parameter uniformly convergent exponential spline difference scheme for singularly perturbed semilinear reaction diffusion problems abstract we consider a dirichlet boundary value problem for singularly perturbed semilinear reaction diffusion equation the problem is discretized using an exponential spline difference scheme derived on the basis of splines in tension on piecewise uniform shishkin type mesh the convergence analysis is given and the method is shown to be almost second order accurate in the discrete maximum norm uniformly in the perturbation parameter e numerical experiments are conducted to demonstrate the theoretical results

parameter uniform convergence of a numerical method for a coupled system of singularly perturbed semilinear reaction diffusion equations with boundary and interior layers abstract in this paper we consider a coupled system of m 2 singularly perturbed semilinear reaction diffusion equations with a discontinuous source term having a discontinuity at a point in the interior of the domain the diffusion term of each equation is multiplied by small singular perturbation parameter but these parameters are assumed to be different in magnitude a numerical method is constructed on a variant of shishkin mesh the approximations generated by this method are shown to be almost second order uniformly convergent with respect to all perturbation parameters numerical results are in support of the theoretical results

stability and error estimates of local discontinuous galerkin methods with implicit explicit time marching for advection diffusion problems the main purpose of this paper is to analyze the stability and error estimates of the local discontinuous galerkin ldg methods coupled with carefully chosen implicit explicit imex runge kutta time discretization up to third order accuracy for solving one dimensional linear advection diffusion equations in the time discretization the advection term is treated explicitly and the diffusion term implicitly there are three highlights of this work the first is that we establish an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient in the ldg methods the second is that by aid of the aforementioned relationship and the energy method we show that the imex ldg schemes are unconditionally stable for the linear problems in the sense that the time step tau is only required to be upper bounded by a constant which depends on the ratio of the diffusion and the square of the advection coefficients and is independent

error estimators for advection reaction diffusion equations based on the solution of local problems this paper deals with a posteriori error estimates for advection reaction diffusion equations in particular error estimators based on the solution of local problems are derived for a stabilized finite element method these estimators are proved to be equivalent to the error with equivalence constants eventually depending on the physical parameters numerical experiments illustrating the performance of this approach are reported

corner singularities and boundary layers in a simple convection diffusion problem abstract a singularly perturbed convection diffusion problem posed on the unit square is considered its solution may have exponential and parabolic boundary layers and corner singularities may also be present pointwise bounds on the solution and its derivatives are derived the dependence of these bounds on the small diffusion coefficient on the regularity of the data and on the compatibility of the data at the corners of the domain are all made explicit the bounds are derived by decomposing the solution into a sum of solutions of elliptic boundary value problems posed on half planes then analyzing these simpler problems

finite volume solutions of convection diffusion test problems the cell vertex formulation of the finite volume method has been developed and widely used to model inviscid flows in aerodynamics more recently one of us has proposed an extension for viscous flows the purpose of the present paper is two fold first we have applied this scheme to a well known convection diffusion model problem involving flow round a 180 degrees bend which highlights some of the issues concerning the application of the boundary conditions in such cell based schemes the results are remarkably good when the boundary conditions are applied in an appropriate manner in our efforts to explain the high quality of the results we were led to a detailed analysis of the corresponding one dimensional problem our second purpose is thus to gather together various approaches to the analysis of this problem and to draw attention to the supra convergence phenomena enjoyed by the proposed methods 25 refs 9 figs 1 tab

n widths and singularly perturbed boundary value problems using the theory of n widths the approximability of solutions of singularly perturbed linear reaction diffusion problems in two dimensions is quantified

on the non existence of e uniform finite difference methods on uniform meshes for semilinear two point boundary value problems in this paper fitted finite difference methods on a uniform mesh with internodal spacing h are considered for a singularly perturbed semilinear two point boundary value problem it is proved that a scheme of this type with a frozen fitting factor cannot converge e uniformly in the maximum norm to the solution of the differential equation as the mesh spacing h goes to zero numerical experiments are presented which show that the same result is true for a number of schemes with variable fitting factors

a second order scheme for a time dependent singularly perturbed convection diffusion equation we consider a numerical scheme for a one dimensional time dependent singularly perturbed convection diffusion problem the problem is discretized in space by a standard finite element method on a bakhvalov shishkin type mesh the space error is measured in an l2 norm for the time integration the implicit midpoint rule is used the fully discrete scheme is shown to be convergent of order 2 in space and time uniformly in the singular perturbation parameter

a generalized bridge number for links in 3 manifolds in sch schubert introduced a new invariant of knots in the 3 sphere called the bridge number and showed that when reduced by 1 it is an additive invariant under the connected sum operation of knots several generalizations of schubert s theorem can be imagined indeed from schubert s perspective the behaviour of bridge number under addition is only a corollary of a more general theorem about bridge number of satellite knots boileau and lickorish have independently observed b l that the following generalization of schubert s theorem would be useful bridge number as defined by schubert begins with a projection of the knot or link onto a sphere in s 3 project instead onto a higher genus unknotted surface in s 3 one advantage of this approach is that it allows bridge number to be defined for a link in a 3 manifold other than 5 3 by projecting the link onto a heegaard surface for the 3 manifold here we explore how such an invariant behaves under connected sum using a line of argument similar to jaco s proof of haken s theorem that heegaard genus is additive ja a crucial additional ingredient needed is lemma4 6 some of the arguments were motivated by no in which as an aside a different generalization of schubert s theorem is attempted they want to show that the standard bridge number behaves well under knot decomposition even when the knot is decomposed by spheres intersecting the knot more than twice e g under tangle sums l besides recovering a new proof of schubert s theorem more in the spirit of jaco s proof of the haken lemma we are also able to prove an analogous theorem for bridge number defined with respect to genus one heegaard splittings of 5 3 or the lens space and also with respect to irreducible heegaard splittings of closed nonhaken 3 manifolds

annuli in generalized heegaard splittings and degeneration of tunnel number abstract we analyze how a family of essential annuli in a compact 3 manifold will induce from a strongly irreducible generalized heegaard splitting of the ambient manifold generalized heegaard splittings of the complementary components there are specific applications to the subadditivity of tunnel number of knots improving somewhat bounds of kowng kw for example in the absence of 2 bridge summands the tunnel number of the sum of n knots is no less than frac 2 5 the sum of the tunnel numbers

on instantiating pairing based protocols with elliptic curves of embedding degree one since the discovery of identity based encryption schemes in 2000 bilinear pairings have been used in the design of hundreds of cryptographic protocols the most commonly used pairings are constructed from elliptic curves over finite fields with small embedding degree these pairings can have different security performance and functionality characteristics and were therefore classified into types 1 2 3 and 4 in this paper we observe that this conventional classification is not applicable to pairings from elliptic curves with embedding degree one it is important to understand the security efficiency and functionality of these pairings in light of recent attacks on certain pairings constructed from elliptic curves with embedding degree greater than one we define three kinds of pairings from elliptic curves with embedding degree one discuss some subtleties with using them to implement pairing based protocols and provide an estimated cost of implementing them on modern processors

continuation of periodic solutions in the waveguide array mode locked laser abstract we apply the adjoint continuation method to construct highly accurate periodic solutions that are observed to play a critical role in the multi pulsing transition of mode locked laser cavities the method allows for the construction of solution branches and the identification of their bifurcation structure supplementing the adjoint continuation method with a computation of the floquet multipliers allows for explicit determination of the stability of each branch this method reveals that when gain is increased the multi pulsing transition starts with a hopf bifurcation followed by a period doubling bifurcation and a saddle node bifurcation for limit cycles finally the system exhibits chaotic dynamics and transitions to the double pulse solutions although this method is applied specifically to the waveguide array mode locking model the multi pulsing transition is conjectured to be ubiquitous and these results agree with experimental and computational results from other models

a multigrid preconditioner for stabilised discretisations of advection diffusion problems abstract this paper is concerned with the iterative solution of linear systems arising from stabilised discretisations of advection diffusion problems on stretched finite element grids using nonuniform grids of this type leads in general to very badly conditioned matrix problems we therefore consider using gmres in conjunction with a multigrid mg preconditioning strategy in particular we show that in order to achieve the grid size independent convergence which is characteristic of mg methods it is essential to use an effective stabilisation strategy at each level of the mg structure

error estimates for a class of partial functional differential equation with small dissipation this article deals with a class of partial functional differential equation with a small dissipating parameter on a rectangular domain classical numerical methods for solving this type of problems reveal disappointing behavior or are tremendously expensive in computer memory and processor time this arises because the precision of an approximate solution depends inversely on perturbation parameter values and thus it deteriorates as a parameter decreases therefore it is of particular interest to develop numerical methods whose error estimates would be independent of the perturbation parameter contaminating the solution in order to overcome the said difficulty we derive robust parameter uniform error estimates for a class of partial functional differential equations the analysis presented in this paper uses a suitable decomposition of the error into smooth and a singular component combined with the appropriate barrier functions and comparison principle

canonical euler splitting method for nonlinear composite stiff evolution equations in this paper a new splitting method called canonical euler splitting method ces is constructed and studied which can be used for the efficient numerical solution of general nonlinear composite stiff problems in evolution equations of various type such as ordinary differential equations odes semi discrete unsteady partial differential equations pdes and ordinary or partial volterra functional differential equations vfdes and can significantly improve the computing speed on the basis of ensuring the computing quality stability consistency and convergence theories of this method are established a series of numerical experiments are given which check the efficiency of ces method and confirm our theoretical results

on richardson extrapolation for fitted operator finite difference methods abstract recently there has been a great interest towards the higher order methods for singularly perturbed problems as compared to their lower order counterparts they provide better accuracy with fewer mesh points construction and or implementation of direct higher order methods is usually very complicated thus a natural choice is to use some convergence acceleration techniques e g richardson extrapolation however as we see in this article such techniques do not perform equally well on all type of methods to investigate this we consider two fitted operator finite difference methods fofdms developed by patidar k c patidar high order fitted operator numerical method for self adjoint singular perturbation problems appl math comput 171 1 2005 547 566 and lubuma and patidar j lubuma k c patidar uniformly convergent non standard finite difference methods for self adjoint singular perturbation problems j comput appl math 191 2006 228 238 referred to as fofdm i and fofdm ii respectively the fofdm i is fourth and second order accurate for moderate and smaller values of e respectively unfortunately richardson extrapolation does not improve the order of this method the fofdm ii is second order uniformly convergent and we show that its order can be improved up to four by using richardson extrapolation both the methods are analyzed for convergence and comparative numerical results supporting theoretical estimates are provided

high order fitted operator numerical method for self adjoint singular perturbation problems we consider self adjoint singularly perturbed two point boundary value problems in conservation form highest possible order of uniform convergence for such problems achieved hitherto via fitted operator methods was one see e g doolan et al uniform numerical methods for problems with initial and boundary layers boole press dublin 1980 p 121 reducing the original problem into the normal form and then using the theory of inverse monotone matrices a fitted operator finite difference method is derived via the standard numerov s method the scheme thus derived is fourth order accurate for moderate values of the perturbation parameter e whereas for very small values of this parameter the method is e uniformly convergent with order two numerical examples are given in support of the theory

a finite element analysis of a coupled system of singularly perturbed reaction diffusion equations we consider a system of two coupled reaction diffusion equations when the parameters multiplying the second order derivatives in the equations are small their solutions exhibit boundary layers moreover when the parameters are of different magnitudes two distinct but overlapping boundary layers are present we study a finite element discretization on general layer adapted meshes including the frequently studied shishkin mesh and the bakhvalov mesh supporting numerical results are presented

efficient linearly implicit methods for nonlinear multidimensional parabolic problems in this paper we present and analyze new methods to integrate multidimensional parabolic problems with nonlinear reaction terms we consider a first standard spatial semidiscretization stage obtaining a family of stiff nonlinear initial value problems the totally discrete schemes are obtained by numerical integration in time of such problems using new additive runge kutta schemes we show that the resulting algorithms which are only linearly implicit reach unconditional convergence if the additive methods used have suitable properties of linear absolute stability besides they have a computational cost per time step with the same order as the explicit methods finally three numerical experiences are shown in order to illustrate the behavior of our methods

implicit explicit runge kutta methods for financial derivatives pricing models abstract implicit explicit runge kutta methods are investigated for application to financial derivatives pricing models in the partial differential equations approach the methods are showed to be an alternative to other existing procedures for the numerical valuation of american type contracts we follow the method of lines in order to have a numerical method that can be used with a variety of state variable discretizations including finite elements finite differences and finite volume methods some numerical experiments are presented

a posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection diffusion equations this paper is devoted to the study of a posteriori and a priori error estimates for the scalar nonlinear convection diffusion equation c t nabla cdot u f c varepsilon delta c 0 the estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the l 1 norm in the situation where the diffusion parameter varepsilon is smaller or comparable to the mesh size numerical experiments underline the theoretical results

an error analysis for a difference scheme of exponential type applied to a nonlinear singular perturbation problem without turning points abstract a difference scheme of exponential type given in 5 is analyzed again for solving a nonlinear singular perturbation problem without turning points it is shown that a solution of the scheme converges uniformly in a perturbation parameter to an exact solution of the continuous problem with order one in l1 sense

on supraconvergence phenomenon for second order centered finite differences on non uniform grids in the present note we consider an example of a boundary value problem for a simple second order ordinary differential equation which may exhibit a boundary layer phenomenon we show that usual central finite differences which are second order accurate on a uniform grid can be substantially upgraded to the fourth order by a suitable choice of the underlying non uniform grid this example is quite pedagogical and may give some ideas for more complex problems

finite element methods for convection diffusion problems using exponential splines on triangles abstract a new family of petrov galerkin finite element methods on triangular grids is constructed for singularly perturbed elliptic problems in two dimensions it uses divergence free trial functions that form a natural generalization of one dimensional exponential trial functions this family includes an improved version of the divergence free finite element method used in the pltmg code numerical results show that the new method is able to compute strikingly accurate solutions on coarse meshes an analysis of the use of slotboom variables shows that they are theoretically unsatisfactory and explains why certain petrov galerkin methods lose stability when generalized from one to two dimensions

a fault attack on pairing based cryptography current fault attacks against public key cryptography focus on traditional schemes such as rsa and ecc and to a lesser extent on primitives such as xtr however bilinear maps or pairings have presented theorists with a new and increasingly popular way of constructing cryptographic protocols most notably this has resulted in efficient methods for identity based encryption ibe since identity based cryptography seems an ideal partner for identity aware devices such as smart cards in this paper we examine the security of concrete pairing instantiations in terms of fault attack

numerical methods for evolutionary reaction diffusion problems with nonlinear reaction terms we have developed new methods to integrate efficiently reaction diffusion parabolic problems with non linear reaction terms in order to obtain uniform and unconditional convergence such methods combine the advantages of alternating direction methods the additive runge kutta methods designed by cooper and sayfy for nonlinear stiff problems as well as the use of shishkin meshes in the singularly perturbed case the resulting algorithms are only linearly implicit and they have the same order of computational complexity per time step that any explicit method we show some numerical experiences which illustrate the good properties of our schemes predicted by the theoretical results

difference methods for solving convection diffusion equations abstract difference methods for solving the convection diffusion equation are discussed the superiority of allen s approximation over central or upwind differences for one dimensional problems is confirmed the superiority being greatest when the boundary layer is very thin higher order methods give improved accuracy with negligible increase in cost a new iterative scheme is proposed for the two dimensional problem which requires orders of magnitude fewer arithmetic operations than existing procedures currently available in the literature

rooted tree analysis of runge kutta methods with exact treatment of linear terms we investigate a class of time discretization schemes called etd runge kutta methods where the linear terms of an ordinary differential equation are treated exactly while the other terms are numerically integrated by a one step method these schemes proposed by previous authors can be regarded as modified runge kutta methods whose coefficients are matrices instead of scalars from this viewpoint we reexamine the notion of consistency convergence and order to provide a mathematical foundation for new methods applying the rooted tree analysis expansion theorems of both the strict and numerical solutions are proved and two types of order conditions are defined several classes of formulas with up to four stages that satisfy the conditions are constructed and it is shown that the power series of matrices employed as their coefficients can be determined using the order conditions

singularly perturbed convection diffusion problems with boundary and weak interior layers in this paper a singularly perturbed convection diffusion equation with a discontinuous source term is examined boundary and weak interior layers appear in the solution a numerical method is constructed for this problem which involves an appropriate piecewise uniform mesh the method is shown to be uniformly convergent with respect to the singular perturbation parameter

the number field sieve for integers of low weight we define the weight of an integer n to be the smallest w such that n can be represented as w i 1 i 2 ci with 1 w 1 1 since arithmetic modulo a prime of low weight is particularly efficient it is tempting to use such primes in cryptographic protocols in this paper we consider the difficulty of the discrete logarithm problem modulo a prime n of low weight as well as the difficulty of factoring an integer n of low weight we describe a version of the number field sieve which handles both problems in the case that w 2 the method is the same as the special number field sieve which runs conjecturally in time exp 32 9 1 3 o 1 log n 1 3 log log n 2 3 for n oo for fixed w 2 we conjecture that there is a constant less than 32 9 1 3 2w 3 w 1 1 3 such that the running time of the algorithm is at most exp o 1 log n 1 3 log log n 2 3 for n we further conjecture that no less than 32 9 1 3 2w 2 2 1 w 1 2 3 has this property our analysis reveals that on average the method performs significantly better than it does in the worst case we consider all the examples given in a recent paper of koblitz and menezes and demonstrate that in every case but one our algorithm runs faster than the standard versions of the number field sieve

a shishkin mesh for a singularly perturbed riccati equation in this paper a singularly perturbed riccati initial value problem is examined parameter explicit bounds on the solution and its derivatives are given a numerical method composed of an implicit difference operator and a piecewise uniform shishkin mesh is constructed a theoretical parameter independently bound on the errors in the numerical approximations is established numerical results are presented which are in agreement with the theoretical error bound

coupling stabilized finite element methods with finite difference time integration for advection diffusion reaction problems abstract we present some numerical schemes for the unsteady advection diffusion reaction linear problem in one space dimension we investigate two possible different ways of combining the discretization in time and in space where the sequence of the discretizations is interchanged discretization in time is performed by using the crank nicolson finite difference scheme while for the space discretization we consider three classical stabilized finite element schemes and the more recent link cutting bubble strategy proposed in f brezzi g hauke l d marini g sangalli link cutting bubbles for the stabilization of convection diffusion reaction problems math models methods appl sci 13 2003 445 461 numerical experiments are presented to assess and valuate the capabilities of the proposed methods an l 1 error analysis of the link cutting bubble strategy for solving the steady problem is included

global convergence method for singularly perturbed boundary value problems we give uniformly convergent splines difference scheme for singularly perturbed boundary value problems eu p x u q x u f x u a 0 u b 1 1 by using splines fitted with delta sequence due to the very stiff nature of the problem under consideration we prove the o min h2 e2 order of uniform convergence with respect to small parameter e at nodes on uniform mesh and o min h e order of uniform global convergence with respect to the approximate solution given by s x i 1n s i x h xi x where h is the heaviside function which is the approximation for the closed form of the exact solution

some new additive runge kutta methods and their applications we propose some new additive runge kutta methods of orders ranging from 2 to 4 that may be used for solving some nonlinear system of odes especially for the temporal discretization of some nonlinear systems of pdes with constraints only linear odes or pdes need to be solved at each time step with these new methods

on a singular perturbation problem with two second order turning points in this paper we study the singular perturbation problem ep x uxx 2 x a b x q x ux 2su 0 0 x 1 0 a b 1 where 0 el1 is a small positive parameter p x and q x are sufficiently smooth and strictly positive functions the main feature of this equation is that there are two second order turning points in the interval 0 1 based on the rigorous results on singular perturbation problems with one second order turning point in our previous work we obtain a uniform asymptotic approximation for the general solution of the above equation by means of a matching technique

additive runge kutta methods for stiff ordinary differential equations certain pairs of runge kutta methods may be used additively to solve a system of n differential equations x j t x g t x pairs of methods of order p 4 where one method is semiexplicit and a stable and the other method is explicit are obtained these methods require the lu factorization of one n x n matrix and p evaluations of g in each step it is shown that such methods have a stability property which is similar to a stability property of perturbed linear differential equations 1 introduction in a recent article 2 the authors showed that certain pairs of methods may be used in an additive fashion to solve an initial value problem for a system of n differential equations x f t x x a xo a t b

a robust finite element method for a singularly perturbed elliptic problem with two small parameters abstract in this paper we consider a bilinear finite element method fem for a singularly perturbed elliptic problem with two small parameters by using butuzov asymptotic expansion 1 and the technique we developed in 2 4 we prove that our fem on a special piecewise uniform mesh converges independently of small parameters numerical results show that our method perform much better than the classical fem published by elsevier science ltd

identity based encryption from the weil pairing we propose a fully functional identity based encryption ibe scheme the scheme has chosen ciphertext security in the random oracle model assuming a variant of the computational diffie hellman problem our system is based on bilinear maps between groups the weil pairing on elliptic curves is an example of such a map we give precise definitions for secure ibe schemes and give several applications for such systems

a class of singularly perturbed semilinear differential equations with interior layers in this paper singularly perturbed semilinear differential equations with a discontinuous source term are examined a numerical method is constructed for these problems which involves an appropriate piecewise uniform mesh the method is shown to be uniformly convergent with respect to the singular perturbation parameter numerical results are presented that validate the theoretical results

iterative domain decomposition algorithms for a convection diffusion problem abstract this paper deals with iterative algorithms for domain decomposition applied to the solution of a singularly perturbed convection diffusion problem with regular boundary layers these algorithms are based on a finite difference domain decomposition approach and on a modified schwarz alternating method convergence properties of the algorithm are established and their parallel implementations are discussed numerical results for a test singularly perturbed problem are presented

b spline collocation method for the singular perturbation problem using artificial viscosity in this paper we develop a b spline collocation method using artificial viscosity for solving singularly perturbed equations given by eu x a x u x b x u x f x a x a 0 b x b 0 0 1 u 0 a u 1 b we use the artificial viscosity to capture the exponential features of the exact solution on a uniform mesh and use b spline collocation method which leads to a tridiagonal linear system the convergence analysis is given and the method is shown to have uniform convergence of second order the design of artificial viscosity parameter is confirmed to be a crucial ingredient for simulating the solution of the problem known test problems have been studied to demonstrate the accuracy of the method numerical results show the behaviour of the method with emphasis on treatment of boundary conditions results shown by the method are found to be in good agreement with the exact solution

a new stabilized finite element formulation for scalar convection diffusion problems the streamline and approximate upwind petrov galerkin method abstract a new stabilized and accurate finite element formulation for convection dominated problems is herein developed the basis of the new formulation is the choice of a new upwind function the upwind function chosen for the new method provokes its degeneration into the supg or cau methods depending on the approximate solution s regularity the accuracy and stability of the new formulation for the linear and scalar advection diffusion equation is demonstrated in several numerical examples

a uniformly convergent finite difference scheme for a singularly perturbed semilinear equation boundary value problems for singularly perturbed semilinear elliptic equations are considered special piecewise uniform meshes are constructed which yield accurate numerical solutions irrespective of the value of the small parameter numerical methods composed of standard monotone finite difference operators and these piecewise uniform meshes are shown theoretically to be uniformly with respect to the singular perturbation parameter convergent numerical results are also presented which indicate that in practice the method is first order accurate

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

on the super additivity of tunnel number of knots abstract in the present paper we show a necessary and sufficient condition for knots k 1 k 2 in s 3 with some side condition to have the super additivity of tunnel number under connected sum i e t k 1 k 2 t k 1 t k 2 1

a class of singularly perturbed quasilinear differential equations with interior layers a class of singularly perturbed quasilinear differential equations with discontinuous data is examined in general interior layers will appear in the solutions of problems from this class a numerical method is constructed for this problem class which involves an appropriate piecewise uniform mesh the method is shown to be a parameter uniform numerical method with respect to the singular perturbation parameter numerical results are presented which support the theoretical results

asymptotic behaviour of solutions near a turning point the example of the brusselator equation abstract the brusselator equation is an example of a singularly perturbed differential equation with an additional parameter it has two turning points at x 0 and x 1 we study some properties of so called canard solutions that remain bounded in a full neighbourhood of 0 and in the largest possible domain the main goal is the complete asymptotic expansion of the difference between two values of the additional parameter corresponding to such solutions for this purpose we need a study of behaviour of the solutions near a turning point here we prove that for a large class of equations if 0 is a turning point of order p any solution y not exponentially large has in some sector centred at 0 an asymptotic behaviour when e 0 of the form y n x e e n where e p 1 e for x e x with x large enough but independent of e in the brusselator case we moreover compute a stokes constant for a particular nonlinear differential equation

a priori estimates and analysis of a numerical method for a turning point problem bounds are obtained for the derivatives of the solution of a turning point problem these results suggest a modification of the el mistikawy werle finite difference scheme at the turning point a uniform error estimate is obtained for the resulting method and illustrative numerical results are given i introduction we will examine the following two point boundary value problem with dirichlet data at the endpoints l la ly eyx x p x yx x q x y x f x for i x 1

how accurate is the streamline diffusion finite element method we investigate the optimal accuracy of the streamline diffusion finite element method applied to convection dominated problems for linear bilinear elements the theoretical order of convergence given in the literature is either o h 3 2 for quasi uniform meshes or o h 2 for some uniform meshes the determination of the optimal order in general was an open problem by studying a special type of meshes it is shown that the streamline diffusion method may actually converge with any order within this range depending on the characterization of the meshes

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

a generalized structure approach to additive runge kutta methods this work considers a general structure of the additively partitioned runge kutta methods by allowing for different stage values as arguments of different components of the right hand side an order conditions theory is developed for the new formulation of generalized additive methods and stability and monotonicity investigations are carried out the paper discusses the construction and properties of implicit explicit and implicit implicit methods in the new framework the new approach named gark introduces additional flexibility when compared to traditional partitioned runge kutta formalism and therefore offers additional opportunities for the development of flexible solvers for systems with multiple scales or driven by multiple physical processes

inadequacy of first order upwind difference schemes for some recirculating flows spurious numerical solutions of problems with closed sub characteristics by upwind difference schemes in particular problems of recirculating incompressible flow at high reynolds numbers are proved to be due to the anisotropy of the artificial viscosity numerical examples are presented to show that even very simple problems including basic problems in fluid dynamics are not approximated well by schemes with anisotropic artificial viscosity regardless of numerical parameters 1991 academic press inc numerical methods for solving incompressible fluid flow equations are frequently tested on problems of recirculating flow quite often results obtained by different methods differ from each other significantly reliable experimental data is usually difficult to obtain and discrepancies between numerical two dimensional solutions and experimental data may be the result of the influence of walls in the experiments the problems in solving recirculating incompressible flows at high reynolds numbers are well known several researchers have pointed out that some of the methods employed to obtain stable discretization e g upwind differencing may lead to spurious results such results have been attributed to excessive artificial viscosity in the numerical scheme to multiple solutions of the non linear set of algebraic equations that is obtained from the numerical scheme and to poor resolution by grids that are too coarse 2 4 51 but spurious results occur even when none of these reasons are valid indeed a single linear partial differential equation may exhibit such behavior when the sub characteristics are closed this is true even when the mesh size is small enough to easily resolve the solution and truncation errors are genuinely small everywhere and even when the differential solution does not depend on the

fault attacks on pairing based protocols revisited several papers have studied fault attacks on computing a pairing value e p q where p is a public point and q is a secret point in this paper we observe that these attacks are in fact effective only on a small number of pairing based protocols and that too only when the protocols are implemented with specific symmetric pairings we demonstrate the effectiveness of the fault attacks on a public key encryption scheme an identity based encryption scheme and an oblivious transfer protocol when implemented with a symmetric pairing derived from a supersingular elliptic curve with embedding degree 2

singularly perturbed problems in partial differential equations a survey this survey paper contains a surprisingly large amount of material on singularly perturbed partial differential equations and indeed can serve as an introduction to some of the ideas and methods of the singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the methods developed by numerous researchers after 1980 2000 in this review we have covered singularly perturbed partial differential equations about odes the survey has already been done by us see m k kadalbajoo k c patidar appl math comput 130 2002 457 510

convergence analysis of finite element methods for singularly perturbed problems abstract in this paper a unified convergence analysis is presented for solving singularly perturbed problems by using the standard galerkin finite element method on a nontraditional shishkin type mesh which separates the boundary layers totally from other subregions the results obtained show that the error estimates on such nontraditional shishkin type mesh are much easier to prove than on the traditional shishkin type mesh however both meshes give comparable error estimates which justifies the conjecture of roos 1 the generality of our techniques is showed by investigations of high order problems steady and nonsteady semilinear problems

spectral fractional step runge kutta discretizations for initial boundary value problems with time dependent boundary conditions in this paper we develop a technique for avoiding the order reduction caused by nonconstant boundary conditions in the methods called splitting alternating direction or more generally fractional step methods such methods can be viewed as the combination of a semidiscrete in time procedure with a special type of additive runge kutta method which is called the fractional step runge kutta method and a standard space discretization which can be of type finite differences finite elements or spectral methods among others spectral methods have been chosen here to complete the analysis of convergence of a totally discrete scheme of this type of improved fractionary steps the numerical experiences performed also show the increase of accuracy that this technique provides

newton like method with modification of the right hand side vector this paper proposes a new newton like method which defines new iterates using a linear system with the same coefficient matrix in each iterate while the correction is performed on the right hand side vector of the newton system in this way a method is obtained which is less costly than the newton method and faster than the fixed newton method local convergence is proved for nonsingular systems the influence of the relaxation parameter is analyzed and explicit formulae for the selection of an optimal parameter are presented relevant numerical examples are used to demonstrate the advantages of the proposed method

additive runge kutta methods for the resolution of linear parabolic problems we study the consistency for general additive runge kutta methods in the integration of linear nonhomogeneous problems obtaining necessary and sufficient conditions of order p for arbitrary values of p we use this result joined to some a stability conditions for developing a third order additive runge kutta method of type fractional steps and we show its efficiency in the numerical integration of a two dimensional evolutionary convection diffusion problem

a tetrahedral mixed finite element method for the stationary semiconductor continuity equations a petrov galerkin mixed finite element method based on tetrahedral elements for the stationary semiconductor device continuity equations is presented this method can be regarded as a natural extension to three dimensions of the well known scharfetter gummel one dimensional scheme existence uniqueness and stability of the approximate solution are proved and an error estimate is given for an arbitrary delaunay mesh and corresponding dirichlet tessellation the associated linear system is a symmetric and positive definite m matrix the evaluation of the terminal currents associated with the method is also discussed and it is shown that the computed terminal currents are convergent and conservative numerical results for a model three dimensional device are presented to validate the theoretical results

symplectic methods based on decompositions we consider methods that integrate systems of differential equations dy dt f y by taking advantage of a decomposition of the right hand side f sum f nu we derive a general necessary and sufficient condition for those methods to be symplectic for hamiltonian problems special attention is given to the case of additive runge kutta methods

structure preservation for constrained dynamics with super partitioned additive runge kutta methods a broad class of partitioned differential equations with possible algebraic constraints is considered including hamiltonian and mechanical systems with holonomic constraints for mechanical systems a formulation eliminating the coriolis forces and closely related to the euler lagrange equations is presented a new class of integrators is defined the super partitioned additive runge kutta spark methods this class is based on the partitioning of the system into different variables and on the splitting of the differential equations into different terms a linear stability and convergence analysis of these methods is given spark methods allowing the direct preservation of certain properties are characterized different structures and invariants are considered the manifold of constraints symplecticness reversibility contractivity dilatation energy momentum and quadratic invariants with respect to linear stability and structure preservation the class of s stage lobatto iiia b c c spark methods is of special interest controllable numerical damping can be introduced by the use of additional parameters some issues related to the implementation of a reversible variable stepsize strategy are discussed

a time adaptive finite volume method for the cahn hilliard and kuramoto sivashinsky equations this paper presents a complete finite volume method for the cahn hilliard and kuramoto sivashinsky type of equations the spatial discretization is high order accurate and suitable for general unstructured grids the time integration is addressed by means of implicit an implicit explicit fourth order runge kutta schemes with error control and adaptive time stepping the outcome is a practical accurate and efficient simulation tool which has been successfully applied to accuracy tests and representative simulations the use of adaptive time stepping is of paramount importance in problems governed by the cahn hilliard model an adaptive method may be several orders of magnitude more efficient than schemes using constant or heuristic time steps in addition to driving the simulations efficiently the time adaptive procedure provides a quantitative not just qualitative characterization of the rich temporal scales present in phase separation processes governed by the cahn hilliard phase field model

maximum norm error analysis of a 2d singularly perturbed semilinear reaction diffusion problem a semilinear reaction diffusion equation with multiple solutions is considered in a smooth two dimensional domain its diffusion parameter e 2 is arbitrarily small which induces boundary layers constructing discrete sub and super solutions we prove existence and investigate the accuracy of multiple discrete solutions on layer adapted meshes of bakhvalov and shishkin types it is shown that one gets second order convergence with in the case of the shishkin mesh a logarithmic factor in the discrete maximum norm uniformly in e for 0 is the maximum side length of mesh elements while the number of mesh nodes does not exceed ch 2 numerical experiments are performed to support the theoretical results

an analysis of a defect correction method for a model convection diffusion equation the error in a defect correction method for a model one dimensional convection diffusion problem without turning points is analyzed although the analysis is limited to the one dimensional problem the scheme is a computationally attractive way to increase the accuracy in smooth regions of first order schemes for general multidimensional problems it is shown that in one dimension the kth approximation converges uniformly in varepsilon in regions bounded away from the layer with rate o varepsilon 0 varepsilon k h 2 varepsilon 0 o h the estimates degrade to o 1 near layers numerical experiments confirm that this is precisely the behavior of the method

some numerical experiments with multigrid methods on shishkin meshes piecewise uniform meshes introduced by shishkin are a very useful tool to construct robust and efficient numerical methods to approximate the solution of singularly perturbed problems for small values of the diffusion coefficient the step size ratios in this kind of grids can be very large in this case standard multigrid methods are not convergent to avoid this troublesome in this paper we propose a modified multigrid algorithm which works fine on shishkin meshes we show some numerical experiments confirming that the proposed multigrid method is convergent and it has similar properties that standard multigrid for classical elliptic problems