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

the moser trudinger inequality on kahler einstein manifolds we prove the conjecture of tian on the strong form of the moser trudinger inequality for k ahler einstein manifolds with positive first chern class when there are no holomorphic vector fields and more generally when the setting is invariant under a maximal compact subgroup of the automorphism group

truncations of level 1 of elements in the loop group of a reductive group the aim of this article is to dene and study a new invariant of elements of loop groups that is invariant under conjugation by a hyperspecial maximal open subgroup and that we call the truncation of level 1 we classify truncations of level 1 and describe their specialization behavior furthermore we prove group theoretic conditions for the set of conjugacy classes obtained from elements of a given truncation of level 1 and in particular for the generic conjugacy class in any given truncation stratum in the last section we relate our invariant to the ekedahl oort stratication of the siegel moduli space and to generalizations to other pel shimura varieties

the supersingular locus of the shimura variety of gu 1 n 1 ii we complete the study of the supersingular locus mathcal m mathrm ss in the fiber at p of a shimura variety attached to a unitary similitude group gu 1 n 1 over in the case that p is inert this was started by the first author in can j math 62 668 720 2010 where complete results were obtained for n 2 3 the supersingular locus mathcal m mathrm ss is uniformized by a formal scheme mathcal n which is a moduli space of so called unitary p divisible groups it depends on the choice of a unitary isocrystal n we define a stratification of mathcal n indexed by vertices of the bruhat tits building attached to the reductive group of automorphisms of n we show that the combinatorial behavior of this stratification is given by the simplicial structure of the building the closures of the strata and in particular the irreducible components of mathcal n mathrm red are identified with generalized deligne lusztig varieties we show that the bruhat tits stratification is a refinement of the ekedahl oort stratification and also relate the ekedahl oort strata to deligne lusztig varieties we deduce that mathcal m mathrm ss is locally a complete intersection that its irreducible components and each ekedahl oort stratum in every irreducible component is isomorphic to a deligne lusztig variety and give formulas for the number of irreducible components of every ekedahl oort stratum of mathcal m mathrm ss

minimal p divisible groups a p divisible group x can be seen as a tower of building blocks each of which is isomorphic to the same finite group scheme x p clearly if x1 and x2 are isomorphic then x1 p x2 p however conversely x1 p x2 p does in general not imply that x1 and x2 are isomorphic can we give over an algebraically closed field in characteristic p a condition on the p kernels which ensures this converse here are two known examples of such a condition consider the case that x is ordinary or the case that x is superspecial x is the p divisible group of a product of supersingular elliptic curves in these cases the p kernel uniquely determines x

scalar curvature and projective embeddings i we prove that a metric of constant scalar curvature on a polarised kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings satisfying a condition introduced by h luo this gives as a corollary the uniqueness of constant scalar curvature kahler metrics in a given rational cohomology class the proof uses results in the literature on the asymptotics of the bergman kernel the arguments are presented in a general framework involving moment maps for two different group actions

a new method to deduce high order compact difference schemes for two dimensional poisson equation abstract this paper introduces a novel method for deducing high order compact difference schemes for the two dimensional 2d poisson equation like finite volume method a dual partition is introduced combining simpson integral formula and parabolic interpolation a family of fourth order and sixth order compact difference schemes are obtained based on three different types of dual partitions moreover several new fourth order compact schemes are gained and numerical experiments are shown two of them are much better than almost any other fourth order schemes which have been presented in others work the outline for the nonlinear poisson equation is also given numerical experiments are presented to verify the feasibility of this new method and the high accuracy of these fourth order and sixth order compact difference schemes

the space of kahler metrics this paper the second of a series deals with the function space mathcal h of all smooth kahler metrics in any given n dimensional closed complex manifold v these metrics being restricted to a given fixed real cohomology class called a polarization of v this function space is equipped with a pre hilbert metric structure introduced by t mabuchi 10 who also showed that formally this metric has nonpositive curvature in the first paper of this series 4 the second author showed that the same space is a path length space he also proved that mathcal h is geodesically convex in the sense that for any two points of mathcal h there is a unique geodesic path joining them which is always length minimizing and of class c1 1 this partially verifies two conjectures of donaldson 8 on the subject in the present paper we show first of all that the space is as expected a path length space of nonpositive curvature in the sense of a d aleksandrov a second result is related to the theory of extremal kahler metrics namely that the gradient flow in mathcal h of the k energy of v has the property that it strictly decreases the length of all paths in mathcal h except those induced by one parameter families of holomorphic automorphisms of m

the g stable pieces of the wonderful compactification let g be a connected simple algebraic group over an algebraically closed field there is a partition of the wonderful compactification of into finite many g stable pieces which was introduced by lusztig in this paper we will investigate the closure of any g stable piece in we will show that the closure is a disjoint union of some g stable pieces which was first conjectured by lusztig we will also prove the existence of cellular decomposition if the closure contains finitely many orbits

single cell discretizations of order two and four for biharmonic problems abstract new difference formulas are derived for solving the biharmonic problem in two dimensions over a rectangular domain these methods use only the nine grid points of a single mesh cell and do not require fictitious points in order to approximate the boundary conditions derivatives of the solution are obtained as a by product of the methods second order formulas are derived for both the first and second biharmonic problems in numerical experiments the new second order formulas compare favourably with the standard second order methods extensions to fourth order formulas are given the method of deriving these formulas can be used to derive similar formulas for arbitrarily shaped regions

foliations in moduli spaces of abelian varieties in this paper we study abelian varieties and p divisible groups in characteristic p even though a non trivial deformation of an abelian variety can produce a non trivial galois representation say on the tate group of the generic fiber in any characteristic the geometric generic fiber has a constant tate group we en counter the same phenomenon for the p structure in positive characteristic any two ordinary abelian varieties of the same dimension over an algebraically closed field have isomorphic p divisible groups however for non ordinary abelian varieties this seems to break down the fas cinating structure which comes out of this is that the maximal locus where a given geometric isomorphism class of a p divisible group is realized e g in a family of abelian varieties is a locally closed set the locus defined by the geometric iso morphism type of a p divisible group will be called a central leaf see 2 1 and 3 4 this gives rise to a foliation of the open stratum attached to a newton polygon the dimension of any leaf in the same newton polygon stratum solely depends on in extreme cases either the leaf is the whole stratum as in the ordinary case or in the almost ordinary case the p rank equals g 1 or a leaf is zero dimensional as in the supersingular case in intermediate cases a leaf can be a proper subset and still be positive dimensional we have worked out the example of g 4 in 8 1

newton polygons as lattice points let m be the reduction modulo p of a shimura variety and let n m be the partially ordered set predicted to be the set of all types occurring in the stratification of m by newton polygons in the context of f isocrystals with additional structures we prove that n m is ranked or catenary in the sense that any two maximal chains have the same length we also give a lie theoretic formula for the distance between two comparable elements in n m this formula can be used to give the expected dimension of the newton strata of m

simple p kernels of p divisible groups a p divisible group x determines its p kernel x p g we show that g determines x uniquely if g is minimal and that there are infinitely many possibilities for x if g is not minimal the indecomposable minimal g are precisly those which are simple

skew hermitian based iterations for nine point approximations of convection diffusion problems we study skew hermitian based splitting methods for the iterative solution of nonsymmetric linear systems arising from high order compact hoc approximations of two dimensional convection diffusion problems such discretisations lead to nine point system matrices with block tridiagonal structures the nonsymmetric discretisation matrix for a constant coefficient grid aligned flow problem is shown to be positive definite and thus hermitian and skew hermitian splitting hss methods can be considered for the solution of the corresponding linear system for the solution of this linear system using the hss iteration we derive an analytical expression for the optimal value for the upper bound of the contraction factor comparison of the hss iteration with triangular and skew hermitian splitting tss methods and their corresponding block variants btss are carried out for various test problems cpu timings for the different splitting methods using experimentally determined optimal acceleration parameters are also given our results show that the combination of high order compact discretisations and skew hermitian based iterative methods for solving the corresponding linear systems provide efficient procedures for approximating the solutions of two dimensional convection diffusion equations

on the lower order terms of the asymptotic expansion of tian yau zelditch we compute the first three coefficients of the asymptotic expansion of tian yau zelditch we also prove that in general the fcth coefficient is a polynomial of the curvature and its derivative of weight k

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

equidistribution and convergence speed for zeros of holomorphic sections of singular hermitian line bundles abstract we establish the equidistribution of zeros of random holomorphic sections of powers of a semipositive singular hermitian line bundle with an estimate of the convergence speed

numerical simulation of singularly perturbed non linear elliptic boundary value problems using finite element method the present paper presents the finite element solution of two dimensional non linear singularly perturbed elliptic partial differential equation subject to appropriate dirichlet boundary conditions a new fifth order convergent newton type iterative method has been described and used to linearize the non linear problem the inclusion of this newton s method of fifth order convergence in finite element method for solving non linear system of equations reduces the number of iterations and hence the cost of computation to demonstrate the usefulness of the proposed scheme a non convex variational ginzburg landau equation is considered

bergman kernels and equilibrium measures for line bundles over projective manifolds let l be a holomorphic line bundle over a compact complex projective hermitian manifold x any fixed smooth hermitian metric h on l induces a hilbert space structure on the space of global holomorphic sections with values in the k th tensor power of l in this paper various convergence results are obtained for the corresponding bergman kernels i e orthogonal projection kernels the convergence is studied in the large k limit and is expressed in terms of the equilibrium metric h e associated to h as well as in terms of the monge ampere measure of h on a certain support set it is also shown that the equilibrium metric h e is in the class c 1 1 on the complement of the augmented base locus of l for l ample these results give generalizations of well known results concerning the case when the curvature of h is globally positive then h e h in general the results can be seen as local metrized versions of fujita s approximation theorem for the volume of l

ekedahl oort and newton strata for shimura varieties of pel type we study the ekedahl oort stratification for good reductions of shimura varieties of pel type these generalize the ekedahl oort strata defined and studied by oort for the moduli space of principally polarized abelian varieties the siegel case they are parameterized by certain elements w in the weyl group of the reductive group of the shimura datum we show that for every such w the corresponding ekedahl oort stratum is smooth quasi affine and of dimension ell w and in particular non empty some of these results have previously been obtained by moonen vasiu and the second author using different methods we determine the closure relations of the strata we give a group theoretical definition of minimal ekedahl oort strata generalizing oort s definition in the siegel case and study the question whether each newton stratum contains a minimal ekedahl oort stratum as an interesting application we determine which newton strata are non empty this criterion proves conjectures by fargues and by rapoport generalizing a conjecture by manin for the siegel case we give a necessary criterion when a given ekedahl oort stratum and a given newton stratum meet

a class of efficient quadrature based predictor corrector methods for solving nonlinear systems we extend a class of quadrature based predictor corrector techniques for root finding to multivariate systems they are found to have a rate of convergence of 1 2 regardless of the degree of precision for the quadrature technique from which they are derived provided it is at least one by reusing the linear system from the previous iterate this class incorporates a significant improvement in computational time relative to the standard class through the inclusion of an lu decomposition during the iteration complexity is equivalent to newton s method as they only require knowledge of f x and f x

high order variable mesh off step discretization for the solution of 1 d non linear hyperbolic equation abstract in this paper we propose a new high order three level implicit method based on off step discretization on a non uniform mesh for the solution of 1 d non linear hyperbolic partial differential equation of the form u tt u xx g x t u u x u t subject to appropriate initial and dirichlet boundary conditions we use only three evaluations of the function g and three grid points at each time level in a compact cell our method is directly applicable to the wave equation in polar coordinates and we do not require any special technique to handle singular coefficients of the differential equation the method is convergent for uniform mesh numerical results are provided to justify the usefulness of the proposed method

high order implicit finite difference schemes for the two dimensional poisson equation abstract in this paper a new family of high order finite difference schemes is proposed to solve the two dimensional poisson equation by implicit finite difference formulas of 2 m 1 operator points the implicit formulation is obtained from taylor series expansion and wave plane theory analysis and it is constructed from a few modifications to the standard finite difference schemes the approximations achieve 2 m 4 order accuracy for the inner grid points and up to eighth order accuracy for the boundary grid points using a successive over relaxation method the high order implicit schemes have faster convergence as m is increased compensating the additional computation of more operator points thus the proposed solver results in an attractive method easy to implement with higher order accuracy but nearly the same computation cost as those of explicit or compact formulation in addition particular case m 1 yields a new compact finite difference schemes of sixth order of accuracy numerical experiments are presented to verify the feasibility of the proposed method and the high accuracy of these difference schemes

on the cohomology of compact unitary group shimura varieties at ramified split places in this article we prove results about the cohomology of compact unitary group shimura varieties at split places in nonendoscopic cases we are able to give a full description of the cohomology after restricting to integral hecke operators at p on the automorphic side we allow arbitrary ramification at p even the pel data may be ramified this gives a description of the semisimple local hasse weil zeta function in these cases we also treat cases of nontrivial endoscopy for this purpose we give a general stabilization of the expression given in 38 following the stabilization given by kottwitz in 25 this introduces endoscopic transfers of the functions h introduced in 38 we state a general conjecture relating these endoscopic transfers with langlands parameters we verify this conjecture in all cases of el type and deduce new results about the endoscopic part of the cohomology of shimura varieties this allows us to simplify the construction of galois representations attached to conjugate self dual regular algebraic cuspidal automorphic representations of gln as previously constructed by one of us 41

uniqueness in e x ω abstract we prove uniqueness for the dirichlet problem for the complex monge ampere equation on compact kahler manifolds in the case of probability measures vanishing on pluripolar sets the proof uses the mass concentration technique due to kolodziej coupled with inequalities for mixed monge ampere measures and the comparison principle

la filtration canonique des mathcal o modules p divisibles in this article we associate to g a truncated p divisible mathcal o module of given signature where mathcal o is a finite unramified extension of mathbb z p a filtration of g by sub mathcal o modules under the condition that its hasse mu invariant is smaller than an explicit bound this filtration generalise the one given when g is mu ordinary the construction of the filtration relies on a precise study of the crystalline periods of a p divisible mathcal o module we then apply this result to families of such groups in particular to strict neighbourhoods of the mu ordinary locus inside some pel shimura varieties

the weighted monge ampere energy of quasiplurisubharmonic functions abstract we study degenerate complex monge ampere equations on a compact kahler manifold x we show that the complex monge ampere operator d d c n is well defined on the class e x of plurisubharmonic functions with finite weighted monge ampere energy the class e x is the largest class of psh functions on which the monge ampere operator is well defined and the comparison principle is valid it contains several functions whose gradient is not square integrable we give a complete description of the range of the operator d d c n on e x as well as on some of its subclasses we also study uniqueness properties extending calabi s result to this unbounded and degenerate situation and we give applications to complex dynamics and to the existence of singular kahler einstein metrics

bruhat strata for shimura varieties of pel type we study the bruhat stratification for shimura varieties of pel type in the siegel case this stratification is a scheme theoretic variant of the stratification by the a number we show that all bruhat strata are smooth and determine their dimensions we also prove that the closure of a bruhat stratum is a union of bruhat strata and describe which bruhat strata are contained in the closure of a given bruhat stratum

fundamental elements of an affine weyl group fundamental elements are certain special elements of affine weyl groups introduced by gortz haines kottwitz and reuman they play an important role in the study of affine deligne lusztig varieties in this paper we obtain characterizations of the fundamental elements and their natural generalizations we also derive an inverse to a version of newton hodge decomposition in affine flag varieties as an application we obtain a group theoretic generalization of oort s results on minimal p divisible groups and we show that in certain good reduction of pel shimura datum each newton stratum contains a minimal ekedahl oort stratum this generalizes a result of viehmann and wedhorn

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

the mabuchi geometry of finite energy classes abstract we introduce different finsler metrics on the space of smooth kahler potentials that will induce a natural geometry on various finite energy classes e x motivated by questions raised by r berman v guedj and y rubinstein we characterize the underlying topology of these spaces in terms of convergence in energy and give applications of our results to existence of kahler einstein metrics on fano manifolds

the geometry of newton strata in the reduction modulo p of shimura varieties of pel type in this thesis we study the newton stratification on the reduction of shimura varieties of pel type with hyperspecial level structure the main result is a formula for the dimension of newton strata and the description of their closure a key ingredient of its proof is the calculation of the dimension of some rapoport zink spaces as an interesting application to deformation theory we determine the dimension and closure of newton strata on the algebraisation of the deformation space of a barsotti tate group with p el structure

strata hasse invariants hecke algebras and galois representations we construct group theoretical generalizations of the hasse invariant on strata closures of the stacks mathop g hbox mathtt zip nolimits mu restricting to zip data of hodge type we obtain a group theoretical hasse invariant on every ekedahl oort stratum closure of a general hodge type shimura variety a key tool is the construction of a stack of zip flags mathop g hbox mathtt zipflag nolimits mu fibered in flag varieties over mathop g hbox mathtt zip nolimits mu it provides a simultaneous generalization of the classical case homogeneous complex manifolds studied by griffiths schmid and the flag space for siegel varieties studied by ekedahl van der geer four applications are obtained 1 pseudo representations are attached to the coherent cohomology of hodge type shimura varieties modulo a prime power 2 galois representations are associated to many automorphic representations with non degenerate limit of discrete series archimedean component 3 it is shown that all ekedahl oort strata in the minimal compactification of a hodge type shimura variety are affine thereby proving a conjecture of oort 4 part of serre s letter to tate on mod p modular forms is generalized to general hodge type shimura varieties

dynamical analysis and perturbation solution of an seir epidemic model abstract in this paper we present a kind of technique to obtain the asymptotical solution of an seir where s is the susceptible population e is the exposed population i is the infectious population and r is the recovered population epidemic model by employing the method of perturbation at first we investigate the epidemic model by analysing its dynamical behavior then we use the method of perturbation to obtain the analytical solution of the model we assign values for parameters and draw figures to observe the magnitude of error of the perturbation method in a macro view finally we analyse macroscopically the two comparison chart on analytical solution and the exact solution and know that it is feasible to analyse the solution of the epidemic model by using the perturbation method

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

canonical sections of the hodge bundle over ekedahl oort strata of shimura varieties of hodge type abstract we construct canonical non vanishing global sections of powers of the hodge bundle on each ekedahl oort stratum of a hodge type shimura variety in particular we recover the quasi affineness of the ekedahl oort strata in the projective case this gives a very short proof of non emptiness of ekedahl oort strata it follows that the newton strata are also nonempty by a result of s nie from the canonicity of our construction we deduce the fact that the ordinary locus is determined by the ekedahl oort strata of its image under any embedding of shimura varieties

a new fast algorithm based on half step discretization for one space dimensional quasilinear hyperbolic equations abstract in this article we describe a new compact three level implicit method of order four in time and space based on half step discretization for one space dimensional quasilinear hyperbolic equation u tt a x t u u xx f x t u u x u t defined in the semi infinite solution region where a 0 we require only nine grid points for the unknown variable u x t and two extra half step points each for x and t variables the proposed method is directly applicable to wave equation with singular coefficients which is main attraction of our work we do not require extra grid points for computation we describe the derivation of the method in detail the proposed method when applied to damped wave equation is shown to be unconditionally stable many benchmark problems are solved to confirm the fourth order convergence of the proposed method

geometric and homological properties of affine deligne lusztig varieties this paper studies ane deligne lusztig varieties x w b in the ane ag variety of a quasi split tamely ramied group we describe the geometric structure of x w b for a minimal length element w in the conjugacy class of an extended ane weyl group we then provide a reduction method that relates the structure of x w b for arbitrary elements w in the extended ane weyl group to those associated with minimal length elements based on this reduction we establish a connection between the dimension of ane deligne lusztig varieties and the degree of the class polynomial of ane

order h 4 difference methods for a class of singular two space elliptic boundary value problems abstract in this article we derive difference methods of o h4 for solving the system of two space nonlinear elliptic partial differential equations with variable coefficients having mixed derivatives on a uniform square grid using nine grid points we obtain two sets of fouth order difference methods one in the absence of mixed derivatives second when the coefficients of uxy are not equal to zero and the coefficients of uxx and uyy are equal there do not exist fourth order schemes involving nine grid points for the general case the method having two variables has been tested on two dimensional viscous incompressible steady state navier stokes model equations in polar coordinates the proposed difference method for scalar equation is also applied to the poisson s equation in polar coordinates some numerical examples are provided to illustrate the fourth order convergence of the proposed methods

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

poisson equation solver with fourth order accuracy by using interpolated differential operator scheme abstract a poisson equation solver with fourth order spatial accuracy has been developed by using the interpolated differential operator ido scheme the number of grids required for obtaining the results of numerical computation with the same accuracy as that by the second order center finite difference method is drastically reduced the consistency is confirmed to the use of the multigrid method and the red black algorithm the decrease of the cpu time by the multigrid method and by the parallel computing indicates the applicability of the ido poisson solver to large scale problems

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

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

the smart blage algorithm for singularly perturbed 2d elliptic partial differential equations we discuss a new smart block alternating group explicit smart blage algorithm for solving the symmetric block linear systems derived from nine point fourth order method based on arithmetic average discretizations to solve the singularly perturbed two dimensional elliptic equations the smart blage algorithm is predicted to save time as the idea involved is to eliminate evaluating two similar terms appearing in the algorithm the error analysis of the smart blage algorithm is discussed briefly numerical tests are performed to demonstrate the usefulness of the proposed smart blage method and compared with the corresponding block successive over relaxation bsor method by considering two test problems