high order finite difference methods for a second order dual phase lagging models of microscale heat transfer two adi solvers are given for 2d and 3d heat conduct equations at microscale they are uniquely solvable it is shown that they are of order 2 in time and order 4 in space numerical results exhibit the efficiency of the solvers in this paper a compact alternating direction implicit adi method which combines the fourth order compact difference for the approximations of the second spatial derivatives and the approximation factorizations of difference operators is firstly presented for solving two dimensional 2d second order dual phase lagging models of microscale heat transfer by the discrete energy method it is shown that it is second order accurate in time and fourth order accurate in space with respect to l2 norms additionally the compact adi method is successfully generalized to solve corresponding three dimensional 3d problem also the convergence result of the solver for 3d case is given rigorously finally numerical examples are carried out to testify the computational efficiency of the algorithms and exhibit the correctness of theoretical results

bilum block versions of multielimination and multilevel ilu preconditioner for general sparse linear systems we introduce block versions of the multielimination incomplete lu ilum factorization preconditioning technique for solving general sparse unstructured linear systems these preconditioners have a multilevel structure and for certain types of problems may exhibit properties that are typically enjoyed by multigrid methods several heuristic strategies for forming blocks of independent sets are introduced and their relative merits are discussed the advantages of block ilum over point ilum include increased robustness and efficiency we compare several versions of the block ilum point ilum and the dual threshold based ilut preconditioners in particular tests with some convection diffusion problems show that it may be possible to obtain convergence that is nearly independent of the reynolds number as well as of the grid size

a mixed collocation finite difference method for 3d microscopic heat transport problems three dimensional time dependent initial boundary value problems of a novel microscopic heat equation are solved by the mixed collocation finite difference method in and on the boundaries of a particle when the thickness is much smaller than both the length and width the collocation method on fixed grid size is used to approximate the space operator whereas the finite difference scheme is used for time discretization this new mixed method is applied to a novel heat problem in a particle in order to compute the temperature distribution in and on the particle s surface the second derivatives of the basis functions for the spectral approximation are derived direct substitution of derivatives in the model transforms the differential equation into a linear system of equations that is solved by the specific preconditioned conjugate gradient method the high order accuracy and resolution achieved by the proposed method allows one to obtain engineering accuracy solution on coarse meshes the consistency stability and convergence analysis are provided and numerical results are presented

a general meshsize fourth order compact difference discretization scheme for 3d poisson equation a fourth order compact difference scheme with unrestricted general meshsizes in different coordinate directions is derived to discretize three dimensional poisson equation on a regular cubic domain the difference scheme derivation procedure makes use of the symbolic representation of the finite difference schemes and is easier to understand in such complex three dimensional manipulations we use a preconditioned conjugate gradient method to solve the resulting sparse linear systems and verify the formal order of convergence of the derived fourth order finite difference scheme

unconditionally stable finite difference scheme and iterative solution of 2d microscale heat transport equation abstract a two dimensional time dependent heat transport equation at the microscale is derived a second order finite difference scheme in both time and space is introduced and the unconditional stability of the finite difference scheme is proved a computational procedure is designed to solve the discretized linear system at each time step by using a preconditioned conjugate gradient method numerical results are presented to validate the accuracy of the finite difference scheme and the efficiency of the proposed computational procedure

an alternating direction implicit method for a second order hyperbolic diffusion equation with convection a numerical method is presented to solve a two dimensional hyperbolic diffusion problem where is assumed that both convection and diffusion are responsible for flow motion since direct solutions based on implicit schemes for multidimensional problems are computationally inefficient we apply an alternating direction method which is second order accurate in time and space the stability of the alternating direction method is analyzed using the energy method numerical results are presented to illustrate the performance in different cases

equational logic of circular data type specification abstract iteration theories introduced by bloom elgot and wright in 1980 formalize the equational properties of the strong behaviors of flowchart algorithms we show that the same equational properties are shared by the functors used to specify circular data types lehmann and smyth 1981 have shown how to specify circular data types such as stacks as fixed points of certain functors the called gw functors for example the set of stacks of elements in the set a is a solution to the equation in the variable x x f a x where f x x 1 a x is the functor on set taking the pair a x to the disjoint union of the singleton set 1 and the product a x such equations have initial solutions f a which in turn determine a solution functor f the equational properties of the operation f f are precisely captured by the axioms for iteration theories more precisely we show how the structure th c of all functors c n c p n p 0 on the category c forms an iteration theory and conversely any identity valid in all such structures is valid in the class of all iteration theories

a spacetime discontinuous galerkin method for hyperbolic heat conduction non fourier conduction models remedy the paradox of infinite signal speed in the traditional parabolic heat equation for applications involving very short length or time scales hyperbolic conduction models better represent the physical thermal transport processes this paper reviews the maxwell cattaneo vernotte modification of the fourier conduction law and describes its implementation within a spacetime discontinuous galerkin sdg finite element method that admits jumps in the primary variables across element boundaries with arbitrary orientation in space and time a causal advancing front meshing procedure enables a patch wise solution procedure with linear complexity in the number of spacetime elements an h adaptive scheme and a special sdg shock capturing operator accurately resolve sharp solution features in both space and time numerical results for one spatial dimension demonstrate the convergence properties of the sdg method as well as the effectiveness of the shock capturing method simulations in two spatial dimensions demonstrate the proposed method s ability to accurately resolve continuous and discontinuous thermal waves in problems where rapid and localized heating of the conducting medium takes place

on the classification of nuclear c algebras two of the most influential works on c algebras from the mid seventies brown douglas and fillmore s 6 and elliott s 21 both contain uniqueness and existence results in the now standard sense which we shall outline below these papers served as keystones for two separate theories kk theory and the classification program which for many years parted ways with only moderate interaction but with this common origin in mind it is not surprising that recent years have seen a fruitful interaction which has been one of the main engines behind rapid progress in the classification program in the present paper we take this interaction even further combining a concept of quasidiagonality for representations and a new characterization of equivalence in the cuntz picture of kk theory we achieve a general uniqueness result and from the quasidiagonality notion we derive a corresponding general existence result by comparing well known realizations of the kk groups these results are then employed to obtain new classification results for certain classes of quasidiagonal c algebras introduced by h lin an important novel feature of these classes is that they are defined by a certain local approximation property rather than by an inductive limit construction we emphasize that our fundamental uniqueness result does not depend on the universal coefficient theorem uct nor on nuclearity of the c algebras involved on the other hand when we refine the uniqueness result for use in classification we need to gradually add such conditions on the c algebras it is to be expected that requirements of this type are necessary but we have found that holding them back as long as possible leads to more conceptual proofs while clarifying the role of the uct and nuclearity in classification further since uniqueness results answer fundamental questions about kk and k theory they are of interest in themselves we have pursued this theme in 14 and defer to this paper the proof of the new characterization of kk theory which lies behind our uniqueness results our existence and uniqueness results are in the spirit of the classical ext theory from 6 the main complication overcome in the paper is to control the stabilization which is necessary when one works with finite c algebras in the infinite case where programs of this type have already been successfully carried out stabilization is unnecessary yet our methods are sufficiently versatile to allow us to reprove from a handful of basic results the classification of purely infinite nuclear c algebras of kirchberg and phillips

an unconditionally stable three level explicit difference scheme for the schro dinger equation with a variable coefficient a three level explicit difference scheme for the schrodinger equation with a variable coefficient is established it is proved by the discrete energy method that the scheme is unconditionally stable

an alternative discretization and solution procedure for the dual phase lag equation we describe an alternative numerical treatment of the dual phase lag equation often used to account for microscale short time heat transport the approach consists of an undecomposed formulation of the partial differential equation resulting from taylor expansion with respect to lag times of the original delay partial differential equation trapezoidal integration in time and centered differencing in space provide an accurate discretization as demonstrated by comparisons with analytical and experimental results in one dimension and via grid function convergence tests in three dimensions for relatively fine 3 d grids the approach is approximately six times faster than a standard explicit scheme and nearly three times faster than an implicit method employing conjugate gradient iteration at each time step

symmetries of the car algebra a automorphism of period two of the uhf c algebra of type 2 is constructed for which the fixed point algebra is not approximately finite dimensional the construction also yields a diagonal maximal commutative c subalgebra whose spectrum is not totally disconnected variations and consequences of the construction are discussed

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

varieties of iteration theories the equational properties of iteration when combined with composition and pairing are captured by the notion of iteration theory which was introduced in siam j comput 9 1980 pp 26 45 9 1980 pp 525 540 we believe although all of our evidence will not be exhibited here that every iterative construction satisfies at least the properties of iteration theories in this paper axiomatizations are given for several varieties of iteration theories which occur naturally in the semantics of programming languages i e those generated by theories of trees j comput sys sci 16 1978 pp 362 399 theories of sequacious functions proc 1973 colloquium vol 80 studies in logic north holland amsterdam 1975 pp 175 230 theories of partial functions and theories of both sequacious and partial functions with distinguished predicates we show which additional equations must be added to the axioms for iteration theories comput ling comput lang 14 1980 pp 183 207 in order to

a new two level implicit discretization of o k 2 kh 2 h 4 for the solution of singularly perturbed two space dimensional non linear parabolic equations we propose a new two level implicit difference method of o k2 kh2 h4 for the solution of singularly perturbed non linear parabolic differential equation e uxx uyy f x y t u ux u y ut 0 0 subject to appropriate initial and dirichlet boundary conditions where k 0 and h 0 are grid sizes in time and space directions respectively and e 0 is a small parameter we also develop new methods of o kh2 h4 for the estimates of u x and u y in all cases we use 9 spatial grid points and a single computational cell the proposed methods are directly applicable to singular problems we do not require any special scheme to solve singular problems we also discuss alternating direction implicit adi method for solving diffusion equation in polar cylindrical coordinates this method permits multiple use of the one dimensional tri diagonal algorithm with a considerable saving in computing time and produces a very efficient solver it is shown that the adi method is unconditionally stable numerical experiments are conducted to test the high accuracy of the proposed methods and compared with the exact solutions

approximation of complex symmetric operators an operator t on a complex hilbert space mathcal h is called a complex symmetric operator if there exists a conjugate linear isometric involution c mathcal h rightarrow mathcal h so that ctc t in this paper we study the approximation of complex symmetric operators by virtue of an intensive analysis of compact operators in singly generated c algebras we obtain a complete characterization of norm limits of complex symmetric operators and provide a classification of complex symmetric operators up to approximate unitary equivalence this gives a general solution to the norm closure problem for complex symmetric operators as an application we provide a concrete description of partial isometries which are norm limits of complex symmetric operators

a new high accuracy two level implicit off step discretization for the system of two space dimensional quasi linear parabolic partial differential equations abstract in this article we propose a two level compact implicit off step discretization for the solution of the system of 2d quasi linear parabolic partial differential equations subject to suitable initial and boundary conditions new methods for the estimates of first order space derivatives of the solution are also derived these methods are fourth order accurate in space and second order accurate in time and involve only nine spatial grid points of a single compact cell we further develop the alternating direction implicit adi scheme for a general linear parabolic equation which is shown to be unconditionally stable for the heat equation in polar coordinates the proposed methods are directly applicable to singular problems without the need of any special technique which is the main advantage of this work the method is effectively applied to the time dependent navier stokes model equations in polar coordinates numerical examples are provided to illustrate the accuracy of the methods

high accuracy finite difference scheme for three dimensional microscale heat equation a compact finite difference scheme is developed to the three dimensional microscale heat transport equation this new scheme is fourth order in space and second order in time it is proved to be unconditionally stable with respect to initial values numerical results are provided for comparison testing purpose