analysis of dual consistency for discontinuous galerkin discretizations of source terms the effects of dual consistency on discontinuous galerkin discretizations of solution and solution gradient dependent source terms are examined two common discretizations are analyzed the standard weighting technique for source terms and the mixed formulation it is shown that if the source term depends on the first derivative of the solution the standard weighting technique leads to a dual inconsistent scheme a straightforward procedure for correcting this dual inconsistency and arriving at a dual consistent discretization is demonstrated the mixed formulation where the solution gradient in the source term is replaced by an additional variable that is solved for simultaneously with the state leads to an asymptotically dual consistent discretization numerical results for a one dimensional test problem confirm that the dual consistent and asymptotically dual consistent schemes achieve higher asymptotic convergence rates with grid refinement than a similar dual inconsistent scheme for both the primal and adjoint solutions as well as a simple functional output

a comparison of two entropy stable discontinuous galerkin spectral element approximations for the shallow water equations with non constant topography in this work we compare and contrast two provably entropy stable and high order accurate nodal discontinuous galerkin spectral element methods applied to the one dimensional shallow water equations for problems with non constant bottom topography of particular importance for numerical approximations of the shallow water equations is the well balanced property the well balanced property is an attribute that a numerical approximation can preserve a steady state solution of constant water height in the presence of a bottom topography numerical tests are performed to explore similarities and differences in the two high order schemes

energy stable and high order accurate finite difference methods on staggered grids for wave propagation over distances of many wavelengths high order finite difference methods on staggered grids are widely used due to their excellent dispersion properties however the enforceme

discrete dynamical systems associated with root systems of indefinite type a geometric charactrization of the equation found by hietarinta and viallet which satisfies the singularity confinement criterion but which exhibits chaotic behavior is presented it is shown that this equation can be lifted to an automorphism of a certain rational surface and can therefore be considered to be a realization of a cremona isometry on the picard group of the surface it is also shown that the group of cremona isometries is isomorphic to an extended weyl group of indefinite type a method to construct the mappings associated with some root systems of indefinite type is also presented

superconvergence of the velocity in mimetic finite difference methods on quadrilaterals superconvergence of the velocity is established for mimetic finite difference approximations of second order elliptic problems over h 2 uniform quadrilateral meshes the superconvergence result holds for a full tensor coefficient the analysis exploits the relation between mimetic finite differences and mixed finite element methods via a special quadrature rule for computing the scalar product in the velocity space the theoretical results are confirmed by numerical experiments

common fixed point and invariant approximation results for noncommuting generalized f g nonexpansive maps we present common fixed point results for noncommuting generalized f g nonexpansive maps as application invariant approximation results are obtained our results unify and generalize various known results existing in the literature

linearisable ultradiscrete systems with sign variables and the confinement of singularities we present the singularity analysis of the ultradiscrete analogue of linearisable mappings of quispel roberts thompson qrt type the ultradiscretisation method used here is one which keeps track of signs and thus can be applied without the positivity restrictions of the classical ultradiscretisation approach we show that in all cases the mappings possess confined singularities the same is true for two non autonomous equations which are equally linearisable in the discrete case we construct explicitly the solutions of the ultradiscrete mappings analysed here

third order integrable difference equations generated by a pair of second order equations we show that the third order difference equations proposed by hirota kimura and yahagi are generated by a pair of second order difference equations in some cases the pair of the second order equations are equivalent to the quispel robert thomson qrt system but in the other cases they are irrelevant to the qrt system we also discuss an ultradiscretization of the equations

analytical and numerical treatment of jungck type iterative schemes in this paper we introduce a new and general jungck type iterative scheme for a pair of nonself mappings and study its strong convergence stability and data dependence it is exhibited that our iterative scheme has much better convergence rate than those of jungck mann jungck ishikawa jungck noor and jungck cr iterative schemes numerical examples in support of validity and applications of our results are provided our results are extension improvement and generalization of many known results in the literature of fixed point theory

grid stabilization of high order one sided differencing ii second order wave equations we demonstrate the stable boundary closure of difference methods of order up through 16 for the solution of wave equations in second order form our method combines the introduction of 1 2 judiciously placed subcell grid points near the boundary with minimal stencil one sided difference operators of the same order as the interior scheme the method is tested on a variety of problems including the scalar wave equation discretized on mapped grids and overlapping composite grids as well as an integrable nonlinear system

invariant approximations and r subweakly commuting maps the existence of invariant approximations for r subweakly commuting mappings is proved our results extend most of the known results to a new class of noncommuting mappings

superconvergent functional output for time dependent problems using finite differences on summation by parts form finite difference operators satisfying the summation by parts sbp rules can be used to obtain high order accurate energy stable schemes for time dependent partial differential equations when the boundary conditions are imposed weakly by the simultaneous approximation term sat in general an sbp sat discretization is accurate of order p 1 with an internal accuracy of 2p and a boundary accuracy of p despite this it is shown in this paper that any linear functional computed from the time dependent solution will be accurate of order 2p when the boundary terms are imposed in a stable and dual consistent way the method does not involve the solution of the dual equations and superconvergent functionals are obtained at no extra computational cost four representative model problems are analyzed in terms of convergence and errors and it is shown in a systematic way how to derive schemes which gives superconvergent functional outputs

on the complexity of some birational transformations using three different approaches we analyse the complexity of various birational maps constructed from simple operations inversions on square matrices of arbitrary size the first approach comprises the study of the images of lines and relies mainly on univariate polynomial algebra the second approach is a singularity analysis and the third method is more numerical using integer arithmetics these three methods have their own domain of application but they give corroborating results and lead us to a conjecture on the algebraic entropy of a class of maps constructed from matrix inversions

on the algebraic structure of rational discrete dynamical systems we show how singularities shape the evolution of rational discrete dynamical sys tems the stabilisation of the form of the iterates suggests a description providing among other things generalised hirota form exact evaluation of the algebraic entropy as well as remarkable polynomial factorisation properties we illustrate the phenomenon explicitly with examples covering a wide range of models

a well balanced and entropy conservative discontinuous galerkin spectral element method for the shallow water equations in this work we design an arbitrary high order accurate nodal discontinuous galerkin spectral element type method for the one dimensional shallow water equations the novel method uses a skew symmetric formulation of the continuous problem we prove that this discretisation exactly preserves the local mass and momentum furthermore we show that combined with a special numerical interface flux function the method exactly preserves the entropy which is also the total energy for the shallow water equations finally we prove that the surface fluxes the skew symmetric volume integrals and the source term are well balanced numerical tests are performed to demonstrate the theoretical findings

common fixed point and approximation results for h operator pair with applications abstract the existence of common fixed point results for new classes of noncommuting selfmaps satisfying generalized i contraction or i nonexpansive type conditions is established as applications several invariant best approximation results are proved which unify extend and complement the well known results

algebraic entropy and the space of initial values for discrete dynamical systems a method to calculate the algebraic entropy of a mapping which can be lifted to an isomorphism of a suitable rational surface the space of initial values is presented it is shown that the degree of the nth iterate of such a mapping is given by its action on the picard group of the space of initial values it is also shown by construction that the degree of the nth iterate of every painleve equation in sakai s list is o n2 and therefore its algebraic entropy is zero

banach operator pairs and common fixed points in hyperconvex metric spaces the purpose of this paper is to establish demarr s well known theorem for an arbitrary family of symmetric banach operator pairs in hyperconvex metric spaces without the compactness assumption we also give necessary and sufficient criteria for the existence of a common fixed point of a semigroup of isometric mappings as an application several results on the invariant best approximation are proved

on common fixed points in g metric spaces using e a property in this paper we introduce some new types of pairs of mappings f g on g metric spaces called g weakly commuting of type g f and g r weakly commuting of type g f we obtain also several common fixed point results by using the e a property

superconvergence of discontinuous galerkin and local discontinuous galerkin schemes for linear hyperbolic and convection diffusion equations in one space dimension in this paper we study the superconvergence property for the discontinuous galerkin dg and the local discontinuous galerkin ldg methods for solving one dimensional time dependent linear conservation laws and convection diffusion equations we prove superconvergence towards a particular projection of the exact solution when the upwind flux is used for conservation laws and when the alternating flux is used for convection diffusion equations the order of superconvergence for both cases is proved to be k frac 3 2 when piecewise p k polynomials with k geq1 are used the proof is valid for arbitrary nonuniform regular meshes and for piecewise p k polynomials with arbitrary k geq1 improving upon the results in y cheng and c w shu j comput phys 227 2008 pp 9612 9627 y cheng and c w shu computers and structures 87 2009 pp 630 641 in which the proof based on fourier analysis was given only for uniform meshes with periodic boundary condition and piecewise p 1 polynomials

singularity confinement test for ultradiscrete equations with parity variables we present a new ultradiscretization method which does not require that the solutions of the discrete equation have a fixed sign we construct an ultradiscrete analogue of the singularity confinement test using this method and thereby propose an integrability test for ultradiscrete equations

superconvergent functional estimates from summation by parts finite difference discretizations diagonal norm summation by parts sbp operators can be used to construct time stable high order accurate finite difference schemes however to achieve both stability and accuracy these operators must use s order accurate boundary closures when the interior scheme is 2s order accurate the boundary closure limits the solution to s 1 order global accuracy despite this bound on solution accuracy we show that functional estimates can be constructed that are 2s order accurate this superconvergence requires dual consistency which depends on the sbp operators the boundary condition implementation and the discretized functional the theory is developed for scalar hyperbolic and elliptic partial differential equations in one dimension in higher dimensions we show that superconvergent functional estimates remain viable in the presence of curvilinear multiblock grids with interfaces the generality of the theoretical results is demonstrated using a two dimensional poisson problem and a nonlinear hyperbolic system the euler equations of fluid mechanics

common fixed point results for noncommuting mappings without continuity in generalized metric spaces using the setting of a generalized metric space a fixed point theorem is proved for one map and several fixed point theorems are proved for two maps these results generalize several well known comparable results in the literature

elementary exact calculations of degree growth and entropy for discrete equations second order discrete equations are studied over the field of rational functions c z where z is a variable not appearing in the equation the exact degree of each iterate as a function of z can b

discretizing singular point sources in hyperbolic wave propagation problems we develop high order accurate source discretizations for hyperbolic wave propagation problems in first order formulation that are discretized by finite difference schemes by studying the fourier series expansions of the source discretization and the finite difference operator we derive sufficient conditions for achieving design accuracy in the numerical solution only half of the conditions in fourier space can be satisfied through moment conditions on the source discretization and we develop smoothness conditions for satisfying the remaining accuracy conditions the resulting source discretization has compact support in physical space and is spread over as many grid points as the number of moment and smoothness conditions in numerical experiments we demonstrate high order of accuracy in the numerical solution of the 1 d advection equation both in the interior and near a boundary the 3 d elastic wave equation and the 3 d linearized euler equations

common fixed points in best approximation for banach operator pairs with ciric type i contractions abstract the common fixed point theorems similar to those of ciric lj b ciric on a common fixed point theorem of a gregus type publ inst math beograd n s 49 1991 174 178 lj b ciric on diviccaro fisher and sessa open questions arch math brno 29 1993 145 152 lj b ciric on a generalization of gregus fixed point theorem czechoslovak math j 50 2000 449 458 fisher and sessa b fisher s sessa on a fixed point theorem of gregus internat j math math sci 9 1986 23 28 jungck g jungck on a fixed point theorem of fisher and sessa internat j math math sci 13 1990 497 500 and mukherjee and verma r n mukherjee v verma a note on fixed point theorem of gregus math japon 33 1988 745 749 are proved for a banach operator pair as applications common fixed point and approximation results for banach operator pair satisfying ciric type contractive conditions are obtained without the assumption of linearity or affinity of either t or i our results unify and generalize various known results to a more general class of noncommuting mappings

energy conservation issues in the numerical solution of the semilinear wave equation in this paper we discuss energy conservation issues related to the numerical solution of the semilinear wave equation as is well known this problem can be cast as a hamiltonian system that may be autonomous or not depending on the prescribed boundary conditions we relate the conservation properties of the original problem to those of its semi discrete version obtained by the method of lines subsequently we show that the very same properties can be transferred to the solutions of the fully discretized problem obtained by using energy conserving methods in the hbvms hamiltonian boundary value methods class similar arguments hold true for different types of hamiltonian partial differential equations e g the nonlinear schrodinger equation

well posed boundary conditions for the navier stokes equations in this article we propose a general procedure that allows us to determine both the number and type of boundary conditions for time dependent partial differential equations with those well posedness can be proven for a general initial boundary value problem the procedure is exemplified on the linearized navier stokes equations in two and three space dimensions on a general domain

common fixed point theorems for families of weakly compatible maps in this paper the existence and approximation of a unique common fixed point of two families of weakly compatible self maps on a complete metric space are investigated an example is presented to show that our results for the mappings considered satisfying non linear contractive type conditions are genuine generalizations of the recent result for metric spaces b singh s jain a fixed point theorem in menger space through weak compatibility j math anal appl 301 2005 439 448 theorem 3 3 and many other known results

rational surfaces associated with affine root systems and geometry of the painleve equations we present a geometric approach to the theory of painleve equations based on rational surfaces our starting point is a compact smooth rational surface x which has a unique anti canonical divisor d of canonical type we classify all such surfaces x to each x there corresponds a root subsystem of e 1 8 inside the picard lattice of x we realize the action of the corresponding affine weyl group as the cremona action on a family of these surfaces we show that the translation part of the affine weyl group gives rise to discrete painleve equations and that the above action constitutes their group of symmetries by backlund transformations the six painleve differential equations appear as degenerate cases of this construction in the latter context x is okamoto s space of initial conditions and d is the pole divisor of the symplectic form defining the hamiltonian structure

best simultaneous approximations asymptotically nonexpansive mappings and variational inequalities in banach spaces abstract the existence of common fixed points is established for three mappings where t is either generalized f g nonexpansive or asymptotically f g nonexpansive on a nonempty subset of a banach space as applications the invariant best simultaneous approximation results are proved and the existence of solution of variational inequalities is obtained our results unify and substantially improve several recent results existing in the current literature

summation by parts projections and stability ii we have derived stability results for high order finite difference approximations of mixed hyperbolic parabolic initial boundary value problems ibvp the results are obtained using summation by parts and a new way of representing general linear boundary conditions as an orthogonal projection by slightly rearranging the analytic equations we can prove strict stability for hyperbolic parabolic ibvp furthermore we generalize our technique so as to yield strict stability on curvilinear non smooth domains in two space dimensions finally we show how to incorporate inhomogeneous boundary data while retaining strict stability using the same procedure one can prove strict stability in higher dimensions as well

continuous families of rational surface automorphisms with positive entropy for any k we construct k parameter families of rational surface automorphisms with positive entropy these are automorphisms of surfaces mathcal x which are constructed from iterated blowups over the projective plane in certain cases we are able to determine the exact automorphism group of mathcal x as well as when two of these surfaces are inequivalent

grid stabilization of high order one sided differencing i first order hyperbolic systems we construct stable maximal order boundary closures for high order central difference methods the stability is achieved by adding a small number of additional subcell nodes near the boundaries at experimentally determined locations we find that methods up through 8th order can be stabilized by the addition of a single node up through 16th order by the addition of two nodes and up through 22nd order with three extra nodes we also consider the application of the technique to dispersion relation preserving methods and we construct and test artificial dissipation operators

noncommuting selfmaps and invariant approximations abstract common fixed point results are established for a new class of noncommuting selfmaps we apply them to obtain several invariant approximation results which unify extend and complement well known results

compatible maps and invariant approximations the existence of invariant best approximations for compatible maps is proved our results unify and generalize various known results to a more general class of noncommuting mappings

common fixed point theorems for pd operator pairs under relaxed conditions with applications in this paper we define pd operator pair of single valued mappings and obtain some common fixed point theorems for this class of maps under relaxed conditions our theorem generalizes results of bhatt et al a bhatt h chandra d r sahu common fixed point theorems for occasionally weakly compatible mappings under relaxed conditions nonlinear anal 73 2010 176 182 jungck and rhoades g jungck b e rhoades fixed point theorems for occasionally weakly compatible mappings fixed point theory 7 2 2006 287 296 and hussain et al n hussain m a khamsi a latif common fixed points for jh operators and occasionally weakly biased pairs under relaxed conditions nonlinear anal 74 2011 2133 2140 and others as applications we also study the solution of functional equations arising from dynamic programming and variational inequalities arises in two point obstacle problem we also discuss some illustrative examples to highlight the realized improvements

a coupled coincidence point result in partially ordered metric spaces for compatible mappings abstract in this paper we introduce the notion of compatibility of mappings in a partially ordered metric space and use this notion to establish a coupled coincidence point result our work extends the work of bhaskar and lakshmikantham t gnana bhaskar v lakshmikantham fixed point theorems in partially ordered metric spaces and applications nonlinear anal tma 65 2006 1379 1393 an example is also given

common fixed points for banach operator pairs with applications the existence of common fixed point results for a banach operator pair under certain generalized contractions is established the invariant best approximation results are proved as applications and the existence of solutions of variational inequalities is obtained we also study the solution of functional equations arising from dynamic programming 2007 elsevier ltd all rights reserved

singularity confinement and chaos in discrete systems we present a number of second order maps which pass the singularity confinement test commonly used to identify integrable discrete systems but which nevertheless are non integrable as a more sensitive integrability test we propose the analysis of the complexity algebraic entropy of the map using the growth of the degree of its iterates integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems

algebraic entropy of an extended hietarinta viallet equation we introduce a series of discrete mappings which is considered to be an extension of the hietarinta viallet mapping with one parameter we obtain the algebraic entropy for this mapping by obtaining the recurrence relation for the degrees of the iterated mapping for some parameter values the mapping has a confined singularity in which case the mapping is equivalent to a recurrence relation between six irreducible polynomials for other parameter values the mapping does not pass the singularity confinement test the properties of irreducibility and co primeness of the terms play crucial roles in the discussion

output error estimation for summation by parts finite difference schemes the paper develops a posteriori error estimates of integral output functionals for summation by parts finite difference methods the error estimates are based on the adjoint weighted residual method and take advantage of a variational interpretation of summation by parts discretizations the estimates are computed on a fixed grid and do not require an embedded grid or explicit interpolation operators for smooth boundary value problems containing first and second derivatives the error estimates converge to the exact error as the mesh is refined the theory is verified using linear boundary value problems and the euler equations

minimal repetition dynamic checkpointing algorithm for unsteady adjoint calculation adjoint equations of differential equations have seen widespread applications in optimization inverse problems and uncertainty quantification a major challenge in solving adjoint equations for time dependent systems has been the need to use the solution of the original system in the adjoint calculation and the associated memory requirement in applications where storing the entire solution history is impractical checkpointing methods have frequently been used however traditional checkpointing algorithms such as revolve require a priori knowledge of the number of time steps making these methods incompatible with adaptive time stepping we propose a dynamic checkpointing algorithm applicable when the number of time steps is a priori unknown our algorithm maintains a specified number of checkpoints on the fly as time integration proceeds for an arbitrary number of time steps the resulting checkpoints at any snapshot during the time integration have the optimal repetition number the efficiency of our algorithm is demonstrated both analytically and experimentally in solving adjoint equations this algorithm also has significant advantage in automatic differentiation when the length of execution is variable

high order finite difference methods for wave propagation in discontinuous media high order finite difference approximations are derived for the second order wave equation with discontinuous coefficients on rectangular geometries the discontinuity is treated by splitting the domain at the discontinuities in a multi block fashion each sub domain is discretized with compact second derivative summation by parts operators and the blocks are patched together to a global domain using the projection method this guarantees a conservative strictly stable and high order accurate scheme the analysis is verified by numerical simulations in one and two spatial dimensions

common fixed points for jℋ operators and occasionally weakly biased pairs under relaxed conditions abstract some common fixed point theorems due to bhatt et al a bhatt et al common fixed point theorems for occasionally weakly compatible mappings under relaxed conditions nonlinear anal 73 2010 176 182 jungck and rhoades g jungck and b e rhoades fixed point theorems for occasionally weakly compatible mappings fixed point theory 7 2 2006 287 296 fixed point theory 9 2008 383 384 erratum and imdad and soliman m imdad a h soliman some common fixed point theorems for a pair of tangential mappings in symmetric spaces appl math lett 23 2010 351 355 are extended to two new classes of non commuting selfmaps which contain the occasionally weakly compatible and weakly biased selfmaps as proper subclasses some illustrative examples are also provided to highlight the realized improvements

stable and accurate wave propagation in discontinuous media a time stable discretization is derived for the second order wave equation with discontinuous coefficients the discontinuity corresponds to inhomogeneity in the underlying medium and is treated by splitting the domain each homogeneous sub domain is discretized using narrow diagonal summation by parts operators and then patched to its neighbors by using a penalty method leading to fully explicit time integration this discretization yields a time stable and efficient scheme the analysis is verified by numerical simulations in one dimension using high order finite difference discretizations and in three dimensions using an unstructured finite volume discretization

summation by parts operators and high order quadrature summation by parts sbp operators are finite difference operators that mimic integration by parts the sbp operator definition includes a weight matrix that is used formally for discrete integration however the accuracy of the weight matrix as a quadrature rule is not explicitly part of the sbp definition we show that sbp weight matrices are related to trapezoid rules with end corrections whose accuracy matches the corresponding difference operator at internal nodes for diagonal weight matrices the accuracy of sbp quadrature extends to curvilinear domains provided the jacobian is approximated with the same sbp operator used for the quadrature this quadrature has significant implications for sbp based discretizations in particular the diagonal norm accurately approximates the l 2 norm for functions and multi dimensional sbp discretizations accurately approximate the divergence theorem

common fixed points for p operator pair with applications the existence of common fixed point results for new classes of noncommuting selfmaps satisfying generalized i contraction or i nonexpansive type conditions is established as applications several invariant best approximation results are proved which unify extend and complement the well known results

do integrable cellular automata have the confinement property we analyse a criterion introduced by joshi and lafortune for the integrability of cellular automata obtained from discrete systems through the ultradiscretization procedure we show that while this criterion can be used in order to single out integrable ultradiscrete systems there do exist cases where the system is nonintegrable and still the criterion is satisfied conversely we show that for ultradiscrete systems that are derived from linearizable mappings the criterion is not satisfied we investigate this phenomenon further in the case of a mapping which includes a linearizable subcase and show how the violation of the criterion comes to be finally we comment on the growth properties of ultradiscrete systems

a geometric approach to singularity confinement and algebraic entropy a geometric approach to the equation found by hietarinta and viallet which satisfies the singularity confinement criterion but exhibits chaotic behaviour is presented it is shown that this equation can be lifted to an automorphism of a certain rational surface and can therefore be considered to be the action of an extended weyl group of indefinite type a method to calculate its algebraic entropy by using the theory of intersection numbers is presented

optimal time splitting for two and three dimensional navier stokes equations with mixed derivatives a new explicit time splitting algorithm has been developed for finite difference modelling of the full two and three dimensional time dependent compressible viscous navier stokes equations of fluid mechanics the scheme is optimal in the sense that the split operators achieve their maximum allowable time step i e the corresponding courant number the algorithm allows a conservation form formulation stability is proven analytically and verified numerically in proving stability it was shown that all nine matrix coefficients of the navier stokes equations are simultaneously symmetrizable by a similarity transformation two such transformations and their resulting symmetric matrix coefficients are presented explicitly

a systematic method for constructing discrete painleve equations in the degeneration cascade of the e8 group we present a systematic and quite elementary method for constructing discrete painleve equations in the degeneration cascade for e8 1 starting from the invariant for the autonomous limit of the e8 1 equation one wishes to study the method relies on choosing simple homographies that will cast this invariant into certain judiciously chosen canonical forms these new invariants lead to mappings the deautonomisations of which allow us to build up the entire degeneration cascade of the original mapping we explain the method on three examples two symmetric mappings and an asymmetric one and we discuss the link between our results and the known geometric structure of these mappings