analysis of an explicit and matrix free fractional step method for incompressible flows in this paper we present the analysis of an explicit and matrix free fractional step method for incompressible flows the presented method can be employed in either conservation or non conservation form the stabilization convergence and conservation aspects of the presented method are discussed a procedure for eliminating the first order error in time introduced by the split is proposed some benchmark steady and unsteady state examples are presented to demonstrate the proposed new aspects of the matrix free scheme

fast adaptive 2d vortex methods we present a new approach to vortex methods for the 2d euler equations we obtain long time high order accuracy at almost optimal cost by using three tools fast adaptive quadrature rules a free lagrangian formulation and a useful new analysis of the consistency error our error analysis halves the order of differentiability required of the flow and suggests an efficient new balance of smoothing parameters which works well with fast summation schemes numerical experiments with our methods confirm our theoretical predictions and display excellent long time accuracy

multi domain spectral collocation method for variable order nonlinear fractional differential equations abstract spectral and spectral element methods using galerkin type formulations are efficient for solving linear fractional pdes fpdes of constant order but are not efficient in solving nonlinear fpdes and cannot handle fpdes with variable order in this paper we present a multi domain spectral collocation method that addresses these limitations we consider fpdes in the riemann liouville sense and employ jacobi lagrangian interpolants to represent the solution in each element we provide variable order differentiation formulas which can be computed efficiently for the multi domain discretization taking into account the nonlocal interactions we enforce the interface continuity conditions by matching the solution values at the element boundaries via the lagrangian interpolants and in addition we minimize the jump in integer fluxes using a penalty method we analyze numerically the effect of the penalty parameter on the condition number of the global differentiation matrix and on the stability and convergence of the penalty collocation scheme we demonstrate the effectiveness of the new method for the fractional helmholtz equation of constant and variable order using h p refinement for different values of the penalty parameter we also solve the fractional burgers equation with constant and variable order and compare with solutions obtained with a single domain spectral collocation method

an improved unsteady unstructured artificial compressibility finite volume scheme for viscous incompressible flows part i theory and implementation a robust artificial compressibility scheme has been developed for modelling laminar steady state and transient incompressible flows over a wide range of reynolds and rayleigh numbers artificial compressibility is applied in a consistant manner resulting in a system of preconditioned governing equations a locally generalized preconditioner is introduced designed to be robust and offer good convergence rates free artificial compressibility parameters in the equations are automated to allow ease of use while facilitating improved or comparable convergence rates as compared with the standard artificial compressibility scheme memory efficiency is achieved through a multistage pseudo time explicit time marching solution procedure a node centred dual cell edge based finite volume discretization technique suitable for unstructured grids is used due to its computational efficiency and high resolution spatial accuracy in the interest of computational efficiency and ease of implementation stabilization is achieved via a scalar valued artificial dissipation scheme temporal accuracy is facilitated by employing a second order accurate dual time stepping method in this part of the paper the theory and implementation details are discussed in part 11 the scheme will be applied to a number of example problems to solve flows over a wide range of reynolds and rayleigh numbers

a generalized spectral collocation method with tunable accuracy for variable order fractional differential equations we generalize existing jacobi gauss lobatto collocation methods for variable order fractional differential equations using a singular approximation basis in terms of weighted jacobi polynomials of the form 1 pm x mu p j a b x where mu 1 in order to derive the differentiation matrices of the variable order fractional derivatives we develop a three term recurrence relation for both integrals and derivatives of these weighted jacobi polynomials hence extending the three term recurrence relationship of jacobi polynomials the new spectral collocation method is applied to solve fractional ordinary and partial differential equations with endpoint singularities we demonstrate that the singular basis enhances greatly the accuracy of the numerical solution by properly tuning the parameter mu even for cases where we do not know explicitly the form of singularity in the solution at the boundaries

wellposedness and regularity of the variable order time fractional diffusion equations abstract it was proved that the solution to a time fractional partial differential equation lacks regularity at the initial time t 0 which is often inconsistent with the physical problem being modeled even though the nature of the physical problem away from t 0 is well captured by a time fractional partial differential equation we argue that the fundamental reason why this phenomenon occurs lies in the incompatibility between the nonlocality of the time fractional partial differential equations and the locality of the classical initial condition at t 0 we show that the variable order time fractional partial differential equations in which the order varies in time as t goes to 0 to accommodate the impact of the local initial condition at time t 0 turn out to be a natural candidate to eliminate the nonphysical singularity of the solutions to the constant order time fractional partial differential equations and open up opportunities for modeling multiphysics phenomena from nonlocal to local dynamics and vice versa we prove the wellposedness of a variable order linear time fractional diffusion partial differential equation in multiple space dimensions we also prove that the regularity of its solution depends on the behavior of the variable order and its derivatives at time t 0 in addition to the usual smoothness assumptions more precisely we prove that its solution has full regularity like its integer order analogue if the variable order has an integer limit at t 0 or exhibits singular behaviors at t 0 like in the case of the constant order time fractional partial differential equations if the variable order has a non integer value at time t 0

non linear free vibrations of a rotating flexible arm abstract the non linear moderately large amplitude flexural free vibrations of an arm clamped with a setting angle to a rigid rotating hub are studied the shear deformation and rotary inertia effects are assumed to be negligible but account is taken of axial inertia non linear curvature and the inextensibility condition the lagrangian approach in conjunction with the assumed modes method assuming constant hub rotation speed is used in a consistent manner to obtain the third order non linear uni modal temporal problem because of the strength of the non linearities in the temporal problem which includes elastic and inertial geometric stiffening as well as inertial softening terms a time transformation method is employed to obtain an approximate solution to the frequency amplitude relation of arm free oscillation results in non dimensional form are presented graphically for the effect of hub rotation speed blade setting angle and hub radius on the variation of the natural frequency with vibration amplitude

the characteristic based split cbs scheme a unified approach to fluid dynamics this paper presents a comprehensive overview of the characteristic based methods and characteristic based split cbs scheme the practical difficulties of employing the original characteristic schemes are discussed the important features of the cbs scheme are brought out by studying several problems of compressible and incompressible flows all special consideration necessary for solving these problems are thoroughly discussed the cbs scheme is presented in such a way that any interested researcher should be able to develop a code using the information provided several invicid and viscous flow examples are also provided to demonstrate the unified cbs approach for sample two dimensional codes input files and instructions the readers are referred to www nithiarasu co uk copyright 2006 john wiley sons ltd

a cut cell non conforming cartesian mesh method for compressible and incompressible flow this paper details a multigrid accelerated cut cell non conforming cartesian mesh methodology for the modelling of inviscid compressible and incompressible flow this is done via a single equation set that describes sub trans and supersonic flows cut cell technology is developed to furnish body fitted meshes with an overlapping mesh as starting point and in a manner which is insensitive to surface definition inconsistencies spatial discretization is effected via an edge based vertex centred finite volume method an alternative dual mesh construction strategy similar to the cell centred method is developed incompressibility is dealt with via an artificial compressibility algorithm and stabilization achieved with artificial dissipation in compressible flow shocks are captured via pressure switch activated upwinding the solution process is accelerated with full approximation storage fas multigrid where coarse meshes are generated automatically via a volume agglomeration methodology this is the first time that the proposed discretization and solution methods are employed to solve a single compressible incompressible equation set on cut cell cartesian meshes the developed technology is validated by numerical experiments the standard discretization and alternative methods were found equivalent in accuracy and computational cost the multigrid implementation achieved decreases in cpu time of up to one order of magnitude copyright 2007 john wiley sons ltd

explicit and semi implicit cbs procedures for incompressible viscous flows this paper presents a numerical comparison of two different numerical procedures based on the characteristic based split cbs algorithm for the solution of incompressible flows the first procedure is based on the original semi implicit version of the cbs introduced in the early days of the scheme development the second was proposed more recently in order to have a fully explicit version of the scheme and is based on the introduction of an artificial compressibility type of formulation the interest in fully explicit schemes has increased because of the easy and efficient parallelization the main differences between the two versions of the schemes are highlighted in this work and a numerical comparison is presented on the basis of several benchmark solutions both steady and unsteady flows are considered with the explicit version of the scheme a dual time stepping procedure is needed to recover unsteady solutions copyright 2006 john wiley sons ltd

a dual time method for two dimensional unsteady incompressible flow calculations a method for computing unsteady incompressible viscous flows on moving or deforming meshes is described it uses a well established time marching finite volume flow solver developed for steady compressible flows past rigid bodies time marching methods cannot be applied directly to incompressible flows because the governing equations are not hyperbolic such methods can be extended to steady incompressible flows using an artificial compressibility scheme a time accurate scheme for unsteady incompressible flows is achieved by using an implicit real time discretization and a dual time approach which uses a technique similar to the artificial compressibility scheme results are presented for test cases on both fixed and deforming meshes experimental numerical and theoretical data have been included for comparison where available and reasonable agreement has been achieved 1998 john wiley sons ltd

global two dimensional stability of the staggered cavity flow with an hoc approach in this paper we have carried out a global stability analysis of the flow inside a two sided staggered lid driven cavity for both the parallel and anti parallel motions of the lids a fourth order spatially accurate and second order temporally accurate compact scheme is used to discretize the two dimensional navier stokes equations governing the flow as well as the perturbed equations leading to the generalized eigenvalue problem estimates of the critical parameters are found for several grids and extensive grid convergence study is performed to accurately identify them computation is performed for a wide range of reynolds numbers re on either side of the critical values in the range 50 re 12 000 for flows below the critical reynolds number re c our numerical results are compared with established steady state results and an excellent agreement is obtained in all the cases for reynolds numbers above re c phase plane and spectral density analysis confirmed the existence of periodic and chaotic flow patterns

a numerical study of the two dimensional navier stokes equations in vorticity velocity variables abstract solution methods for compact finite difference schemes are applied to the vorticity velocity form of the two dimensional unsteady navier stokes equations numerical experiments for stagnation point and driven cavity flows are described

an efficient artificial compressibility ac scheme based on the characteristic based split cbs method for incompressible flows in this paper an artificial compressibility scheme using the finite element method is introduced 2002 zienkiewicz silver medal and prize winning paper the multi purpose cbs scheme is implemented in its fully explicit form to solve incompressible fluid dynamics problems it is important to note that the scheme developed here includes split and velocity correction the proposed method takes advantage of good features from both velocity correction and standard artificial compressibility schemes unlike many other artificial compressibility schemes the proposed one works on a variety of grids and gives results for a wide range of reynold s numbers the paper presents some bench mark two and three dimensional steady and unsteady incompressible flow solutions obtained from the proposed scheme copyright 2003 john wiley sons ltd

the natural vibration of a conical shell with an annular end plate in this paper natural frequencies and mode shapes for a conical shell with an annular end plate or a round end plate are investigated in detail by combining the vibration theory with the transfer matrix method the governing equations of vibration for this structure are expressed in terms of a matrix differential equation and a novel recurrence formula method that can close in on analytical solution of transfer matrix is presented once the transfer matrix of single component has been determined and the product of each component matrix and the joining matrix can obtain entire structure matrix the frequency equations and mode shape functions are represented in terms of the elements of the structural transfer matrices the 3d finite element numerical simulations have verified the present formulas of natural frequencies and mode shapes the conclusions show that the transfer matrix method can accurately obtain natural vibration characteristics of the conical shell with an annular end plate

second order numerical methods for multi term fractional differential equations smooth and non smooth solutions abstract starting with the asymptotic expansion of the error equation of the shifted grunwald letnikov formula we derive a new modified weighted shifted grunwald letnikov wsgl formula by introducing appropriate correction terms we then apply one special case of the modified wsgl formula to solve multi term fractional ordinary and partial differential equations and we prove the linear stability and second order convergence for both smooth and non smooth solutions when the regularity of the solutions is known we show theoretically and numerically that numerical solutions with good accuracy can be obtained with only a few correction terms moreover the correction terms can be tuned according to the fractional derivative orders without explicitly knowing the analytical solutions numerical simulations verify the theoretical results and demonstrate that the new wsgl formula leads to better performance compared to other known numerical approximations with similar resolution

vibration analysis of a rotating timoshenko beam the governing equations for linear vibration of a rotating timoshenko beam are derived by the d alembert principle and the virtual work principle in order to capture all inertia e ect and coupling between extensional and exural deformation the consistent linearization of the fully geometrically non linear beam theory is used the e ect of coriolis force on the natural frequency of the rotating beam is considered a method based on the power series solution is proposed to solve the natural frequency of the rotating timoshenko beam numerical examples are studied to verify the accuracy of the proposed method and to investigate the e ect of coriolis force on the natural frequency of rotating beams with di erent angular velocity hub radius and slenderness ratio 2001 academic press

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

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

cavity flow dynamics at higher reynolds number and higher aspect ratio abstract a primitive variable unsteady viscous incompressible navier stokes flow simulation in a higher aspect ratio a depth width 2 driven cavity at a higher reynold s number re 10 000 exhibits dynamical features not apparent in previous studies of stationary solutions or at lower reynold s number or in the more usual unit cavity investigations the initial dynamics e g between t 0 sec and t 70 sec reveals transient bifurcations between states including two three and four interacting vortices in an anterior separation region following this there is an intermediate interval e g between t 70 sec and t 140 sec characterized by a vertical oscillation of the primary vortex related to the general activities of the secondary features of the flow the long time behavior e g between t 140 seconds and t 360 seconds is one of qualitative smoothing of all secondary features except a persistent oscillation indicating a hopf bifurcation

vibration analysis of rotating twisted and open conical shells abstract based on a non linear strain displacement relationship of a non rotating twisted and open conical shell on thin shell theory a numerical method for free vibration of a rotating twisted and open conical shell is presented by the energy method where the effect of rotation is considered as initial deformation and initial stress resultants which are obtained by the principle of virtual work for steady deformation due to rotation then an energy equilibrium of equation for vibration of a twisted and open conical shell with the initial conditions is also given by the principle of virtual work in the two numerical processes the rayleigh ritz procedure is used and the two in plane and a transverse displacement functions are assumed to be algebraic polynomials in two elements the effects of characteristic parameters with respect to rotation and geometry such as an angular velocity and a radius of rotating disc a setting angle a twist angle curvature and a tapered ratio of cross section on vibration performance of rotating twisted and open conical shells are studied by the present method

an improved unsteady unstructured artificial compressibility finite volume scheme for viscous incompressible flows part ii application in part i of this paper a preconditioned artificial compressibility scheme was developed for modelling laminar steady state and transient incompressible flows for a wide range of reynolds and rayleigh numbers in this part several examples of laminar incompressible problems are solved and discussed the influence of various ac parameters on robustness and convergence rates are assessed for a complex category of problems it is shown that the scheme developed in part i is an accurate robust and easy to use method for solving incompressible laminar flow problems over a wide range of flow regimes copyright 2002 john wiley sons ltd

dsc solution for flow in a staggered double lid driven cavity a benchmark quality solution is presented for flow in a staggered double lid driven cavity obtained by using the wavelet based discrete singular convolution dsc the proposed wavelet based algorithm combines local methods flexibility and global methods accuracy and hence is a promising approach for achieving the high accuracy solution of the navier stokes equations block structured grids with pseudo overlapping subdomains are employed in the present simulation a third order runge kutta scheme is used for the temporal discretization quantitative results are presented apart from the qualitative fluid flow patterns the prevalence of rich features of flow morphology such as two primary vortex patterns merged single primary vortex patterns and secondary eddies makes this problem very attractive and interesting the problem is quite challenging for the possible existence of numerically induced asymmetric flow patterns and elliptic instability important computational issues like consistence convergence and reliability of the numerical scheme are examined the dsc algorithm is tested on the single lid driven cavity flow and the taylor problem with a closed form solution the double lid driven cavity simulations are cross validated with the standard second order finite volume method copyright 2003 john wiley sons ltd

steady and unsteady incompressible flow in a double driven cavity using the artificial compressibility ac based characteristic based split cbs scheme in this paper the explicit characteristic based split cbs scheme has been employed to solve both steady and unsteady flows inside a non rectangular double driven cavity this problem is recently suggested as a benchmark problem for incompressible flows both unstructured and structured meshes have been employed in the present study to make sure that the predicted results are as close to reality as possible the results obtained show the existence of steady state at lower reynolds numbers 1000 and transient states at higher reynolds numbers the flow approaches a turbulent state as the reynolds number is increased to 10 000 copyright 2005 john wiley sons ltd

adaptive finite element method for fractional differential equations using hierarchical matrices abstract a robust and fast solver for the fractional differential equation fdes involving the riesz fractional derivative is developed using an adaptive finite element method it is based on the utilization of hierarchical matrices h matrices for the representation of the stiffness matrix resulting from the finite element discretization of the fdes we employ a geometric multigrid method for the solution of the algebraic system of equations we combine it with an adaptive algorithm based on a posteriori error estimation a posteriori error estimation based adaptive algorithm is used to deal with general type singularities arising in the solution of the fdes through various test examples we demonstrate the efficiency of the method and the high accuracy of the numerical solution even in the presence of singularities the proposed technique has been verified effectively through fundamental examples including riesz left right riemann liouville fractional derivative and furthermore it can be readily extended to more general fractional differential equations with different boundary conditions and low order terms

semi implicit and explicit finite element schemes for coupled fluid thermal problems a comparative investigation based on a series of numerical tests of two purely explicit and one semi implicit finite element methods used for incompressible flow computation is presented the segregated approach is followed and the equations of motion are considered sequentially the fundamental concepts and characteristics of the formulations and the solution methodology used are described in technical detail various modifications to chorin s projection algorithm are investigated particularly with respect to their effects on stability and accuracy the stability of the semi implicit method is shown to be less restrictive when compared to the explicit methods as the reynolds number increases at large time steps the artificial viscosity is also reduced and higher accuracy is obtained the performance of the methods discussed in this paper is illustrated by the numerical solutions obtained for the cavity flow and flow past a rearward facing step problems at high reynolds numbers and free convection flow problem at high rayleigh numbers it is shown that the semi implicit method needs fewer iterations than the explicit methods and the accuracy of the present methods is guaranteed by comparison with the existing methods

a novel vector potential formulation of 3d navier stokes equations with through flow boundaries by a local meshless method an alternative vector potential formulation is used to solve the navier stokes n s equations in 3d incompressible viscous flow problems with and without through flow boundaries difficulties of the vector potential formulation include the implementation of boundary conditions for through flow boundaries and the numerical treatment of fourth order partial differential equations the advantages on the other hand are the automatic satisfaction of the continuity equation and pressure is decoupled from the velocity the objective of this paper is to introduce the appropriate gauge and boundary conditions on the vector potential formulation by a localized meshless method to handle the divergence free property a coulomb gauge condition is enforced on the vector potential to ensure its existence and uniqueness mathematically we further improve the algorithm to through flow problems for the boundary conditions of vector potential by introducing the concept of stokes theorem based on this innovation there is no need to include an additional variable to tackle the through flow fields this process will greatly simplify the imposition of boundary conditions by the vector potential approach under certain conditions the coupled fourth order partial differential equations can be easily solved by using this meshless local differential quadrature ldq method due to the ldq capability to deal with the high order differential equations this algorithm is very attractive to solve this fourth order vector potential formulation for the n s equations as comparing to the conventional numerical schemes such as finite element or finite difference methods the proposed vector potential formulation is simpler and has improved accuracy and efficiency compared to other pressure free or pressure coupled algorithms this investigation can be regarded as the first complete study to obtain the n s solutions by vector potential formulation through a ldq method two classic 3d benchmark problems lid driven cavity and backward facing step flows are numerically solved to examine the feasibility of the improved algorithm results show the flexibility of the proposed vector potential formulation for the 3d navier stokes equations

a fractional phase field model for two phase flows with tunable sharpness algorithms and simulations abstract we develop a fractional extension of a mass conserving allen cahn phase field model that describes the mixture of two incompressible fluids the fractional order controls the sharpness of the interface which is typically diffusive in integer order phase field models the model is derived based on an energy variational formulation an additional constraint is employed to make the allen cahn formulation mass conserving and comparable to the cahn hilliard formulation but at reduced cost the spatial discretization is based on a petrov galerkin spectral method whereas the temporal discretization is based on a stabilized adi scheme both for the phase field equation and for the navier stokes equation we demonstrate the spectral accuracy of the method with fabricated smooth solutions and also the ability to control the interface thickness between two fluids with different viscosity and density in simulations of two phase flow in a pipe and of a rising bubble we also demonstrate that using a formulation with variable fractional order we can deal simultaneously with both erroneous boundary effects and sharpening of the interface at no extra resolution

a spectral method of exponential convergence for singular solutions of the diffusion equation with general two sided fractional derivative an open problem in the numerical solution of fractional partial differential equations fpdes is how to obtain high order accuracy for singular solutions even for smooth right hand sides solutions of fpdes are singular here we consider the one dimensional diffusion equation with general two sided fractional derivative characterized by a parameter p in 0 1 for p 1 2 we recover the riesz fractional derivative while for p 1 0 we obtain the one sided fractional derivative we employ a petrov galerkin projection in a properly weighted sobolev space with two sided jacobi polyfracnomials as basis and test functions in particular we derive these two sided jacobi polyfractonomials as eigenfunctions of a sturm liouville problem with weights uniquely determined by the parameter p we provide a rigorous analysis and obtain optimal error estimates that depend on the regularity of the forcing term i e for smooth data corresponding to singular solutions we obtain exponential convergence wh

preconditioned methods for solving the incompressible low speed compressible equations abstract acceleration methods are presented for solving the steady state incompressible equations these systems are preconditioned by introducing artificial time derivatives which allow for a faster convergence to the steady state we also consider the compressible equations in conservation form with slow flow two arbitrary functions and are introduced in the general preconditioning an analysis of this system is presented and an optimal value for is determined given a constant it is further sown that the resultant incompressible equations form a symmetric hyperbolic system and so are well posed several generalizations to the compressible equations are presented which extend previous results

a tunable finite difference method for fractional differential equations with non smooth solutions in this work a finite difference method of tunable accuracy for fractional differential equations fdes with end point singularities is developed modified weighted shifted grunwald letnikov wsgl formulas are proposed to approximate the left and right riemann liouville fractional operators which show better accuracy than the original wsgl formulas due to the use of the correction terms finite difference schemes are constructed to solve two fractional boundary value problems and a space fractional allen cahn equation even if the singularity of the considered fdes is unknown satisfactory numerical solutions can still be obtained by suitably tuning the correction terms various numerical examples are presented to verify the effectiveness of the present method and comparisons with other known methods are also made that demonstrate higher accuracy of the current method

finite element vibration analysis of rotating timoshenko beams abstract the stiffness and mass matrices of a rotating twisted and tapered beam element are derived the angle of twist breadth and depth are assumed to vary linearly along the length of beam the effects of shear deformation and rotary inertia are also considered in deriving the elemental matrices the first four natural frequencies and mode shapes in bending bending mode are calculated for cantilever beams the effects of twist offset speed of rotation and variation of depth and breadth taper ratios are studied

numerical algorithm for two dimensional fractional stokes first problem for a heated generalized second grade fluid with smooth and non smooth solution abstract in this paper we consider the numerical solution of the two dimensional fractional stokes first problem for a heated generalized second grade fluid the proposed method is based on the l1 finite difference scheme for the temporal direction while the legendre spectral method for the spatial direction numerical stability and convergence of the method are rigorously established at the mean time the case of non smooth solution is considered by adding some correction terms numerical experiments are included which verify the theoretical predictions

comparison of pressure based and artificial compressibility methods for solving 3d steady incompressible viscous flows pressure based and artificial compressibility methods for calculating three dimensional steady incompressible viscous flows are compared in this work each method is applied to the prediction of three dimensional laminar flows in strongly curved ducts of square and circular cross sections numerical predictions from each method are compared with available experimental data and previously reported predictions using a multigrid numerical method the accuracy grid independence convergence behavior and computational efficiency of each method are critically examined hence some of the merits and demerits of each method are revealed in this study

the accuracy and stability of cn wsgd schemes for space fractional diffusion equation abstract in this paper we discuss the accuracy and the stability of the cn wsgd when it is applied to the initial boundary value problem for a space fractional diffusion equation in the cn wsgd scheme the weighted and shifted grunwald difference wsgd scheme is used to discretize the space fractional derivative and the crank nicolson cn scheme is used to discretize the temporal derivative there is a free parameter in a cn wsgd scheme that affects the stability and the accuracy of the scheme in general a cn wsgd scheme has temporally and spatially second order accuracy and there exists a scheme that has spatially third order accuracy however the third order accuracy scheme is not always stable in this paper we present and prove some theoretical results old and new on the stability of several cn wsgd schemes and consider choosing a value for the parameter such that the corresponding cn wsgd scheme is unconditioned stable and has optimal upper bound for the accuracy numerical results are presented to verify the theoretical results

hopf bifurcation in the driven cavity abstract incompressible two dimensional calculations are reported for the impulsively started lid driven cavity with aspect ratio two the algorithm is based on the time dependent stream function equation with a crank nicolson differencing scheme for the diffusion terms and with an adams bashforth scheme for the convection terms a multigrid method is used to solve the linear implicit equations at each time step periodic asymptotic solutions have been found for re 10000 and for re 5000 the re 5000 results are validated by grid refinement calculations the solutions are shown to be precisely periodic and care is taken to demonstrate that asymptotic states have been reached a discussion is included about the indicators that are used to show that an asymptotic state has been reached and to show that the asymptotic state is indeed periodic

a two grid mmoc finite element method for nonlinear variable order time fractional mobile immobile advection diffusion equations abstract a fully discrete two grid modified method of characteristics mmoc scheme is proposed for nonlinear variable order time fractional advection diffusion equations in two space dimensions the mmoc is used to handle the advection dominated transport and the two grid method is designed for efficiently solving the resulting nonlinear system optimal l 2 error estimates are derived for both the mmoc scheme and the corresponding two grid mmoc scheme numerical experiments are presented to demonstrate the accuracy and the efficiency of the proposed method

modeling low reynolds number incompressible flows using sph the method of smoothed particle hydrodynamics sph is extended to model incompressible flows of low reynolds number for such flows modification of the standard sph formalism is required to minimize errors associated with the use of a quasi incompressible equation of state treatment of viscosity state equation kernel interpolation and boundary conditions are described simulations using the method show close agreement with series solutions for couette and poiseuille flows furthermore comparison with finite element solutions for flow past a regular lattice of cylinders shows close agreement for the velocity and pressure fields the sph results exhibit small pressure fluctuations near curved boundaries further improvements to the boundary conditions may be possible which will reduce these errors a similar method to that used here may permit the simulation of other flows at low reynolds numbers using sph further development will be needed for cases involving free surfaces or substantially different equations of state