stability analysis of lattice boltzmann methods the lattice boltzmann equation describes the evolution of the velocity distribution function on a lattice in a manner that macroscopic fluid dynamical behavior is recovered although the equation is a derivative of lattice gas automata it may be interpreted as a lagrangian finite difference method for the numerical simulation of the discrete velocity boltzmann equation that makes use of a bgk collision operator as a result it is not surprising that numerical instability of lattice boltzmann methods have been frequently encountered by researchers we present an analysis of the stability of perturbations of the particle populations linearized about equilibrium values corresponding to a constant density uniform mean flow the linear stability depends on the following parameters the distribution of the mass at a site between the different discrete speeds the bgk relaxation time the mean velocity and the wavenumber of the perturbations this parameter space is too large to compute the complete stability characteristics we report some stability results for a subset of the parameter space for a 7 velocity hexagonal lattice a 9 velocity square lattice and a 15 velocity cubic lattice results common to all three lattices are 1 the bgk relaxation time must be greater than 12 corresponding to positive shear viscosity 2 there exists a maximum stable mean velocity for fixed values of the other parameters and 3 as is increased from 12 the maximum stable velocity increases monotonically until some fixed velocity is reached which does not change for larger

an error based approach to complementary formulations of static field solutions the constitutive relationship between fields in electromagnetic media is cast in error form for well posed static problems the error splits systematically into complementary functionals error minimization provides a universal variational principle that generates complementary and dual solution formulations the error which is wholly attributed to the numerical constraints in approximate solutions is always defined and computable locally and globally

a tensor artificial viscosity using a mimetic finite difference algorithm abstract we have developed a two dimensional tensor artificial viscosity for finite difference shock wave computations the discrete viscosity tensor is formed by multiplying the gradient of velocity tensor by a scalar term the scalar term is based on the form of viscosity first presented by kurapatanko and also contains a limiter designed to switch off the viscosity for shockless compression and rigid body rotation mimetic discretizations are used to derive the form of the momentum and energy equations for a nonorthogonal grid where the viscosity tensor is evaluated at the zone edges the advantage of the tensor viscosity is a reduction of the dependence of the solution on the relation of the grid to the flow structure

lattice boltzmann model for the one dimensional nonlinear dirac equation in this paper a lattice boltzmann model for one dimensional nonlinear dirac equation is presented by using double complex valued distribution functions and carefully selected equilibrium distribution functions the effects of space and time resolutions and relaxation time on the accuracy and stability of the model are numerically investigated in detail it is found that the model is of second order accuracy in both space and time and the order of accuracy is near 3 0 at lower grid resolution which shows that the lattice boltzmann method is an effective numerical scheme for the nonlinear dirac equation

lattice boltzmann model for two dimensional unsteady burgers equation in this paper a special lattice boltzmann model is proposed to simulate two dimensional unsteady burgers equation the maximum principle and the stability are proved the model has been verified by several test examples excellent agreement is obtained between numerical predictions and exact solutions the cases of steep oblique shock waves are solved and compared with the two point compact scheme results the study indicates that lattice boltzmann model is highly stable and efficient even for the problems with severe gradients

the modified extended tanh function method for solving burgers type equations abstract by means of the modified extended tanh function metf method the multiple travelling wave solutions of some different kinds of nonlinear partial differential equations are presented and implemented in a computer algebraic system solutions for the nonlinear equations such as one dimensional burgers kdv burgers coupled burgers and two dimensional burgers equations are obtained precisely and so the efficiency of the method can be demonstrated

the solution of coupled burgers equations using adomian pade technique abstract the purpose of this study is to implement adm pade madm pade technique which is a combination of adomian decomposition method modified adomian decomposition method and pade approximation for solving homogeneous inhomogeneous two dimensional parabolic equation also we apply adm pade technique to solve coupled burgers equations the results obtained by adm pade madm pade techniques are compared to those obtained by using adm madm alone the numerical results demonstrate that adm pade madm pade technique gives the approximate solution with faster convergence rate and higher accuracy than using adm madm

a discrete geometric approach to solving time independent schrodinger equation the time independent schrodinger equation stems from quantum theory axioms as a partial differential equation this work aims at providing a novel discrete geometric formulation of this equation in terms of integral variables associated with precise geometric elements of a pair of three dimensional interlocked grids one of them based on tetrahedra we will deduce in a purely geometric way a computationally efficient discrete counterpart of the time independent schrodinger equation in terms of a standard symmetric eigenvalue problem moreover boundary and interface conditions together with non homogeneity and anisotropy of the media involved are accounted for in a straightforward manner this approach yields to a sensible computational advantage with respect to the finite element method where a generalized eigenvalue problem has to be solved instead such a modeling tool can be used for analyzing a number of quantum phenomena in modern nano structured devices where the accounting of the real 3d geometry is a crucial issue

discrete geometric formulation of admittance boundary conditions for frequency domain problems over tetrahedral dual grids in this work it is shown how admittance boundary conditions for electromagnetic boundary value problems in the frequency domain can be formulated for the discrete geometric approach the details are presented for primal grids composed of tetrahedra and barycentric dual grids

correction to the application of the conjugate gradient method for the solution of electromagnetic scattering from arbitrarily oriented wire antennas the method of conjugate gradients is applied to the analysis of radiation from thin wire antennas with this iterative technique it is possible to solve electrically large arbitrarily oriented wire structures without storing any matrices as is conventionally done in the method of moments the basic difference between the proposed method and galerkin s method for the same expansion functions is that for the iterative technique we are solving a least squares problem hence as the order of the approximation is increased the proposed technique guarantees a monotonic decrease of the least squared error parallel ai y parallel 2 whereas galerkin s method does not even though the method converges for any initial guess a good one may significantly reduce the time of computation also explicit error formulas are given for the rate of convergence of this method hence any problem can be solved to a prespecified degree of accuracy it is shown that the method has the advantage of a direct solution as the final solution is obtained in a finite number of steps the method is also suitable for solving singular operator equations in which case the method monotonically converges to the least squares solution with minimum norm numerical results are presented for the thin wire antennas and are compared with the solution obtained by the method of moments

formulation of the eddy current problem in multiply connected regions in terms of h in this paper various formulations for the eddy current problem are presented the formulations are based on solving directly for the magnetic field h and they differ from each other mainly by how the field on the boundary is treated the electromagnetic problem is studied in connection with the fivefold decomposition of the space of square integrable vector fields within a bounded region this provides us with numerical approaches with clear signposts about how to solve the eddy current problem in multiply connected domains besides the fivefold decomposition another essential tool in our approach is whitney elements as they provide the structure needed to retain consistency between the continuous and discrete problems the paper demonstrates the usefulness of these mathematical tools in solving electromagnetic field problems 1998 john wiley sons ltd

geometric interpretation of finite dimensional eddy current formulations this paper gives an explicit geometric interpretation of finite element formulations of the eddy current problem the paper shows step by step how the eddy current problem can be implemented in a finite element kind of software system exploiting familiar geometric ideas such as lengths of edges areas of faces volumes of tetrahedra and the mutual interconnections between the simplices of a mesh the approach is a specific case of the so called geometric techniques copyright 2006 john wiley sons ltd

on geometric discretization of elasticity this paper presents a geometric discretization of elasticity when the ambient space is euclidean this theory is built on ideas from algebraic topology exterior calculus and the recent developments of discrete exterior calculus we first review some geometric ideas in continuum mechanics and show how constitutive equations of linearized elasticity similar to those of electromagnetism can be written in terms of a material hodge star operator in the discrete theory presented in this paper instead of referring to continuum quantities we postulate the existence of some discrete scalar valued and vector valued primal and dual differential forms on a discretized solid which is assumed to be a triangulated domain we find the discrete governing equations by requiring energy balance invariance under time dependent rigid translations and rotations of the ambient space there are several subtle differences between the discrete and continuous theories for example power of tractions in the discrete theory is written on a layer of cells with a nonzero volume we obtain the compatibility equations of this discrete theory using tools from algebraic topology we study a discrete cosserat medium and obtain its governing equations finally we study the geometric structure of linearized elasticity and write its governing equations in a matrix form we show that in addition to constitutive equations balance of angular momentum is also metric dependent all the other governing equations are topological

acceleration of fast multipole method for large scale periodic structures with finite sizes using sub entire domain basis functions an acceleration technique to the fast multipole method fmm has been proposed to handle large scale problems of periodic structures in free space with finite sizes based on the accurate sub entire domain basis functions in the proposed algorithm only nine or 27 elements in the whole impedance matrix are required to be computed and stored for a two dimensional or three dimensional periodic structure and the matrix vector multiply can be performed efficiently using the combination of fast fourier transform and fmm the theoretical analysis and numerical results show that both the memory requirement and computational complexity are only of the order of o n with small constants where n is the total number of unknowns

lattice boltzmann method for the generalized kuramoto sivashinsky equation abstract in this paper a lattice boltzmann model with an amending function is proposed for the generalized kuramoto sivashinsky equation that has the form u t u u x u x x u x x x u x x x x 0 with the chapman enskog expansion the governing evolution equation is recovered correctly from the continuous boltzmann equation it is found that the numerical results agree well with the analytical solutions

computation of stationary 3d halo currents in fusion devices with accuracy control this paper addresses the calculation of the resistive distribution of halo currents in three dimensional structures of large magnetic confinement fusion machines a neumann electrokinetic problem is solved on a geometry so complicated that complementarity is used to monitor the discretization error an irrotational electric field is obtained by a geometric formulation based on the electric scalar potential whereas three geometric formulations are compared to obtain a solenoidal current density a formulation based on the electric vector potential and two geometric formulations inspired from mixed and mixed hybrid finite elements the electric vector potential formulation is usually considered impractical since an enormous computing power is wasted by the topological pre processing it requires to solve this challenging problem we present novel algorithms based on lazy cohomology generators that enable to save orders of magnitude computational time with respect to all other state of the art solutions proposed in literature believing that our results are useful in other fields of scientific computing the proposed algorithm is presented as a detailed pseudocode in such a way that it can be easily implemented

fast algorithms for large scale periodic structures using subentire domain basis functions two efficient algorithms are proposed to analyze a large scale periodic structure with finite size using the subentire domain sed basis functions and the conjugate gradient fast fourier transform cg fft the sed basis function is defined on the support of each single element of the periodic structure in a simplified sed ssed cg fft algorithm all elements of the periodic structure share the same sed basis function as a consequence ssed cg fft can be performed in the whole periodic structure however ssed cg fft becomes less accurate if the gap between two unit elements is very small where the single sed basis function cannot capture the strong mutual coupling in order to consider the mutual coupling an accurate sed ased cg fft algorithm is proposed in this algorithm nine types of sed basis functions are employed to distinguish interior cells edge cells and corner cells as a consequence ased cg fft can be performed in all interior cells of the periodic structure comparing with the conventional method of moments with subdomain basis functions the proposed algorithms are more efficient in both the computational complexity and the memory requirement numerical results are given to test the validity and efficiency of the proposed methods

fast algorithm for electromagnetic scattering by buried conducting plates of large size this letter presents a fast algorithm for electromagnetic scattering by buried conducting plates of large size and arbitrary shape using the conjugate gradient cg method combined with the fast fourier transform fft due to the use of fft in handling the cyclic convolutions related to toeplitz matrices the sommerfeld integrals evaluation for the buried scattering problem which is usually time consuming has been reduced to a minimum the memory required for this algorithm is of the order n the number of unknowns and the computational complexity is of order n sub iter nlogn n sub iter is the iteration number n sub iter spl lt n for large problems

analysis of arbitrary metallic surfaces conformed to a circular cylinder using the conjugate gradient fast fourier transform cg fft method an efficient scheme for analyzing electrodynamic problems involving arbitrary metallic surfaces conformed to a circular cylinder using the conjugate gradient fast fourier transform cg fft method is presented the numerical method outlined is efficient in cpu time and minimizes the memory storage requirements several results are presented and compared with numerical or measured values that appear in the literature

a lattice boltzmann equation for waves we propose a lattice boltzmann model for the wave equation using a lattice boltzmann equation and the chapman enskog expansion we get 1d and 2d wave equations with truncation error of order two the numerical tests show the method can be used to simulate the wave motions

convergence analysis of the high order mimetic finite difference method we prove second order convergence of the conservative variable and its flux in the high order mfd method the convergence results are proved for unstructured polyhedral meshes and full tensor diffusion coefficients for the case of non constant coefficients we also develop a new family of high order mfd methods theoretical result are confirmed through numerical experiments

a surface patch model for polygonal plates a method of moments surface patch model for polygon plates is presented three methods for segmenting polygon plates are considered also the choice between full surface testing and point matching is discussed as it affects the efficiency and generality of the solution numerical results are compared with measurements and independent calculations for polygonal plates of three different shapes including a wire to plant junction

constitutive equations for discrete electromagnetic problems over polyhedral grids in this paper a novel approach is proposed for constructing discrete counterparts of constitutive equations over polyhedral grids which ensure both consistency and stability of the algebraic equations discretizing an electromagnetic field problem the idea is to construct discrete constitutive equations preserving the thermodynamic relations for constitutive equations in this way consistency and stability of the discrete equations are ensured at the base a purely geometric condition between the primal and the dual grids has to be satisfied for a given primal polyhedral grid by properly choosing the dual grid numerical experiments demonstrate that the proposed discrete constitutive equations lead to accurate approximations of the electromagnetic field

comments on application of fft and the conjugate gradient method for the solution of electromagnetic radiation from electrically large and small conducting bodies in the original paper ibid vol ap 34 no 5 p 635 40 1986 t k sarkar et al describe how the fast fourier transform fft and the conjugate gradient method cgm can be used to efficiently make use of the convolutional form of the electric field integral equation for straight wire antennas the commentators agree that the cgm converges more rapidly than the previously used method of steepest descent and the spectral iteration technique they also point out that experience with the dyadic green s function encountered in the two dimensional transverse electric te and three dimensional equations for dielectric bodies is not so encouraging it is found that for these cases the cgm convergence can be quite slow degrading the efficiency of the algorithm in addition the solution obtained using the pulse basis point matching method to discretize the integral equation are very inaccurate for the two dimensional te and three dimensional cases a reply to these comments from the authors of the original paper is included

an a posteriori driven adaptive mixed high order method with application to electrostatics in this work we propose an adaptive version of the recently introduced mixed high order method and showcase its performance on a comprehensive set of academic and industrial problems in computational electromagnetism the latter include in particular the numerical modeling of comb drive and mems devices mesh adaptation is driven by newly derived residual based error estimators the resulting method has several advantageous features it supports fairly general meshes it enables arbitrary approximation orders and has a moderate computational cost thanks to hybridization and static condensation the a posteriori driven mesh refinement is shown to significantly enhance the performance on problems featuring singular solutions allowing to fully exploit the high order of approximation

an analysis of finite volume finite element and finite difference methods using some concepts from algebraic topology in this paper we apply the ideas of algebraic topology to the analysis of the finite volume and finite element methods illuminating the similarity between the discretization strategies adopted by the two methods in the light of a geometric interpretation proposed for the role played by the weighting functions in finite elements we discuss the intrinsic discrete nature of some of the factors appearing in the field equations underlining the exception represented by the constitutive term the discretization of which is maintained as the key issue for numerical methods devoted to field problems we propose a systematic technique to perform this task present a rationale for the adoption of two dual discretization grids and point out some optimization opportunities in the combined selection of interpolation functions and cell geometry for the finite volume method finally we suggest an explanation for the intrinsic limitations of the classical finite difference method in the construction of accurate high order formulas for field problems

variational iteration method for solving burger s and coupled burger s equations by means of variational iteration method the solutions of burger s equation and coupled burger s equations are exactly obtained comparison with the adomian decomposition method is made showing that the former is more effective than the later in this paper he s variational iteration method is introduced to overcome the difficulty arising in calculating adomian polynomials

2 d validation of a formulation for the analysis of 3 d bodies that are periodic in one direction and finite sized in the others a numerical scheme to analyze three dimensional bodies that are periodic in one direction z and finite sized in the other ones x y is presented the geometry and material composition of the body can be arbitrary a new formulation using the conjugate gradient fast fourier transform method cg fft has been developed the formulation is based on the discretization and resolution of the electric field integral equation efie in both the real and spectral domains and leads to an efficient and accurate numerical procedure results are presented for rcs equivalent currents and fields inside 3 d periodic structures infinitely long cylinders with arbitrary shape and material composition these results are compared with analytical solutions and the agreement is found to be good

critical analysis of the spanning tree techniques two algorithms based upon a tree cotree decomposition called in this paper spanning tree technique stt and generalized spanning tree technique gstt have been shown to be useful in computational electromagnetics the aim of this paper is to give a rigorous description of the gstt in terms of homology and cohomology theories together with an analysis of its termination in particular the authors aim to show by concrete counterexamples that various problems related with both stt and gstt algorithms exist the counterexamples clearly demonstrate that the failure of stt and gstt is not an exceptional event but something that routinely occurs in practical applications

lattice boltzmann model for wave propagation a lattice boltzmann model for two dimensional wave equation is proposed by using the higher order moment method the higher order moment method is based on the solution of a series of partial differential equations obtained by using multiscale technique and chapman enskog expansion in order to obtain the lattice boltzmann model for the wave equation with higher order accuracy of truncation errors we removed the second order dissipation term and the third order dispersion term by employing the moments up to fourth order the reversibility in time appears owing to the absence of the second order dissipation term and the third order dispersion term as numerical examples some classical examples such as interference diffraction and wave passing through a convex lens are simulated the numerical results show that this model can be used to simulate wave propagation

convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes the stability and convergence properties of the mimetic finite difference method for diffusion type problems on polyhedral meshes are analyzed the optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved

parallel simulation of particle suspensions with the lattice boltzmann method a description of the steps taken to produce a massively parallel code for particle suspension problems using the lattice boltzmann method is presented a number of benchmarks based on a binary fluid lattice boltzmann model are used to assess the performance of the code in terms of the computational overhead required for the particle problem compared with the fluid only problem and for the scaling of the code to large processor numbers on the blue gene l architecture the additional computational cost of particle suspensions of up to 40 solid volume fraction here over a million particles is negligible compared with the fluid only code

construction of a finite element basis of the first de rham cohomology group and numerical solution of 3d magnetostatic problems we devise an efficient algorithm for the finite element approximation of harmonic fields and the numerical solution of three dimensional magnetostatic problems in particular we construct a finite element basis of the first de rham cohomology group of the computational domain the proposed method works for general topological configurations and does not need the determination of cutting surfaces

generators of h_1 gamma_ h mathbbz for triangulated surfaces construction and classification we consider a bounded lipschitz polyhedron omega subset mathbb r 3 of general topology equipped with a tetrahedral triangulation that induces a mesh gamma h of the surface partial omega we seek a maximal set of surface edge cycles that are independent in h 1 gamma h mathbb z and bounding with respect to the exterior of omega we present an algorithm for constructing suitable 1 cycles in gamma h first representatives of a basis of the homology group h 1 gamma h mathbb z are constructed merely using the combinatorial description of the surface mesh gamma h then a duality pairing based on linking numbers is used to determine those combinations that are bounding with respect to mathbb r 3 setminus omega this is the key to circumventing a triangulation of the exterior region mathbb r 3 setminus omega in the computations for shape regular quasi uniform families of meshes the asymptotic complexity of the algorithm is shown to be o n2 where n is the number of edges of gamma h the scheme provides an essential preprocessing step for all boundary element methods for eddy current simulation which rely on discrete divergence free vectorfields and their description through stream functions

a technique for organizing large moment calculations for use with iterative solution methods through a specific choice of evenly spaced basis and testing functions one can construct a moment matrix formulation wherein significant redundancy is embodied in the matrix a scheme that exploits this redundancy to effect substantial computer storage reduction in an iterative field solution such as the conjugate gradient method is described the expense incurred is a modest increase in computation time the redundancy results from translational similarity features of specific classes of geometries and leads to matrix equations that may be interpreted as discrete convolutions for illustration the analysis here is carried out on a planar scatterer but the strategy can be applied to spherical or cylindrical scatterers

a weak form of the conjugate gradient fft method for plate problems a number of electromagnetic field problems for planar structures can be formulated in terms of a hypersingular integral equation in which a grad div operator acts on a vector potential the vector potential is a convolution of the free space green s function and some surface current density over the domain of interest a weak form of this integral equation is obtained by testing it with subdomain basis functions defined over the plate domain only as a next step the vector potential is expanded in a sequence of subdomain basis functions and the grad div operator is integrated analytically over the plate domain only for the problem of electromagnetic scattering by a plate the method shows excellent numerical performance the numerical difficulties encountered in some previous conjugate gradient fast fourier transform cgfft methods have been eliminated

a residual based error estimator for the mimetic finite difference method we present a local error indicator for the mimetic finite difference method for diffusion type problems on polyhedral meshes under essentially the same general hypotheses used in siam j numer anal 43 1872 1896 2005 to show the convergence of the method we prove the global reliability and local efficiency of the proposed estimator

numerical solution of the coupled viscous burgers equations by chebyshev legendre pseudo spectral method in this paper we consider chebyshev legendre pseudo spectral clps method for solving coupled viscous burgers vb equations a leapfrog scheme is used in time direction while clps method is used for space direction chebyshev gauss lobatto cgl nodes are used for practical computation the error estimates of semi discrete and fully discrete of clps method for coupled vb equations are obtained by energy estimation method the numerical results of the present method are compared with the exact solution for two test problems

electromagnetic scattering from ducts with irregular edges ii noncircular case a formulation is developed for electromagnetic scattering from finite circular ducts terminated with irregular edges the analysis is based on the solution of the electric field integral equation using an entire domain galerkin expansion for both the axial and the circumferential variation of the currents defined in terms of an edge slope dependent vector field that provides simplifying symmetry properties for the method of moments system matrix comparisons are made with edge slope independent formulations the analysis is general and applicable for cases in which the functional variation of the edge irregularities is specified by either a deterministic or a random process circumferential modal decoupling occurs when the irregularities are specified by a stationary stochastic process having a periodic correlation function numerical results are given for edge irregularities governed by a gaussian random process and are compared for various limiting cases with results for right circular cylindrical ducts

simulations of complex fluids by mixed lattice boltzmann finite difference methods abstract we present the numerical results of simulations of complex fluids under shear flow we employ a mixed approach which combines the lattice boltzmann method for solving the navier stokes equation and a finite difference scheme for the convection diffusion equation the evolution in time of shear banding phenomenon is studied this is allowed by the presented numerical model which takes into account the evolution of local structures and their effect on fluid flow

numerical study of the solution of the burgers and coupled burgers equations by a differential transformation method in this paper the differential transformation method dtm is employed to obtain the numerical analytical solutions of the burgers and coupled burgers equations we begin by showing how the differential transformation method applies to the linear and nonlinear parts of any pde and give some examples to illustrate the sufficiency of the method for solving such nonlinear partial differential equations we also compare it against three famous methods namely the homotopy perturbation method the homotopy analysis method and the variational iteration method these results show that the technique introduced here is accurate and easy to apply

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

a geometric integral formulation for eddy currents summary in the computational electromagnetism community it is known how the differential formulation of an eddy current problem can be translated into a finite dimensional system of equations involving circulations and fluxes by means of the so called discrete geometric approach this is done by exploiting the geometric structure behind maxwell s equations in this paper we will show how the same discrete geometric approach can be profitably used also to discretize an eddy current problem formulated in an integral way we rely on a purely geometric definition of a novel set of face vector basis functions that we use to construct the discrete counterparts matrices of both the ohm s constitutive relation and of the integral relation between the magnetic vector potential and the eddy current density vector the symmetry and positive definiteness of such matrices will be demonstrated and their geometric structure will be apparent copyright q 2010 john wiley sons ltd

rcs computation of large inhomogeneous objects using a fast integral equation solver in this paper we apply a fast fourier transform fft accelerated volume integral equation solver to compute the radar cross section of large scale inhomogeneous objects this method is related to bojarski s k space method and the subsequent conjugate cg and biconjugate gradient bcg fft algorithms the method developed here combines the weak form discretization with the bcg and stabilized bcg bcgs solvers we show that this method has marked improvements over related algorithms the numerical method has been validated by mie series for multilayer spheres and applied to some practical problems with this method we are currently able to solve a three dimensional problem with a volume of size 3648 spl lambda sup 3 21 23 million unknowns on a single workstation

a high order compact finite difference scheme and precise integration method based on modified hopf cole transformation for numerical simulation of n dimensional burgers system abstract this paper modifies a n dimensional hopf cole transformation to the n dimensional burgers system we obtain the n dimensional heat conduction equation through the modification of the hopf cole transformation then the fourth order precise integration method pim in combination with a spatially global sixth order compact finite difference cfd scheme is presented to solve the equation with high accuracy moreover coupling with the strang splitting method the scheme is extended to multi dimensional two three dimensional burgers system numerical results show that the proposed method appreciably improves the computational accuracy compared with the existing numerical method moreover the two dimensional and three dimensional examples demonstrate excellent adaptability and the numerical simulation results also have very high accuracy in medium reynolds numbers

a scheme to analyze conducting plates of resonant size using the conjugate gradient method and the fast fourier transform a scheme for analyzing electrodynamic problems involving conducting plates of resonant size using the conjugate gradient cg method and the fast fourier transform fft is presented in detail the problems are analyzed by solving their corresponding electric field integral equation the procedure is made easy and systematic by using a sampling process with rooftop functions to represent the induced current and pulses to average the fields these functions have been widely used in moment method mm applications the scheme is an efficient numerical tool benefiting from the good convergence and low memory requirements of the cg and the low cpu time consumed in performing convolutions with the fft in comparison with the mm the scheme avoids the storage of large matrices and reduces the computer time by an order of magnitude several results are presented and compared with analytical numerical or measured values that appear in the literature

lattice electromagnetic theory from a topological viewpoint the language of differential forms and topological concepts are applied to study classical electromagnetic theory on a lattice it is shown that differential forms and their discrete counterparts cochains provide a natural bridge between the continuum and the lattice versions of the theory allowing for a natural factorization of the field equations into topological field equations i e invariant under homeomorphisms and metric field equations the various potential sources of inconsistency in the discretization process are identified distinguished and discussed a rationale for a consistent extension of the lattice theory to more general situations such as to irregular lattices is considered

appropriate stabilized galerkin approaches for solving two dimensional coupled burgers equations at high reynolds numbers abstract this paper aims to seek proper stabilized galerkin methods for solving the two dimensional coupled burgers equations at high reynolds numbers the stabilization techniques employed here include the streamline upwind petrov galerkin supg method the spurious oscillations at layers diminishing sold method and the characteristic galerkin cg method the first two methods are combined with the crank nicolson scheme for time discretization and the last one is applied in its semi implicit version different from most of the studies on the equations which are usually devoted to improving the accuracy of computed solution in the case of low reynolds numbers this paper mainly focuses on keeping the stability of the solution at high reynolds numbers which is significant in practical applications and also challenging in numerical computation we study two problems equipped with mixed boundary conditions and only dirichlet boundary conditions respectively numerical experiments reveal that the supg method is optimal for the former problem and the sold method is more appropriate for the latter one in addition the performances of these methods demonstrate the difference between the two problems which is seldom mentioned previously and might be helpful to other conventional methods intending to solve the problems at high reynolds numbers and last since sold methods have rarely been utilized to solve nonlinear unsteady problems before this study also indicates the potential of this class of methods to solve nonlinear unsteady convection dominated problems

lattice boltzmann model for nonlinear convection diffusion equations a lattice boltzmann model for convection diffusion equation with nonlinear convection and isotropic diffusion terms is proposed through selecting equilibrium distribution function properly the model can be applied to the common real and complex valued nonlinear evolutionary equations such as the nonlinear schrodinger equation complex ginzburg landau equation burgers fisher equation nonlinear heat conduction equation and sine gordon equation by using a real and complex valued distribution function and relaxation time detailed simulations of these equations are performed and it is found that the numerical results agree well with the analytical solutions and the numerical solutions reported in previous studies

complementary geometric formulations for electrostatics summary the simultaneous use of a pair of complementary discrete formulations for electrostatic boundary value problems bvps allows to accurately compute electromagnetic quantities such as capacitance or electrostatic force with a minimum computational effort in fact the two formulations provide the upper and lower bounds for these quantities and their averages result quite close to the exact solution even for extremely coarse meshes despite the potential benefit to the many three dimensional large scale applications taking advantage of this feature is not exploited in practice due to theoretical difficulties in the potential design the aim of this paper is to fill this gap by rigorously introducing a pair of three dimensional complementary geometric formulations to solve electrostatic bvps on domains covered by conformal polyhedral meshes in particular an original formulation based on a vector potential is introduced by using cohomology theory with integer coefficients it is shown how the so called thick links are needed which are representatives of the second cohomology group generators of the dielectric region two easy to implement graph theoretic algorithms to automatically find such a basis with optimal computational complexity are described some benchmark problems are presented to show how the simultaneous use of both formulations yields to a sensible computational advantage therefore solvers based on complementary formulations should be embedded in the next generation of electromagnetic computer aided engineering cae softwares copyright 2010 john wiley sons ltd

a chebyshev spectral collocation method for solving burgers type equations in this paper we elaborated a spectral collocation method based on differentiated chebyshev polynomials to obtain numerical solutions for some different kinds of nonlinear partial differential equations the problem is reduced to a system of ordinary differential equations that are solved by runge kutta method of order four numerical results for the nonlinear evolution equations such as 1d burgers kdv burgers coupled burgers 2d burgers and system of 2d burgers equations are obtained the numerical results are found to be in good agreement with the exact solutions numerical computations for a wide range of values of reynolds number show that the present method offers better accuracy in comparison with other previous methods moreover the method can be applied to a wide class of nonlinear partial differential equations

a lattice boltzmann model for the generalized burgers huxley equation abstract in this paper we develop a lattice boltzmann model for the generalized burgers huxley equation gbhe by choosing the proper time and space scales and applying the chapman enskog expansion the governing equation is recovered correctly from the lattice boltzmann equation and the local equilibrium distribution functions are obtained excellent agreement with the exact solution is observed and better numerical accuracy is obtained than the available numerical result the results indicate the present model is satisfactory and efficient the method can also be applied to the generalized burgers fisher equation and be extended to multidimensional cases

lattice boltzmann simulation of some nonlinear convection diffusion equations in this work we proposed a lattice boltzmann model for the nonlinear convection diffusion equation ncde with anisotropic diffusion the constraints on the model for correctly recovering macroscopic equation are also carefully analyzed which are ignored in some existing work detailed simulations of some 1d 2d ncdes including the nonlinear schrodinger equation nlse buckley leverett equation with discontinuous initial data ncde with anisotropic diffusion and generalized zakharov system are performed the numerical results obtained by the proposed model agree well with the analytical solutions and or the numerical solutions reported in previous studies it is also found that for complex valued nlse the model using a complex distribution function is superior to that using two real distribution functions for the real and imaginary parts of the nlse separately

lattice boltzmann model for generalized nonlinear wave equations in this paper a lattice boltzmann model is developed to solve a class of the nonlinear wave equations through selecting equilibrium distribution function and an amending function properly the governing evolution equation can be recovered correctly according to our proposed scheme in which the chapman enskog expansion is employed we validate the algorithm on some problems where analytic solutions are available including the second order telegraph equation the nonlinear klein gordon equation and the damped driven sine gordon equation it is found that the numerical results agree well with the analytic solutions which indicates that the present algorithm is very effective and can be used to solve more general nonlinear problems

convergence of the conjugate gradient method when applied to matrix equations representing electromagnetic scattering problems an iterative procedure based on the conjugate gradient method is used to solve a variety of matrix equations representing electromagnetic scattering problems in an attempt to characterize the typical rate of convergence of that method it is found that this rate depends on the cell density per wavelength used in the discretization the presence of symmetries in the solution and the degree to which mixed cell sizes are used in the models assuming cell densities used in the discretization are in the range of ten per linear wavelength the iterative algorithm typically requires n 4 to n 2 steps to converge to necessary accuracy where n is the order of the matrix under consideration

aggregation based algebraic multigrid for convection diffusion equations we consider the iterative solution of large sparse linear systems arising from the upwind finite difference discretization of convection diffusion equations the system matrix is then an m matrix with nonnegative row sum and further when the convective flow has zero divergence the column sum is also nonnegative possibly up to a small correction term we investigate aggregation based algebraic multigrid methods for this class of matrices a theoretical analysis is developed for a simplified two grid scheme with one damped jacobi postsmoothing step an uncommon feature of this analysis is that it applies directly to problems with variable coefficients e g to problems with recirculating convective flow on the basis of this theory we develop an approach in which a guarantee is given on the convergence rate thanks to an aggregation algorithm that allows an explicit control of the location of the eigenvalues of the preconditioned matrix some issues that remain beyond the analysis are discussed in the

automatic generation of cuts on large sized meshes for the t ω geometric eddy current formulation in this paper a geometric t x formulation to solve eddy current problems on a tetrahedral mesh is presented when non simply connected conducting regions are considered the formulation requires the socalled thick cuts while in the literature more attention is usually given to the so called thin cuts while the automatic construction of thin cuts has been theoretically solved many years ago no implementation of an algorithm to compute the thick cuts which can be used in practice exists so far in this paper we propose how to fill this gap by introducing an algorithm to automatically compute the thick cuts on real sized meshes based on a belted tree and a tree cotree decomposition the belted tree is constructed by means of a homology computation by exploiting efficient reduction methods a number of benchmarks are presented to demonstrate the generality and the robustness of the algorithm a rigorous definition of thick cuts which necessarily has to rely on cohomology is presented in addition

mimetic finite difference method for the stokes problem on polygonal meshes various approaches to extend finite element methods to non traditional elements general polygons pyramids polyhedra etc have been developed over the last decade the construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis the mimetic finite difference mfd method works on general polygonal meshes and has many similarities with low order finite element methods both schemes try to preserve the fundamental properties of the underlying physical and mathematical models the essential difference between the two schemes is that the mfd method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices since no extension of basis functions inside the mesh elements is required practical implementation of the mfd method is simple for polygonal meshes that may include degenerate and non convex elements in this article we present a new mfd method for the stokes problem on arbitrary polygonal meshes and analyze its stability the method is developed for the general case of tensor coefficients which allows us to apply it to a linear elasticity problem as well numerical experiments show for the velocity variable second order convergence in a discrete l 2 norm and first order convergence in a discrete h 1 norm for the pressure variable first order convergence is shown in the l 2 norm

a new lattice boltzmann model for solving the coupled viscous burgers equation abstract in this paper a new lattice boltzmann model for the coupled nonlinear system of viscous burgers equation is proposed by using the double evolutionary equations through selecting equilibrium distribution functions and amending functions properly the governing evolution system can be recovered correctly according to our proposed scheme in which the chapman enskog expansion is employed the effects of space and time resolutions on the accuracy and stability of the model are numerically investigated in detail the numerical solutions for various initial and boundary conditions are calculated and validated against analytic solutions or other numerical solutions reported in previous studies it is found that the numerical results agree well with the analytic solutions which indicates the potential of the present algorithm for solving the coupled nonlinear system of viscous burgers equation

a combined finite element boundary integral formulation for solution of two dimensional scattering problems via cgfft a novel technique is presented for computing the scattering by two dimensional structures of arbitrary inhomogeneity the proposed approach combines the usual finite element fe method with the boundary integral equation to formulate a discrete system this is subsequently solved via the conjugate gradient cg algorithm a particular characteristic of the method is the use of rectangular boundaries to enclose the scatterer several of the resulting boundary integrals are then convolutions and can be evaluated via the fast fourier transform fft in the implementation of the cg algorithm the solution approach presented here offers the principal advantage of having o n memory demand and employs a one dimensional fft as against the two dimensional fft required in a traditional implementation of the proposed cg fft algorithm the speed of the proposed solution method is compared with that of the traditional cg fft algorithm results are presented for several rectangular composite cylinders and one perfectly conducting cylinder these are shown to be in excellent agreement with the moment method

lattice boltzmann models for two dimensional coupled burgers equations abstract in this paper two lattice boltzmann models for two dimensional coupled burgers equations are proposed through treating the part or all of convection items as the source term where the spatial gradient can be calculated by the distribution function the models can exactly recover the burgers equations without any assumptions some numerical tests are also performed to validate the present models it is found that the proposed models are more accurate and efficient in solving two dimensional coupled burgers equations

phase separation of incompressible binary fluids with lattice boltzmann methods we introduce new versions of lattice boltzmann methods lbm for incompressible binary mixtures where fluctuations of total density are inhibited as a test for the improved algorithms we consider the problem of phase separation of two dimensional binary mixtures quenched from a disordered state into the coexistence region we find that the stability properties of lbm are greatly improved the control of density fluctuations and the possibility of running longer simulations allow a more precise evaluation of the growth exponent

numerical study of the ordering properties of lamellar phase the quench of a binary mixture into a stable lamellar phase is numerically investigated we present a novel approach to study the phenomenological equations governing the system dynamics basedon the combination of lattice boltzmann and1nite d equations we can access large systems on time scales long enough to 1nd evidence of dynamical scaling regime in the ordering process when hydrodynamics is operating c

two dimensional lattice boltzmann model for compressible flows with high mach number abstract in this paper we present an improved lattice boltzmann model for compressible navier stokes system with high mach number the model is composed of three components i the discrete velocity model by m watari and m tsutahara phys rev e 67 2003 036306 ii a modified lax wendroff finite difference scheme where reasonable dissipation and dispersion are naturally included iii artificial viscosity the improved model is convenient to compromise the high accuracy and stability the included dispersion term can effectively reduce the numerical oscillation at discontinuity the added artificial viscosity helps the scheme to satisfy the von neumann stability condition shock tubes and shock reflections are used to validate the new scheme in our numerical tests the mach numbers are successfully increased up to 20 or higher the flexibility of the new model makes it suitable for tracking shock waves with high accuracy and for investigating nonlinear nonequilibrium complex systems

a new set of basis functions for the discrete geometric approach by exploiting the geometric structure behind maxwell s equations the so called discrete geometric approach allows to translate the physical laws of electromagnetism into discrete relations involving circulations and fluxes associated with the geometric elements of a pair of interlocked grids the primal grid and the dual grid to form a finite dimensional system of equations discrete counterparts of the constitutive relations must be introduced in addition they are referred to as constitutive matrices which must comply with precise properties symmetry positive definiteness consistency in order to guarantee the stability and consistency of the overall finite dimensional system of equations the aim of this work is to introduce a general and efficient set of vector functions associated with the edges and faces of a polyhedral primal grids or of a dual grid obtained from the barycentric subdivision of the boundary of the primal grid these vector functions comply with precise specifications which allow to construct stable and consistent discrete constitutive equations for the discrete geometric approach in the framework of an energetic method

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

on a class of finite step iterative methods conjugate directions for the solution of an operator equation arising in electromagnetics a class of finite step iterative methods for the solution of linear operator equations is presented specifically the basic principles of the method of conjugate directions are developed gaussian elimination and the method of conjugate gradients are then presented as two special cases with an arbitrary initial guess the method of conjugate gradient always converges to the solution in at most n iterations where n is the number of independent eigenvalues for the operator in the finite dimensional space in which the problem is being solved the conjugate gradient method requires much less storage sim 5n than the conventional matrix methods sim n 2 in the solution of problems of higher complexity also after each iteration the quality of the solution is known in the conjugate gradient method the conjugate gradient method is also superior to the spectral iterative method as the latter does not always converge and it doubles the complexity of a given problem unnecessarily four versions of the conjugate gradient method are presented in detail and numerical results for a thin wire scatterer are given to illustrate various properties of each version

piecewise uniform bases and energetic approach for discrete constitutive matrices in electromagnetic problems in the paper we introduce piecewise uniform edge and face vector functions on a simplicial primal cell complex having geometric structure common to whitney s vector functions we also introduce piecewise uniform bases functions in the barycentric dual complex where the analogous of whitney s functions does not exist by using these piecewise uniform bases functions and by exploiting an energetic approach we construct symmetric positive definite constitutive matrices for discrete maxwell s equations we also prove that these constitutive matrices are deeply related to those of the finite elements with whitney s bases functions copyright 2005 john wiley sons ltd

base functions and discrete constitutive relations for staggered polyhedral grids an electromagnetic problem can be discretized on a pair of interlocked primal dual grids according to discrete geometric approaches like the finite integration technique fit or the cell method cm the critical aspect is however the construction of the discrete counterparts of the constitutive relations assuring stability and consistency of the overall discrete system of algebraic equations initially only orthogonal cartesian grids where considered more recently primal grids of tetrahedra and oblique prisms with triangular base can be handled with this paper a novel set of edge and face vector functions for general polyhedral primal grids is presented complying with precise specifications which allow to construct stable and consistent discrete constitutive equations in the framework of an energetic approach

efficient solution of the schroedinger poisson equations in layered semiconductor devices in this paper we present several mathematical models that can be used to create approximate solutions of the three dimensional schroedinger poisson equation in layered semiconductor devices a general algorithmic strategy that can be used to create efficient solution procedures for each of these models is described computational results demonstrating the accuracy and efficiency that can be obtained with the use of these models is presented

homology and cohomology computation in finite element modeling a homology and cohomology solver for finite element meshes is represented it is an integrated part of the finite element mesh generator gmsh we demonstrate the exploitation of the cohomology computation results in a finite element solver and use an induction heating problem as a working example the homology and cohomology solver makes the use of a vector scalar potential formulation straightforward this gives better overall performance than a vector potential formulation cohomology computation also clarifies the lumped parameter coupling of the problem and enables the user to obtain useful postprocessing data as a part of the finite element solution

a novel lattice boltzmann model for the coupled viscous burgers equations abstract in this work a new lattice boltzmann model for coupled burgers equations is proposed through selecting proper distribution functions unlike the previous models the present model can exactly recover the coupled equations without any assumptions a detailed numerical study on several coupled burgers equations is performed to validate the present model and the results show that the present model not only has a second order convergence rate in space but also is more accurate than the previous model

mimetic discretizations for maxwell s equations we have constructed reliable finite difference methods for approximating the solution to maxwell s equations using accurate discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus in discrete form the numerical approximation does not have spurious modes and mimics many fundamental properties of the underlying physical problem including conservation laws symmetries in the solution and the nondivergence of particular vector fields numerical examples demonstrate the high quality of the method when the medium is strongly discontinuous and for nonorthogonal nonsmooth computational grids

a numerical treatment of the coupled viscous burgers equation in the presence of very large reynolds number abstract a numerical investigation of the coupled viscous burgers equation for very large reynolds numbers do not exist in the literature coupled viscous burgers equations are solved numerically in the presence of very large reynolds numbers for this case numerical approach to the coupled viscous burgers equation via contributions of two effective methods is used the first component of the mixed method is finite difference method and the second one is differential quadrature method for this process the third order modified cubic b spline functions are used as base function to display the effectiveness of the present mixed method four different test problems have been investigated for various values of the coefficients more particularly for very large reynolds numbers in other words for the very small value of kinematic viscosity parameters solutions are obtained error norms are calculated and compared with analytical results and also with numerical results of the related literature present results display that the present mixed method obtains high accurate solutions and in compatibility with both of the analytical and numerical results

second order lattice boltzmann methods for pdes of asian option pricing with regime switching this paper establishes lattice boltzmann models with five amending functions for solving system of partial differential equations pdes arising in asian options pricing with regime switching with the chapman enskog multi scale expansion the pdes are recovered correctly from the continuous boltzmann equation and then the lattice boltzmann method lbm is proposed in the lbm the coefficients of equilibrium distribution and amending functions are taken as polynomials instead of constants in the traditional lbms the lbm has second order convergence rate in space and first order convergence rate in time the stability convergence rates and computational cost of lbms are studied and verified by numerical examples