an edge based method for the incompressible navier stokes equations on polygonal meshes abstract a pressure based method is presented for discretizing the unsteady incompressible navier stokes equations using hybrid unstructured meshes the edge based data structure and assembly procedure adopted lead naturally to a strictly conservative discretization which is valid for meshes composed of n sided polygons particular attention is given to the construction of a pressure velocity coupling procedure which is supported by edge data resulting in a relatively simple numerical method that is consistent with the boundary and initial conditions required by the incompressible navier stokes equations edge formulas are presented for assembling the momentum equations which are based on an upwind biased linear reconstruction of the velocity field similar formulas are presented for assembling the pressure equation the method is demonstrated to be second order accurate in space and time for two navier stokes problems admitting an exact solution results for several other well known problems are also presented including lid driven cavity flow impulsively started cylinder flow and unsteady vortex shedding from a circular cylinder although the method is by construction minimalist it is shown to be accurate and robust for the problems considered

a fast poisson solver for complex geometries robust fast solvers for the poisson equation have generally been limited to regular geometries where direct methods based on fourier analysis or cyclic reduction and multigrid methods can be used while multigrid methods can be applied in irregular domains and to a broader class of partial differential equations they are difficult to implement in a robust fashion since they require an appropriate hierarchy of coarse grids which are not provided in many practical situations in this paper we present a new fast poisson solver based on potential theory rather than on direct discretization of the partial differential equation our method combines fast algorithms for computing volume integrals and evaluating layer potentials on a grid with a fast multipole accelerated integral equation solver the amount of work required is o m log m n where m is the number of interior grid points and n is the number of points on the boundary asymptotically the cost of our method is just twice that of a standard poisson solver on a rectangular domain in which the problem domain can be embedded independent of the complexity of the geometry

application of subival in solving initial value problems with fractional derivatives abstract the application of a numerical method for the approximation of the fractional derivative in riemann liouville and caputo definitions in initial value problems is discussed the method previously known as the subinterval based method is now referred to by its acronym subival for simpler future references it is dependent on subinterval partitions performed according to a proposed algorithm interpolations using selected time axis nodes i e nodes where solutions have been computed and analytical monomial integrodifferentiation formulae two exemplary circuit problems have been introduced as a test for the method these problems have analytical solutions available in literature the evaluations of these solutions have been compared with results obtained through an adaptive step size predictor corrector scheme where the core computations relied on the proposed numerical method subival has been implemented into an activex dynamic link library dll the paper contains instructions on how the predictor corrector algorithm can be implemented into a computer algebra system where the subival library is applied for the fractional derivative approximation examples of this are given in the form of scripts in matlab and mathematica these scripts generally allow to solve systems of fractional order state equations to which the introduced circuit examples can be brought to

a second order accurate symmetric discretization of the poisson equation on irregular domains in this paper we consider the variable coefficient poisson equation with dirichlet boundary conditions on an irregular domain and show that one can obtain second order accuracy with a rather simple discretization moreover since our discretization matrix is symmetric it can be inverted rather quickly as opposed to the more complicated nonsymmetric discretization matrices found in other second order accurate discretizations of this problem multidimensional computational results are presented to demonstrate the second order accuracy of this numerical method in addition we use our approach to formulate a second order accurate symmetric implicit time discretization of the heat equation on irregular domains then we briefly consider stefan problems

a fractional step solution method for the unsteady incompressible navier stokes equations in generalized coordinate systems abstract a fractional step method is developed for solving the time dependent three dimensional incompressible navier stokes equations in generalized coordinate systems the primitive variable formulation uses the pressure defined at the center of the computational cell and the volume fluxes across the faces of the cells as the dependent variables instead of the cartesian components of the velocity this choice is equivalent to using the contravariant velocity components in a staggered grid multiplied by the volume of the computational cell the governing equations are discretized by finite volumes using a staggered mesh system the solution of the continuity equation is decoupled from the momentum equations by a fractional step method which enforces mass conservation by solving a poisson equation this procedure combined with a consistent approximation of the geometric quantities is done to satisfy the discretized mass conservation equation to machine accuracy as well as to gain the favorable convergence properties of the poisson solver the momentum equations are solved by an approximate factorization method and a novel zebra scheme with four color ordering is devised for the efficient solution of the poisson equation several two and three dimensional laminar test cases are computed and compared with other numerical and experimental results to validate the solution method good agreement is obtained in all cases

large eddy simulation in complex geometric configurations using boundary body forces anumericalmethodispresentedthatallowslargeeddysimulation les ofturbulente owsincomplexgeometric cone gurations with moving boundaries and that retains the advantages of solving the navier stokes equations on e xed orthogonal grids the boundary conditions are applied independently of the grid by assigning body forces over surfaces that need not coincide with coordinate lines the use of orthogonal nondeforming grids simplie es grid generation facilitates theimplementation of high order nondissipativediscretization schemes andminimizes the spatial and temporal variations in e lter width that complicate unstructured deforming grid les dynamic subgrid scaleturbulence models areparticularly appealing in combination with the body forceprocedure because the dynamic model accounts automatically for the presence of solid walls without requiring damping functions the method is validated by simulations of the turbulent e ow in a motored axisymmetric piston cylinder assembly for which detailed experimental measurements are available computed mean and rms velocity proe les show very good agreement with measured ensemble averages thepresent numerical code runs on small personal computerlike workstations for a comparable level of accuracy computational requirements memory and cpu time are at least a factor of 10 lower compared to published simulations for the same cone guration obtained using an unstructured boundary e tted deforming grid approach

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

approximate solution of abel integral equation this paper presents a new stable approximate inversion of abel integral equation by using the taylor expansion of the unknown function abel equation is approximately transformed to a system of linear equations for the unknown function together with its derivatives a desired solution can be determined by solving the resulting system according to cramer s rule this method gives a simple and closed form of approximate abel inversion which can be performed by symbolic computation the nth order approximation is exact for a polynomial of degree up to n abel integral equation is approximately expressed in terms of integrals of input data so the suggested approach is stable for experimental data with random noise an error analysis of this approach is given finally several numerical examples are given to illustrate the accuracy and stability of this method

status of the differential transformation method abstract further to a recent controversy on whether the differential transformation method dtm for solving a differential equation is purely and solely the traditional taylor series method it is emphasized that the dtm is currently used often only as a technique for analytically calculating the power series of the solution in terms of the initial value parameters sometimes a piecewise analytic continuation process is implemented either in a numerical routine e g within a shooting method or in a semi analytical procedure e g to solve a boundary value problem emphasized also is the fact that at the time of its invention the currently used basic ingredients of the dtm that transform a differential equation into a difference equation of same order that is iteratively solvable were already known for a long time by the traditional taylor method users notably in the elaboration of software packages numerical routines for automatically solving ordinary differential equations at now the defenders of the dtm still ignore the though much better developed studies of the traditional taylor method users who in turn seem to ignore similarly the existence of the dtm the dtm has been given an apparent strong formalization set on the same footing as the fourier laplace or mellin transformations though often used trivially it is easily attainable and easily adaptable to different kinds of differentiation procedures that has made it very attractive hence applications to various problems of the taylor method and more generally of the power series method including noninteger powers has been sketched it seems that its potential has not been exploited as it could be after a discussion on the reasons of the misunderstandings which have caused the controversy the preceding topics are concretely illustrated it is concluded that for the sake of clarity when the dtm is applied to odes it should be mentioned that the dtm exactly coincides with the traditional taylor method contrary to what is currently written

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

practical aspects of formulation and solution of moving mesh partial differential equations moving mesh partial differential equations mmpdes are used in the mmpde moving mesh method to generate adaptive moving meshes for the numerical solution of time dependent problems how mmpdes are formulated and solved is crucial to the efficiency and robustness of the method in this paper several practical aspects of formulating and solving mmpdes are studied they include spatial balance scaling invariance effective control of mesh concentration bounds on time steps multiple sub steps and two level mesh movement numerical results are also given c 2001 academic press

iterative refinement for a system of linear integro differential equations of fractional type a novel technique based on iterative refinement is developed to approximate the analytical solution of a system of linear fractional integro differential equations while the study focuses mainly on fredholm type equations adaptation to the volterra type is also presented a comparison is made with the method of successive approximations on the basis of convergence speed and accuracy several numerical examples are given to demonstrate the efficacy of our algorithm the authors also present formulations of some error bounds

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

numerical simulation for coupled systems of nonlinear fractional order integro differential equations via wavelets method abstract in this paper a new method for solving coupled systems of nonlinear fractional order integro differential equations is proposed the idea is to use bernoulli wavelets and operational matrix the main purpose of the technique is to transform the studied systems of fractional order integro differential equations into systems of algebraic equations which can be solved easily illustrative examples and comparisons with haar wavelets and legendre wavelets are included to reveal the effectiveness of the method and the accuracy of the convergence analysis

analytical solution of the bagley torvik equation by adomian decomposition method abstract the fractional derivative has been occurring in many physical problems such as frequency dependent damping behavior of materials motion of a large thin plate in a newtonian fluid creep and relaxation functions for viscoelastic materials the pi d controller for the control of dynamical systems etc phenomena in electromagnetics acoustics viscoelasticity electrochemistry and material science are also described by differential equations of fractional order the solution of the differential equation containing fractional derivative is much involved instead of application of the existing methods an attempt has been made in the present analysis to obtain the solution of bagley torvik equation r l bagley p j torvik on the appearance of the fractional derivative in the behavior of real materials asme j appl mech 51 1984 294 298 i podlubny fractional differential equations academic press san diego ca usa 1999 by the relatively new adomian decomposition method the results obtained by this method are then graphically represented and then compared with those available in the work of podlubny i podlubny fractional differential equations academic press san diego ca usa 1999 a good agreement of the results is observed

the iterated galerkin method for linear integro differential equations abstract the iterated galerkin method of i h sloan which yields superconvergence for the approximate solution of integral equations is generalized to get similar results for the larger class of integro differential equations two approaches are investigated to obtain superconvergence in different norms moreover some additional aspects are discussed

a differentially interpolated direct forcing immersed boundary method for predicting incompressible navier stokes equations in time varying complex geometries a dispersion relation preserving dual compact scheme developed in cartesian grids is applied together with the immersed boundary method to solve the flow equations in irregular and time varying domains the artificial momentum forcing term applied at certain points in cells containing fluid and solid allows an imposition of velocity condition to account for the motion of solid body we develop in this study a differential based interpolation scheme which can be easily extended to three dimensional simulation the results simulated from the proposed immersed boundary method agree well with other numerical and experimental results for the chosen benchmark problems the accuracy and fidelity of the ib flow solver developed to predict flows with irregular boundaries are therefore demonstrated

hydrodynamics of confined colloidal fluids in two dimensions we apply a hybrid molecular dynamics and mesoscopic simulation technique to study the dynamics of two dimensional colloidal disks in confined geometries we calculate the velocity autocorrelation functions and observe the predicted t 1 long time hydrodynamic tail that characterizes unconfined fluids as well as more complex oscillating behavior and negative tails for strongly confined geometries because the t 1 tail of the velocity autocorrelation function is cut off for longer times in finite systems the related diffusion coefficient does not diverge but instead depends logarithmically on the overall size of the system the langevin equation gives a poor approximation to the velocity autocorrelation function at both short and long times

a cartesian grid embedded boundary method for poisson s equation on irregular domains we present a numerical method for solving poisson s equation with variable coefficients and dirichlet boundary conditions on two dimensional regions the approach uses a finite volume discretization which embeds the domain in a regular cartesian grid we treat the solution as a cell centered quantity even when those centers are outside the domain cells that contain a portion of the domain boundary use conservative differencing of second order accurate fluxes on each cell volume the calculation of the boundary flux ensures that the conditioning of the matrix is relatively unaffected by small cell volumes this allows us to use multigrid iterations with a simple point relaxation strategy we have combined this with an adaptive mesh refinement amr procedure we provide evidence that the algorithm is second order accurate on various exact solutions and compare the adaptive and nonadaptive calculations

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

structural non linearities and the nature of the classic flutter instability abstract the paper presents an investigation of the subcritical supercritical nature of the flutter hopf bifurcation of a two degree of freedom with linear quadratic and cubic restoring forces under certain conditions the instability gives rise to stable limit cycle oscillations whereas unstable periodic orbits appear for other conditions providing a potentially dangerous non linear instability mechanism in regimes for which linear analyses predict stability the analytic approach employed herein allows for a more thorough investigation of the high dimensional parameter space than that afforded by others experimental and computational efforts counterexamples to previously held conjectures are readily found favorable comparisons are made to predictions based on less rigorous describing function or harmonic balance techniques

mathematical modeling and simulation of aquatic and aerial animal locomotion in this paper we investigate the locomotion of fish and birds by applying a new unsteady flexible wing theory that takes into account the strong nonlinear dynamics semi analytically we also make extensive comparative study between the new approach and the modified vortex blob method inspired from chorin s and krasny s work we first implement the modified vortex blob method for two examples and then discuss the numerical implementation of the nonlinear analytical mathematical model of wu we will demonstrate that wu s method can capture the nonlinear effects very well by applying it to some specific cases and by comparing with the experiments available in particular we apply wu s method to analyze wagner s result for a wing abruptly undergoing an increase in incidence angle moreover we study the vorticity generated by a wing in heaving pitching and bending motion in both cases we show that the new method can accurately represent the vortex structure behind a flying wing and its influence on the bound vortex sheet on the wing

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

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

scw method for solving the fractional integro differential equations with a weakly singular kernel in this paper based on the second chebyshev wavelets scw operational matrix of fractional order integration a numerical method for solving a class of fractional integro differential equations with a weakly singular kernel is proposed by using the operational matrix the fractional integro differential equations with weakly singular kernel are transformed into a system of algebraic equations the upper bound of the error of the second chebyshev wavelets expansion is investigated finally some numerical examples are shown to illustrate the efficiency and accuracy of the approach

boundary conditions for viscous vortex methods this paper presents a neumann type vorticity boundary condition for the vorticity formulation of the navier stokes equations the vorticity creation process at the boundary due to the no slip condition is expressed in terms of a vorticity flux the scheme is incorporated then into a lagrangian vortex blob method that uses a particle strength exchange algorithm for viscous diffusion the no slip condition is not enforced by the generation of new vortices at the boundary but instead by modifying the strength of the vortices in the vicinity of the boundary

vortex blob methods applied to interfacial motion we develop a boundary integral method for computing the motion of an interface separating two incompressible inviscid fluids the velocity integral is regularized so that the vortex sheet on the interface is replaced by a sum of blobs of vorticity the regularization allows control of physical instabilities we design a class of high order blob methods and analyze the errors numerical tests suggest that the blob size should be scaled with the local spacing of the interfacial markers for a vortex sheet in one fluid with a first order kernel we obtain a spiral roll up similar to krasny j comput phys 65 1986 292 but the higher order kernels lead to more detailed structure we verify the accuracy of the new method by computing a liquid gas interface with rayleigh taylor instability we then apply the method to the more difficult case of rayleigh taylor flow separating two fluids of positive density a case for which the regularization appears to be essential as found by kerr and tryggvason both j comput phys 76 1988 48 75 1988 253 we use a blob regularization in certain local terms in the evolution equations as well as in the velocity integral we find strong evidence that improved spatial resolution with fixed blob size leads to a converged regularized solution without numerical instabilities however it is not clear that there is a weak limit as the regularization is decreased

reconstruction of exponentially rate of convergence to legendre collocation solution of a class of fractional integro differential equations in this paper legendre collocation method an easy to use variant of the spectral methods for the numerical solution of a class of fractional integro differential equations fide s is researched in order to obtain high order accuracy for the approximation the integral term in the resulting equation is approximated by using legendre gauss quadrature formula an efficient convergence analysis of the proposed method is given and rate of convergence is established in the l 2 norm due to the fact that the solutions of fide s usually have a weak singularity at origin we use a variable transformation to change the original equation into a new equation with a smooth solution we prove that after this regularization technique numerical solution of the new equation by adopting the legendre collocation method has exponentially rate of convergence numerical results are presented which clarify the high accuracy of the proposed method

preconditioned multigrid methods for unsteady incompressible flows a highly efficient numerical approach based on multigrid and preconditioning methods is developed for modeling 3d steady and time dependent incompressible flows thek turbulence model is used to estimate the effects of turbulence the model equations are solved together with the n s equations in a strongly coupled way and acceleration techniques like the multigrid method are also used for the turbulence model equations for unsteady problems a dual time stepping procedure is adopted to satisfy the divergence free constraint and to obtain a time accurate solution to improve the performance of this approach for small physical time steps a modification to residual smoothing parameters is proposed the numerical algorithm and the turbulence model are validated first by calculating unsteady inviscid flow around an oscillating cylinder unsteady laminar flow past a circular cylinder and steady high reynolds number turbulent flow over a 6 1 prolate spheroid then the three dimensional time dependent turbulent flow over a spheroid when it is undergoing a pitch up maneuver is calculated and compared with experimental data

a cartesian grid method for solving the two dimensional streamfunction vorticity equations in irregular regions we describe a method for solving the two dimensional navier stokes equations in irregular physical domains our method is based on an underlying uniform cartesian grid and second order finite difference finite volume discretizations of the streamfunction vorticity equations geometry representing stationary solid obstacles in the flow domain is embedded in the cartesian grid and special discretizations near the embedded boundary ensure the accuracy of the solution in the cut cells along the embedded boundary we determine a distribution of vorticity sources needed to impose the no slip flow conditions this distribution appears as a right hand side term in the discretized fluid equations and so we can use fast solvers to solve the linear systems that arise to handle the advective terms we use the high resolution algorithms in clawpack we show that our stokes solver is second order accurate for steady state solutions and that our full navier stokes solver is between first and second order accurate and reproduces results from well studied benchmark problems in viscous fluid flow finally we demonstrate the robustness of our code on flow in a complex domain

the blob projection method for immersed boundary problems a new finite difference numerical method for modeling the interaction between flexible elastic membranes and an incompressible fluid in a two dimensional domain is presented the method differs from existing methods in the way the forces exerted by the membranes on the fluid are modeled these are described by a collection of regularized point forces and the velocity field they induce is computed directly on a regular cartesian grid via a smoothed dipole potential comparisons between this method and the immersed boundary method of peskin and mcqueen are presented the results show that the method proposed here preserves volumes better and has a higher order of convergence

transport coefficients for stochastic rotation dynamics in three dimensions explicit expressions for the transport coefficients of a recently introduced stochastic model for simulating fluctuating fluid dynamics are derived in three dimensions by means of green kubo relations and simple kinetic arguments the results are shown to be in excellent agreement with simulation data two collision rules are considered and their computational efficiency is compared

taylor polynomial solutions of nonlinear volterra fredholm integral equations the kanwal and liu method for the solution of fredholm integral equations is applied to certain nonlinear volterra fredholm integral equations in addition examples that illustrate the pertinent features of the method are presented and the results of study are discussed

eigenanalysis of unsteady flow about airfoils cascades and wings a general technique for constructing reduced order models of unsteady aerodynamic flows about two dimensional isolated airfoils cascades of airfoils and three dimensional wings is developed the starting point is a the domain computational model of the unsteady small disturbance flow for illustration purposes we apply the technique to an unsteady incompressible vortex lattice model the eigenmodes of the system which may be thought of as aerodynamic states are computed and subsequently used to construct computationally efficient reduced order models of the unsteady flowfield only a handful of the most dominant eigenmodes are retained in the reduced order model the effect of the remaining eigenmodes is included approximately using a static correction technique an important advantage of the present method is that once the eigenmode information has been computed reduced order models can be constructed for any number of arbitrary modes of airfold motion very inexpensively numerical examples are presented that demonstrate the accuracy and computational efficiency of the present method finally we show how the reduced order model may be incorporated into an aeroelastic flutter model

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

interpolation correction for collocation solutions of fredholm integro differential equations in this paper we discuss the collocation method for a large class of fredholm linear integro differential equations it will be shown that when a certain higher order interpolation operation is added to the collocation solution of this equation the new approximations will under suitable assumptions admit a multiterm error expansion in even powers of the step size h based on this expansion ideal multilevel correction results of this collocation solution are obtained

an efficient and stable lagrangian matrix approach to abel integral and integro differential equations abstract this article studies abel integral equations aies and singular integro differential equations sides and aims to develop two numerical schemes for them it also emphasises on the comparative analysis of both aies sides which is based on mainly two process namely gauss legendre roots as collocation node points and random node points over the domain 0 1 for generating interpolating basis functions ibf we used lagrangian interpolating polynomial and for orthonormal lagrangian basis functions olbf we used gram schmidt orthogonalization algorithm respectively firstly we introduced the function approximation by using generated ibf and olbf then established the error bounds of these approximations the constructed approximations by both the schemes convert the aies and sides into the system of algebraic equations we have also established error bounds stability and convergence analysis of the proposed schemes by considering several mild mathematical conditions moreover the stability of schemes is also established numerically finally the test functions with the support of graphs clearly show the reliability and computational efficiency of the proposed methods

discrete compatibility in finite difference methods for viscous incompressible fluid flow thom s vorticity condition for solving the incompressible navier stokes equations is generally known as a first order method since the local truncation error for the value of boundary vorticity is first order accurate in the present paper it is shown that convergence in the boundary vorticity is actually second order for steady problems and for time dependent problems whent 0 the result is proved by looking carefully at error expansions for the discretization which have been previously used to show second order convergence of interior vorticity numerical convergence studies confirm the results att 0 the computed boundary vorticity is first order accurate as predicted by the local truncation error using simple model problems for insight we predict that the size of the second order error term in the boundary condition blows up likec tast 0 this is confirmed by careful numerical experiments a similar phenomenon is observed for boundary vorticity computed using a primitive method based on the staggered marker and cell grid

an immersed boundary method based on discrete stream function formulation for two and three dimensional incompressible flows an immersed boundary method is proposed in the framework of discrete stream function formulation for incompressible flows in order to impose the non slip boundary condition the forcing term is determined implicitly by solving a linear system the number of unknowns of the linear system is the same as that of the lagrangian points representing the body surface thus the extra cost in force calculation is negligible if compared with that in the basic flow solver in order to handle three dimensional flows at moderate reynolds numbers a parallelized flow solver based on the present method is developed using the domain decomposition strategy to verify the accuracy of the immersed boundary method proposed in this work flow problems of different complexity decaying vortices flows over stationary and oscillating cylinders and a stationary sphere and flow over low aspect ratio flat plate are simulated and the results are in good agreement with the experimental or computational data in previously published literatures

numerical solution of fractional integro differential equations by collocation method abstract this paper deals with the numerical solution of fractional integro differential equations of the type d q y t p t y t f t 0 t k t s y s d s t i 0 1 by polynomial spline functions we derive a system of equations that characterizing the numerical solution some numerical examples are also provided to illustrated our results

computational algorithm for solving fredholm time fractional partial integrodifferential equations of dirichlet functions type with error estimates abstract numerical modeling of partial integrodifferential equations of fractional order shows interesting properties in various aspects of science which means increased attention to fractional calculus this paper is concerned with a feasible and accurate technique for obtaining numerical solutions for a class of partial integrodifferential equations of fractional order in hilbert space within appropriate kernel functions the algorithm relies on the reproducing kernel hilbert space method that provides the solutions in rapidly convergent series representations for the reproducing kernel based upon the fourier coefficients of orthogonalization process the caputo fractional derivatives are introduced to address these issues moreover the error estimate of the generated solutions is established as well as the convergence of the iterative method is investigated under some theoretical assumptions the superiority and applicability of the present technique is illustrated by handling linear and nonlinear numerical examples the outcomes obtained are compared with exact solutions and existing methods to confirm the effectiveness of the reproducing kernel method

vortex design problem in this investigation we propose a computational approach for the solution of optimal control problems for vortex systems with compactly supported vorticity the problem is formulated as a pde constrained optimization in which the solutions are found using a gradient based descent method recognizing such euler flows as free boundary problems the proposed approach relies on shape differentiation combined with adjoint analysis to determine cost functional gradients in explicit tracking of interfaces vortex boundaries this method offers an alternative to grid based techniques such as the level set methods and represents a natural optimization formulation for vortex problems computed using the contour dynamics technique we develop and validate this approach using the design of 2d equilibrium euler flows with finite area vortices as a model problem it is also discussed how the proposed methodology can be applied to euler flows featuring other vorticity distributions such as vortex sheets and to time dependent phenomena

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

an implicit forcing immersed boundary method for simulating viscous flows in irregular domains abstract we present a method for solving the incompressible navier stokes equations in irregular domains these equations are discretized using finite difference method in a uniform cartesian grid stationary rigid boundaries are embedded in the cartesian grid and singular forces are applied at the rigid boundaries to impose the no slip conditions the singular forces are then distributed to the nearby cartesian grid points using the immersed boundary method in the present work the singular forces are computed implicitly by solving a small system of equations at each time step this system of equations is derived from a second order projection method the main advantage of this method is that it imposes the no slip boundary condition exactly and avoids the need for small time step to maintain stability the ability of the method to simulate viscous flows in irregular domains is demonstrated by applying to 2 dimensional flows past a circular cylinder multiple rigid obstacles and 3 dimensional flow past a sphere

computational steering of cluster formation in brownian suspensions we simulate cluster formation of model colloidal particles interacting via dlvo derjaguin landau vervey overbeek potentials the interaction potentials can be related to experimental conditions defined by the ph value the salt concentration and the volume fraction of solid particles suspended in water the system shows different structural properties for different conditions including cluster formation a glass like repulsive structure or a liquid suspension typically many simulations are required to explore the parameter space in order to reduce the computational effort and data storage requirements we developed a steering approach to control a running simulation and to detect interesting transitions from one region in the configuration space to another the advantages of the steering approach and the restrictions of its applicability due to physical constraints are illustrated by several example cases

a new abel inversion by means of the integrals of an input function with noise abel s integral equations arise in many areas of natural science and engineering particularly in plasma diagnostics this paper proposes a new and effective approximation of the inversion of abel transform this algorithm can be simply implemented by symbolic computation and moreover an nth order approximation reduces to the exact solution when it is a polynomial in r 2 of degree less than or equal to n approximate abel inversion is expressed in terms of integrals of input measurement data so the suggested approach is stable for experimental data with random noise an error analysis of the approximation of abel inversion is given finally several test examples used frequently in plasma diagnostics are given to illustrate the effectiveness and stability of this method

combined immersed boundary finite difference methods for three dimensional complex flow simulations a second order accurate highly efficient method is developed for simulating unsteady three dimensional incompressible flows in complex geometries this is achieved by using boundary body forces that allow the imposition of the boundary conditions on a given surface not coinciding with the computational grid the governing equations therefore can be discretized and solved on a regular mesh thus retaining the advantages and the efficiency of the standard solution procedures two different forcings are tested showing that while the quality of the results is essentially the same in both cases the efficiency of the calculation strongly depends on the particular expression a major issue is the interpolation of the forcing over the grid that determines the accuracy of the scheme this ranges from zeroth order for the most commonly used interpolations up to second order for an ad hoc velocity interpolation the present scheme has been used to simulate several flows whose results have been validated by experiments and other results available in the literature finally in the last example we show the flow inside an ic piston cylinder assembly at high reynolds number to our knowledge this is the first example in which the immersed boundary technique is applied to a full three dimensional complex flow with moving boundaries and with a reynolds number high enough to require a subgrid scale turbulence model

numerical solution of a special type of integro differential equations the aim of this paper is to present a solutions to special type integro differential equation of the first and second kind practical examples are illustrated rashed method is very useful in computing a numerical solution of linear integro differential equation of first and second order this paper describe three methods to treat a special type of linear integro differential equation of the first and second kind

numerical solution of fractional integro differential equations by a hybrid collocation method fractional integro differential equations have been recently solved by many methods such as adomian decomposition method differential transform method collocation method and taylor expansion approach in this paper a hybrid collocation method is used which combines a non polynomial collocation used on the first subinterval and graded piecewise polynomial collocation used on the rest of the interval a theoretical analysis for the convergence order of the method is presented some numerical examples are given which confirm the theoretical results

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

approximate solution of fractional integro differential equations by taylor expansion method in this paper taylor expansion approach is presented for solving approximately a class of linear fractional integro differential equations including those of fredholm and of volterra types by means of the mth order taylor expansion of the unknown function at an arbitrary point the linear fractional integro differential equation can be converted approximately to a system of equations for the unknown function itself and its m derivatives under initial conditions this method gives a simple and closed form solution for a linear fractional integro differential equation in addition illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method

the rapid evaluation of volume integrals of potential theory on general regions abstract we present a new method for the rapid high order accurate evaluation of certain volume integrals in potential theory on general irregular regions the kernels of the integrals are either a fundamental solution or a linear combination of the derivatives of a fundamental solution of a second order linear elliptic differential equation instead of using a standard quadrature formula or the exact evaluation of any integral the methods rely on rapid methods of solving the differential equation which the kernel is the solution of therefore the number of operations needed to evaluate the volume integral is essentially equal to the number of operations needed to solve the differential equation on a rectangular region with a regular grid and the method requires no evaluation of the kernel