterminant multi soliton solutions obtained by means of darboux transformation with those obtained by the dressing method we also discuss the model based on the lie group su n and obtain explicit soliton solutions of the model based on su 2

driven cavity flows by efficient numerical techniques abstract efficient and reliable numerical techniques of high order accuracy are presented for solving problems of steady viscous incompressible flow in the plane and are used to obtain accurate solutions for the driven cavity a solution is obtained at reynolds number 10 000 on a 180 180 grid the numerical methods combine an efficient linear system solver an adaptive newton like method for nonlinear systems and a continuation procedure for following a branch of solutions over a range of reynolds numbers

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

quasideterminant multisoliton solutions of a supersymmetric chiral field model in two dimensions the darboux transformation of a supersymmetric principal chiral field model in two dimensions is studied we obtain exact superfield multisoliton solutions of the model by means of iterated darboux transformation and express them in terms of quasideterminants

vorticity boundary conditions and boundary vorticity generation for two dimensional viscous incompressible flows abstract in this paper we present boundary conditions appropriate for the vorticity formulation of the two dimensional incompressible viscous navier stokes equations these boundary conditions are incorporated into a finite difference scheme and the resulting method is of the vorticity creation type i e vorticity is generated at the boundary to ensure that the tangential velocity boundary condition is satisfied the results of computations with this finite difference method are presented for flow past a circular cylinder a difference scheme and computational results for a model problem the prandtl boundary layer equations describing flow over a semi infinite flat plate are also presented

an immersed interface method for simulating the interaction of a fluid with moving boundaries in the immersed interface method boundaries are represented as singular force in the navier stokes equations which enters a numerical scheme as jump conditions recently we systematically derived all the necessary spatial and temporal jump conditions for simulating incompressible viscous flows subject to moving boundaries in 3d with second order spatial and temporal accuracy near the boundaries sheng xu z jane wang systematic derivation of jump conditions for the immersed interface method in three dimensional flow simulation siam j sci comput 2006 in press in this paper we implement the immersed interface method to incorporate these jump conditions in a 2d numerical scheme we study the accuracy efficiency and robustness of our method by simulating taylor couette flow flow induced by a relaxing balloon flow past single and multiple cylinders and flow around a flapping wing our results show that 1 our code has second order accuracy in the infinity norm for both the velocity and the pressure 2 the addition of an object introduces relatively insignificant computational cost 3 the method is equally effective in computing flow subject to boundaries with prescribed force or boundaries with prescribed motion

meshfree point collocation method for the stream vorticity formulation of 2d incompressible navier stokes equations abstract meshfree point collocation method is developed for the stream vorticity formulation of two dimensional incompressible navier stokes equations particular emphasis is placed on the novel formulation of effective vorticity condition on no slip boundaries the moving least square approximation is employed to construct shape functions in conjunction with the framework of point collocation method the derivatives of an arbitrary function can be obtained by the linear combination of these shape functions which enables the vorticity boundary condition to vary linearly with the boundary velocity and the stream function together with the second order partial differential equations for the stream and vorticity functions this boundary condition of vorticity provides the efficient meshfree point collocation scheme for the two dimensional incompressible flow the accuracy and stability for the proposed scheme are demonstrated through a new type of application problem with complex geometry in addition to several typical examples of steady state flow simulation

the controversial stability analysis in this review we present different techniques for obtaining stability limits for a finite difference scheme the forward time and space centered numerical scheme applied to the convection diffusion equation a survey of past attempts to state stability conditions for this scheme illustrates the difficulties in stability analysis that arise as soon as a scheme becomes more complex and illuminates the concepts of necessary and sufficient conditions for stability

an algebraic method to construct the darboux matrix we present an effective procedure to construct the 1 soliton darboux matrix our approach based on the zakharov shabat mikhailov s dressing method is especially useful in the case of non canonical normalization and for non isospectral linear problems the construction is divided into two steps first we represent a given linear problem as a system of some algebraic constraints on two matrices in this context we introduce and discuss invariants of the darboux matrix second we derive the darboux matrix demanding that it preserves the algebraic constraints in particular we consider in details the restrictions imposed by various reduction groups on the form of the darboux matrix

boundary conditions for open boundaries for the incompressible navier stokes equation abstract in this paper we investigate new boundary conditions for the incompressible time dependent navier stokes equation especially inflow and outflow conditions are considered the equations are linearized around a constant flow so that we can use laplace fourier technique to investigate the strength of boundary layers at open boundaries such layers are unphysical and new boundary conditions are proposed so that these boundary layers are suppressed we also show that the boundary conditions we propose do not produce divergence furthermore they give solutions that do not grow in time as long as the forcing function in the system does not we also discuss the numerical treatment of boundaries when fourth order accurate finite difference operators are used to approximate spatial derivatives using higher order methods introduces eigensolutions with boundary layer thickness of the same order of magnitude as the grid size these eigensolutions have to be suppressed in order to not destroy the fourth order accuracy of the method numerical results for the non linear navier stokes equation together with the new boundary conditions are presented these calculations confirm that the results for the linearized problem hold for the non linear problem as well

accurate stable and efficient navier stokes solvers based on explicit treatment of the pressure term we present numerical schemes for the incompressible navier stokes equations based on a primitive variable formulation in which the incompressibility constraint has been replaced by a pressure poisson equation the pressure is treated explicitly in time completely decoupling the computation of the momentum and kinematic equations the result is a class of extremely efficient navier stokes solvers full time accuracy is achieved for all flow variables the key to the schemes is a neumann boundary condition for the pressure poisson equation which enforces the incompressibility condition for the velocity field irrespective of explicit or implicit time discretization of the viscous term in the momentum equation the explicit time discretization of the pressure term does not affect the time step constraint indeed we prove unconditional stability of the new formulation for the stokes equation with explicit treatment of the pressure term and first or second order implicit treatment of the viscous term systematic numerical experiments for the full navier stokes equations indicate that a second order implicit time discretization of the viscous term with the pressure and convective terms treated explicitly is stable under the standard cfl condition additionally various numerical examples are presented including both implicit and explicit time discretizations using spectral and finite difference spatial discretizations demonstrating the accuracy flexibility and efficiency of this class of schemes in particular a galerkin formulation is presented requiring only c0 elements to implement

finite difference methods for 3d viscous incompressible flows in the vorticity vector potential formulation on nonstaggered grids simple efficient and accurate finite difference methods are introduced for 3d unsteady viscous incompressible flows in the vorticity vector potential formulation on nonstaggered grids two different types of methods are discussed they differ in the implementation of the normal component of the vorticity boundary condition and consequently the enforcement of the divergence free condition for vorticity both second order and fourth order accurate schemes are developed a detailed accuracy test is performed revealing the structure of the error and the effect of how the convective terms are discretized near the boundary the influence of the divergence free condition for vorticity to the overall accuracy is studied results on the cubic driven cavity flow at reynolds number 500 and 3200 are shown and compared with that of the mac scheme

a compact streamfunction velocity scheme on nonuniform grids for the 2d steady incompressible navier stokes equations in this paper an effective compact finite difference approximation which carries streamfunction and its first derivatives velocities as the unknown variables for the streamfunction velocity formulation of the steady two dimensional incompressible navier stokes equation is developed on non uniform orthogonal cartesian grids to solve the resulting system of equations a multigrid iterative strategy on nonuniform grids is introduced by using the interpolation techniques numerical experiments involving two test problems with analytical solutions and the lid driven square cavity flow problem are carried out to display the superiority of the currently developed method on nonuniform grid numerical results show that the present method on nonuniform grids gets as similarly efficient convergence rate as on uniform grids viz second order accuracy and the resolution of the computed solutions for the problems with the sharp changes can be significantly improved when the nonuniform grid strategy is utilized the backward facing step flow is also calculated by the present method to exhibit the capability to simulate the distant field using fewer grid points the solution for the natural convection problem reveals further the wide applications of the present method not only in the flow problems but also in the heat transfer problems all of these numerical results demonstrate the accuracy and efficiency of the currently proposed schemes

a discrete generalized coupled dispersionless integrable system and its multisoliton solutions abstract an integrable discretization of generalized coupled dispersionless dgcd integrable system via lax pair is presented a lax pair for the dgcd system is defined a darboux transformation is used on the lax pair to obtain multi soliton solutions of the dgcd system the solutions are expressed in terms of quasideterminants explicit expressions of discrete one and two soliton solutions are obtained for the s u 2 case by using properties of quasideterminants we also study continuous analogue of the dgcd system by applying continuum limit

finite difference schemes for incompressible flow based on local pressure boundary conditions in this paper we discuss the derivation and use of local pressure boundary conditions for finite difference schemes for the unsteady incompressible navier stokes equations in the velocity pressure formulation their use is especially well suited for the computation of moderate to large reynolds number flows we explore the similarities between the implementation and use of local pressure boundary conditions and local vorticity boundary conditions in the design of numerical schemes for incompressible flow in 2d in their respective formulations when these local numerical boundary conditions are coupled with a fully explicit convectively stable time stepping procedure the resulting methods are simple to implement and highly efficient unlike the vorticity formulation the use of the local pressure boundary condition approach is readily applicable to 3d flows the simplicity of the local pressure boundary condition approach and its easy adaptation to more general flow settings make the resulting scheme an attractive alternative to the more popular methods for solving the navier stokes equations in the velocity pressure formulation we present numerical results of a second order finite difference scheme on a nonstaggered grid using local pressure boundary conditions stability and accuracy of the scheme applied to stokes flow is demonstrated using normal mode analysis also described is the extension of the method to variable density flows

a three point combined compact difference scheme a new three point combined compact difference ccd scheme is developed for numerical models the major features of the ccd scheme are three point implicit sixth order accuracy and inclusion of boundary values due to its combination of the first and second derivatives the ccd scheme becomes more compact and more accurate than normal compact difference schemes the efficient twin tridiagonal for calculating derivatives and triple tridiagonal for solving partial difference equation with the ccd scheme methods are also presented besides the ccd scheme has sixth order accuracy at periodic boundaries and fifth order accuracy at nonperiodic boundaries the possibility of extending to a three point eighth order scheme is also included

discrete analogues of the generalized coupled integrable dispersionless equations we study the integrable discretization of the generalized coupled integrable dispersionless equations two semi discrete and one fully discrete versions of the system are given via hirota s bilinear method soliton solutions for the derived discrete systems are also presented

a central difference scheme for a pure stream function formulation of incompressible viscous flow we present a numerical scheme for incompressible viscous flow formulated as an equation for the stream function the pure stream function formulation obviates the difficulty associated with vorticity boundary conditions the resulting biharmonic equation is discretized with a compact scheme and solved with an algebraic multigrid solver the advection of vorticity is implemented with a high resolution central scheme that remains stable and accurate in the presence of large gradients the accuracy and robustness of the method are demonstrated for high reynolds number flows in a lid driven cavity

backlund transformations and the atiyah ward ansatz for non commutative anti self dual yang mills equations we present backlund transformations for the non commutative anti self dual yang mills equation where the gauge group is g gl 2 and use it to generate a series of exact solutions from a simple seed solution the solutions generated by this approach are represented in terms of quasi determinants we also explain the origins of all the ingredients of the backlund transformations within the framework of non commutative twistor theory in particular we show that the generated solutions belong to a non commutative version of the atiyah ward ansatz

recursive solution of lowner vandermonde systems of equations ii abstract an o n 2 algorithm for the solution of a linear system the n n coefficient matrix of which consists of a vandermonde and a lowner part is presented in the first part of the algorithm the corresponding vandermonde system is solved in the second part the last rows of the vandermonde matrix are successively replaced by rows of the lowner structure the algorithm is convenient if the vandermonde part is dominant the main tool is the connection of such systems with rational interpolation problems

darboux transformations for the nonlinear schrodinger equations darboux transformations for the akns zs system are constructed in terms of grammian type determinants of vector solutions of the associated lax pairs with an operator spectral parameter a study of the reduction of the darboux transformation for the nonlinear schrodinger equations with standard and anomalous dispersion is presented two different families of new solutions for a given seed solution of the nonlinear schrodinger equation are given being one family related to a new vector lax pair for it in the first family and associated to diagonal matrices we present topological solutions with different asymptotic argument for the amplitude and nonzero background for the anomalous dispersion case they represent continuous deformations of the bright n soliton solution which is recovered for zero background in particular these solutions contain the combination of multiple homoclinic orbits of the focusing nonlinear schrodinger equation associated with jordan blocks we find rational deformations of the just described solutions as well as pure rational solutions the second family contains not only the solutions mentioned above but also broader classes of solutions for example in the standard dispersion case we are able to obtain the dark soliton solutions

a novel velocity vorticity formulation of the navier stokes equations with applications to boundary layer disturbance evolution a novel velocity vorticity formulation of the unsteady three dimensional navier stokes equations is presented the formulation is particularly suitable for simulating the evolution of three dimensional disturbances in boundary layers a key advantage is that there are only three governing equations for three primary dependent variables another advantage is that no wall boundary conditions are needed for the vorticity instead the conditions placed on the velocity are linked to the vorticity field through integral constraints based on the definition of vorticity numerical methods are presented in the context of application to the three dimensional boundary layer over a rotating disc the discretization scheme uses spectral expansions in the wall normal and azimuthal or spanwise directions and compact finite differences in the radial or streamwise direction the scheme is implemented so that the advantages of spectral convergence can be combined with the use of an efficient line iteration solution procedure the linearized form of the new velocity vorticity method is validated for the case of convective instabilities evolving over both rigid and compliant discs

boundary layer separation and adverse pressure gradient for 2 d viscous incompressible flow we study the detailed process of bifurcation in the flow s topological structure for a two dimensional 2 d incompressible flow subject to no slip boundary conditions and its connection with boundary layer separation the boundary layer separation theory of m ghil t ma and s wang based on the structural bifurcation concept is translated into vorticity form the vorticity formulation of the theory shows that structural bifurcation occurs whenever a degenerate singular point for the vorticity appears on the boundary this singular point is characterized by nonzero tangential second order derivative and nonzero time derivative of the vorticity furthermore we prove the presence of an adverse pressure gradient at the critical point due to reversal in the direction of the pressure force with respect to the basic shear flow at this point a numerical example of 2 d driven cavity flow governed by the navier stokes equations is presented boundary layer separation occurs the bifurcation criterion is satisfied and an adverse pressure gradient is shown to be present

an implicit method for coupled flow body dynamics we propose an efficient method for computing coupled flow body dynamics the time stepping is implicit and uses an iterative method preconditioned gmres to solve the flow body equations the preconditioner solves a decoupled version of the equations which involves only the inversion of banded matrices and requires a small number of iterations per time step we use the method to probe the instability to horizontal motions of an elliptical body with simple vertical motions flapping and rising in both cases a linear instability to horizontal motion sets in above a critical reynolds number leading to a stable oscillatory state the pressure forces play a destabilizing role against the stabilizing viscous forces with oscillatory time scales set by either external flapping or the intrinsic flow body coupling the latter lowers the instability threshold in reynolds number

darboux transformation of the generalized coupled dispersionless integrable system the darboux transformation on matrix solutions to the generalized coupled dispersionless integrable system based on a non abelian lie group is studied and the solutions are shown to be expressed in terms of quasideterminants as an explicit example the darboux transformation on scalar solutions to the system based on the lie group su 2 is discussed in detail and the solutions are shown to be expressed as ratios of determinants

surface pressure poisson equation formulation of the primitive equations numerical schemes numerical methods for the primitive equations pes of oceanic flow are presented in this paper first a two dimensional poisson equation with a suitable boundary condition is derived to solve the surface pressure consequently we derive a new formulation of the pes in which the surface pressure poisson equation replaces the nonlocal incompressibility constraint which is known to be inconvenient to implement based on this new formulation backward euler and crank nicolson schemes are presented the marker and cell scheme which gives values of physical variables on staggered mesh grid points are chosen as spatial discretization the convergence analysis of the fully discretized scheme is established in detail the accuracy check for the scheme is also shown

natural vorticity boundary conditions on solid walls abstract we derive a new kind of boundary conditions for the vorticity equation with solid wall boundaries for fluid flow problems the formulation uses a dirichlet condition for the normal component of vorticity and neumann type conditions for the tangential components in a galerkin integral formulation the tangential condition is natural i e it is enforced by a right hand side functional and does not impose a boundary constraint on trial and test spaces the functional involves the pressure variable and we discuss several velocity vorticity formulations where the proposed condition is appropriate several numerical experiments are given that illustrate the validity of the new approach

an explicit divergence free dg method for incompressible flow abstract we present an explicit divergence free dg method for incompressible flow based on velocity formulation only an h div conforming and globally divergence free finite element space is used for the velocity field and the pressure field is eliminated from the equations by design the resulting ode system can be discretized using any explicit time stepping methods we use the third order strong stability preserving runge kutta method in our numerical experiments our spatial discretization produces the identical velocity field as the divergence conforming dg method of cockburn et al 2007 based on a velocity pressure formulation when the same dg operators are used for the convective and viscous parts due to the global nature of the divergence free constraint and its interplay with the boundary conditions it is very hard to construct local bases for our finite element space here we present a key result on the efficient implementation of the scheme by identifying the equivalence of the mass matrix inversion of the globally divergence free finite element space to a standard hybrid mixed poisson solver hence in each time step a hybrid mixed poisson solver is used which reflects the global nature of the incompressibility condition in the actual implementation of this fully discrete scheme the pressure field is also computed via the hybrid mixed poisson solver hence the scheme can be interpreted as a velocity pressure formulation that treats the incompressibility constraint and pressure forces implicitly but the viscous and convective part explicitly since we treat viscosity explicitly for the navier stokes equation our method shall be best suited for unsteady high reynolds number flows so that the cfl constraint is not too restrictive

vorticity boundary condition and related issues for finite difference schemes this paper discusses three basic issues related to the design of finite difference schemes forunsteadyviscous incompressible flows using vorticity formulations the boundary condition for vorticity an efficient time stepping procedure and the relation between these schemes and the ones based on velocity pressure formulation we show that many of the newly developed global vorticity boundary conditions can actually be written as some local formulas derived earlier we also show that if we couple a standard centered difference scheme with third or fourth order explicit runge kutta methods the resulting schemes havenocell reynolds number constraints for high reynolds number flows these schemes are stable under the cfl condition given by the convective terms finally we show that the classical mac scheme is the same as thom s formula coupled with second order centered differences in the interior in the sense that one can define discrete vorticity in a natural way for the mac scheme and get the same values as the ones computed from thom s formula we use this to derive an efficient fourth order runge kutta time discretization for the mac scheme from the one for thom s formula we present numerical results for driven cavity flow at high reynolds number 105

quasideterminant solutions of an integrable chiral model in two dimensions the darboux transformation is used to obtain multisoliton solutions of the chiral model in two dimensions the matrix solutions of the principal chiral model and its lax pair are expressed in terms of quasideterminants the iteration of the darboux transformation gives the quasideterminant multisoliton solutions of the model it has been shown that the quasideterminant multisoliton solution of the chiral model is the same as obtained by zakharov and mikhailov using the dressing method based on the matrix riemann hilbert problem