fast high order ader schemes for linear hyperbolic equations a reformulation of the ader approach arbitrary high order schemes using derivatives for linear hyperbolic pde s is presented this reformulation leads to a drastic decrease of the computational effort a formula for the construction of ader schemes that are arbitrary high order accurate in space and time is given the accuracy for some selected schemes is shown numerically for the two dimensional linearized euler equations as a mathematical model for noise propagation in the time domain in aeroacoustics

accurate computation of stokes flow driven by an open immersed interface we present numerical methods for computing two dimensional stokes flow driven by forces singularly supported along an open immersed interface two second order accurate methods are developed one for accurately evaluating boundary integral solutions at a point and another for computing stokes solution values on a rectangular mesh we first describe a method for computing singular or nearly singular integrals such as a double layer potential due to sources on a curve in the plane evaluated at a point on or near the curve to improve accuracy of the numerical quadrature we add corrections for the errors arising from discretization which are found by asymptotic analysis when used to solve the stokes equations with sources on an open immersed interface the method generates second order approximations for both the pressure and the velocity and preserves the jumps in the solutions and their derivatives across the boundary we then combine the method with a mesh based solver to yield a hybrid method for computing stokes solutions at n 2 grid points on a rectangular grid numerical results are presented which exhibit second order accuracy to demonstrate the applicability of the method we use the method to simulate fluid dynamics induced by the beating motion of a cilium the method preserves the sharp jumps in the stokes solution and their derivatives across the immersed boundary model results illustrate the distinct hydrodynamic effects generated by the effective stroke and by the recovery stroke of the ciliary beat cycle

computations of multi fluid flows abstract full numerical simulations of three dimensional flows of two or more immiscible fluids of different densities and viscosities separated by a sharp interface with finite surface tension are discussed the method used is based on a finite difference approximation of the full navier stokes equations and explicit tracking of the interface between the fluids preliminary simulations of the rayleigh taylor instability and the motion of bubbles are shown

homogenization of the acoustic transmission through a perforated layer the paper deals with homogenized transmission conditions imposed on an interface plane separating two halfspaces occupied by an acoustic medium the conditions are obtained as the two scale homogenization limit of a standard acoustic problem imposed on the layer perforated by a sieve like obstacle with a periodic structure both the characteristic scale of the perforations and the layer thickness are parametrized by e 0 the limit model involving some homogenized coefficients governs the interface discontinuity of the acoustic pressure associated with the two halfspaces and the magnitude of the transverse acoustic velocity this novel approach allows for the treatment of complicated designs of perforations and presents an alternative to the usual description of acoustic impedance which relies on a rough averaging of the quasi experimental data

numerical treatment of two dimensional interfaces for acoustic and elastic waves we present a numerical method to take into account 2d arbitrary shaped interfaces in classical finite difference schemes on a uniform cartesian grid this work extends the explicit simplified interface method esim previously proposed in 1d j comput phys 168 2001 227 248 the physical problem under study concerns the linear hyperbolic systems of acoustics and elastodynamics with stationary interfaces our method maintains near the interfaces properties of the schemes in homogeneous medium such as the order of accuracy and the stability limit moreover it enforces the numerical solution to satisfy the exact interface conditions lastly it provides subcell geometrical features of the interface inside the meshing the esim can be coupled automatically with a wide class of numerical schemes lax wendroff flux limiter schemes etc for a negligible additional computational cost throughout the paper we focus on the challenging case of an interface between a fluid and an elastic solid in numerical experiments we provide comparisons between numerical solutions and original analytic solutions showing the efficiency of the method

integral criteria for a classification of solutions of linear differential equations are positive continuous function on the interval 0 co by solution we always mean a continuously differentiable function such that r t x t has a continuous derivative satisfying e when the func tion r is continuous but does not have a continuous derivative e can be interpreted as the first order system for the vector x y x rx 381

the fixed mesh ale approach for the numerical approximation of flows in moving domains in this paper we propose a method to approximate flow problems in moving domains using always a given grid for the spatial discretization and therefore the formulation to be presented falls within the category of fixed grid methods even though the imposition of boundary conditions is a key ingredient that is very often used to classify the fixed grid method our approach can be applied together with any technique to impose approximately boundary conditions although we also describe the one we actually favor our main concern is to properly account for the advection of information as the domain boundary evolves to achieve this we use an arbitrary lagrangian eulerian framework the distinctive feature being that at each time step results are projected onto a fixed background mesh that is where the problem is actually solved

high order gradients with the shifted boundary method an embedded enriched mixed formulation for elliptic pdes abstract we propose an extension of the embedded boundary method known as shifted boundary method to elliptic diffusion equations in mixed form e g darcy flow heat diffusion problems with rough coefficients etc our aim is to obtain an improved formulation that for linear finite elements is at least second order accurate for both flux and primary variable when either dirichlet or neumann boundary conditions are applied following previous work of nishikawa and mazaheri in the context of residual distribution methods we consider the mixed form of the diffusion equation i e with darcy type operators and introduce an enrichment of the primary variable this enrichment is obtained exploiting the relation between the primary variable and the flux variable which is explicitly available at nodes in the mixed formulation the proposed enrichment mimics a formally quadratic pressure approximation although only nodal unknowns are stored similar to a linear finite element approximation we consider both continuous and discontinuous finite element approximations and present two approaches a non symmetric enrichment which as in the original references only improves the consistency of the overall method and a symmetric enrichment which enables a full error analysis in the classical finite element context combined with the shifted boundary method these two approaches are extended to high order embedded computations and enable the approximation of both primary and flux gradient variables with second order accuracy independently on the type of boundary conditions applied we also show that the the primary variable is third order accurate when pure dirichlet boundary conditions are embedded

fictitious domain approach with hp finite element approximation for incompressible fluid flow we consider the application of fictitious domain approach combined with least squares spectral elements for the numerical solution of fluid dynamic incompressible equations fictitious domain methods allow problems formulated on a complicated shaped domain to be solved on a simpler domain containing least squares spectral element method has been used to develop the discrete model as this scheme combines the generality of finite element methods with the accuracy of spectral methods moreover the least squares methods have theoretical and computational advantages in the algorithmic design and implementation this paper presents the formulation and validation of the fictitious domain least squares spectral element approach for the steady incompressible navier stokes equations the convergence of the approximated solution is verified solving two dimensional benchmark problems demonstrating the predictive capability of the proposed formulation

limit point type solutions of nonlinear differential equations abstract we are concerned with the nonexistence of l 2 solutions of a nonlinear differential equation x a t x f t x by applying technique similar to that exploited by hallam siam j appl math 19 1970 430 439 for the study of asymptotic behavior of solutions of this equation we establish nonexistence of solutions from the class l 2 t 0 under milder conditions on the function a t which as the examples show can be even square integrable therefore the equation under consideration can be classified as of limit point type at infinity in the sense of the definition introduced by graef and spikes nonlinear anal 7 1983 851 871 we compare our results to those reported in the literature and show how they can be extended to third order nonlinear differential equations

time domain numerical modeling of wave propagation in 2d heterogeneous porous media this paper deals with the numerical modeling of wave propagation in porous media described by biot s theory the viscous efforts between the fluid and the elastic skeleton are assumed to be a linear function of the relative velocity which is valid in the low frequency range the coexistence of propagating fast compressional wave and shear wave and of a diffusive slow compressional wave makes numerical modeling tricky to avoid restrictions on the time step the biot s system is splitted into two parts the propagative part is discretized by a fourth order ader scheme while the diffusive part is solved analytically near the material interfaces a space time mesh refinement is implemented to capture the small spatial scales related to the slow compressional wave the jump conditions along the interfaces are discretized by an immersed interface method numerical experiments and comparisons with exact solutions confirm the accuracy of the numerical modeling the efficiency of the approach is illustrated by simulations of multiple scattering

on the deficiency indices and self adjointness of symmetric hamiltonian systems abstract the main purpose of this paper is to investigate the formal deficiency indices n of a symmetric first order system jf bf h f on an interval i where i r or i r here j b h are n n matrix valued functions and the hamiltonian h 0 may be singular even everywhere we obtain two results for such a system to have minimal numbers n 0 if i r resp n n if i r and a criterion for their maximality n 2n for i r as well as the quasi regularity this covers the kac krein and de branges trans amer math soc 99 1961 118 theorems on 2 2 canonical systems and some results from kogan and rofe beketov proc roy soc edinburgh sect a 74 1974 75 5 some conditions for a canonical system to have intermediate formal deficiency indices are presented too we also obtain a generalization of the well known titchmarsh sears theorem for second order sturm liouville type equations this contains results due to lidskii and krein as special cases we present two approaches to the above problems one dealing with formal deficiency indices and one dealing with ordinary deficiency indices our main non formal approach is based on the investigation of a symmetric linear relation smin which is naturally associated to a first order system this approach works in the framework of extension theory and therefore we investigate in detail the domain d s min of s min in particular we prove the so called regularity theorem for d s min as a byproduct of the regularity result we obtain very short proofs of generalizations of the main results of the paper by kogan and rofe beketov 1974 75

positive solutions of nonlinear differential equations with prescribed decay of the first derivative an existence and uniqueness result for bounded positive solutions x t of the equation u f t u u 0 t greater than or equal to t 0 greater than or equal to 0 is established by means of the banach contraction principle for such a solution it is shown that alpha t less than or equal to x t less than or equal to beta t t greater than or equal to t 0 where alpha beta are given nonnegative continuous functions which are integrable over t 0 infinity the result complements others known in the literature

analytical calculation of two leading exponents of the dilute potts model a potts model on a square lattice with two and four spin interaction and site and bond dilution is shown to be dual to itself the model is mapped onto a vertex problem which in turn is equivalent to a solid on solid model by means of these mappings the dilute potts model can be written as a gaussian like model with staggered and direct periodic fields the leading and next to leading exponents of the potts model are calculated subject to the validity of certain assumptions

on the l2 nature of solutions of nth order symmetric differential equations and mcleod s conjecture we investigate the integrable square properties of solutions of linear symmetric differential equations of arbitrarily large order 2 m whose coefficients involve a real multiple r of certain positive real powers of the independent variable x information on the l 2 nature is obtained by variation of parameters from meijer function solutions of an associated homogeneous equation of hypergeometric type when the coefficients of the differential expressions are positive it is possible by a suitable choice of r and m to obtain between m and 2 m 1 linearly independent solutions in l 2 0 this proves a conjecture of j b mcleod that the deficiency index can take values between m and 2 m 1 for such operators

numerical modeling of elastic waves across imperfect contacts a numerical method is described for studying how elastic waves interact with imperfect contacts such as fractures or glue layers existing between elastic solids these contacts have been classicaly modeled by interfaces using a simple rheological model consisting of a combination of normal and tangential linear springs and masses the jump conditions satisfied by the elastic fields along the interfaces are called the spring mass conditions by tuning the stiffness and mass values it is possible to model various degrees of contact from perfect bonding to stress free surfaces the conservation laws satisfied outside the interfaces are integrated using classical finite difference schemes the key problem arising here is how to discretize the spring mass conditions and how to insert them into a finite difference scheme this was the aim of the present paper for this purpose we adapted an interface method previously developed for use with perfect contacts j comput phys 195 2004 pp 90 116 this numerical method also describes closely the geometry of arbitrarily shaped interfaces on a uniform cartesian grid at negligible extra computational cost comparisons with original analytical solutions show the efficiency of this approach

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

interaction of oscillating filaments a computational study abstract the immersed boundary technique is used to model the interaction of swimming filaments in a viscous incompressible fluid fluid quantities are represented on a grid and the immersed filaments are each represented by a discrete collection of moving points the effects of phase differences and proximity of the filaments on energy dissipation and swimming speeds are studied the computed results are compared with previous asymptotic analysis furthermore we present case studies which exhibit phase un locking phenomena undetectable using previous asymptotic analysis

on lp solutions for a class of sequential fractional differential equations abstract under some simple conditions on the coefficient a t we establish that the initial value problem 0 d t x a t x 0 t 0 lim t 0 t 1 x t 0 has no solution in l p 1 r where p 1 p 1 p and 0 d t designates the riemann liouville derivative of order our result might be useful for developing a non integer variant of h weyl s limit circle limit point classification of differential equations

new cartesian grid methods for interface problems using the finite element formulation new finite element methods based on cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients the triangulations in these methods do not need to fit the interfaces the basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately both non conforming and conforming finite element spaces are considered corresponding interpolation functions are proved to be second order accurate in the maximum norm the conforming finite element method has been shown to be convergent with cartesian triangulations these new methods can be used as finite difference methods numerical examples are provided to support the methods and the theoretical analysis

weak imposition of essential boundary conditions in the finite element approximation of elliptic problems with non matching meshes this is the accepted version of the following article codina r and baiges j 2015 weak imposition of essential boundary conditions in the finite element approximation of elliptic problems with non matching meshes int j numer meth engng 104 624 654 doi 10 1002 nme 4815 which has been published in final form at http onlinelibrary wiley com doi 10 1002 nme 4815 abstract

a decoupled augmented iim strategy for incompressible two phase flows with interfaces on irregular domains a decoupled augmented immersed interface method for solving incompressible two phase flows involving both irregular domains and interfaces is presented in order to impose the prescribed velocity at the boundary of the irregular domain singular force as one set of augmented variables is introduced the velocity components at the two fluid interface as another set of augmented variables are introduced to satisfy the continuity condition of the velocity across the interface so that the jump conditions for the velocity and pressure are decoupled across the interface the augmented variables and or the forces along the interface boundary are related to the jumps in both pressure and velocity and the jumps in their derivatives across the interface boundary and applied to the fluid through jump conditions the resulting augmented equation is a couple system of these two sets of augmented variables and the direct application of the gmres is impractical due to larger iterations in this work the novel decoupling of two sets of the augmented variables is proposed and the decoupled augmented equation is then solved by the lu or the gmres method the stokes equations are discretized via the finite difference method with the incorporation of jump contributions on a staggered cartesian grid and solved by the conjugate gradient uzawa type method the numerical results show that second order accuracy for the velocity is confirmed the present method has also been applied to solve for incompressible two phase navier stokes flow with interfaces on irregular domains copyright 2011 john wiley sons ltd

immersed interface methods for stokes flow with elastic boundaries or surface tension a second order accurate interface tracking method for the solution of incompressible stokes flow problems with moving interfaces on a uniform cartesian grid is presented the interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids the interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed the stokes equations are then discretized using the second order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper siam j numer anal 31 1994 pp 1019 1044 the resulting velocities are interpolated to the interface to determine the motion of the interface an implicit quasi newton method is developed that allows reasonable time steps to be used

shape topology optimization for navier stokes problem using variational level set method we consider the shape topology optimization of the navier stokes problem a new algorithm is proposed based on the variational level set method by this algorithm a relatively smooth evolution can be maintained without re initialization and drastic topology change can be handled easily finally the promising features of the proposed method are illustrated by two benchmark examples

static and dynamic critical phenomena of the two dimensional q state potts model theq state potts model on the square lattice is studied by monte carlo simulation forq 3 4 5 6 very good agreement is obtained with exact results of kiharaet al and baxter for energy and free energy at the critical point critical exponent estimates forq 3 are 0 4 0 1 1 45 in rough agreement with high temperature series extrapolation and real space renormalization group methods the transition forq 5 6 is found to be a very weakly first order transition i e pronounced pseudocritical phenomena occur specific heat susceptibility etc nearly diverge at the first order transition temperature dynamics is associated to the model in the same way as for the kinetic ising model and the nonlinear slowing down of the order parameter and of the energy is studied the dynamic exponent is estimated to be zv 1 9 within our accuracy noq dependence is detected the relaxation is found to be consistent with dynamic scaling predictions and dynamic scaling functions associated with the nonlinear relaxation are estimated

dilute potts model duality and site bond percolation a number of results are obtained for potts models on an arbitrary planar lattice it is shown that the dual of a dilute potts model is a graph generating function and that the dual of a constrained dilute potts model is an undiluted potts model it is also shown that the dilute potts model generates a correlated site bond percolation for uncorrelated site bond percolations the analysis determines the percolation threshold from a knowledge of the critical point of a potts model this generalises a recent result of kondor 1980 who uses a star triangle transformation to derive this relationship for the honeycomb and triangle lattices

simulation of elastic filaments interacting with a viscous pulsatile flow abstract motivated by the need to understand the motion of ruptured atherosclerosis plaque fragments in a flowing blood we simulated flexible elastic filaments interacting with a viscous pulsatile flow in two dimensions using the immersed boundary method our simulations show that 1 a flexible horizontal filament positioned perpendicularly to the flow is a stable configuration and 2 hydrodynamical forces prevent adhesion of filaments but mutual interaction of vortical fields yields clustering of filaments into groups that may be called aggregation

an immersed interface method for viscous incompressible flows involving rigid and flexible boundaries we present an immersed interface method for the incompressible navier stokes equations capable of handling both rigid and flexible boundaries the immersed boundaries are represented by a number of lagrangian control points in order to ensure that the no slip condition on the rigid boundary is satisfied singular forces are applied on the fluid the forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity and are interpolated using cubic splines the strength of the singular forces at the rigid boundary is determined by solving a small system of equations at each timestep for flexible boundaries the forces that the boundary exerts on the fluid are computed from the constitutive relation of the flexible boundary and are applied to the fluid through the jump conditions the position of the flexible boundary is updated implicitly using a quasi newton method bfgs within each timestep the navier stokes equations are discretized on a staggered cartesian grid by a second order accurate projection method for pressure and velocity and the overall scheme is second order accurate

the shifted boundary method for embedded domain computations part ii linear advection diffusion and incompressible navier stokes equations abstract we propose a new embedded finite element method for the linear advection diffusion equation and the laminar and turbulent incompressible navier stokes equations the proposed method belongs to the class of surrogate approximate boundary algorithms and is based on the idea of shifting the location where boundary conditions are applied from the true to a surrogate boundary accordingly boundary conditions enforced weakly are appropriately modified to preserve optimal error convergence rates we include the full analysis of stability and convergence of the method in the linear advection diffusion equation and a battery of tests for the case of laminar and turbulent incompressible navier stokes equations we also discuss the conservation properties of the proposed method in all cases

immersed interface methods for neumann and related problems in two and three dimensions we develop and apply a finite difference method to discretize the laplacian operator with neumann boundary conditions on an irregular domain using a regular cartesian grid the method is an extension of the immersed interface method developed by leveque and li siam j numer anal 31 1994 1019 1044 with careful selection of stencils the method is second order accurate and produces a matrix that is stable diagonally semidominant the method is illustrated on several two dimensional problems and one three dimensional problem

a cartesian grid projection method for the incompressible euler equations in complex geometries many problems in fluid dynamics require the representation of complicated internal or external boundaries of the flow here we present a method for calculating time dependent incompressible inviscid flow which combines a projection method with a cartesian grid approach for representing geometry in this approach the body is represented as an interface embedded in a regular cartesian mesh the advection step is based on a cartesian grid algorithm for compressible flow in which the discretization of the body near the flow uses a volume of fluid representation a redistribution procedure is used to eliminate time step restrictions due to small cells where the boundary intersects the mesh the projection step uses an approximate projection based on a cartesian grid method for potential flow the method incorporates knowledge of the body through volume and area fractions along with certain other integrals over the mixed cells convergence results are given for the projection itself and for the time dependent algorithm in two dimensions the method is also demonstrated on flow past a half cylinder with vortex shedding

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

a fictitious domain method for dirichlet problem and applications abstract in this article we discuss the solution of the dirichlet problem for a class of elliptic operators by a lagrange multiplier fictitious domain method this approach allows the use of regular grids and therefore of fast specialized solvers for problems on complicated geometries the resulting saddle point system can be solved by an uzawa conjugate gradient algorithm in the case of two dimensional problems a quasi optimal preconditioner has been found by fourier analysis and numerical experiments confirm its nice scaling properties the resulting methodology is applied to a nonlinear time dependent problem namely the flow of a viscous plastic medium in a cylindrical pipe showing the potential of this methodology for some classes of nonlinear problems

a note on immersed interface method for three dimensional elliptic equations abstract the immersed interface method proposed by leveque and li 1 is extended to three dimensional elliptic equations of the form x u x x u x f x we study the situation in which there is an irregular interface surface s contained in the solution domain across which and f may be discontinuous or even singular as a result the solution u will usually be nonsmooth or even discontinuous a finite difference approach with a uniform cartesian grid is used in the discretization local truncation error analysis is performed to estimate the accuracy of the numerical solution

the fast solution of poisson s and the biharmonic equations on irregular regions we present fast methods for solving laplace s and the biharmonic equations on irregular regions with smooth boundaries the methods used for solving both equations make use of fast poisson solvers on a rectangular region in which the irregular region is embedded they also both use an integral equation formulation of the problem where the integral equations are fredholm integral equations of the second kind the main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region fast solvers are then used to compute the extended solution aside from solving the equations we have also been able to compute derivatives of the solutions with little loss of accuracy when the data was sufficiently smooth

a fictitious domain distributed lagrange multiplier based fluid structure interaction scheme with hierarchical b spline grids abstract we present a numerical scheme for fluid structure interaction based on hierarchical b spline grids and fictitious domain distributed lagrange multipliers the incompressible navier stokes equations are solved over a cartesian grid discretised with b splines the fluid grid near the immersed solids is refined locally using hierarchical b splines the immersed solid is modelled as geometrically exact beam discretised with standard linear lagrange shape functions the kinematic constraint at the fluid solid interface is enforced with distributed lagrange multipliers the unconditionally stable and second order accurate generalised method is used for integration in time for both the fluid and solid domains a fully implicit and fully coupled solution scheme is developed by using the newton raphson method to solve the non linear system of equations obtained with galerkin weak formulation first the spatial and temporal convergence of the proposed scheme is assessed by studying steady and unsteady flow past a fixed cylinder then the scheme is applied to several benchmark problems to demonstrate the efficiency and robustness of the proposed scheme the results obtained with the present scheme are compared with the reference values

a regularization method for the numerical solution of periodic stokes flow we introduce a regularization method that gives a smooth formulation for the fundamental solution to stokes flow driven by an infinite triply periodic array of point forces with this formulation the velocity at any spatial location may be calculated including at and very near the point forces these locations typically lead to numerical difficulties due to the singularity within the stokeslet when using other methods for computational efficiency we build upon previous methods in which the periodic stokeslet is split into two rapidly decaying sums one in physical space and one in reciprocal or fourier space we present two validation studies of our method first we compute the drag coefficient for periodic arrays of spheres with a variety of concentrations of sphere packings and second we prescribe a force density onto a periodic array of spheres compute the resulting nearby velocity field and compare these velocities to those computed using an immersed boundary method formulation the drag coefficients computed with our method are within 0 63 of previously published values the velocity field comparison shows a relative error of about 0 18 in the l 2 norm we then apply our numerical method to a periodic arrangement of sinusoidal swimmers by systematically varying their spacing in three directions we are able to explore how their spacing affects their collective swimming speed

a fictitious domain method for external incompressible viscous flow modeled by navier stokes equations abstract in this article we discuss a fictitious domain method for the numerical solutions of three dimensional elliptic problems with dirichlet boundary conditions and also of the navier stokes equations modeling incompressible viscous flow the methodology for the navier stokes equations described here takes a systematic advantage of time discretization by operator splitting in order to treat separately advection imbedding and incompressibility due to the decoupling fast elliptic solvers can be used to treat the incompressibility condition even if the original problem is taking place on a nonregular geometry the resulting methodology is applied to two dimensional unsteady external incompressible viscous flow problems and three dimensional stokes problems

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

the numerical approximation of a delta function with application to level set methods it is shown that a discrete delta function can be constructed using a technique developed by anita mayo the fast solution of poisson s and the biharmonic equations on irregular regions siam j sci comput 21 1984 285 299 for the numerical solution of elliptic equations with discontinuous source terms this delta function is concentrated on the zero level set of a continuous function in two space dimensions this corresponds to a line and a surface in three space dimensions delta functions that are first and second order accurate are formulated in both two and three dimensions in terms of a level set function the numerical implementation of these delta functions achieves the expected order of accuracy

multicritical behaviour in the q state potts lattice gas the phase diagram of the q state potts lattice gas is determined exactly on a cayley tree for q 2 the authors use this result to make some conjectures concerning the lower dimensional cases it is argued that there should be a tetracritical point different from the transition point where ising droplets diverge in the case of three dimensional correlated site bond percolation q 1 in two dimensions the critical exponent gamma p of the mean cluster size at the ising critical point is derived from the extended den nijs conjectures gamma p 91 48 1 896

locally corrected semi lagrangian methods for stokes flow with moving elastic interfaces we present a new method for computing two dimensional stokes flow with moving interfaces that respond elastically to stretching the interface is moved by semi lagrangian contouring a distance function is introduced on a tree of cells near the interface transported by a semi lagrangian time step and then used to contour the new interface the velocity field in a periodic box is calculated as a potential integral resulting from interfacial and body forces using a technique based on ewald summation with analytically derived local corrections the interfacial stretching is found from a surprisingly natural formula a test problem with an exact solution is constructed and used to verify the speed accuracy and robustness of the approach

nonlinear limit point type solutions ofnth order differential equations sufficient conditions are given under which the n th order nonlinear differential equation is of the nonlinear limit point limit circle type

analysis of a one dimensional model for the immersed boundary method numerical methods are studied for the one dimensional heat equation with a singular forcing term u t u xx c t delta x alpha t the delta function delta x is replaced by a discrete approximation d h x and the resulting equation is solved by a crank nicolson method on a uniform grid the accuracy of this method is analyzed for various choices of d h the case where c t is specified and also the case where c is determined implicitly by a constraint on the solution at the point a are studied these problems serve as a model for the immersed boundary method of peskin for incompressible flow problems in irregular regions some insight is gained into the accuracy that can be achieved and the importance of choosing appropriate discrete delta functions

limit point type results for nonlinear fourth order differential equations sufficient conditons are given under which the nonlinear fourth order differential equation is of the nonlinear limit point type

a computational model of the cochlea using the immersed boundary method this thesis describes a two dimensional computational model of the cochlea inner ear solved using a modified form of peskin s immersed boundary method this method is an extension of peskin s method that was originally applied to solving a model of the heart 1972 this method solves the time dependent incompressible navier stokes equations in the presence of immersed boundaries the fluid equations are specified on a fixed eulerian grid while the immersed boundaries are specified on a moving lagrangian grid the immersed boundaries are used to represent the structures of the cochlea the rigid bone the actively moving stapes and the flexible membranes the immersed boundaries exert forces locally on the fluid these local forces are seen by the fluid as external forces that are added to the other forces pressure and viscous acting on the fluid the computational method divides into three separate parts first the forces exerted by the points of the immersed boundary are calculated and extended to the fluid grid second the fluid equations are solved and the velocities are restricted to the immersed boundary points finally the immersed boundary points are moved to their new locations at the new local velocities the immersed boundary forces are either specified such that the boundary moves at a given velocity zero or non zero or they are specified such that the forces obey some given force law for example hooke s law these forces can be calculated either explicitly or implicitly or a combination of both for the fluid the navier stokes equations are solved on a doubly periodic rectangular grid in a second order accurate manner using a projection method developed by bell colella and glaz 1988 the extension and restriction of the boundary forces and fluid velocities to and from the fluid grid are also done in a second order accurate manner the cochlea is modelled as two fluid chambers separated by a flexible partition whose stiffness varies exponentially along its length the stapes is represented by a moving piston and the bony outer walls are allowed to be either straight or tapered a travelling wave propagates along the flexible partition under the influence of the moving piston and the dependence of this wave on cochlea geometry and partition stiffness is studied also included are studies of bone conduction and transient signal analysis finally comparisons of the model results are made to experimental data and asymptotic results

numerical modeling of the acoustic wave propagation across a homogenized rigid microstructure in the time domain homogenization of a thin micro structure yields effective jump conditions that incorporate the geometrical features of the scatterers these jump conditions apply across a thin but nonzero thickness interface whose interior is disregarded this paper aims i to propose a numerical method able to handle the jump conditions in order to simulate the homogenized problem in the time domain ii to inspect the validity of the homogenized problem when compared to the real one for this purpose we adapt an immersed interface method originally developed for standard jump conditions across a zero thickness interface doing so allows us to handle arbitrary shaped interfaces on a cartesian grid with the same efficiency and accuracy of the numerical scheme than those obtained in an homogeneous medium numerical experiments are performed to test the properties of the numerical method and to inspect the validity of the homogenization problem

an interface fitted subspace projection method for finite element simulations of particulate flows abstract a novel finite element method for the direct numerical simulation of particles in a newtonian carrier fluid is presented the proposed method is based on a fictitious or one domain formulation and a subspace projection method to account for the rigid body motion of the particles underlying equations are posed in an ale arbitrary lagrangian eulerian formulation allowing for moving computational meshes the mesh is adapted to the particles by a novel mesh smoothing approach guaranteeing both mesh optimality and a sharp representation of the particles boundaries we show that by using quadratic taylor hood finite elements for the discretization of the navier stokes equations and isoparametric elements to represent the geometry second order convergence with respect to the energy norm can be achieved as opposed to a sharp h 1 2 result which holds if no mesh adaptation is performed we present numerical examples in 2d confirming the theoretical convergence results furthermore we validate the approach by simulating the sedimentation process of a single particle and show the potential of the method by simulating a lid driven cavity flow with 100 particles

removing the stiffness from interfacial flows with surface tension a new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in two dimensional irrotational and incompressible fluids through the laplace young condition at the interface surface tension introduces high order terms both nonlinear and nonlocal into the dynamics this leads to severe stability constraints for explicit time integration methods and makes the application of implicit methods difficult this new formulation has all the nice properties for time integration methods that are associated with having a linear constant coefficient highest order term that is using this formulation we give implicit time integration methods that have no high order time step stability constraint associated with surface tension and are explicit in fourier space the approach is based on a boundary integral formulation and applies more generally even to problems beyond the fluid mechanical context here they are applied to computing with high resolution the motion of interfaces in hele shaw flows and the motion of free surfaces in inviscid flows governed by the euler equations one hele shaw computation shows the behavior of an expanding gas bubble over long time as the interface undergoes successive tip splittings and finger competition a second computation shows the formation of a very ramified interface through the interaction of surface tension with an unstable density stratification in euler flows the computation of a vortex sheet shows its roll up through the kelvin helmholtz instability this motion culminates in the late time self intersection of the interface creating trapped bubbles of fluid this is we believe a type of singularity formation previously unobserved for such flows in 2d finally computations of falling plumes in an unstably stratified boussinesq fluid show a very similar behavior

partially implicit motion of a sharp interface in navier stokes flow we develop a numerical method for the coupled motion of navier stokes flow with an elastic interface of zero thickness which exerts tension and bending forces on the fluid the interface motion is made partially implicit by approximating a backward euler step in the high wavenumbers as in the small scale decomposition method of hou lowengrub and shelley this modified step is combined with the method of beale and layton j t beale a t layton a velocity decomposition approach for moving interfaces in viscous fluids j comput phys 228 2009 3358 67 the fluid velocity is found by computing the stokes velocity and a more regular remainder the resulting scheme is second order in space and first order in time it can be made second order in time by extrapolation the discontinuities in the pressure and velocity gradient are preserved the partially implicit method allows much larger time steps than an explicit method with negligible added effort the formulas in the fourier transform for the implicit approximation in high wavenumbers are similar to those derived in hou and shi t y hou z shi an efficient semi implicit immersed boundary method for the navier stokes equations j comput phys 227 2008 9138 69 in a different context

non limit circle criteria for singular hamiltonian differential systems non limit circle criteria for singular hamiltonian differential expressions with complex coefficients are obtained the main results are extensions of the previous limit point criterion due to h weyl for second order differential equations

a front tracking method for viscous incompressible multi fluid flows abstract a method to simulate unsteady multi fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described the flow field is discretized by a conservative finite difference approximation on a stationary grid and the interface is explicitly represented by a separate unstructured grid that moves through the stationary grid since the interface deforms continuously it is necessary to restructure its grid as the calculations proceed in addition to keeping the density and viscosity stratification sharp the tracked interface provides a natural way to include surface tension effects both two and three dimensional full numerical simulations of bubble motion are presented

smoothed profile method for particulate flows error analysis and simulations abstract we re formulate and analyze a new method for particulate flows the so called smoothed profile method spm first proposed in y nakayama r yamamoto simulation method to resolve hydrodynamic interactions in colloidal dispersions phys rev e 71 2005 036707 which uses a fixed computational mesh without conformation to the geometry of the particles the method represents the particles by certain smoothed profiles to construct a body force term added into the navier stokes equations spm imposes accurately and efficiently the proper conditions at the particle fluid interface in particular while the original method employs a fully explicit time integration scheme we develop a high order semi implicit splitting scheme which we implement in the context of spectral hp element discretization first we show that the modeling error of spm has a non monotonic dependence on the time step size t it is a function of t where is the kinematic viscosity of fluid and is the interface thickness of the smoothed profile subsequently we present several steady and unsteady simulations including flow past 3d complex shaped particles and compare against direct numerical simulations and the force coupling method fcm

global existence of solutions with prescribed asymptotic behavior for second order nonlinear differential equations where the nonlinearity f satis9es certain monotonicity conditions speci9ed in the sequel we always assume without mentioning that the function f t u v is continuous in the domain d t u v t t0 u v r where t0 1 it is supposed also that all equations and inequalities involving t hold for t t0 unless otherwise speci9ed the symbols o and o have their standard meanings as t and we use the notation r def 0 eq 1 is often used for mathematical modelling of various physical chemical and biological systems and attracts constant interest of researchers a great deal of papers published during the last three decades are concerned with local and global existence of solutions of eq 1 and its numerous particular cases uniqueness of solutions

on classification of second order differential equations with complex coefficients this paper is concerned with the deficiency index problem of second order differential equations with complex coefficients it is known that this class of equations is classified into cases i ii and iii according to the number of linearly independent solutions in suitable weighted square integrable spaces in this study the original equation is reformulated into a new formally self adjoint differential system by introducing a new spectral parameter and the relationship between the classifications of the equation and the system is obtained moreover the exact dependence of cases ii and iii on the corresponding half planes is given and some criteria of the three cases are established

improved volume conservation in the computation of flows with immersed elastic boundaries this paper introduces a recipe for the construction of a finite difference divergence operator the coefficients of which are completely determined once an interpolation scheme has been chosen substitution of this divergence operator for the previously used divergence based on central differences makes a dramatic improvement in the overall volume conservation that is observed in an immersed boundary computation this improvement is particularly important for computations in which an elastic boundary separates chambers containing fluid at substantially different pressures a situation which is prominent in cardiac fluid dynamics during the contraction of the ventricles

least squares spectral method for the two dimensional stokes interface problems abstract this paper deals with the pseudo spectral least squares method to approximate the solution of the stokes interface problem that is the stokes equations with discontinuous viscosity and singular source term the stokes interface equations are adjusted in each sub domain along with the interface conditions then the equations are reformulated as a first order system by introducing stress tensor as a new independent variable the continuous and discrete norm least squares functionals using legendre and chebyshev weights for the first order system are defined and shown to be equivalent to appropriate product norms the spectral convergence of the proposed method is presented numerical examples are also given to demonstrate the method and analysis

q state potts models in d dimensions migdal kadanoff approximation the first and second order phase transitions of the q state potts models are obtained in arbitrary dimension d critical and tricritical behaviours merge and annihilate at qc d clearing the way to first order transitions at q qc d by the condensation of effective vacancies the value of qc d decreases with increasing d from diverging as exp 2 d 1 at d to 1 to qc 2 3 81 cf exact value of 4 to lower values at d 2 for given d a changeover in critical behaviour occurs at q1 d as the critical fixed points merge from the potts lattice gas region to the undiluted potts limit it is suggested that the power law singularities of the percolation problem q to 1 have logarithmic corrections

a simplified reproducing kernel method for 1 d elliptic type interface problems abstract in this paper an efficient numerical method is proposed for the 1d elliptic type interface problems we first construct a broken reproducing kernel space and then apply the simplified reproducing kernel method to the interface problems the convergence of the proposed method is discussed finally some representative numerical examples are presented to illustrate the accuracy and robustness of the proposed method