a finite difference galerkin formulation for the incompressible navier stokes equations abstract the development of a new computational method for solving the incompressible navier stokes equations in primitive variable form is presented it is found that certain finite difference approximations for these equations can be transformed into an equivalent system which efficiently determines the discrete velocity field and which completely eliminates the pressure two such difference schemes for two dimensional problems are examined and some preliminary numerical results are discussed for the steady driven cavity problem

an unsplit higher order godunov method for scalar conservation laws in multiple dimensions abstract in this paper we develop an unsplit higher order godunov method for scalar conservation laws in two dimensions the method represents an extension for the special case being considered of methods developed by colella and by van leer in our method we begin with a piecewise bilinear representation of the solution on each grid cell a new piecewise bilinear representation at the next time level is then obtained from a two step procedure in the first step a conservative predictor corrector scheme derived from the integral form of the differential equation and characteristic considerations is used to obtain average values of the solution over grid cells at the new time level next these new average values are then used to construct a limited piecewise bilinear profile for each cell at the new time level the resulting method is shown to satisfy a maximum principle for constant coefficient linear advection computational results are presented comparing the new method to colella s method for linear advection the method is also applied to two model problems from porous media flow miscible and immiscible displacement the new scheme provides accurate resolution of sharp fronts without any significant distortion

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

stable and accurate pressure approximation for unsteady incompressible viscous flow how to properly specify boundary conditions for pressure is a longstanding problem for the incompressible navier stokes equations with no slip boundary conditions an analytical resolution of this issue stems from a recently developed formula for the pressure in terms of the commutator of the laplacian and leray projection operators here we make use of this formula to a improve the accuracy of computing pressure in two kinds of existing time discrete projection methods implicit in viscosity only and b devise new higher order accurate time discrete projection methods that extend a slip correction idea behind the well known finite difference scheme of kim and moin we test these schemes for stability and accuracy using various combinations of c 0 finite elements for all three kinds of time discretization one can obtain third order accuracy for both pressure and velocity without a time step stability restriction of diffusive type furthermore two kinds of projection methods are found stable using piecewise linear elements for both velocity and pressure

a quadratic spline based interface quasi reconstruction algorithm for accurate tracking of two phase flows a new quadratic spline based interface quasi reconstruction algorithm is presented which provides an accurate and continuous representation of the interface in a multiphase domain and facilitates the direct estimation of local interfacial curvature the fluid interface in each of the mixed cells is represented by piecewise parabolic curves and an initial discontinuous plic approximation of the interface is progressively converted into a smooth quadratic spline made of these parabolic curves the conversion is achieved by a sequence of predictor corrector operations enforcing function c 0 and derivative c 1 continuity at the cell boundaries using simple analytical expressions for the continuity requirements the efficacy and accuracy of the current algorithm has been demonstrated using standard test cases involving reconstruction of known static interface shapes and dynamically evolving interfaces in prescribed flow situations these benchmark studies illustrate that the present algorithm performs excellently as compared to the other interface reconstruction methods available in literature quadratic rate of error reduction with respect to grid size has been observed in all the cases with curved interface shapes only in situations where the interface geometry is primarily flat the rate of convergence becomes linear with the mesh size the flow algorithm implemented in the current work is designed to accurately balance the pressure gradients with the surface tension force at any location as a consequence it is able to minimize spurious flow currents arising from imperfect normal stress balance at the interface this has been demonstrated through the standard test problem of an inviscid droplet placed in a quiescent medium finally the direct curvature estimation ability of the current algorithm is illustrated through the coupled multiphase flow problem of a deformable air bubble rising through a column of water

the accuracy of the fractional step method we analyze the accuracy of the fractional step method of kim and moin j comput phys 59 1985 pp 308 323 for the incompressible navier stokes equations we show that the boundary conditions cannot be exactly satisfied in the projection step and that this limits the accuracy of the method we also show that the pressure in any projection method can be at best first order accurate our analysis is simpler and more direct than the previous analyses of this method we also show that there is no numerical boundary layer for velocity or pressure but there is one for the auxiliary pressure variable

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

stationary random fields arising from second order partial differential equations on compact lie groups wide sense stationary processes are a mainstay of classical signal processing it is well known that they can be obtained by solving ordinary differential equations with constant coefficients whose right hand side is a white noise this paper addresses the extension of this construction to random fields defined on compact lie groups on an underlying compact lie group the paper studies left invariant second order elliptic partial differential equations whose right hand side is a spatial white noise quite often the solution of a partial differential equation is not defined as a function but as a distribution to adapt to this situation the paper introduces a definition of wide sense stationary distributions on a compact lie group this is shown to be consistent with the more restricted definition of wide sense stationary fields given in a classic paper by yaglom it is proved that the solution of a partial differential equation of the kind being studied is a wide sense stationary distribution whose covariance structure is determined by the fundamental solution of the equation as a concrete example this paper describes the fundamental solution of the helmholtz equation on the rotation group and the resulting covariance structure

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

a cartesian grid embedded boundary method for hyperbolic conservation laws we present a second order godunov algorithm to solve time dependent hyperbolic systems of conservation laws on irregular domains our approach is based on a formally consistent discretization of the conservation laws on a finite volume grid obtained from intersecting the domain with a cartesian grid we address the small cell stability problem associated with such methods by hybridizing our conservative discretization with a stable nonconservative discretization at irregular control volumes and redistributing the difference in the mass increments to nearby cells in a way that preserves stability and local conservation the resulting method is second order accurate in l 1 for smooth problems and is robust in the presence of large amplitude discontinuities intersecting the irregular boundary

a new family of 5 5 cc 4oc schemes applicable for unsteady navier stokes equations in this paper a new family of implicit compact finite difference schemes for computation of unsteady convection diffusion equation with variable convection coefficient is proposed the schemes which are fourth order accurate in space and second or lower order accurate in time depending on the choice of weighted time average parameter are then applied to unsteady navier stokes system the proposed schemes where transport variable and its first derivatives are carried as the unknowns combine virtues of compact discretization and pade scheme for spatial derivative these schemes which are based on a five point stencil with constant coefficients named as 5 5 constant coefficient 4th order compact 5 5 cc 4oc give rise to a diagonally dominant system of equations and shows higher accuracy and better phase and amplitude error characteristics than some of the standard methods these schemes are capable of using a grid aspect ratio other than unity and are unconditionally stable they efficiently capture both transient and steady solutions of linear and nonlinear convection diffusion equations with dirichlet as well as neumann boundary conditions subsequently the proposed schemes are applied to problems governed by the incompressible navier stokes equations the results obtained are in excellent agreement with analytical and available numerical results in all cases establishing efficiency and accuracy of the proposed schemes

hausdorff and packing dimension of fibers and graphs of prevalent continuous maps abstract the notions of shyness and prevalence generalize the property of being zero and full haar measure to arbitrary not necessarily locally compact polish groups the main goal of the paper is to answer the following question what can we say about the hausdorff and packing dimension of the fibers of prevalent continuous maps let k be an uncountable compact metric space we prove that a prevalent f c k r d has many fibers with almost maximal hausdorff dimension this generalizes a theorem of dougherty and yields that a prevalent f c k r d has graph of maximal hausdorff dimension generalizing a result of bayart and heurteaux we obtain similar results for the packing dimension we show that for a prevalent f c 0 1 m r d the set of y f 0 1 m for which dim h f 1 y m contains a dense open set having full measure with respect to the occupation measure m f 1 where dim h and m denote the hausdorff dimension and the m dimensional lebesgue measure respectively we also prove an analogous result when 0 1 m is replaced by any self similar set satisfying the open set condition we cannot replace the occupation measure with lebesgue measure in the above statement we show that the functions f c 0 1 for which positively many level sets are singletons form a non shy set in c 0 1 in order to do so we generalize a theorem of antunovic burdzy peres and ruscher as a complementary result we prove that the functions f c 0 1 for which dim h f 1 y 1 for all y min f max f form a non shy set in c 0 1 we also prove sharper results in which large hausdorff dimension is replaced by positive measure with respect to generalized hausdorff measures which answers a problem of fraser and hyde

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

on the hausdorff dimension of some graphs consider the functions wb x b cn 1 bnx on 1 0n n oo where b 1 0 0 such that if b is large enough then the hausdorff dimension of the graph of wb is bounded below by 2a c ln b we also show that if a function f is convex lipschitz of order a then the graph of f has a finite measure with respect to hausdorff s measure in dimension 2 a the convex lipschitz functions of order a include zygmund s class a our analysis shows that the graph of the classical van der waerdentagaki nowhere differentiable function has a finite measure with respect to h t t in 1 t we consider the hausdorff dimension of the graphs of various continuous functions we introduce a new geometric property of a function convex lipschitz of some order and obtain an upper bound on the dimension of a graph with this property in particular our analysis includes functions of the form

deconvolution density estimation on so n this paper develops nonparametric deconvolution density estimation over so n the group of n x n orthogonal matrices of determinant 1 the methodology is to use the group and manifold structures to adapt the euclidean deconvolution techniques to this lie group environment this is achieved by employing the theory of group representations explicit to so n general consistency results are obtained with specific rates of convergence achieved under sufficient smoothness conditions application to empirical bayes prior estimation and inference is also discussed

fractal random series generated by poisson voronoi tessellations in this paper we construct a new family of random series defined on r d indexed by one scaling parameter and two hurst like exponents the model is close to takagi knopp functions save for the fact that the underlying partitions of r d are not the usual dyadic meshes but random voronoi tessellations generated by poisson point processes this approach leads us to a continuous function whose random graph is shown to be fractal with explicit and equal box and hausdorff dimensions the proof of this main result is based on several new distributional properties of the poisson voronoi tessellation on the one hand an estimate of the oscillations of the function coupled with an application of a frostman type lemma on the other hand finally we introduce two related models and provide in particular a box dimension calculation for a derived deterministic takagi knopp series with hexagonal bases

levy processes and fourier analysis on compact lie groups we study the fourier expansion of the distribution density of a levy process in a compact lie group based on the peter weyl theorem

adaptive cartesian immersed boundary method for simulation of flow over flexible geometries a solver for the incompressible navier stokes equations in which solid surfaces are represented using an immersed boundary method is presented adaptive cartesian mesh refinement and coarsening are employed both to resolve immersed boundaries and to dynamically resolve important flow features as they are developing dynamically evolving immersed boundaries are tracked using lagrangian marker points that are governed by direct momentum forcing using either simple force laws or explicitly prescribed motion in a framework that permits simulation of both rigid and flexible lifting surfaces the adaptive cartesian immersed boundary method capability presented here is well suited for low reynolds number flows over flexible lifting surfaces as encountered in modern flapping wing flyers flow over complex geometries undergoing large scale deformations and vortex dominated flows the implementation of both the incompressible navier stokes solver and the immersed boundary method solution algorithm are described in

decompounding an estimation problem for poisson random sums given a sample from a compound poisson distribution we consider estimation of the corresponding rate parameter and base distribution this has applications in insurance mathematics and queueing theory we propose a plug in type estimator that is based on a suitable inversion of the compounding operation asymptotic results for this estimator are obtained via a local analysis of the decompounding functional

differentiability of functions in the zygmund class let f be a function in the zygmund class in the euclidean space it is proved that the hausdorff dimension of the set of points where f has bounded divided differences is bigger or equal to one furthermore if f is in the small zygmund class then the hausdorff dimension of the set of points where f is differentiable is bigger or equal to one the sharpness of these results is also discussed

isotropic multiple scattering processes on hyperspheres this paper presents several results about isotropic random walks and multiple scattering processes on hyperspheres mathbb s p 1 it allows one to derive the fourier expansions on mathbb s p 1 of these processes a result of unimodality for the multiconvolution of symmetrical probability density functions on mathbb s p 1 is also introduced such processes are then studied in the case where the scattering distribution is von mises fisher vmf asymptotic distributions for the multiconvolution of vmfs on mathbb s p 1 are obtained both fourier expansion and asymptotic approximation allow us to compute estimation bounds for the parameters of compound cox processes on mathbb s p 1

an immersed boundary method with formal second order accuracy and reduced numerical viscosity a formally second order accurate immersed boundary method is presented and tested in this paper we apply this new scheme to simulate the flow past a circular cylinder and study the effect of numerical viscosity on the accuracy of the computation by comparing the numerical results with those of a first order method the numerical evidence shows that the new scheme has less numerical viscosity and is therefore a better choice for the simulation of high reynolds number flows with immersed boundaries

similarity and fractality in the modelling of roughness by a multilevel profile with hierarchical structure some modern scaling and fractal approaches to the characterization of surface roughness are under consideration a new mathematical model of a multilevel profile suitable for flexible simulation of roughness is presented as a development from these approaches the iterative construction of the model is described it is shown that the profile is a hierarchical structure with self affine and fractal features its geometrical properties are studied both analytically and numerically and some of them are proved to be of the renormalization type the profile contour length can be finite or infinite depending on the chosen values of the structural parameters some characteristics including the bearing curve and the upper volume function of the profile and the corresponding prestructures are calculated

efficient implementation of thinc scheme in this paper we improve the multi dimensional thinc tangent of hyperbola for interface capturing scheme f xiao y honma t kono a simple algebraic interface capturing scheme using hyperbolic tangent function int j numer meth fluid 48 2005 1023 the thinc scheme is a vof volume of fluid type method in the original thinc scheme one dimensional thinc scheme was straightforwardly used for multi dimensional cases in this paper we propose the wlic weighed line interface calculation method to extend the thinc scheme to multi dimension in the wlic method the interface is reconstructed by taking an average of line interfaces along x y and z coordinates with the weights calculated from surface normal the wlic method can extend the thinc scheme to multi dimension while maintaining simplicity of implementation and achieve a higher accuracy than the original thinc scheme the wlic method can readily extend the thinc scheme to three dimension

asymptotic minimax bounds for stochastic deconvolution over groups this paper examines stochastic deconvolution over noncommutative compact lie groups this involves fourier analysis on compact lie groups as well as convolution products over such groups an observation process consisting of a known impulse response function convolved with an unknown signal with additive white noise is assumed data collected through the observation process then allow us to construct an estimator of the signal signal recovery is then assessed through integrated mean squared error for which the main results show that asymptotic minimaxity depends on smoothness properties of the impulse response function thus if the fourier transform of the impulse response function is bounded polynomially then the asymptotic minimax signal recovery is polynomial while if the fourier transform of the impulse response function is exponentially bounded then the asymptotic minimax signal recovery is logarithmic such investigations have been previously considered in both the engineering and statistics literature with applications in among others medical imaging robotics and polymer science

an immersed interface method for incompressible navier stokes equations the method developed in this paper is motivated by peskin s immersed boundary ib method and allows one to model the motion of flexible membranes or other structures immersed in viscous incompressible fluid using a fluid solver on a fixed cartesian grid the ib method uses a set of discrete delta functions to spread the entire singular force exerted by the immersed boundary to the nearby fluid grid points our method instead incorporates part of this force into jump conditions for the pressure avoiding discrete dipole terms that adversely affect the accuracy near the immersed boundary this has been implemented for the two dimensional incompressible navier stokes equations using a high resolution finite volume method for the advective terms and a projection method to enforce incompressibility in the projection step the correct jump in pressure is imposed in the course of solving the poisson problem this gives sharp resolution of the pressure across the interface and also gives better volume conservation than the traditional ib method comparisons between this method and the ib method are presented for several test problems numerical studies of the convergence and order of accuracy are included

an immersed boundary finite volume method for simulations of flow in complex geometries abstract a new immersed boundary method for simulating flows over or inside complex geometries is developed by introducing a mass source sink as well as a momentum forcing the present method is based on a finite volume approach on a staggered mesh together with a fractional step method both momentum forcing and mass source are applied on the body surface or inside the body to satisfy the no slip boundary condition on the immersed boundary and also to satisfy the continuity for the cell containing the immersed boundary in the immersed boundary method the choice of an accurate interpolation scheme satisfying the no slip condition on the immersed boundary is important because the grid lines generally do not coincide with the immersed boundary therefore a stable second order interpolation scheme for evaluating the momentum forcing on the body surface or inside the body is proposed three different flow problems decaying vortices and flows over a cylinder and a sphere are simulated using the immersed boundary method proposed in this study and the results agree very well with previous numerical and experimental results verifying the accuracy of the present method

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

a smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations the effects of complex boundary conditions on flows are represented by a volume force in the immersed boundary methods the problem with this representation is that the volume force exhibits non physical oscillations in moving boundary simulations a smoothing technique for discrete delta functions has been developed in this paper to suppress the non physical oscillations in the volume forces we have found that the non physical oscillations are mainly due to the fact that the derivatives of the regular discrete delta functions do not satisfy certain moment conditions it has been shown that the smoothed discrete delta functions constructed in this paper have one order higher derivative than the regular ones moreover not only the smoothed discrete delta functions satisfy the first two discrete moment conditions but also their derivatives satisfy one order higher moment condition than the regular ones the smoothed discrete delta functions are tested by three test cases a one dimensional heat equation with a moving singular force a two dimensional flow past an oscillating cylinder and the vortex induced vibration of a cylinder the numerical examples in these cases demonstrate that the smoothed discrete delta functions can effectively suppress the non physical oscillations in the volume forces and improve the accuracy of the immersed boundary method with direct forcing in moving boundary simulations

a new fractal dimension the topological hausdorff dimension we introduce a new concept of dimension for metric spaces the so called topological hausdorff dimension it is defined by a very natural combination of the definitions of the topological dimension and the hausdorff dimension the value of the topological hausdorff dimension is always between the topological dimension and the hausdorff dimension in particular this new dimension is a non trivial lower estimate for the hausdorff dimension we examine the basic properties of this new notion of dimension compare it to other well known notions determine its value for some classical fractals such as the sierpinski carpet the von koch snowflake curve kakeya sets the trail of the brownian motion etc as our first application we generalize the celebrated result of chayes chayes and durrett about the phase transition of the connectedness of the limit set of mandelbrot s fractal percolation process they proved that certain curves show up in the limit set when passing a critical probability and we prove that actually thick families of curves show up where roughly speaking the word thick means that the curves can be parametrized in a natural way by a set of large hausdorff dimension the proof of this is basically a lower estimate of the topological hausdorff dimension of the limit set for the sake of completeness we also give an upper estimate and conclude that in the non trivial cases the topological hausdorff dimension is almost surely strictly below the hausdorff dimension finally as our second application we show that the topological hausdorff dimension is precisely the right notion to describe the hausdorff dimension of the level sets of the generic continuous function in the sense of baire category defined on a compact metric space

a new generalization of the takagi function abstract we consider a one parameter family of functions f t x t on 0 1 and partial derivatives t k f t x with respect to the parameter t each function of the class is defined by a certain pair of two square matrices of order two the class includes the lebesgue singular functions and other singular functions our approach to the takagi function is similar to hata and yamaguti the class of partial derivatives t k f t x includes the original takagi function and some generalizations we consider real analytic properties of t k f t x as a function of x specifically differentiability the hausdorff dimension of the graph the asymptotics around dyadic rationals variation a question of local monotonicity and a local modulus of continuity our results are extensions of some results for the original takagi function and some generalizations

a stable and convergent scheme for viscoelastic flow in contraction channels we present a new algorithm to simulate unsteady viscoelastic flows in abrupt contraction channels in our approach we split the viscoelastic terms of the oldroyd b constitutive equation using duhamel s formula and discretize the resulting pdes using a semi implicit finite difference method based on a lax wendroff method for hyperbolic terms in particular we leave a small residual elastic term in the viscous limit by design to make the hyperbolic piece well posed a projection method is used to impose the incompressibility constraint we are able to compute the full range of unsteady elastic flows in an abrupt contraction channel from the viscous limit to the elastic limit in a stable and convergent manner we demonstrate the range of our method for unsteady flow of a maxwell fluid with and without viscosity in planar contraction channels we also demonstrate stable and convergent results for benchmark high weissenberg number problems at we 1 and we 10

an unusual self adjoint linear partial differential operator hhin an american mathematical society memoir published in 2003 the authors everitt and markus apply their prior theory of symplectic algebra to the study of symmetric linear partial differential expressions and the generation of self adjoint differential operators in sobolev hilbert spaces in the case when the differential expression has smooth coefficients on the closure of a bounded open region in euclidean space and when the region has a smooth boundary this theory leads to the construction of certain self adjoint partial differential operators which cannot be defined by applying classical or generalized conditions on the boundary of the open region this present paper concerns the spectral properties of one of these unusual self adjoint operators sometimes called the harmonic operator the boundary value problems considered in the memoir see above and in this paper are called regular in that the cofficients of the differential expression do not have singularities within or on the boundary of the region also the region is bounded and has a smooth boundary under these and some additional technical conditions it is shown in the memoir and emphasized in this present paper that all the self adjoint operators considered are explicitly determined on their domains by the partial differential expression this property makes a remarkable comparison with the case of symmetric ordinary differential expressions in the regular ordinary case the spectrum of all the self adjoint operators is discrete in that it consists of a countable number of eigenvalues with no finite point of accumulation and each eigenvalue is of finite multiplicity thus the essential spectrum of all these operators is empty this spectral property extends to the present partial differential case for the classical dirichlet and neumann operators but not to the harmonic operator it is shown in this paper that the harmonic operator has an eigenvalue of infinite multiplicity at the origin of the complex spectral plane thus the essential spectrum of this operator is not empty both the weak and strong formulations of the harmonic boundary value problem are considered these two formulations are shown to be equivalent in the final section of the paper examples are considered which show that the harmonic operator defined by the methods of symplectic algebra has a domain that cannot be determined by applying either classical or generalized local conditions on the boundary of the region

a high order immersed interface method for simulating unsteady incompressible flows on irregular domains immersed boundary methods and immersed interface methods are becoming increasingly popular for the computation of unsteady flows around complex geometries using a cartesian grid while good results both qualitative and quantitative have been obtained most of the methods rely on low order corrections to account for the immersed boundary the objective of the present work is to present as an alternative a high order modified immersed interface method for the 2d unsteady incompressible navier stokes equations in stream function vorticity formulation the method employs an explicit fourth order runge kutta time integration scheme fourth order compact finite differences for computation of spatial derivatives and a nine point fourth order compact discretization of the poisson equation for computation of the stream function corrections to the finite difference schemes are used to maintain high formal accuracy at the immersed boundary as confirmed by analytical tests to validate the method in its application to incompressible flows several physically relevant test cases are computed including uniform flow past a circular cylinder and tollmien schlichting waves in a boundary layer

discrete exterior calculus discretization of incompressible navier stokes equations over surface simplicial meshes a conservative discretization of incompressible navier stokes equations is developed based on discrete exterior calculus dec a distinguishing feature of our method is the use of an algebraic discretization of the interior product operator and a combinatorial discretization of the wedge product the governing equations are first rewritten using the exterior calculus notation replacing vector calculus differential operators by the exterior derivative hodge star and wedge product operators the discretization is then carried out by substituting with the corresponding discrete operators based on the dec framework numerical experiments for flows over surfaces reveal a second order accuracy for the developed scheme when using structured triangular meshes and first order accuracy for otherwise unstructured meshes by construction the method is conservative in that both mass and vorticity are conserved up to machine precision the relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step

physically motivated domain decomposition for singularly perturbed equations a domain decomposition algorithm suitable for the efficient and accurate solution of a parabolic reaction convection diffusion equation with small parameter multiplying the diffusion term is presented convergence is established via maximum principle arguments the equation arises in the modeling of laminar transonic flow decomposition into subdomains is accomplished via singular perturbation analysis which dictates regions where certain reduced equations may be solved in place of the full equation effectively preconditioning the problem this paper concentrates on the theoretical basis of the method establishing local and global a priori error bounds

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

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

a 3d immersed interface method for fluid solid interaction abstract in immersed interface methods solids in a fluid are represented by singular forces in the navier stokes equations and flow jump conditions induced by the singular forces directly enter into numerical schemes this paper focuses on the implementation of an immersed interface method for simulating fluid solid interaction in 3d the method employs the mac scheme for the spatial discretization the rk4 scheme for the time integration and an fft based poisson solver for the pressure poisson equation a fluid solid interface is tracked by lagrangian markers intersections of the interface with mac grid lines identify finite difference stencils on which jump contributions to finite difference schemes are needed to find the intersections and to interpolate jump conditions from the lagrangian markers to the intersections parametric triangulation of the interface is used the velocity of the lagrangian markers is interpolated directly from surrounding mac grid nodes with interpolation schemes accounting for jump conditions numerical examples demonstrate that 1 the method has near second order accuracy in the infinity norm for velocity and the accuracy for pressure is between first and second order 2 the method conserves the volume enclosed by a no penetration boundary and 3 the method can efficiently handle multiple moving solids with ease

takagi functions and approximate midconvexity abstract let v be a convex subset of a normed space and let e 0 p 0 be given constants a function f v r is called e p midconvex if f x y 2 f x f y 2 e x y p for all x y v we consider the case p 1 2 and investigate the relations between continuous e p midconvex functions and takagi like functions given by p x k 0 2 2 k p dist 2 k x z for x r it occurs that functions p are optimal 1 p midconvex functions this gives us sharp estimations for continuous e p midconvex functions we also compute the maximum of the function p for a certain set of parameter values

escape rate for 2 dimensional brownian motion conditioned to be transient with application to zygmund functions the escape rate of a 2 dimensional brownian motion conditioned to be transient is determined to be p x t f t i o as t t oo 0 or 1 according as ezre n log f ee or oo the result is used to construct a complex valued zygmund function as a lacunary series whose graph does not have a finite linear hausdorff measure this contrasts the result of mauldin and williams that the graphs of all real valued zygmund functions have a finite linear hausdorff measure

self affine fractal functions and wavelet series abstract we consider functions represented by series g g c g g 1 x of wavelet type where g is a group generated by affine functions l 1 l n and is piecewise affine by means of those functions we characterize the class of self affine fractal functions previously studied by barnsley et al we compute their global and local holder exponents and investigate points of non differentiability wavelet representations for various continuous nowhere differentiable and singular functions are presented another application is the construction of functions with prescribed local holder exponents at each point

decompounding on compact lie groups noncommutative harmonic analysis is used to solve a nonparametric estimation problem stated in terms of compound poisson processes on compact lie groups this problem of decompounding is a generalization of a similar classical problem the proposed solution is based on a characteristic function method the treated problem is important to recent models of the physical inverse problem of multiple scattering

numerical simulation of two dimensional flows over a circular cylinder using the immersed boundary method in this work a virtual boundary method is applied to the numerical simulation of a uniform flow over a cylinder the force source term added to the two dimensional navier stokes equations guarantees the imposition of the no slip boundary condition over the body fluid interface these equations are discretized using the finite differences method the immersed boundary is represented with a finite number of lagrangian points distributed over the solid fluid interface a cartesian grid is used to solve the fluid flow equations the key idea is to propose a method to calculate the interfacial force without ad hoc constants that should usually be adjusted for the type of flow and the type of the numerical method when this kind of model is used in the present work this force is calculated using the navier stokes equations applied to the lagrangian points and then distributed over the eulerian grid the main advantage of this approach is that it enables calculation of this force field even if the interface is moving or deforming it is unnecessary to locate the eulerian grid points near this immersed boundary the lift and drag coefficients and the strouhal number calculated for an immersed cylinder are compared with previous experimental and numerical results for different reynolds numbers

stochastic deconvolution over groups in this paper we address a class of inverse problems that are formulated as group convolutions this is a rich area of research with applications to radon transform inversion for tomography wide band and narrow band signal processing inverse rendering in computer graphics and channel estimation in communications as well as robotics and polymer science we present a group theoretic framework for signal modeling and analysis for such problems and propose a minimum mean square error mmse deconvolution method in a probabilistic setting key components of our approach are group representation theory and the concept of group stationarity the proposed deconvolution method incorporates a priori information and noise statistics into the inversion process which leads to a natural regularized solution we present recovery of self similar processes that are blurred and embedded in noise as a demonstration example the method is applicable to a wide range of inverse problems involving both commutative and noncommutative groups including finite compact and majority of well behaved locally compact groups

fractal dimensions and singularities of the weierstrass type functions a new type of fractal measures k s 1 s 2 defined on the subsets of the graph of a continuous function is introduced the k dimension defined by this measure is closer to the hausdorff dimension than the other fractal dimensions in recent literatures for the weierstrass type functions defined by w x 0 i g i x where 1 0 1 and g is an almost periodic lipschitz function of order greater than it is shown that the k dimension of the graph of w equals to 2 this conclusion is also equivalent to certain rate of the local oscillation of the function some problems on the knot points and the nondifferentiability of w are also discussed

extrinsic mean of brownian distributions on compact lie groups this paper studies brownian distributions on compact lie groups these are defined as the marginal distributions of brownian processes and are intended as a natural extension of the well known normal distributions to compact lie groups it is shown that this definition preserves key properties of normal distributions in particular brownian distributions transform in a nice way under group operations and satisfy an extension of the central limit theorem brownian distributions on a compact lie group g belong to one of two parametric families nl g c and nr g c g g and c a positive definite symmetric matrix in particular the parameter g appears as a location parameter an approach based on the extrinsic mean for estimation of the parameters g and c is studied in detail it is shown that g is the unique extrinsic mean for a brownian distribution nl g c or nr g c resulting estimates are proved to be consistent and asymptotically normal while they may also be used to simultaneously estimate g and c it is seen this requires that g be embedded into a higher dimensional matrix lie group going beyond brownian distributions it is shown the extrinsic mean can be used to recover the location parameter for a wider class of distributions arising more generally from levy processes the compact lie group structure places limitations on the analogy between normal distributions and brownian distributions this is illustrated by the study of multivariate brownian distributions these are introduced as brownian distributions on some product group e g g g this paper describes their covariance structure and considers its transformation under group operations

gerris a tree based adaptive solver for the incompressible euler equations in complex geometries an adaptive mesh projection method for the time dependent incompressible euler equations is presented the domain is spatially discretised using quad octrees and a multilevel poisson solver is used to obtain the pressure complex solid boundaries are represented using a volume of fluid approach second order convergence in space and time is demonstrated on regular statically and dynamically refined grids the quad octree discretisation proves to be very flexible and allows accurate and efficient tracking of flow features the source code of the method implementation is freely available

regular article an accurate cartesian grid method for viscous incompressible flows with complex immersed boundaries a cartesian grid method has been developed for simulating two dimensional unsteady viscous incompressible flows with complex immersed boundaries a finite volume method based on a second order accurate central difference scheme is used in conjunction with a two step fractional step procedure the key aspects that need to be considered in developing such a solver are imposition of boundary conditions on the immersed boundaries and accurate discretization of the governing equation in cells that are cut by these boundaries a new interpolation procedure is presented which allows systematic development of a spatial discretization scheme that preserves the second order spatial accuracy of the underlying solver the presence of immersed boundaries alters the conditioning of the linear operators and this can slow down the iterative solution of these equations the convergence is accelerated by using a preconditioned conjugate gradient method where the preconditioner takes advantage of the structured nature of the underlying mesh the accuracy and fidelity of the solver is validated by simulating a number of canonical flows and the ability of the solver to simulate flows with very complicated immersed boundaries is demonstrated

the limited rademacher functions and bernoulli convolutions associated with pisot numbers abstract in this paper we give a systematical study of the local structures and fractal indices of the limited rademacher functions and bernoulli convolutions associated with pisot numbers for a given pisot number in the interval 1 2 we construct a finite family of non negative matrices maybe non square such that the corresponding fractal indices can be re expressed as some limits in terms of products of these non negative matrices we are especially interested in the case that the associated pisot number is a simple pisot number i e the unique positive root of the polynomial x k x k 1 x 1 k 2 3 in this case the corresponding products of matrices can be decomposed into the products of scalars based on which the precise formulas of fractal indices as well as the multifractal formalism are obtained

the immersed interface method for the navier stokes equations with singular forces abstract peskin s immersed boundary method has been widely used for simulating many fluid mechanics and biology problems one of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain however the immersed boundary method is known to be first order accurate and usually smears out the solutions in this paper we propose an immersed interface method for the incompressible navier stokes equations with singular forces along one or several interfaces in the solution domain the new method is based on a second order projection method with modifications only at grid points near or on the interface from the derivation of the new method we expect fully second order accuracy for the velocity and nearly second order accuracy for the pressure in the maximum norm including those grid points near or on the interface this has been confirmed in our numerical experiments furthermore the computed solutions are sharp across the interface nontrivial numerical results are provided and compared with the immersed boundary method meanwhile a new version of the immersed boundary method using the level set representation of the interface is also proposed in this paper

cantor boundary behavior of analytic functions let mathbb d be the open unit disc and let partial mathbb d be the boundary of mathbb d for f z analytic in mathbb d and continuous on overline mathbb d it follows from the open mapping theorem that partial f mathbb d subset f partial mathbb d these two sets have very rich and intrigue geometric properties when f z is univalent then they are equal and there is a large literature to study their boundary behaviors our interest is on the class of analytic functions f z for which the image curves f partial mathbb d form infinitely many loops everywhere they are not univalent of course we formulate this as the cantor boundary behavior we give sufficient conditions for such property making use of the distribution of the zeros of f and the mean growth rate of f examples includes the complex weierstrass functions and the cauchy transform of the canonical hausdorff measure on the sierpiski gasket

the graph and range singularity spectra of b adic independent cascade functions abstract with the iso holder sets of a function we naturally associate subsets of the graph and the range of the function we compute the hausdorff dimension of these subsets for a class of statistically self similar multifractal functions namely the b adic independent cascade functions

improving the speed and accuracy of projection type incompressible flow solvers abstract superseding so called first generation incompressible flow solvers of the projection type based on taylor galerkin advection second order pressure damping and element based data structures the current second generation solvers based on high order upwind advection fourth order pressure damping and edge based data structures have now been in use for half a decade and have proven remarkably robust and efficient for many large scale problems in order to achieve higher accuracy and speed these solvers have recently been enhanced in a variety of ways a substepping for advection b implicit treatment of advective terms via sgs and gmres lu sgs iterative solvers c fully implicit time accurate advancement of pressure and velocities and d linelet preconditioning for the pressure poisson equation the combined effect of these third generation improvements leads to speedups of the order of o 1 5 1 10 with similar or even better temporal accuracy as demonstrated on a variety of academic and industrial problems

application of multi block approach in the immersed boundary lattice boltzmann method for viscous fluid flows the immersed boundary lattice boltzmann method was presented recently to simulate the rigid particle motion it combines the desirable features of the lattice boltzmann and immersed boundary methods it uses a regular eulerian grid for the flow domain and a lagrangian grid for the boundary for the lattice boltzmann method as compared with the single relaxation time collision scheme the multi relaxation time collision scheme has better computational stability due to separation of the relaxations of various kinetic models especially near the geometric singularity so the multi relaxation time collision scheme is used to replace the single relaxation time collision scheme in the original immersed boundary lattice boltzmann method in order to obtain an accurate result very fine lattice grid is needed near the solid boundary to reduce the computational effort local grid refinement is adopted to offer high resolution near a solid body and to place the outer boundary far away from the body so the multi block scheme with the multi relaxation time collision model is used in the immersed boundary lattice boltzmann method in each block uniform lattice spacing can still be used in order to validate the multi block approach for the immersed boundary lattice boltzmann method with multi relaxation time collision scheme the numerical simulations of steady and unsteady flows past a circular cylinder and airfoil are carried out and good results are obtained

graphs of continuous functions and packing dimension abstract we prove that for a typical continuous function f c x over an uncountable compact metric space x the packing dimension of its graph g f x f x x x is dim p x 1 we also consider decompositions of functions in c 0 1 in terms of upper box dimension as well as packing dimension which are quite different from the case of hausdorff dimension

a new algorithm for solving some mechanical problems this paper explores the utility of a discrete singular convolution algorithm for solving certain mechanical problems benchmark mechanical systems including plate vibrations and incompressible flows are employed to illustrate the robustness and to test accuracy of the present algorithm numerical results indicate that the present approach is very accurate efficient and reliable for solving the aforementioned problems

on the dimension of the graph of the classical weierstrass function abstract this paper examines dimension of the graph of the famous weierstrass non differentiable function w b x n 0 n cos 2 b n x for an integer b 2 and 1 b 1 we prove that for every b there exists explicitly given b 1 b 1 such that the hausdorff dimension of the graph of w b is equal to d 2 log log b for every b 1 we also show that the dimension is equal to d for almost every on some larger interval this partially solves a well known thirty year old conjecture furthermore we prove that the hausdorff dimension of the graph of the function f x n 0 n b n x for an integer b 2 and 1 b 1 is equal to d for a typical z periodic c 3 function

a remark on decomposition of continuous functions abstract in this note we show that given a real 1 2 any real valued continuous function on the unit interval can be decomposed as a sum of two real valued continuous functions whose graphs have hausdorff dimension which answers affirmatively a question raised by bayart and heurteaux 2012 2

fractional derivatives of weierstrass type functions for a special type of fractional differentiation formulas for the weierstrass and the weierstrass mandelbrot functions are shown for integer valued parameters a well conditioned numerical procedure for computing the derivatives in the mean of these functions is derived and used to compute some values

uncoupled temporally second order accurate implicit solver of incompressible navier stokes equations an efficient method with temporal second order accuracy for solving the incompressible navier stokes equations in generalized coordinate systems is developed efficiency is obtained by extending an existing fractional step semi implicit solver into an implicit one with implicit approximation of the convection terms a novel three level linearization scheme is proposed to decouple the momentum equations from one another in addition to the fractional step approach that decouples the continuity equation from the momentum equations consequently all of the governing equations are uncoupled without degrading temporal accuracy or stability and without the need for iterations to solve the nonlinear system of equations at each time level the proposed decoupling technique can be used for other systems of nonlinear and time dependent partial differential equations moreover only minor modifications are required to apply the method to existing two level schemes several test cases confirm that the proposed decoupled scheme has temporal second order accuracy and allows the use of large time steps

a well conditioned augmented system for solving navier stokes equations in irregular domains an augmented method based on a cartesian grid is proposed for the incompressible navier stokes equations in irregular domains the irregular domain is embedded into a rectangular one so that a fast poisson solver can be utilized in the projection method unlike several methods suggested in the literature that set the force strengths as unknowns which often results in an ill conditioned linear system we set the jump in the normal derivative of the velocity as the augmented variable the new approach improves the condition number of the system for the augmented variable significantly using the immersed interface method we are able to achieve second order accuracy for the velocity numerical results and comparisons to benchmark tests are given to validate the new method a lid driven cavity flow with multiple obstacles and different geometries are also presented

the hausdorff dimension of the level sets of takagi s function abstract recently maddock 2006 12 has conjectured that the hausdorff dimension of each level set of takagi s function is at most 1 2 we prove this conjecture using the self affinity of the function of takagi and the existing relationship between the hausdorff and box counting dimensions

weierstrass functions with random phases consider the function f te x sum n 0 infty b n al g b nx te n where b 1 0 al 1 and g is a non constant 1 periodic lipschitz function the phases te n are chosen independently with respect to the uniform probability measure on 0 1 we prove that with probability one we can choose a sequence of scales delta k searrow 0 such that for every interval i of length abs i delta k the oscillation of f te satisfies osc f te i geq c abs i al moreover the inequality osc f te i geq c abs i al varepsilon is almost surely true at every scale when b is a transcendental number these results can be improved the minoration osc f te i geq c abs i al is true for every choice of the phases te n and at every scale

a hybrid particle level set method for improved interface capturing in this paper we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field our method uses lagrangian marker particles to rebuild the level set in regions which are underresolved this is often the case for flows undergoing stretching and tearing the overall method maintains a smooth geometrical description of the interface and the implementation simplicity characteristic of the level set method our method compares favorably with volume of fluid methods in the conservation of mass and purely lagrangian schemes for interface resolution the method is presented in three spatial dimensions

on the box dimensions of graphs of typical continuous functions abstract let x r be a bounded set we emphasize that we are not assuming that x is compact or borel we prove that for a typical in the sense of baire uniformly continuous function f on x the lower box dimension of the graph of f is as small as possible and the upper box dimension of the graph of f is as big as possible we also prove a local version of this result namely we prove that for a typical uniformly continuous function f on x the lower local box dimension of the graph of f at all points x x is as small as possible and the upper local box dimension of the graph of f at all points x x is as big as possible