symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients we present a symbolic computation procedure for deriving various high order compact difference approximation schemes for certain three dimensional linear elliptic partial differential equations with variable coefficients based on the maple software package we approximate the leading terms in the truncation error of the taylor series expansion of the governing equation and obtain a 19 point fourth order compact difference scheme for a general linear elliptic partial differential equation a test problem is solved numerically to validate the derived fourth order compact difference scheme this symbolic derivation method is simple and can be easily used to derive high order difference approximation schemes for other similar linear elliptic partial differential equations

a new coupled approach high accuracy numerical method for the solution of 3d non linear biharmonic equations in this paper we derive a new fourth order finite difference approximation based on arithmetic average discretization for the solution of three dimensional non linear biharmonic partial differential equations on a 19 point compact stencil using coupled approach the numerical solutions of unknown variable u x y z and its laplacian 2u are obtained at each internal grid point the resulting stencil algorithm is presented which can be used to solve many physical problems the proposed method allows us to use the dirichlet boundary conditions directly and there is no need to discretize the derivative boundary conditions near the boundary we also show that special treatment is required to handle the boundary conditions the new method is tested on three problems and the results are compared with the corresponding second order approximation which we also discuss using coupled approach

high order compact scheme with multigrid local mesh refinement procedure for convection diffusion problems we derive a new fourth order compact finite difference scheme which allows different meshsize in different coordinate directions for the two dimensional convection diffusion equation a multilevel local mesh refinement strategy is used to deal with the local singularity problem a corresponding multilevel multigrid method is designed to solve the resulting sparse linear system numerical experiments are conducted to show that the local mesh refinement strategy works well with the high order compact discretization scheme to recover high order accuracy for the computed solution our solution method is also shown to be effective and robust with respect to the level of mesh refinement and the anisotropy of the problems

solution of high order compact discretized 3d elliptic partial differential equations by an accelerated multigrid method abstract a robust and efficient numerical solver for three dimensional linear elliptic partial differential equations without cross derivative terms but variable coefficients is presented the proposed algorithm is based on a general meshsize fourth order compact finite difference scheme and an accelerated geometric multigrid solver the former not only increases the accuracy of the discretization but also does not affect the bandwidth of the common second order finite difference scheme which is relevant for distributed memory parallel implementations of the multigrid preconditioner the acceleration technique on the other hand allows for a dramatic reduction of the number of cycles and smoothing steps required to achieve convergence furthermore it is found that depending on the coefficients of the elliptic partial differential equation the multigrid preconditioned solver can be the only way to obtain a reliable numerical solution or attain convergence

high order compact computation and nonuniform grids for streamfunction vorticity equations abstract fourth order compact difference schemes for steady streamfunction vorticity formulation of 2d incompressible navier stokes equations on nonuniform grids are derived using maple software package especially we deduce fourth order compact difference scheme of the first partial derivative terms in order to resolve boundary layers grid transformation techniques are used which maps a nonuniform grid onto a uniform one for use with the fourth order compact difference scheme a krylov subspace iterative method with an ilut preconditioning technique is employed to solve the resulting linear system the proposed high accuracy computation method is applied to two model problems computational results are compared with results computed by other schemes in the literature

a higher order compact adi method with monotone iterative procedure for systems of reaction diffusion equations this paper is concerned with an existing compact finite difference adi method published in the paper by liao et al 2002 3 for solving systems of two dimensional reaction diffusion equations with nonlinear reaction terms this method has an accuracy of fourth order in space and second order in time the existence and uniqueness of its solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear reaction terms the convergence of the finite difference solution to the continuous solution is proved an efficient monotone iterative algorithm is presented for solving the resulting discrete system and some techniques for the construction of upper and lower solutions are discussed an application using a model problem gives numerical results that demonstrate the high efficiency and advantages of the method

cheeger muller theorem on manifolds with cusps we prove equality between the renormalized ray singer analytic torsion and the intersection r torsion on a witt manifold with cusps up to an error term determined explicitly by the betti numbers of the cross section of the cusp and the intersection r torsion of a model cone in the first step of the proof we compute explicitly the renormalized ray singer analytic torsion of a model cusp in general dimension and without the witt condition in the second step we establish a gluing formula for renormalized ray singer analytic torsion on a general class of non compact manifolds in any dimension that includes witt manifolds with cusps but also scattering manifolds with asymptotically conical ends in the final step a cheeger muller theorem on cusps follows by a combination of the previous explicit computation and the gluing formula

two level compact implicit schemes for three dimensional parabolic problems we derive a class of two level high order implicit finite difference schemes for solving three dimensional parabolic problems with mixed derivatives the schemes are fourth order accurate in space and second or lower order accurate in time depending on the choice of a weighted average parameter m numerical results with m 0 5 are presented to confirm the high accuracy of the derived scheme and to compare it with the standard second order central difference scheme it is shown that the improvement in accuracy does not come at a higher cost of computation and storage since it is possible to choose the grid parameters so that the present scheme requires less work and memory and gives more accuracy than the standard central difference scheme

minimizing interference in satellite communications using transiently chaotic neural networks the frequency assignment problem fap in satellite communications is solved with transiently chaotic neural networks tcnn the objective of this optimization problem is to minimize cochannel interference between two satellite systems by rearranging the frequency assignments for an n carrier m segment fap problem we construct a tcnn consisting of nxm neurons the performance of the tcnn is demonstrated through solving a set of benchmark problems where the tcnn finds comparative if not better solutions as compared to the existing algorithms

sixth order compact scheme combined with multigrid method and extrapolation technique for 2d poisson equation we develop a sixth order finite difference discretization strategy to solve the two dimensional poisson equation which is based on the fourth order compact discretization multigrid method richardson extrapolation technique and an operator based interpolation scheme we use multigrid v cycle procedure to build our multiscale multigrid algorithm which is similar to the full multigrid method fmg the multigrid computation yields fourth order accurate solution on both the fine grid and the coarse grid a sixth order accurate coarse grid solution is computed by using the richardson extrapolation technique then we apply our operator based interpolation scheme to compute sixth order accurate solution on the fine grid numerical experiments are conducted to show the solution accuracy and the computational efficiency of our new method compared to sun zhang s sixth order richardson extrapolation compact rec discretization strategy using alternating direction implicit adi method and the standard fourth order compact difference foc scheme using a multigrid method

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

a study of a numerical solution of the steady two dimensions navier stokes equations in a constricted channel problem by a compact fourth order method we present a numerical solution for the steady 2d navier stokes equations using a fourth order compact type method the geometry of the problem is a constricted symmetric channel where the boundary can be varied via a parameter from a smooth constriction to one possessing a very sharp but smooth corner allowing us to analyse the behaviour of the errors when the solution is smooth or near singular the set of non linear equations is solved by the newton method results have been obtained for reynolds number up to 500 estimates of the errors incurred have shown that the results are accurate and better than those of the corresponding second order method

multigrid method and fourth order compact difference discretization scheme with unequal meshsizes for 3d poisson equation a fourth order compact difference discretization scheme with unequal meshsizes in different coordinate directions is employed to solve a three dimensional 3d poisson equation on a cubic domain two multgrid methods are developed to solve the resulting sparse linear systems one is to use the full coarsening multigrid method with plane gauss seidel relaxation which uses line gauss seidel relaxation to compute each planewise solution the other is to construct a partial semi coarsening multigrid method with the traditional point or plane gauss seidel relaxations numerical experiments are conducted to test the computed accuracy of the fourth order compact difference scheme and the computational efficiency of the multigrid methods with the fourth order compact difference scheme

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

compact exponential scheme for the time fractional convection diffusion reaction equation with variable coefficients high order compact exponential finite difference scheme for solving the time fractional convection diffusion reaction equation with variable coefficients is considered in this paper the convection diffusion and reaction coefficients can depend on both the spatial and temporal variables we begin with the one dimensional problem and after transforming the original equation to one with diffusion coefficient unity the new equation is discretized by a compact exponential finite difference scheme with a high order approximation for the caputo time derivative we prove the solvability of this fully discrete implicit scheme and analyze its local truncation error for the fractional equation with constant coefficients we use fourier method to prove the stability and utilize matrix analysis as a tool for the error estimate then we discuss the two dimensional problem give the compact adi scheme with the restriction that besides the time variable the convection coefficients can only depend on the corresponding spatial variables respectively numerical results are provided to verify the accuracy and efficiency of the proposed algorithm the time fractional convection diffusion reaction equation is numerically solved compact exponential scheme for the variable coefficients equation is given fourier method and matrix analysis are used for the analysis the compact exponential adi scheme is given for the two dimensional problem high accuracy and good efficiency of the proposed algorithm is demonstrated

analysis of material interface discontinuities and superconvergent fluxes in finite difference theory abstract an analysis of material interface discontinuities is developed and applied in finite difference theory to determine mathematically rigorous averaging techniques for material properties this result is compared with other averaging techniques particularly harmonic averaging which is often applied in practice we also develop a class of formulas of high accuracy for post processing the difference formula to compute derivatives fluxes stresses and conduct supporting numerical studies

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

on the gluing formula for the analytic torsion in this paper we derive the cheeger muller bismut zhang theorem for manifoldswithboundaryandthegluingformulafortheanalytictorsionofflatvectorbundles in full generality i e we do not assume that the hermitian metric on the flat vector bundle is flat nor that the riemannian metric has product structure near the boundary

combined compact difference scheme for the time fractional convection diffusion equation with variable coefficients combined compact finite difference scheme for solving the time fractional convection diffusion reaction equation is given the primary unknown and its derivative are solved simultaneously the coefficients of the fractional partial differential equation are variable both dirichlet and robin boundary conditions are discussed and the algorithm gives high order convergence results fourth order combined compact finite difference scheme is given for solving the time fractional convection diffusion reaction equation with variable coefficients we introduce the flux as a new variable and transform the original equation into a system of two equations compact difference is used as a high order approximation for spatial derivatives of integer order in the coupled partial differential equations the caputo fractional derivative is discretized by a high order approximation both dirichlet and robin boundary conditions are discussed convergence analysis is given for the problem of integer order with constant coefficients under some assumption numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

analytic torsion of complete hyperbolic manifolds of finite volume abstract in this paper we define the analytic torsion for a complete oriented hyperbolic manifold of finite volume it depends on a representation of the fundamental group for manifolds of odd dimension we study the asymptotic behavior of the analytic torsion with respect to certain sequences of representations obtained by restriction of irreducible representations of the group of isometries of the hyperbolic space to the fundamental group

a high order exponential scheme for solving 1d unsteady convection diffusion equations in this paper a high order exponential hoe scheme is developed for the solution of the unsteady one dimensional convection diffusion equation the present scheme uses the fourth order compact exponential difference formula for the spatial discretization and the 2 2 pade approximation for the temporal discretization the proposed scheme achieves fourth order accuracy in temporal and spatial variables and is unconditionally stable numerical experiments are carried out to demonstrate its accuracy and to compare it with analytic solutions and numerical results established by other methods in the literature the results show that the present scheme gives highly accurate solutions for all test examples and can get excellent solutions for convection dominated problems

analysis of a fourth order compact scheme for convection diffusion in 1984 gupta et al introduced a compact fourth order finite difference convection diffusion operator with some very favorable properties in particular this scheme does not seem to suffer excessively from spurious oscillatory behavior and it converges with standard methods such as gauss seidel or sor hence multigrid regardless of the diffusion this scheme has been rederived developed including some variations and applied in both convection diffusion and navier stokes equations by several authors accurate solutions to high reynolds number flow problems at relatively coarse resolutions have been reported these solutions were often compared to those obtained by lower order discretizations such as second order central differences and first order upstream discretizations the latter it was stated achieved far less accurate results due to the artificial viscosity which the compact scheme did not include we show here that while the compact scheme indeed does not suffer from a cross stream artificial viscosity as does the first order upstream scheme when the characteristic direction is not aligned with the grid it does include a streamwise artificial viscosity that is inversely proportional to the natural viscosity this term is not always benign 7 refs 1 fig 1 tab

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

high order compact exponential finite difference methods for convection diffusion type problems a class of high order compact hoc exponential finite difference fd methods is proposed for solving one and two dimensional steady state convection diffusion problems the newly proposed hoc exponential fd schemes have nonoscillation property and yield high accuracy approximation solution as well as are suitable for convection dominated problems the o h 4 compact exponential fd schemes developed for the one dimensional 1d problems produce diagonally dominant tri diagonal system of equations which can be solved by applying the tridiagonal thomas algorithm for the two dimensional 2d problems o h 4 k 4 compact exponential fd schemes are formulated on the nine point 2d stencil and the line iterative approach with alternating direction implicit adi procedure enables us to deal with diagonally dominant tridiagonal matrix equations which can be solved by application of the one dimensional tridiagonal thomas algorithm with a considerable saving in computing time to validate the present hoc exponential fd methods three linear and nonlinear problems mostly with boundary or internal layers where sharp gradients may appear due to high peclet or reynolds numbers are numerically solved comparisons are made between analytical solutions and numerical results for the currently proposed hoc exponential fd methods and some previously published hoc methods the present hoc exponential fd methods produce excellent results for all test problems it is shown that besides including the excellent performances in computational accuracy efficiency and stability the present method has the advantage of better scale resolution the method developed in this article is easy to implement and has been applied to obtain the numerical solutions of the lid driven cavity flow problem governed by the 2d incompressible navier stokes equations using the stream function vorticity formulation

analytic torsion for group actions the reidemeister torsion is a classical topological invariant for nonsimply connected manifolds 14 let m be a closed oriented smooth manifold with the fundamental group m nontrivial let p x m o n be an orthogonal representation of nx m if the twisted real cohomology h m ep vanishes then one can define the reidemeister torsion p gr which is a homeomorphism invariant of m the original interest of p was that it is not a homotopy invariant and so can distinguish spaces which are homotopy equivalent but are not homeomorphic ray and singer asked whether as for many other topological quantities one can compute p by analytic methods 17 given a riemannian metric g on m they defined an analytic torsion t e r as a certain combination of the eigenvalues of the laplacian acting on twisted differential forms they showed that under the above acyclicity condition t is independent of the metric g and they conjectured that the analytic expression t equals the combinatorial expression this conjecture was proven to be true independently by cheeger 4 and m ller 15 one can look at the above situation in the following way the group n m acts freely on the universal cover m and so one has an invariant for free group actions a natural question is whether the reidemeister torsion can be extended to an invariant for more general group actions for a finite group acting not necessarily freely on a closed oriented pl manifold x a reidemeister torsion was defined algebraically by rothenberg 19 one can then ask whether there is a corresponding analytic torsion when x is smooth and whether the analytic torsion equals the combinatorial torsion in 11 we define the analytic torsion tp for a finite group action and show that if the relevant cohomology groups vanish then it is independent of the g invariant metric used in its definition this was shown previously in unpublished work by cheeger 5 the analysis involved to

higher order cartesian grid based finite difference schemes for elliptic equations on irregular domains second and fourth order cartesian grid based finite difference methods are proposed for elliptic and parabolic partial differential equations and associated eigenvalue problems on irregular domains with general boundary conditions our methods are based on the continuation of a solution idea using multivariable taylor s expansion of the solution about selected boundary points and the core ideas of the immersed interface method the methods offer systematic treatment of the general boundary conditions in two and three dimensional domains and are directly applied to semi discretize heat equations on irregular domains convergence analysis and numerical examples are presented the validity and effectiveness of the proposed methods are demonstrated through our numerical results including computations of the eigenvalues of the associated eigenvalue problem

compact finite differencing and filtering procedure applied to the incompressible navier stokes equations a high order discretization and filtering procedure is applied to the solution of the incompressible navier stokes equations in a vorticity stream function formulation which is implemented in curvilinear coordinates compact finite differencing is coupled with the use of a low pass filtering operator to augment the stability of the scheme for high reynolds number flows and or low resolution meshes advancement in time is done through either first or second order backward euler time integrations which are supplemented with newton like subiterations temporal and spatial formal orders of accuracy are examined through exact solutions of the governing equations where the theoretical orders of accuracy are achieved for up to sixth order spatial and second order temporal discretizations the technique is also demonstrated on both steady and unsteady incompressible flow canonical problems including the lid driven cavity and unsteady flow over a cylinder the advantage of the high order scheme coupled with

a two step high order compact scheme for the laplacian operator and its implementation in an explicit method for integrating the nonlinear schrodinger equation we describe and test an easy to implement two step high order compact 2shoc scheme for the laplacian operator and its implementation into an explicit finite difference scheme for simulating the nonlinear schrodinger equation nlse our method relies on a compact double differencing which is shown to be computationally equivalent to standard fourth order non compact schemes through numerical simulations of the nlse using fourth order runge kutta we confirm that our scheme shows the desired fourth order accuracy a computation and storage requirement comparison is made between the 2shoc scheme and the non compact equivalent scheme for both the laplacian operator alone as well as when implemented in the nlse simulations stability bounds are also shown in order to get maximum efficiency out of the method we conclude that the modest increase in storage and computation of the 2shoc schemes is well worth the advantages of having the schemes compact and their ease of implementation makes their use very useful for practical implementations

the zeta functions of ruelle and selberg for hyperbolic manifolds with cusps in this paper we study the ruelle zeta function and the selberg zeta functions attached to the fundamental representations for real hyperbolic manifolds with cusps in particular we show that they have meromorphic extensions to mathbb c and satisfy functional equations we also derive the order of the singularity of the ruelle zeta function at the origin to prove these results we completely analyze the weighted unipotent orbital integrals on the geometric side of the selberg trace formula when test functions are defined for the fundamental representations

a higher order compact finite difference algorithm for solving the incompressible navier stokes equations on the basis of the projection method a higher order compact finite difference algorithm which possesses a good spatial behavior is developed for solving the 2d unsteady incompressible navier stokes equations in primitive variable the present method is established on a staggered grid system and is at least third order accurate in space a third order accurate upwind compact difference approximation is used to discretize the non linear convective terms a fourth order symmetrical compact difference approximation is used to discretize the viscous terms and a fourth order compact difference approximation on a cell centered mesh is used to discretize the first derivatives in the continuity equation the pressure poisson equation is approximated using a fourth order compact difference scheme constructed currently on the nine point 2d stencil new fourth order compact difference schemes for explicit computing of the pressure gradient are also developed on the nine point 2d stencil for the assessment of the effectiveness and accuracy of the method particularly its spatial behavior a problem with analytical solution and another one with a steep gradient are numerically solved finally steady and unsteady solutions for the lid driven cavity flow are also used to assess the efficiency of this algorithm copyright c 2011 john wiley sons ltd

lattice boltzmann method and gas kinetic bgk scheme in the low mach number viscous flow simulations both lattice boltzmann method lbm and the gas kinetic bgk scheme are based on the numerical discretization of the boltzmann equation with collisional models such as the bhatnagar gross krook bgk model lbm tracks limited number of particles and the viscous flow behavior emerges automatically from the intrinsic particle stream and collisions process on the other hand the gas kinetic bgk scheme is a finite volume scheme where the time dependent gas distribution function with continuous particle velocity space is constructed and used in the evaluation of the numerical fluxes across cell interfaces currently lbm is mainly used for low mach number nearly incompressible flow simulation for the gas kinetic scheme the application is focusing on the high speed compressible flows in this paper we are going to compare both schemes in the isothermal low mach number flow simulations the methodology for developing both schemes will be clarified through the introduction of operator splitting boltzmann model and operator averaging boltzmann model from the operator splitting boltzmann model the error rooted in many kinetic schemes which are based on the decoupling of particle transport and collision can be easily understood as to the test case we choose to use the 2d cavity flow since it is one of the most extensively studied cases detailed simulation results with different reynolds numbers as well as the benchmark solutions are presented

a high order compact mac finite difference scheme for the stokes equations augmented variable approach this paper deals with the steady stokes flow on a rectangular domain a high order compact mac finite difference scheme based on the staggered grid is developed for solving stokes equations with a dirichlet boundary condition on the velocity a novel high order boundary treatment is developed via introducing a suitable augmented variable the accuracy of the proposed method is demonstrated in test problems creeping flow solutions for driven cavity problem are obtained numerically and compared with published results

analytic torsion versus reidemeister torsion on hyperbolic 3 manifolds with cusps for a non compact hyperbolic 3 manifold with cusps we prove an explicit formula that relates the regularized analytic torsion associated to the even symmetric powers of the standard representation of mathrm sl 2 mathbb c to the corresponding reidemeister torsion while the analytic torsion is a spectral invariant of the manifold the reidemeister torsion is of combinatorial nature our proof rests on an expression of the analytic torsion in terms of special values of ruelle zeta functions as well as on recent work of pere menal ferrer and joan porti

analytic torsion and ruelle zeta functions for hyperbolic manifolds with cusps abstract in this paper we derive a relationship of the leading coefficient of the laurent expansion of the ruelle zeta function at s 0 and the analytic torsion for hyperbolic manifolds with cusps here the analytic torsion is defined by a certain regularized trace following melrose r b melrose the atiyah patodi singer index theorem res notes math vol 4 a k peters ltd wellesley ma 1993 this extends the result of fried which was proved for the compact case in d fried analytic torsion and closed geodesics on hyperbolic manifolds invent math 84 3 1986 523 540 to a noncompact case

high accuracy iterative solution of convection diffusion equation with boundary layers on nonuniform grids abstract a fourth order compact finite difference scheme and a multigrid method are employed to solve the two dimensional convection diffusion equations with boundary layers the computational domain is first discretized on a nonuniform stretched grid to resolve the boundary layers a grid transformation technique is used to map the nonuniform grid to a uniform one the fourth order compact scheme is applied to the transformed uniform grid a multigrid method is used to solve the resulting linear system numerical experiments are used to show that a graded mesh and a grid transformation are necessary to compute high accuracy solutions for the convection diffusion problems with boundary layers and dicretized by the fourth order compact scheme

an explicit higher order difference scheme on a compact stencil for elliptic equations on curvilinear geometries abstract a fourth order explicit finite difference scheme on a compact stencil is proposed for a general steady convection diffusion equation on an arbitrary curvilinear coordinate system with computationally generated non uniform grid the method is dependent on the governing differential equation and attains higher order through consistent discretization and accurate evaluation of the transformation metrics extensive comparison of accuracy convergence rate and computation time is made with standard explicit and implicit schemes to highlight the relatively high efficiency of using the proposed scheme applicability of the scheme to irregular flows and geometries is validated by solving for i potential flow past a cambered airfoil at an angle of attack and ii full incompressible navier stokes calculations for flow past a cylinder performing rotary oscillations at low reynolds number

a fourth order compact adi method for solving two dimensional unsteady convection diffusion problems n this article an exponential high order compact ehoc alternating direction implicit adi method in which the crank nicolson scheme is used for the time discretization and an exponential fourth order compact difference formula for the steady state 1d convection diffusion problem is used for the spatial discretization is presented for the solution of the unsteady 2d convection diffusion problems the method is temporally second order accurate and spatially fourth order accurate which requires only a regular five point 2d stencil similar to that in the standard second order methods the resulting ehoc adi scheme in each adi solution step corresponds to a strictly diagonally dominant tridiagonal matrix equation which can be inverted by simple tridiagonal gaussian decomposition and may also be solved by application of the one dimensional tridiagonal thomas algorithm with a considerable saving in computing time the unconditionally stable character of the method was verified by means of the discrete fourier or von neumann analysis numerical examples are given to demonstrate the performance of the method proposed and to compare mostly it with the high order adi method of karaa and zhang and the spatial third order compact scheme of note and tan

some experiments with high order compact methods using a computer algebra software part ii non uniform grid abstract we generalize a procedure proposed by mancera and hunt p f a mancera r hunt some experiments with high order compact methods using a computer algebra software part i appl math comput in press doi 10 1016 j amc 2005 05 015 for obtaining a compact fourth order method to the steady 2d navier stokes equations in the streamfunction formulation vorticity using the computer algebra system maple which includes conformal mappings and non uniform grids to analyse the procedure we have solved a constricted stepped channel problem where a fine grid is placed near the re entrant corner by transformation of the independent variables

a computational stream function method for two dimensional incompressible viscous flows this work concerns the development of a numerical method based on the stream function formulation of the navier stokes equations to simulate two dimensional plane or axisymmetric viscous flows the main features of the proposed method are the use of the high order finite difference compact method for the discretization of the stream function equation the implicit pseudo transient newton krylov multigrid matrix free method for the stationary stream function equation and the fourth order runge kutta method for the integration of non stationary flows copyright 2005 john wiley sons ltd

a transformation free hoc scheme for incompressible viscous flows past an impulsively started circular cylinder in this paper we present a higher order compact scheme for the unsteady two dimensional 2d navier stokes equations on nonuniform polar grids specifically designed for the incompressible viscous flows past a circular cylinder the scheme is second order accurate in time and at least third order accurate in space the scheme very efficiently computes both unsteady and time marching steady state flow for a wide range of reynolds numbers re ranging from 10 to 9500 for the impulsively started cylinder the robustness of the scheme is highlighted when it accurately captures the vortex shedding for moderate re represented by the von karman street and the so called and s phenomena for higher re comparisons are made with established numerical and experimental results and excellent agreement is found in all the cases both qualitatively and quantitatively

numerical simulation of 2d square driven cavity using fourth order compact finite difference schemes abstract fourth order compact finite difference schemes are employed with multigrid techniques to simulate the two dimensional square driven cavity flow with small to large reynolds numbers the governing navier stokes equation is linearized in streamfunction and vorticity formulation the fourth order compact approximation schemes are coupled with fourth order approximations for velocities and vorticity boundaries numerical solutions are obtained for square driven cavity flow at high reynolds numbers and are compared with solutions obtained by other researchers using other approximation methods

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

on the derivation of highest order compact finite difference schemes for the one and two dimensional poisson equation with dirichlet boundary conditions the primary aim of this paper is to answer the question what are the highest order five or nine point compact finite difference schemes to answer this question we present several simple derivations of finite difference schemes for the one and two dimensional poisson equation on uniform quasi uniform and nonuniform face to face hyperrectangular grids and directly prove the existence or nonexistence of their highest order local accuracies our derivations are unique in that we do not make any initial assumptions on stencil symmetries or weights for the one dimensional problem the derivation using the three point stencil on both uniform and nonuniform grids yields a scheme with arbitrarily high order local accuracy however for the two dimensional problem the derivation using the corresponding five point stencil on uniform and quasi uniform grids yields a scheme with at most second order local accuracy and on nonuniform grids yields at most first order local accuracy when expanding the five poin

on cyclic reduction and finite difference schemes we investigate a family of finite difference schemes for discretizing the two dimensional poisson equation on both the standard and the reduced grids we study the relation between the cyclic reduction method and the discretization schemes on different grids the spectral radii of the jacobi iteration matrices and the truncation errors of different discretization schemes are compared analytically and numerically

a high order compact exponential scheme for the fractional convection diffusion equation a high order compact exponential finite difference scheme for solving the fractional convection diffusion equation is considered in this paper the convection and diffusion terms are approximated by a compact exponential finite difference scheme with a high order approximation for the caputo time derivative for this fully discrete implicit scheme the local truncation error is analyzed and the fourier method is used to discuss the stability the error estimate is given by the discrete energy method numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

fourth order compact scheme for space fractional advection diffusion reaction equations with variable coefficients abstract in this work a new fourth order compact approximation is derived for riemann liouville space fractional derivatives modified wave numbers are obtained for various approximations of fractional derivatives considering these modified wave numbers fourier analysis of differencing errors is presented to quantify the resolution characteristics of various approximations a novel fourth order compact scheme is developed for space fractional advection diffusion reaction equations with variable coefficients in one and two dimensions in the proposed compact scheme the second derivative approximation of unknowns is approximated using the value of these unknowns and their first derivative approximations this splitting of second derivative approximations allows us to obtain a similar system of linear equations as in zhao and tian 2017 which presents only second order accurate finite difference method for advection diffusion reaction equations in one dimension the stability of the proposed compact scheme is demonstrated numerically furthermore the proposed compact scheme is employed to space fractional black scholes equation for pricing european options as an application of fractional derivatives in mathematical finance numerical illustrations are presented to validate the theoretical claims

a fourth order accurate finite volume compact method for the incompressible navier stokes solutions abstract this paper presents a finite volume fourth order accurate compact scheme for discretization of the incompressible navier stokes equations in primitive variable formulation the numerical method of integrating the navier stokes equations comprises a compact finite volume formulation of the average convective and diffusive fluxes the pressure velocity coupling is achieved via the coupled solution of the resulting system of equations the solution of the coupled set of equations is performed with an implicit newton krylov matrix free method for stationary problems for simulation of unsteady flows a standard fourth order runge kutta method was used for temporal discretization and the velocity pressure coupling was ensured at each stage also using the matrix free method several incompressible viscous steady and unsteady flow problems have been computed to assess the robustness and accuracy of the proposed method

generalized ray singer conjecture i a manifold with a smooth boundary this paper is devoted to a proof of a generalized ray singer conjecture for a manifold with boundary the dirichlet and the neumann boundary conditions are independently given on each connected component of the boundary and the transmission boundary condition is given on the interior boundary the ray singer conjecture rs claims that for a closed manifold the combinatorial and the analytic torsion norms on the determinant of the cohomology are equal for a manifold with boundary the ratio between the analytic torsion and the combinatorial torsion is computed some new general properties of the ray singer analytic torsion are found the proof does not use any computation of eigenvalues and their asymptotic expansions or explicit expressions for the analytic torsions of any special classes of manifolds

some experiments with high order compact methods using a computer algebra software part i abstract we consider a procedure for obtaining a compact fourth order method to the steady 2d navier stokes equations in the streamfunction formulation using the computer algebra system maple the resulting code is short and from it we obtain the fortran program for the method to test the procedure we have solved many cavity type problems which include one with an analytical solution and the results are compared with results obtained by second order central differences to moderate reynolds numbers

a rational high order compact difference method for the steady state stream functionvorticity formulation of the navierstokes equations a rational high order compact rhoc finite difference fd method on the nine point stencil is proposed for solving the steady state two dimensional navierstokes equations in the stream functionvorticity form the resulting system of algebra equations can be solved by using the point successive over or under relaxation sor iteration numerical experiments involving two linear and two nonlinear problems with their analytical solutions and two flow problems including the lid driven cavity and backward facing step flows are carried out to validate the performance of the newly proposed method numerical solutions of the driven cavity problem with different grid mesh sizes maximum being 513513 for reynolds numbers ranging from 0 to 17500 are obtained and compared with some of the accurate results available in the literature

high order adi finite difference schemes for parabolic equations in the combination technique with application in finance in this article we combine high order ho finite difference discretisations with alternating direction implicit adi schemes for parabolic partial differential equations with mixed derivatives in a sparse grid setting in each implicit leg of the adi schemes we propose a high order compact hoc discretisation such that only tridiagonal systems have to be solved with the help of ho spatial discretisations and adi schemes solutions with second order accuracy in time and fourth order accuracy in space can be computed in order to reduce the number of involved grid points we use the combination technique to construct the so called sparse grid solution the theoretical findings are illustrated by numerical examples with european basket options

selberg zeta functions on odd dimensional hyperbolic manifolds of finite volume we study selberg zeta functions z s associated to locally homogeneous vector bundles over the unit sphere bundle of a complete odd dimensional hyperbolic manifold of finite volume we assume a certain condition on the fundamental group of the manifold a priori the selberg zeta functions are defined only for s in some right halfspace of c we will prove that for any locally homogeneous bundle the functions z s have a meromorphic continuation to c and we will give a complete description of their singularities in terms of spectral data of the underlying manifold our work generalizes results of bunke and olbrich to the non compact situation as an application of our results one can compare the normalized reidemeister torsions on hyperbolic 3 manifolds with cusps which were introduced by menal ferrer and porti to the corresponding regularized analytic torsions

multigrid method and fourth order compact scheme for 2d poisson equation with unequal mesh size discretization abstract a fourth order compact difference scheme with unequal mesh sizes in different coordinate directions is employed to discretize a two dimensional poisson equation in a rectangular domain multigrid methods using a partial semicoarsening strategy and line gauss seidel relaxation are designed to solve the resulting sparse linear systems numerical experiments are conducted to test the accuracy of the fourth order compact difference scheme and to compare it with the standard second order difference scheme convergence behavior of the partial semicoarsening and line gauss seidel relaxation multigrid methods is examined experimentally

a new compact difference scheme for second derivative in non uniform grid expressed in self adjoint form a single parameter family of self adjoint compact difference sacd schemes is developed for discretizing the laplacian operator in self adjoint form developed implicit scheme is formally second order accurate with respect to truncation error with a free parameter a which helps control the numerical properties in the spectral plane the sacd scheme is analyzed in the spectral plane for its resolution properties for both periodic and non periodic problems using the matrix spectral analysis t k sengupta g ganeriwal s de analysis of central and upwind schemes j comput phys 192 2 2003 677 694 the major objective here is to identify the advantages of the new scheme over the traditional explicit second order cd2 scheme in discretizing the laplacian operator in self adjoint form this appears in navier stokes equation and in other transport equations and boundary value problems bvp expressed in orthogonal co ordinate systems either in physical or in transformed plane we also compare the developed method with the higher order compact schemes for non uniform grids to demonstrate the accuracy of sacd scheme we have tested it for i bi directional wave propagation problem given by the second order wave equation and ii an elliptic bvp as in the stommel ocean model for the stream function these examples help infer the properties of sacd scheme when solving different types of partial differential equations most importantly the effects of grid stretching and choice of value of the free parameter a are investigated here we also compare the present implicit compact method with explicit compact method known as the higher order compact hoc method also the practical applications of the sacd scheme are explored by solving the navier stokes equation for the cases of a a flow inside a lid driven cavity and b the receptivity and instability of an external adverse pressure gradient flow over a flat plate in the former unsteadiness of the flow is captured and in the latter the receptivity of the flow is studied in causing flow instability by triggering tollmien schlichting waves the new scheme shows a marked improvement over the explicit scheme for low reynolds number steady flow in lid driven cavity whereas for the flow in the same geometry at higher reynolds numbers efficacy of the scheme is established by showing the formation of a triangular vortex and secondary vortical structures presented scheme is perfectly capable of expressing the diffusion operator accurately as shown via the capturing of instability waves inside the shear layer

analytic torsion and r torsion for unimodular representations let m be a closed c manifold of dimension n both r torsion and analytic torsion are smooth invariants of acyclic orthogonal or unitary representations p of the fundamental group 7i m the reidemeister franz torsion or r torsion tm p of p is defined in terms of the combinatorial structure of m given by its smooth triangulations the analytic torsion tm p was introduced by ray and singer rs as an analytic counterpart of r torsion in order to define the analytic torsion one has to choose a riemannian metric on m then tm p is a certain weighted alternating product of regularized determinants of the laplacians on differential qforms of m with values in the flat bundle e defined by p it was conjectured by ray and singer rs that tm p trm p for all acyclic orthogonal or unitary representations p this conjecture was proved independently by cheeger c and the author mu the restriction to orthogonal or unitary representations is certainly a limitation of the applicability of this result if 7i m is infinite because an infinite discrete group will have in general many nonorthogonal finite dimensional representations it is the purpose of the present paper to remove this limitation we call a representation p r m gl e on a finite dimensional real or complex vector space e unimodular if i det p y i 1 for all y e i m then we define r torsion and analytic torsion for unimodular representations and the main result is that for odd dimensional manifolds m the equality of the two torsions extends to all unimodular representations

convergence and performance of iterative methods for solving variable coefficient convection diffusion equation with a fourth order compact difference scheme abstract we conduct convergence analysis on some classical stationary iterative methods for solving the two dimensional variable coefficient convection diffusion equation discretized by a fourth order compact difference scheme several conditions are formulated under which the coefficient matrix is guaranteed to be an m matrix we further investigate the effect of different orderings of the grid points on the performance of some stationary iterative methods multigrid method and preconditioned gmres three sets of numerical experiments are conducted to study the convergence behaviors of these iterative methods under the influence of the flow directions the orderings of the grid points and the magnitude of the convection coefficients

fast and high accuracy multiscale multigrid method with multiple coarse grid updating strategy for the 3d convection diffusion equation an improved multiscale multigrid method with multiple coarse grid updating strategy for a three dimensional convection diffusion equation is presented the novelty of the proposed method lies in a fine grid updating strategy arising from the idea of multiple coarse grid computation the new fine grid updating strategy is able to replace the iterative refinement procedure in the existing multiscale multigrid method wang and zhang 2010 16 to obtain high order solutions since the proposed method needs a fourth order compact scheme with unequal meshsize grids a 19 point fourth order compact difference scheme with unequal meshsize in different coordinate directions is also developed for the three dimensional convection diffusion equation numerical results are given to compare the computed accuracy and the computational efficiency of the multiscale multigrid method with the multiple coarse grid updating strategy against the multiscale multigrid method with the iterative refinement procedure

implicit compact linearized θ methods with factorization for multidimensional reaction diffusion equations an iterative predictor corrector technique for the elimination of the approximate factorization errors which result from the factorization of implicit three point compact linearized q methods in multidimensional reaction diffusion equations is proposed and its convergence and linear stability are analyzed four compact approximate factorization techniques which do not account for the approximate factorization errors and which involve three point stencils for each one dimensional operator are developed the first technique uses the full jacobian matrix of the reaction terms requires the inversion of in general dense matrices and its approximate factorization errors are second order accurate in time the second and third methods approximate the jacobian matrix by diagonal or triangular ones which are easily inverted but their approximate factorization errors are however first order accurate in time the fourth approximately factorized compact implicit method has approximate factorization errors which are second order accurate in time and requires the inversion of lower and upper triangular matrices the techniques are applied to a nonlinear two species two dimensional system of reaction diffusion equations in order to determine the approximate factorization errors and those resulting from the approximations to the jacobian matrix as functions of the allocation of the reaction terms space and time

analytic and topological torsion for manifolds with boundary and symmetry letg be a finite group acting on a riemannian manifoldm by isometries we introduce analytic torsion an m m1 v r z repr g pl torsion pl m m1 v k1 rg poincare torsion pd m m1 v k1 rg and euler characteristic m m1 v repr g for m the disjoint union of m1 and m2 and v an equivariant coefficient system the analytic torsionis defined in terms of the spectrum of the laplace operator the pl torsionis based on the cellular chain complex and poincare torsionmeasures the failure of equivariant poincare duality in the pl setting which does hold in the analytic context denote by repr g the subgroup of repr g generated by the irreducible representations of real or complex type we define an isomorphism

an efficient transient navier stokes solver on compact nonuniform space grids in this paper we propose an implicit higher order compact hoc finite difference scheme for solving the two dimensional 2d unsteady navier stokes n s equations on nonuniform space grids this temporally second order accurate scheme which requires no transformation from the physical to the computational plane is at least third order accurate in space which has been demonstrated with numerical experiments it efficiently captures both transient and steady state solutions of the n s equations with dirichlet as well as neumann boundary conditions the proposed scheme is likely to be very useful for the computation of transient viscous flows involving free and wall bounded shear layers which invariably contain spatial scale variation numerical results are presented and compared with analytical as well as established numerical data excellent comparison is obtained in all the cases

high order compact scheme for the steady stream function vorticity equations a higher order compact scheme that is o h4 on the nine point 2 d stencil is formulated for the steady stream function vorticity form of the navier stokes equations the resulting stencil expressions are presented and hence this new scheme can be easily incorporated into existing industrial software we also show that special treatment of the wall boundary conditions is required the method is tested on representative model problems and compares very favourably with other schemes in the literature

an efficient implementation of fourth order compact finite difference scheme for poisson equation with dirichlet boundary conditions fourth order compact finite difference scheme has been proposed for solving the poisson equation with dirichlet boundary conditions for some time an efficient implementation of such numerical scheme is often desired for practical usage in this paper based on fast discrete sine transform we design an efficient algorithm to implement this scheme to do this poisson equation is first discretized by fourth order compact finite difference method the subsequent discretized system is not solved by the usual method matrix inversion instead it is solved with the fast discrete sine transform by doing this way the computational cost of proposed algorithm for such scheme with large grid numbers can be greatly reduced detailed numerical algorithm of this fast solver for one dimensional two dimensional and three dimensional poisson equation has been presented numerical results in one dimension two dimensions three dimensions and four dimensions have shown that the applied compact finite difference scheme has fourth order accuracy and can be efficiently implemented

analytic torsion and closed geodesics on hyperbolic manifolds pour une variete hyperbolique orientee fermee x et une representation orthogonale de 1 x on definit une fonction zeta de type ruelle r s pour re s grand comme le produit sur les geodesiques closes premieres des facteurs det i exp sl ou l est la longueur de on continue analytiquement r s et on calcule son terme principal en s 0

multigrid method based on the transformation free hoc scheme on nonuniform grids for 2d convection diffusion problems in this paper a multigrid method based on the high order compact hoc difference scheme on nonuniform grids which has been proposed by kalita et al j c kalita a k dass d c dalal a transformation free hoc scheme for steady convection diffusion on non uniform grids int j numer methods fluids 44 2004 33 53 is proposed to solve the two dimensional 2d convection diffusion equation the hoc scheme is not involved in any grid transformation to map the nonuniform grids to uniform grids consequently the multigrid method is brand new for solving the discrete system arising from the difference equation on nonuniform grids the corresponding multigrid projection and interpolation operators are constructed by the area ratio some boundary layer and local singularity problems are used to demonstrate the superiority of the present method numerical results show that the multigrid method with the hoc scheme on nonuniform grids almost gets as equally efficient convergence rate as on uniform grids and the computed solution on nonuniform grids retains fourth order accuracy while on uniform grids just gets very poor solution for very steep boundary layer or high local singularity problems the present method is also applied to solve the 2d incompressible navier stokes equations using the stream function vorticity formulation and the numerical solutions of the lid driven cavity flow problem are obtained and compared with solutions available in the literature

frequency planning as a set partitioning problem abstract some aspects of a new computerized method for automatic generation of frequency plans for radio communication systems are presented the emphasis is on problem formulation where the frequency planning problem is recast as a set partitioning problem the objective is minimization of total system interference a solution algorithm that has been found useful in practical applications is presented an alternative algorithm is also demonstrated

a gradual neural network approach for frequency assignment in satellite communication systems a novel neural network approach called gradual neural network gnn is presented for a class of combinatorial optimization problems of requiring the constraint satisfaction and the goal function optimization simultaneously the frequency assignment problem in the satellite communication system is efficiently solved by gnn as the typical problem of this class the goal of this np complete problem is to minimize the cochannel interference between satellite communication systems by rearranging the frequency assignment so that they can accommodate the increasing demands the gnn consists of n spl times m binary neurons for the n carrier m segment system with the gradual expansion scheme of activated neurons the binary neural network achieves the constrain satisfaction with the help of heuristic methods whereas the gradual expansion scheme seeks the cost optimization the capability of gnn is demonstrated through solving 15 instances in practical size systems where gnn can find far better solutions than the existing algorithm

superconvergent derivatives a taylor series analysis an analysis based on the use of taylor series expansions is developed to determine accuracy estimates for derivatives in one and two dimensions computed by differentiation of a finite element interpolant or approximation the analysis clarifies some issues concerning special points at which the derivatives are believed to be exceptionally accurate with higher convergence rates superconvergence moreover it leads directly to a class of post processing strategies for the derivatives and offers a more direct constructive approach to the subject

flow past an impulsively started circular cylinder using a higher order semicompact scheme in the present investigation an attempt has been made to solve the two dimensional incompressible viscous flow past an impulsively started circular cylinder for reynolds numbers ranging from 20 to 5000 using higher order semicompact scheme hosc unlike conventional higher order compact schemes the hosc scheme has been developed to handle the circular geometry of the chosen problem and the intensive algebraic manipulations have been reduced considerably by relaxing the compactness of the computational stencil for few terms but retained for most of the terms of the discretized equations for the flow past an impulsively started circular cylinder the results obtained at low and moderate reynolds numbers have been validated with the experimental and numerical observations available in the literature for high reynolds number flows the present scheme rightly captures the alpha phenomenon at re 1000 and both beta and alpha phenomena one after the other at re 5000

multiple coarse grid acceleration for multiscale multigrid computation the multiscale multigrid method uses an iterative refinement procedure with the richardson extrapolation technique to obtain a higher order solution the computational cost for the iterative refinement procedure may be significant for some ill conditioned coefficient matrices in this paper we proposed an alternative strategy using multiple coarse grids to eliminate the iterative refinement procedure and thus accelerate the multiscale multigrid computation numerical investigations show that our multiple coarse grid computing strategy is more efficient and scalable than the iterative refinement procedure the multiscale multigrid method with multiple coarse grid strategy is used to solve two dimensional 2d poisson equation and convection diffusion equation but the idea can be used to solve other partial differential equations

an efficient time splitting compact finite difference method for gross pitaevskii equation abstract we propose an efficient time splitting compact finite difference method for gross pitaevskii equation gpe in our method we solve the gpe in time with time splitting technique and in space by the compact finite difference method to find the numerical solution of the resulting discretized system in one dimension 1d two dimensions 2d and three dimensions 3d we apply the fast discrete sine transform in 1d 2d and 3d respectively and get an efficient solver for the discretized system in 1d 2d and 3d respectively our numerical algorithm at every time step does not need linear algebraic equations solver whose computation cost will be much higher when the spatial dimension increases the method also has the merit that it is unconditionally stable and conservative moreover the method can achieve spectral like accuracy in space when high order compact finite difference method is applied extensive numerical tests for the gpe in 1d 2d and 3d are presented to demonstrate the power and accuracy of the proposed numerical method

an exponential compact difference scheme for solving 2d steady magnetohydrodynamic mhd duct flow problems in this article an exponential high order compact ehoc difference scheme on the nine point stencil is developed for the solution of the coupled equations representing the steady incompressible viscous magnetohydrodynamic mhd flow through a straight channel of rectangular section a key property of the ehoc scheme is that it has excellent stability and higher accuracy so that the high gradients near the boundary layer areas can be effectively resolved without refining the mesh numerical experiments are carried out to validate the performance of the currently proposed scheme computation results of the mhd flow in the 2d square channel problems with different wall conductivities are presented for hartmann numbers ranging from 10 to 10 6 the numerical solutions obtained with the newly developed ehoc scheme are also compared with analytic solutions and numerical results by other available methods in the literature

compact mixed methods for convection diffusion type problems abstract this paper presents a class of fourth order compact finite difference technique for solving two dimensional convection diffusion equation the equation is recasted as a first order mixed system introducing a conservation and flux equations since flux appears explicitly in the mixed formulation we search a fourth order compact approximation of the primary solution field and flux based on taylor series expansion the proposed compact mixed formulation generalizes the work of carey and spotz g f carey w f spotz higher order compact mixed methods commun numer meth eng 13 1997 we show that their fourth order formulation corresponds to a particular case of our presented scheme and we extend their work to variable diffusion and convection coefficients some numerical experiments are performed to demonstrate the fourth order effective convergence rate

a high order finite difference method with richardsonextrapolation for 3d convection diffusion equation in this paper we extend the sun and zhang s 24 work on high order finite difference method which is based on the richardson extrapolation technique and an operator interpolation scheme for the one and two dimensional steady convection diffusion equations to the three dimensional case firstly we employ a fourth order compact difference scheme to get the fourth order accurate solution on the fine and the coarse grids then we use the richardson extrapolation technique by combining the two approximate solutions to get a sixth order accurate solution on coarse grid finally we apply an operator interpolation scheme to achieve the sixth order accurate solution on the fine grid during this process we use alternating direction implicit adi method to solve the resulting linear systems numerical experiments are conducted to verify the accuracy and effectiveness of the present method

a stable and accurate convective modelling procedure based on quadratic upstream interpolation a convective modelling procedure is presented which avoids the stability problems of central differencing while remaining free of the inaccuracies of numerical diffusion associated with upstream differencing for combined convection and diffusion the number of operations at each grid point is comparable to that of standard upstream pluscentral differencing however highly accurate solutions can be obtained with a grid spacing much larger than that required by conventional methods for comparable accuracy with obvious practical advantaged in terms of both speed and storage the algorithm is based on a conservative control volume formulation with cell wall values of each field variable written in terms of a quadratic interpolation using in any one coordinate direction the two adjacent nodal values together with the value at the next upstream node this results in a convective differencing scheme with greater formal accuracy than central differencing while retaining the basic stable convective sensitivity property of upstream weighted schemes the consistent treatment of diffusion terms is equivalent to central differencing with careful modelling numerical boundary conditions are not troublesome some idealized problems are studied showing the practical advantages of the method over other schemes in comparison with exact solutions an application to a complex unsteady two dimensional flow is briefly discussed

a new fourth order discretization for singularly perturbed two dimensional non linear elliptic boundary value problems abstract in this article we derive a new difference method of o h4 so called arithmetic average discretization for the solution of two dimensional non linear singularly perturbed elliptic partial differential equation of the form e uxx uyy f x y u ux uy 0 0 is a small parameter we also derive new methods of o h4 for the estimates of u n which are quite often of interest in many physical problems in all cases we require only 9 grid points and a single computational cell the main advantage of the proposed methods is that the methods are directly applicable to singular problems we do not require any special technique or modification to solve singular problems numerical results are provided to demonstrate the usefulness of the methods discussed

compact h4 finite difference approximations to operators of navier stokes type a method of obtaining compact finite difference approximations of h4 accuracy to operators of navier stokes type is considered the basic procedure is developed for operators in one space dimension and subsequently applied to problems in more space dimensions and in time four illustrative numerical examples are given which indicate clearly in various cases that excellent accuracy may be obtained using the methods comparisons with previous results and with the results of h2 accurate computations are made