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

physics compatible discretization techniques on single and dual grids with application to the poisson equation of volume forms this paper introduces the basic concepts for physics compatible discretization techniques the paper gives a clear distinction between vectors and forms based on the difference between forms and pseudo forms and the operator which switches between the two a dual grid description and a single grid description are presented the dual grid method resembles a staggered finite volume method whereas the single grid approach shows a strong resemblance with a finite element method both approaches are compared for the poisson equation for volume forms by defining a suitably weighted inner product for 1 forms this approach can readily be applied to anisotropic diffusion models for volume forms

a penalty method for the vorticity velocity formulation we present a new vorticity velocity formulation and implementation for the unsteady three dimensional navier stokes equations based on a penalty method it relies on an equivalence theorem that employs exact boundary conditions and the vorticity definition on the domain boundary this approach is particularly attractive for high order methods for which the often used influence matrix method fails to converge for t 0 the accuracy and the robustness of the new method is demonstrated in the context of several spectral element simulations of unsteady two and three dimensional internal and external flows in particular the flow past a finite span cylinder attached to end plates is studied in some detail in order to evaluate the effects of the aspect ratio on the formation length

velocity vorticity helicity formulation and a solver for the navier stokes equations for the three dimensional incompressible navier stokes equations we present a formulation featuring velocity vorticity and helical density as independent variables we find the helical density can be observed as a lagrange multiplier corresponding to the divergence free constraint on the vorticity variable similar to the pressure in the case of the incompressibility condition for velocity as one possible practical application of this new formulation we consider a time splitting numerical scheme based on an alternating procedure between vorticity helical density and velocity bernoulli pressure systems of equations results of numerical experiments include a comparison with some well known schemes based on pressure velocity formulation and illustrate the competitiveness on the new scheme as well as the soundness of the new formulation

a transformation free hoc scheme and multigrid method for solving the 3d poisson equation on nonuniform grids a high order compact hoc difference scheme is proposed to solve the three dimensional 3d poisson equation on nonuniform orthogonal cartesian grids involving no coordinate transformation from the physical space to the computational space theoretically the proposed scheme has third to fourth order accuracy its fourth order accuracy is achieved under uniform grid settings then a multigrid method is developed to solve the linear system arising from this hoc difference scheme and the corresponding multigrid restriction and interpolation operators are constructed using the volume law numerical experiments are conducted to show the computed accuracy of the hoc scheme and the computational efficiency of the multigrid method

on conservation laws of navier stokes galerkin discretizations abstract we study conservation properties of galerkin methods for the incompressible navier stokes equations without the divergence constraint strongly enforced in typical discretizations such as the mixed finite element method the conservation of mass is enforced only weakly and this leads to discrete solutions which may not conserve energy momentum angular momentum helicity or vorticity even though the physics of the navier stokes equations dictate that they should we aim in this work to construct discrete formulations that conserve as many physical laws as possible without utilizing a strong enforcement of the divergence constraint and doing so leads us to a new formulation that conserves each of energy momentum angular momentum enstrophy in 2d helicity and vorticity for reference the usual convective formulation does not conserve most of these quantities several numerical experiments are performed which verify the theory and test the new formulation

finite element exterior calculus from hodge theory to numerical stability this article reports on the confluence of two streams of research one emanating from the fields of numerical analysis and scientific computation the other from topology and geometry in it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de rham cohomology and hodge theory are key tools for exploring the well posedness of the continuous problem the discretization methods we consider are finite element methods in which a variational or weak formulation of the pde problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace after a brief introduction to finite element methods we develop an abstract hilbert space framework for analyzing the stability and convergence of such discretizations in this framework the differential complex is represented by a complex of hilbert spaces and stability is obtained by transferring hodgetheoretic structures that ensure well posedness of the continuous problem from the continuous level to the discrete we show stable discretization is achieved if the finite element spaces satisfy two hypotheses they can be arranged into a subcomplex of this hilbert complex and there exists a bounded cochain projection from that complex to the subcomplex in the next part of the paper we consider the most canonical example of the abstract theory in which the hilbert complex is the de rham complex of a domain in euclidean space we use the koszul complex to construct two families of finite element differential forms show that these can be arranged in subcomplexes of the de rham complex in numerous ways and for each construct a bounded cochain projection the abstract theory therefore applies to give the stability and convergence of finite element approximations of the hodge laplacian other applications are considered as well especially the elasticity complex and its application to the equations of elasticity background material is included to make the presentation self contained for a variety of readers

unconditional long time stability of a velocity vorticity method for the 2d navier stokes equations we prove unconditional long time stability for a particular velocity vorticity discretization of the 2d navier stokes equations the scheme begins with a formulation that uses the lamb vector to couple the usual velocity pressure system to the vorticity dynamics equation and then discretizes with the finite element method in space and implicit explicit bdf2 in time with the vorticity equation decoupling at each time step we prove the method s vorticity and velocity are both long time stable in the l 2 l2 and h 1 h1 norms without any timestep restriction moreover our analysis avoids the use of gronwall type estimates which leads us to stability bounds with only polynomial instead of exponential dependence on the reynolds number numerical experiments are given that demonstrate the effectiveness of the method

a modified diffusion coefficient technique for the convection diffusion equation a new modified diffusion coefficient mdc technique for solving convection diffusion equation is proposed the galerkin finite element discretization process is applied on the modified equation rather than the original one for a class of one dimensional convection diffusion equations we derive the modified diffusion coefficient analytically as a function of the equation coefficients and mesh size then prove that the discrete solution of this method coincides with the exact solution of the original equation for every mesh size and or equation coefficients the application of the derived analytic formula of mdc is extended for other classes of convection diffusion equations where the analytic formula is computed locally within each element according to the mesh size and the values of associated coefficients in each direction the numerical results of the proposed approach for two dimensional variable coefficients with boundary layers convection dominated problems show stable and accurate solutions even on coarse grids accordingly multigrid based solvers retain their efficient convergence rates

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

spectral resolutioned exponential compact higher order scheme srechos for convection diffusion equations a spectral resolutioned exponential compact higher order scheme srechos is developed to solve stationary convection diffusion type of differential equations with constant convection and diffusion coefficients the scheme is o h6 for one dimensional problems and produces a tri diagonal system of equations that can be solved efficiently using thomas algorithm for two dimensional problems the scheme produces an o h4 k4 accuracy over a compact nine point stencil which can be solved using any line iterative approach with alternate direction implicit adi procedure the efficiency of the developed scheme is measured using wave number analysis the analysis shows that the srechos has a much better spectral resolution than any of the existing higher order schemes numerical examples are solved and better performance of the srechos in terms of accuracy over the existing schemes in the literature is demonstrated

finite element solution of the 3d compressible navier stokes equations by a velocity vorticity method this paper presents an alternative formulation to the primitive variable form of the navier stokes equations the approach is based on the use of the velocity and vorticity variables as a logical extension of the stream function vorticity method which is quite effective for two dimensional flows second order equations are obtained for the variables and discretization is through a weak galerkin finite element method followed by newton linearization the velocity and vorticity components are solved simultaneously by a direct solver the scheme is demonstrated for two dimensional and three dimensional incompressible and subsonic internal enclosed flow problems

fast and robust sixth order multigrid computation for the three dimensional convection diffusion equation we present a sixth order explicit compact finite difference scheme to solve the three dimensional 3d convection diffusion equation we first use a multiscale multigrid method to solve the linear systems arising from a 19 point fourth order discretization scheme to compute the fourth order solutions on both a coarse grid and a fine grid then an operator based interpolation scheme combined with an extrapolation technique is used to approximate the sixth order accurate solution on the fine grid since the multigrid method using a standard point relaxation smoother may fail to achieve the optimal grid independent convergence rate for solving convection diffusion equations with a high reynolds number we implement the plane relaxation smoother in the multigrid solver to achieve better grid independency supporting numerical results are presented to demonstrate the efficiency and accuracy of the sixth order compact soc scheme compared with the previously published fourth order compact foc scheme

on error analysis for the 3d navier stokes equations in velocity vorticity helicity form we present a rigorous numerical analysis and computational tests for the galerkin finite element discretization of the velocity vorticity helicity formulation of the equilibrium navier stokes equations nses this formulation recently derived by the authors is the first nse formulation that directly solves for helicity and the first velocity vorticity formulation to naturally enforce incompressibility of the vorticity and preliminary computations confirm its potential we present a numerical scheme prove stability existence of solutions uniqueness under a small data condition and convergence and provide numerical experiments to confirm the theory and illustrate the effectiveness of the scheme on a benchmark problem

inclusion of no slip boundary conditions in the meevc scheme abstract this work presents three methods for enforcing tangential velocity boundary conditions for the meevc scheme which was shown to be mass enstrophy energy and vorticity conserving scheme in the case of inviscid flow 1 while the normal velocity component can be strongly imposed in a div conforming formulation for the velocity field inclusion of the tangential velocity needs to be set through an appropriate choice of vorticity boundary conditions three methods to impose the tangential velocity boundary condition will be discussed the kinematic dirichlet formulation the kinematic neumann formulation and the dynamic neumann formulation the conservation properties of each of the resulting schemes are analyzed and numerical results are shown for the taylor green vortex and for the dipole collision test cases these confirm that kinematic neumann vorticity boundary conditions perform best

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

exponential higher order compact scheme for 3d steady convection diffusion problem abstract in this paper the exponential higher order compact ehoc finite difference schemes proposed by tian and dai 2007 10 for solving the one and two dimensional steady convection diffusion equations with constant or variable convection coefficients are extended to the three dimensional case the proposed ehoc scheme has the feature that it provides very accurate solution exact in case of constant convection coefficient in 1d of the homogeneous equation that is responsible for the fundamental singularity of the homogeneous solution while approximates the particular part of the solution by fourth order accuracy over the nineteen point compact stencil the key properties of this scheme are its stability accuracy and efficiency so that high gradients near the boundary layers can be effectively resolved even on coarse uniform meshes to validate the present ehoc method three test problems mostly with boundary or internal layers are solved comparisons are made between numerical results for the present ehoc scheme and other available methods in the literature

symmetry preserving discretization of turbulent flow we propose to perform turbulent flow simulations in such manner that the difference operators do have the same symmetry properties as the underlying differential operators i e the convective operator is represented by a skew symmetric coefficient matrix and the diffusive operator is approximated by a symmetric positive definite matrix mimicking crucial properties of differential operators forms in itself a motivation for discretizing them in a certain manner we give it a concrete form by noting that a symmetry preserving discretization of the navier stokes equations is stable on any grid and conserves the total mass momentum and kinetic energy for the latter the physical dissipation is to be turned off of coarse being stable on any grid the choice of the grid may be based on the required accuracy solely and the main question becomes how accurate is a symmetry preserving discretization its accuracy is tested for a turbulent flow in a channel by comparing the results to those of physical experiments and previous numerical studies the comparison is carried out for a reynolds number of 5600 which is based on the channel width and the mean bulk velocity based on the channel half width and wall shear velocity the reynolds number becomes 180 the comparison shows that with a fourth order symmetry preserving method a 64 64 32 grid suffices to perform an accurate numerical simulation

a matrix inverse eigenvalue problem and its application the problem of generating a matrix a with specified eigenvalues which maps a given set of vectors of another given set is presented an existence theorem is given and proved a stable algorithm for producing the matrix a is discussed the relation between this problem and the pole assignment problem in control theory is investigated the application of this problem in the design of neural networks is discussed

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

assessment of a vorticity based solver for the navier stokes equations we investigate numerically a recently proposed vorticity based formulation of the incompressible navier stokes equations the formulation couples a velocity pressure system with a vorticity helicity system and is intended to provide a numerical scheme with enhanced accuracy and superior conservation properties for a few benchmark problems we study the performance of a finite element method for this formulation and compare it with the commonly used velocity pressure based finite element method it is shown that both steady and unsteady discrete problems in the new formulation admit simple decoupling strategies followed by the application of iterative solves to auxiliary subproblems further we compare several iterative strategies to solve the discrete problems and study the interplay between the choice of stabilization parameters in the finite element method and the efficiency of linear algebra solvers

natural vorticity boundary conditions on solid walls abstract we derive a new kind of boundary conditions for the vorticity equation with solid wall boundaries for fluid flow problems the formulation uses a dirichlet condition for the normal component of vorticity and neumann type conditions for the tangential components in a galerkin integral formulation the tangential condition is natural i e it is enforced by a right hand side functional and does not impose a boundary constraint on trial and test spaces the functional involves the pressure variable and we discuss several velocity vorticity formulations where the proposed condition is appropriate several numerical experiments are given that illustrate the validity of the new approach