the design of local navier stokes preconditioning for compressible flow a family of navier stokes preconditioners is presented that may reduce the stiffness due to complicated interaction between convection and diffusion in viscous flows navier stokes preconditioning is developed based on a fourier analysis of the discretized equations and a dispersion analysis of the differential equations navier stokes preconditioning can be extended from the euler technique with two methods a by using block jacobi preconditioning for the viscous terms b by introducing analytic dependence on the cell reynolds number in the preconditioner with these techniques it is possible to produce a local navier stokes preconditioner effective for all mach and cell reynolds numbers these techniques and a combined method are analyzed with respect to condition number and linear wave propagation and are illustrated with some numerical results

external boundary conditions for three dimensional problems of computational aerodynamics we consider an unbounded steady state flow of viscous fluid over a three dimensional finite body or configuration of bodies for the purpose of solving this flow numerically we discretize the governing equations navier stokes on a finite difference grid prior to the discretization we obviously need to truncate the original unbounded domain by introducing an artificial computational boundary at a finite distance from the body otherwise the number of discrete variables will not be finite this artificial boundary is typically the external boundary of the domain covered by the grid the flow problem both continuous and discretized formulated on the finite computational domain is clearly subdefinite unless supplemented by some artificial boundary conditions abcs at the external computational boundary in this paper we present an innovative approach to constructing highly accurate abcs for three dimensional flow computations the approach extends our previous technique developed for the two dimensional case it employs the finite difference counterparts to calderon s pseudodifferential boundary projections calculated in the framework of the difference potentials method dpm of ryaben kii the resulting abcs appear spatially nonlocal but are particularly easy to implement along with the existing flow solvers the new boundary conditions have been successfully combined with the nasa developed production code tlns3d and used for the analysis of wing shaped configurations in subsonic and transonic flow regimes as demonstrated by the computational experiments and comparison with the standard local methods the dpm based abcs allow one to greatly reduce the size of the computational domain while still maintaining high accuracy of the numerical solution moreover they may provide for a noticeable speedup of multigrid convergence

a solution to the stability issues with block norm summation by parts operators finite difference operators approximating first and second derivatives and satisfying a summation by parts sbp rule have been derived for the 4 t h 6 t h 8 t h and 10 t h order case by using the symbolic mathematics software maple the operators are based on block norms to avoid the curse of losing accuracy at the boundaries that are present for corresponding operators based on diagonal norms the reason why block norm operators have not been used in realistic applications before is related to the well known stability issues on curvilinear grids to avoid this problem an additional boundary stabilization is introduced that removes unstable eigenvalues without interfering with accuracy and stiffness the superior accuracy properties of the newly derived block norm sbp operators will be demonstrated for the second order wave equation in 1 d and for the compressible euler equations in complex 2 d geometries

a fully discrete stable and conservative summation by parts formulation for deforming interfaces we introduce an interface coupling procedure for hyperbolic problems posed on time dependent curved multi domains first we transform the problem from cartesian to boundary conforming curvilinear coordinates and apply the energy method to derive well posed and conservative interface conditions next we discretize the problem in space and time by employing finite difference operators that satisfy a summation by parts rule the interface condition is imposed weakly using a penalty formulation we show how to formulate the penalty operators such that the coupling procedure is automatically adjusted to the movements and deformations of the interface while both stability and conservation conditions are respected the developed techniques are illustrated by performing numerical experiments on the linearized euler equations and the maxwell equations the results corroborate the stability and accuracy of the fully discrete approximations

duality based boundary conditions and dual consistent finite difference discretizations of the navier stokes and euler equations in this paper we derive new far field boundary conditions for the time dependent navier stokes and euler equations in two space dimensions the new boundary conditions are derived by simultaneously considering well posedness of both the primal and dual problems we moreover require that the boundary conditions for the primal and dual navier stokes equations converge to well posed boundary conditions for the primal and dual euler equations we perform computations with a high order finite difference scheme on summation by parts form with the new boundary conditions imposed weakly by the simultaneous approximation term we prove that the scheme is both energy stable and dual consistent and show numerically that both linear and non linear integral functionals become superconvergent

stable galerkin reduced order models for linearized compressible flow the galerkin projection procedure for construction of reduced order models of compressible flow is examined as an alternative discretization of the governing differential equations the numerical stability of galerkin models is shown to depend on the choice of inner product for the projection for the linearized euler equations a symmetry transformation leads to a stable formulation for the inner product boundary conditions for compressible flow that preserve stability of the reduced order model are constructed preservation of stability for the discrete implementation of the galerkin projection is made possible using a piecewise smooth finite element basis stability of the reduced order model using this approach is demonstrated on several model problems where a suitable approximation basis is generated using proper orthogonal decomposition of a transient computational fluid dynamics simulation

adjoint and defect error bounding and correction for functional estimates we present two error estimation approaches for bounding or correcting the error in functional estimates such as lift or drag adjoint methods quantify the error in a particular output functional that results from residual errors in approximating the solution to the partial differential equation defect methods can be used to bound or reduce the error in the entire solution with corresponding improvements to functional estimates both approaches rely on smooth solution reconstructions and may be used separately or in combination to obtain highly accurate solutions with asymptotically sharp error bounds the adjoint theory is presented for both smooth and shocked problems numerical experiments confirm fourth order error estimates for a pressure integral of shocked quasi id euler flow by employing defect and adjoint methods together and accounting for errors in approximating the geometry it is possible to obtain functional estimates that exceed the order of accuracy of the discretization process and the reconstruction approach superconvergent drag estimates are obtained for subsonic euler flow over a lifting airfoil

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

variance reduction through robust design of boundary conditions for stochastic hyperbolic systems of equations we consider a hyperbolic system with uncertainty in the boundary and initial data our aim is to show that dierent boundary conditions gives dierent convergence rates of the variance of the solution this means that we can with the same knowledge of data get a more or less accurate description of the uncertainty in the solution a variety of boundary conditions are compared and both analytical and numerical estimates of the variance of the solution is presented as applications we study the eect of this technique on maxwell s equations as well as on a subsonic outow boundary for the euler equations

boundary conditions and stability of a perfectly matched layer for the elastic wave equation in first order form in computations it is now common to surround artificial boundaries of a computational domain with a perfectly matched layer pml of finite thickness in order to prevent artificially reflected waves from contaminating a numerical simulation unfortunately the pml does not give us an indication about appropriate boundary conditions needed to close the edges of the pml or how those boundary conditions should be enforced in a numerical setting terminating the pml with an inappropriate boundary condition or an unstable numerical boundary procedure can lead to exponential growth in the pml which will eventually destroy the accuracy of a numerical simulation everywhere in this paper we analyze the stability and the well posedness of boundary conditions terminating the pml for the elastic wave equation in first order form first we consider a vertical modal pml truncating a two space dimensional computational domain in the horizontal direction we freeze all coefficients and consider a left half plane problem with linear boundary conditions terminating the pml the normal mode analysis is used to study the stability and well posedness of the resulting initial boundary value problem ibvp the result is that any linear well posed boundary condition yielding an energy estimate for the elastic wave equation without the pml will also lead to a well posed ibvp for the pml second we extend the analysis to the pml corner region where both a horizontal and vertical pml are simultaneously active the challenge lies in constructing accurate and stable numerical approximations for the pml and the boundary conditions third we develop a high order accurate finite difference approximation of the pml subject to the boundary conditions to enable accurate and stable numerical boundary treatments for the pml we construct continuous energy estimates in the laplace space for a one space dimensional problem and two space dimensional pml corner problem we use summation by parts finite difference operators to approximate the spatial derivatives and impose boundary conditions weakly using penalties in order to ensure numerical stability of the discrete pml it is necessary to extend the numerical boundary procedure to the auxiliary differential equations this is crucial for deriving discrete energy estimates analogous to the continuous energy estimates numerical experiments are presented corroborating the theoretical results moreover in order to ensure longtime numerical stability the boundary condition closing the pml or its corresponding discrete implementation must be dissipative furthermore the numerical experiments demonstrate the stable and robust treatment of pml corners

a simplified and flexible variant of gcrot for solving nonsymmetric linear systems there is a need for flexible iterative solvers that can solve large scale 10 6 unknowns nonsymmetric sparse linear systems to a small tolerance among flexible solvers flexible gmres fgmres is attractive because it minimizes the residual norm over a particular subspace in practice fgmres is often restarted periodically to keep memory and work requirements reasonable however like restarted gmres restarted fgmres can suffer from stagnation this has led us to develop a flexible variant of the krylov linear solver gcrot generalized conjugate residual with inner orthogonalization and outer truncation unlike the original gcrot algorithm the proposed gcrot variant uses a simplified truncation strategy similar to loose gmres lgmres this modification is motivated by numerical experiments that suggest the specific subspace retained in the outer iteration of gcrot is less important than its size the flexible gcrot variant appears to be well suited for advection dominated problems in particular when applied to an adjoint problem from computational aerodynamics the proposed gcrot variant is robust and efficient compared with several popular truncated krylov subspace methods finally a flexible version of lgmres is easily constructed by recognizing algorithmic similarities to gcrot

on the stability and convergence of a galerkin reduced order model rom of compressible flow with solid wall and far field boundary treatment a reduced order model rom based on the proper orthogonal decomposition pod galerkin projection method is proposed as an alternative discretization of the linearized compressible euler equations it is shown that the numerical stability of the rom is intimately tied to the choice of inner product used to define the galerkin projection for the linearized compressible euler equations a symmetry transformation motivates the construction of a weighted l 2 inner product that guarantees certain stability bounds satisfied by the rom sufficient conditions for well posedness and stability of the present galerkin projection method applied to a general linear hyperbolic initial boundary value problem ibvp are stated and proven well posed and stable far field and solid wall boundary conditions are formulated for the linearized compressible euler rom using these more general results a convergence analysis employing a stable penalty like formulation of the boundary conditions reveals that the rom solution converges to the exact solution with refinement of both the numerical solution used to generate the rom and of the pod basis an a priori error estimate for the computed rom solution is derived and examined using a numerical test case published in 2010 by john wiley sons ltd

summation by parts in time we develop a new high order accurate time integration technique for initial value problems we focus on problems that originate from a space approximation using high order finite difference methods on summation by parts form with weak boundary conditions and extend that technique to the time domain the new time integration method is global high order accurate unconditionally stable and together with the approximation in space it generates optimally sharp fully discrete energy estimates in particular it is shown how stable fully discrete high order accurate approximations of the maxwells equations the elastic wave equations and the linearized euler and navier stokes equations can obtained even though we focus on finite difference approximations we stress that the methodology is completely general and suitable for all semi discrete energy stable approximations numerical experiments show that the new technique is very accurate and has limited order reduction for stiff problems

adjoint recovery of superconvergent functionals from pde approximations motivated by applications in computational fluid dynamics a method is presented for obtaining estimates of integral functionals such as lift or drag that have twice the order of accuracy of the computed flow solution on which they are based this is achieved through error analysis that uses an adjoint pde to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy numerical results are presented for the poisson equation in one and two dimensions and for the nonlinear quasi one dimensional euler equations the theory is equally applicable to nonlinear equations in complex multi dimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations

hyperbolic systems of equations posed on erroneous curved domains the effect of an inaccurate geometry description on the solution accuracy of a hyperbolic problem is discussed the inaccurate geometry can for example come from an imperfect cad system a faulty mesh generator bad measurements or simply a misconception we show that inaccurate geometry descriptions might lead to the wrong wave speeds a misplacement of the boundary conditions to the wrong boundary operator and a mismatch of boundary data the errors caused by an inaccurate geometry description may affect the solution more than the accuracy of the specific discretization techniques used in extreme cases the order of accuracy goes to zero numerical experiments corroborate the theoretical results

high order narrow stencil finite difference approximations of second order derivatives involving variable coefficients high order accuracy finite difference approximations are developed for problems involving arbitrary variable coefficients in the second order derivatives e g the heat equation or turbulence modeling the methods investigated are discretely conservative use narrow stencils and provide stable approximations for these problems it is known that high order finite difference approximations for these types of equations using the chain rule approach may be inadequate for approximating partial differential equations with certain types of variable coefficients the new approximations are constructed to alleviate this problem by requiring that the operators are stable when the variable coefficients are positive examples in heat transfer problems with variable coefficients are shown to retain the designed order of accuracy and stability with lower error norms than the usual alternative discretizations finally an application of the new stencils is presented for the large eddy simulation of compressible turbulent flows

fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains a time dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered by applying the energy method well posed boundary conditions for the continuous problem are derived summation by parts sbp operators for the space and time discretization together with a weak imposition of boundary and initial conditions using simultaneously approximation terms sats lead to a provable fully discrete energy stable conservative finite difference scheme we show how to construct a time dependent sat formulation that automatically imposes boundary conditions when and where they are required we also prove that a uniform flow field is preserved i e the numerical geometric conservation law ngcl holds automatically by using sbp sat in time and space the developed technique is illustrated by considering an application using the linearized euler equations the sound generated by moving boundaries numerical calculations corroborate the stability and accuracy of the new fully discrete approximations

on the symmetric form of systems of conservation laws with entropy abstract this paper reviews the symmetrizability of systems of conservation laws which possess entropy functions symmetric formulations in conservation form for the equations of gas dynamics are presented

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

aerodynamic optimization algorithm with integrated geometry parameterization and mesh movement an efficient gradient based algorithm for aerodynamic shape optimization is presented the algorithm consists of several components including a novel integrated geometry parameterization and mesh movement a parallel newton krylov flow solver and an adjoint based gradient evaluation to integrate geometry parameterization and mesh movement generalized b spline volumes are used to parameterize both the surface and volume mesh the volume mesh of b spline control points mimics a coarse mesh a linear elasticity mesh movement algorithm is applied directly to this coarse mesh and the fine mesh is regenerated algebraically using this approach mesh movement time is reduced by two to three orders of magnitude relative to a node based movement the mesh adjoint system also becomes smaller and is thus amenable to complex step derivative approximations when solving the flow adjoint equations using restarted krylov subspace methods a nested subspace strategy is shown to be more robust than truncating the entire subspace optimization is accomplished using a sequential quadratic programming algorithm the effectiveness of the complete algorithm is demonstrated using a lift constrained induced drag minimization that involves large changes in geometry

the relation between primal and dual boundary conditions for hyperbolic systems of equations abstract in this paper we study boundary conditions for linear hyperbolic systems of equations and the corresponding dual problem in particular we show that the primal and dual boundary conditions are related by a simple scaling relation it is also shown that the weak dual problem can be derived directly from the weak primal problem based on the continuous analysis we discretize and perform computations with a high order finite difference scheme on summation by parts form with weak boundary conditions it is shown that the results obtained in the continuous analysis lead directly to stability results for the primal and dual discrete problems numerical experiments corroborate the theoretical results

on the convergence rates of energy stable finite difference schemes abstract we consider constant coefficient initial boundary value problems with a first or second derivative in time and a highest spatial derivative of order q and their semi discrete finite difference approximations with an internal truncation error of order p 1 and a boundary error of order r 0 we prove that the convergence rate is min p r q the assumptions needed for these results to hold are i the continuous problem is linear and well posed with a smooth solution ii the numerical scheme is consistent nullspace consistent nullspace invariant and energy stable these assumptions are often satisfied for summation by parts schemes

stable boundary conditions and difference schemas for navier stokes equations the navier stokes equations at high reynolds numbers can be viewed as an incompletely elliptic perturbation of the euler equations by using the entropy function for the euler equations as a measure of energy for the navier stokes equations we are able to obtain nonlinear energy estimates for the mixed initial boundary value problem these estimates are used to derive boundary conditions that guarantee l 2 boundedness even when the reynolds number tends to infinity finally we propose a new difference scheme for modeling the navier stokes equations in multidimensions for which we are able to obtain discrete energy estimates exactly analogous to those we obtained for the differential equation

on pseudo spectral time discretizations in summation by parts form abstract fully implicit discrete formulations in summation by parts form for initial boundary value problems must be invertible in order to provide well functioning procedures we prove that under mild assumptions pseudo spectral collocation methods for the time derivative lead to invertible discrete systems when energy stable spatial discretizations are used

numerical methods based on additive splittings for hyperbolic partial differential equations we derive and analyze several methods for systems of hyperbolic equations with wide ranges of signal speeds these techniques are also useful for problems whose coefficients have large mean values about which they oscillate with small amplitude our methods are based on additive splittings of the operators into components that can be approximated independently on the different time scales some of which are sometimes treated exactly the efficiency of the splitting methods is seen to depend on the error incurred in splitting the exact solution operator this is analyzed and a technique is discussed for reducing this error through a simple change of variables a procedure for generating the appropriate boundary data for the intermediate solutions is also presented

existence and iteration of monotone positive solutions for multipoint boundary value problem with p laplacian operator in the paper we obtain the existence of monotone positive solutions and establish acorresponding iterative scheme for the following multipoint boundary value problem with p laplacian operator f p u t q t f t u t 0 0

multiple positive solutions for some p laplacian boundary value problems this paper deals with the existence of multiple positive solutions for the one dimensional p laplacian p x t q t f t x t x t 0 t 0 1 subject to one of the following boundary conditions p x 0 p x 0 0 p x 1 p x 1 0 or x 0 g1 x 0 0 x 1 g2 x 1 0 where p s s p 2s p 1 by means of a fixed point theorem due to avery and peterson sufficient conditions are obtained that guarantee the existence of at least three positive solutions the interesting point is the nonlinear term f is involved with the first order derivative explicitly

an entropy adjoint approach to mesh refinement this work presents a mesh refinement indicator based on entropy variables with an application to the compressible navier stokes equations the entropy variables are shown to satisfy an adjoint equation an observation that allows recent work in adjoint based error estimation to be leveraged in constructing a relatively cheap but effective adaptive indicator the output associated with the entropy variable adjoint is shown to be the entropy production in the computational domain including physical viscous dissipation when present minus entropy transport out of the domain adaptation using entropy variables which is equivalent to adapting on the integrated residual of the entropy transport equation thus targets areas of the domain responsible for numerical or spurious entropy production adaptive results for inviscid and viscous aerodynamic examples in two and three dimensions demonstrate performance efficiency on par with output based adaptation as measured by errors in various engineering quantities of interest with the comparative advantage of the proposed approach that no adjoint equations need to be solved

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

a dual consistent summation by parts formulation for the linearized incompressible navier stokes equations posed on deforming domains in this article well posedness and dual consistency of the linearized incompressible navier stokes equations posed on time dependent spatial domains are studied to simplify the derivation of the

the role of numerical boundary procedures in the stability of perfectly matched layers in this paper we address the temporal energy growth associated with numerical approximations of the perfectly matched layer pml for maxwell s equations in first order form in the literature several studies have shown that a numerical method which is stable in the absence of the pml can become unstable when the pml is introduced we demonstrate in this paper that this instability can be directly related to numerical treatment of boundary conditions in the pml first at the continuous level we establish the stability of the constant coefficient initial boundary value problem for the pml to enable the construction of stable numerical boundary procedures we derive energy estimates for the pml in the laplace space second we develop a high order accurate and stable numerical approximation for the pml using summation by parts finite difference operators to approximate spatial derivatives and weak enforcement of boundary conditions using penalties by constructing analogous discrete energy estimates we

stable robin solid wall boundary conditions for the navier stokes equations in this paper we prove stability of robin solid wall boundary conditions for the compressible navier stokes equations applications include the no slip boundary conditions with prescribed temperature or temperature gradient and the first order slip flow boundary conditions the formulation is uniform and the transitions between different boundary conditions are done by a change of parameters we give different sharp energy estimates depending on the choice of parameters the discretization is done using finite differences on summation by parts form with weak boundary conditions using the simultaneous approximation term we verify convergence by the method of manufactured solutions and show computations of flows ranging from no slip to almost full slip

algorithm developments for discrete adjoint methods this paper presents a number of algorithm developments for adjoint methods using the discrete approach in which the discretisation of the non linear equations is linearised and the resulting matrix is then transposed with a new iterative procedure for solving the adjoint equations exact numerical equivalence is maintained between the linear and adjoint discretisations the incorporation of strong boundary conditions within the discrete approach is discussed as well as a new application of adjoint methods to linear unsteady flow in turbomachinery

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

the influence of open boundary conditions on the convergence to steady state for the navier stokes equations abstract the question of open boundary conditions of inflow and outflow type for the navier stokes equations and its influence on the convergence to steady state is addressed both the continuous and semi discrete problem are analysed using the energy method and the laplace transform technique the energy method is used to derive well posed boundary conditions for the continuous problem for the semi discrete problem we use the energy method to prove that by using the well posed boundary conditions for the continuous problem and adding a suitable numerical boundary condition well posedness is preserved by employing the laplace transform technique the spectra for different types of boundary conditions are obtained the spectra are analysed and it is shown how the choice of boundary conditions strongly affects the convergence to steady state one dimensional navier stokes calculations are performed and the resulting convergence rates agree well with the theoretical analysis finally the spectra obtained using inflow and outflow types of boundary conditions are compared with spectra obtained using periodic boundary conditions and the choice of a time integration method for the navier stokes equations is discussed

finite difference schemes with transferable interfaces for parabolic problems we derive a method to locally change the order of accuracy of finite difference schemes that approximate the second derivative the derivation is based on summation by parts operators which are co

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

well posed and stable transmission problems we introduce the notion of a transmission problem to describe a general class of problems where different dynamics are coupled in time well posedness and stability is analysed for continuous and d

symmetric surgery and boundary link maps in order to seperate 3 dimensional linking and knotting phenomena john milnor introduced the notion of a link homotopy 14 he allowed selfintersections but did not allow different components to cross during a link homotopy it is clear that any knot is link homotopically trivial but one of the most surprising and unintuitive see the left hand side of fig 1 results of milnor was that arbitrarily many parallel copies of a knot form a homotopically trivial link in fact one knows that any boundary link has vanishing invariants and thus it is homotopically trivial by the main result of milnor by definition in a boundary link the components bound disjointly embedded seifert surfaces for example any knot has a seifert surface and is thus a boundary link similarly parallel copies of a knot bound disjoint parallel copies of this seifert surface the notion of a link homotopy makes clearly sense for links in any dimension in fact the correct category to work in seems to be the following a link map is a continuous map such that connected components in the source are mapped disjointly into the target a link homotopy between two link maps is then an ordinary homotopy through link maps in this context a boundary link map l qi mi n is a link map which is the boundary of a link map qi wi n of oriented manifolds wi the seifert surfaces for l

nested krylov methods based on gcr abstract recently the gmresr method for the solution of linear systems of equations has been introduced by vuik and van der vorst 1991 similar methods have been proposed by axelsson and vassilevski 1991 and saad 1993 fgmres 1 gmresr involves an outer and an inner method the outer method is gcr which is used to compute the optimal approximation over a given set of search vectors in the sense that the residual is minimized the inner method is gmres which computes a new search vector by approximately solving the residual equation this search vector is then used by the outer algorithm to compute a new approximation however the optimality of the approximation over the space of search vectors is ignored in the inner gmres algorithm this leads to suboptimal corrections to the solution in the outer algorithm therefore we propose to preserve the orthogonality relations of gcr in the inner gmres algorithm this gives optimal corrections to the solution and also leads to solving the residual equation in a smaller subspace and with an improved operator which should also lead to faster convergence however this involves using krylov methods with a singular nonsymmetric operator we will discuss some important properties of this we will show by experiments that in terms of matrix vector products this modification almost always leads to better convergence because we do more orthogonalizations it does not always give an improved performance in time this depends on the costs of the matrix vector products relative to the costs of the orthogonalizations of course we can also use methods other than gmres as the inner method methods with short recurrences like bicgstab seem especially interesting the experimental results indicate that especially for such methods it is advantageous to preserve the orthogonality in the inner method

existence of three positive solutions for m point boundary value problems with one dimensional p laplacian abstract in this paper we consider the multipoint boundary value problem for the one dimensional p laplacian p u q t f t u t u t 0 t 0 1 subject to the boundary conditions u 0 0 u 1 i 1 m 2 a i u i where p s s p 2 s p 1 i 0 1 with 0 1 2 m 2 1 and a i 0 1 0 i 1 m 2 a i 1 using a fixed point theorem due to avery and peterson we study the existence of at least three positive solutions to the above boundary value problem the interesting point is that the nonlinear term f explicitly involves a first order derivative

error bounded schemes for time dependent hyperbolic problems in this paper we address the error growth in time for hyperbolic problems in first order form in bounded domains the energy method is used to study when an error growth or a fixed error bound is obtained it is shown that the choice of boundary procedure is a crucial point numerical experiments corroborate the theoretical findings

a multidomain spectral method for supersonic reactive flows this paper has a dual purpose it presents a multidomain chebyshev method for the solution of the two dimensional reactive compressible navier stokes equations and it reports the results of the application of this code to the numerical simulations of high mach number reactive flows in recessed cavity the computational method utilizes a newly derived interface boundary conditions as well as an adaptive filtering technique to stabilize the computations the results of the simulations are relevant to recessed cavity flame holders

three positive fixed points of nonlinear operators on ordered banach spaces we generalize the fixed point theorem of leggett williams which is a theorem giving conditions that imply the existence of three fixed points of an operator defined on a cone in a banach space we then show how to apply our theorem to prove the existence of three positive solutions to a second order discrete boundary value problem 2001 elsevier science ltd all rights reserved

stability analysis of preconditioned approximations of the euler equations on unstructured meshes this paper analyses the stability of a discretisation of the euler equations on 3d unstructured grids using an edge based data structure first order characteristic smoothing a block jacobi preconditioner and runge kutta timemarching this is motivated by multigrid navier stokes calculations in which this inviscid discretisation is the dominant component on coarse grids the analysis uses algebraic stability theory which allows at worst a bounded linear growth in a suitably defined perturbation energy provided the range of values of the preconditioned spatial operator lies within the stability region of the runge kutta algorithm the analysis also includes consideration of the effect of solid wall boundary conditions and the addition of a low mach number preconditioner to accelerate compressible flows in which the mach number is very low in a significant portion of the flow numerical results for both inviscid and viscous applications confirm the effectiveness of the numerical algorithm and show that the analysis provides accurate stability bounds

on link maps and their homotopy classification is called a link map if the two spheres s p and s o have disjoint images f l s p nf2 s 0 two such link maps are called link homotopic if they are homotopic through link maps this type of equivalence relation was introduced by milnor m in the classical case m 3 p q 1 and has recently found much renewed interest see e g mr fr k i l 3 ko2 5 in this paper we study and determine in a number of cases the set lm q of link homotopy classes of link maps f as above

a stable and high order accurate conjugate heat transfer problem this paper analyzes well posedness and stability of a conjugate heat transfer problem in one space dimension we study a model problem for heat transfer between a fluid and a solid the energy method is used to derive boundary and interface conditions that make the continuous problem well posed and the semi discrete problem stable the numerical scheme is implemented using 2nd 3rd and 4th order finite difference operators on summation by parts sbp form the boundary and interface conditions are implemented weakly we investigate the spectrum of the spatial discretization to determine which type of coupling that gives attractive convergence properties the rate of convergence is verified using the method of manufactured solutions

a two dimensional time dependent chebyshev method of collocation for the study of astrophysical flows abstract we present a two dimensional time dependent collocation chebyshev spectral method as a tool for the solution of certain astrophysical flows in this paper we solve the navier stokes equations in spherical coordinates the problem of the propagation of an initial perturbation in an atmosphere of a compact star is used to demonstrate the method and its power in particular viscosity and rotation are included

a new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations a new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested this method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions stability and convergence results are proven for the chebyshev approximation of linear scalar hyperbolic equations the eigenvalues of this method applied to parabolic equations are shown to be real and negative

non weak strong solutions in gas dynamics a c11 p version stlsfef in lagrangian frame of reference using ρ u p primitive variables in this paper we present computations of the non weak solutions of class c11 of transient navier stokes equations for compressible flow in lagrangian frame of reference using space time least squares finite element formulation with primitive variables u p for high speed compressible flows the solutions reported here possess the same orders of continuity as the governing differential equations the role of diffusion i e viscosity physical or artificial and thermal conductivity on shock structure is demonstrated compression of air in a rigid cylinder by a rigid massless and frictionless piston is used as a model problem true time evolutions of class c11 are reported beginning with the first time step until steady shock conditions are achieved comparisons with analytical solutions are presented when possible copyright 2001 john wiley sons ltd

adjoint an approach for the rapid development of discrete adjoint solvers an automatic differentiation tool is used to develop the adjoint code for a three dimensional computational fluid dynamics solver rather than using automatic differentiation to differentiate the entire source code of the computational fluid dynamics solver we have applied it selectively to produce code that computes the flux jacobian matrix and the other partial derivatives that are necessary to compute total derivatives using an adjoint method the resulting linear discrete adjoint system is then solved using the portable extensible toolkit for scientific computation this selective application of automatic differentiation is the central idea behind the automatic differentiation adjoint adjoint approach this approach has the advantage that it is applicable to arbitrary sets of governing equations and cost functions and that it is exactly consistent with the gradients that would be computed by exact numerical differentiation of the original solver furthermore the approach is largely automatic thus avoiding the lengthy development times usually required to develop adjoint solvers for partial differential equations these significant advantages come at the cost of increased memory requirements for the adjoint solver derivatives of drag and lift coefficients are validated and the low computational cost and ease of implementation of the method are shown

stability analysis of a galerkin runge kutta navier stokes discretisation on unstructured tetrahedral grids this paper presents a timestep stability analysis for a class of discretisations applied to the linearised form of the navier stokes equations on a 3d domain with periodic boundary conditions using a suitable definition of the perturbation energy it is shown that the energy is monotonically decreasing for both the original p d e and the semi discrete system of o d e s arising from a galerkin discretisation on a tetrahedral grid using recent theoretical results concerning algebraic and generalised stability sufficient stability limits are obtained for both global and local timesteps for fully discrete algorithms using runge kutta time integration

generalised summation by parts operators and variable coefficients abstract high order methods for conservation laws can be highly efficient if their stability is ensured a suitable means mimicking estimates of the continuous level is provided by summation by parts sbp operators and the weak enforcement of boundary conditions recently there has been an increasing interest in generalised sbp operators both in the finite difference and the discontinuous galerkin spectral element framework however if generalised sbp operators are used the treatment of the boundaries becomes more difficult since some properties of the continuous level are no longer mimicked discretely interpolating the product of two functions will in general result in a value different from the product of the interpolations thus desired properties such as conservation and stability are more difficult to obtain here new formulations are proposed allowing the creation of discretisations using general sbp operators that are both conservative and stable thus several shortcomings that might be attributed to generalised sbp operators are overcome cf nordstrom and ruggiu 2017 38 and manzanero et al 2017 39

multiple positive solutions for p laplacian equations with integral boundary conditions abstract in this paper we consider a class of boundary value problems for p laplacian equation p u h t f t u t u t 0 with integral boundary conditions u 0 u 0 0 1 g 1 s u s d s u 1 u 1 0 1 g 2 s u s d s by using the avery peterson fixed point theorem we obtain the existence of at least three positive solutions

a stable galerkin reduced order model for coupled fluid structure interaction problems summary a stable reduced order model rom of a linear fluid structure interaction fsi problem involving linearized compressible inviscid flow over a flat linear von karman plate is developed separate stable roms for each of the fluid and the structure equations are derived both roms are built using the continuous galerkin projection approach in which the continuous governing equations are projected onto the reduced basis modes in a continuous inner product the mode shapes for the structure rom are the eigenmodes of the governing linear plate equation the fluid rom basis is constructed via the proper orthogonal decomposition for the linearized compressible euler fluid equations a symmetry transformation is required to obtain a stable formulation of the galerkin projection step in the model reduction procedure stability of the galerkin projection of the structure model in the standard l2 inner product is shown the fluid and structure roms are coupled through solid wall boundary conditions at the interface plate boundary an a priori energy linear stability analysis of the coupled fluid structure system is performed it is shown that under some physical assumptions about the flow field the fsi rom is linearly stable a priori if a stabilization term is added to the fluid pressure loading on the plate the stability of the coupled rom is studied in the context of a test problem of inviscid supersonic flow past a thin square elastic rectangular panel that will undergo flutter once the non dimensional pressure parameter exceeds a certain threshold this a posteriori stability analysis reveals that the fsi rom can be numerically stable even without the addition of the aforementioned stabilization term moreover the rom constructed for this problem properly predicts the maintenance of stability below the flutter boundary and gives a reasonable prediction for the instability growth rate above the flutter boundary copyright 2013 john wiley sons ltd

a stable penalty method for the compressible navier stokes equations i open boundary conditions the purpose of this paper is to present asymptotically stable open boundary conditions for the numerical approximation of the compressible navier stokes equations in three spatial dimensions the treatment uses the conservation form of the navier stokes equations and utilizes linearization and localization at the boundaries based on these variables the proposed boundary conditions are applied through a penalty procedure thus ensuring correct behavior of the scheme as the reynolds number tends to infinity the versatility of this method is demonstrated for the problem of a compressible flow past a circular cylinder

convergent cartesian grid methods for maxwell s equations in complex geometries a convergent second order cartesian grid finite difference scheme for the solution of maxwell s equations is presented the scheme employs a staggered grid in space and represents the physical location of the material and metallic boundaries correctly hence eliminating problems caused by staircasing and contrary to the popular yee scheme enforces the correct jump conditions on the field components across material interfaces a detailed analysis of the accuracy of the new embedding scheme is presented confirming its second order global accuracy furthermore the scheme is proven to be a bounded error scheme and thus convergent conditions for fully discrete stability is furthermore established this enables the derivation of bounds for fully discrete stability with cfl restrictions being almost identical to those of the much simpler yee scheme the analysis exposes that the effects of staircasing as well as a lack of properly enforced jump conditions on the field components have significant consequences for the global accuracy it is among other things shown that for cases in which a field component is discontinuous along a grid line as happens at general two and three dimensional material interfaces the yee scheme may exhibit local divergence and loss of global convergence to validate the analysis several one and two dimensional test cases are presented showing an improvement of typically 1 to 2 orders of accuracy at little or no additional computational cost over the yee scheme which in most cases exhibits first order accuracy c 2001 academic press

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

existence of multiple positive solutions for one dimensional p laplacian abstract in this paper we consider the multipoint boundary value problem for one dimensional p laplacian p u f t u 0 t 0 1 subject to the boundary value conditions p u 0 i 1 n 2 a i p u i u 1 i 1 n 2 b i u i where p s s p 2 s p 1 i 0 1 with 0 1 2 n 2 1 and a i b i satisfy a i b i 0 0 i 1 n 2 a i 1 and i 1 n 2 b i 1 using a fixed point theorem for operators on a cone we provide sufficient conditions for the existence of multiple at least three positive solutions to the above boundary value problem

construction of energy stable projection based reduced order models an approach for building energy stable galerkin reduced order models roms for linear hyperbolic or incompletely parabolic systems of partial differential equations pdes using continuous projection is developed this method is an extension of earlier work by the authors specific to the equations of linearized compressible inviscid flow the key idea is to apply to the pdes a transformation induced by the lyapunov function for the system and to build the rom in the transformed variables for linear problems the desired transformation is induced by a special inner product termed the symmetry inner product which is derived herein for several systems of physical interest connections are established between the proposed approach and other stability preserving model reduction methods giving the paper a review flavor more specifically it is shown that a discrete counterpart of this inner product is a weighted l 2 inner product obtained by solving a lyapunov equation first proposed by rowley et al and termed herein the lyapunov inner product comparisons between the symmetry inner product and the lyapunov inner product are made and the performance of roms constructed using these inner products is evaluated on several benchmark test cases

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

the sbp sat technique for initial value problems a detailed account of the stability and accuracy properties of the sbp sattechnique for numerical time integration is presented we show how thetechnique can be used to formulate both global and mu

on the impact of boundary conditions on dual consistent finite difference discretizations in this paper we derive well posed boundary conditions for a linear incompletely parabolic system of equations which can be viewed as a model problem for the compressible navier stokes equations we show a general procedure for the construction of the boundary conditions such that both the primal and dual equations are well posed the form of the boundary conditions is chosen such that reduction to first order form with its complications can be avoided the primal equation is discretized using finite difference operators on summation by parts form with weak boundary conditions it is shown that the discretization can be made energy stable and that energy stability is sufficient for dual consistency since reduction to first order form can be avoided the discretization is significantly simpler compared to a discretization using dirichlet boundary conditions we compare the new boundary conditions with standard dirichlet boundary conditions in terms of rate of convergence errors and discrete spectra it is shown that the scheme with the new boundary conditions is not only far simpler but also has smaller errors error bounded properties and highly optimizable eigenvalues while maintaining all desirable properties of a dual consistent discretization

stable and accurate schemes for the compressible navier stokes equations minimal stencil width discretizations of combined mixed and non mixed second order derivatives are analyzed with respect to accuracy and stability we show that these discretizations lead to stability for cauchy problems with a careful boundary treatment we also show that the stability holds for initial boundary value problems the analysis is verified by numerical simulations of burgers and navier stokes equations in two and three space dimensions

distributed schur complement techniques for general sparse linear systems this paper presents a few preconditioning techniques for solving general sparse linear systems on distributed memory environments these techniques utilize the schur complement system for deriving the preconditioning matrix in a number of ways two of these preconditioners consist of an approximate solution process for the global system which exploits approximate lu factorizations for diagonal blocks of the schur complement another preconditioner uses a sparse approximate inverse technique to obtain certain local approximations of the schur complement comparisons are reported for systems of varying difficulty

the spatial operator in the incompressible navier stokes oseen and stokes equations abstract we investigate the spatial operator in the incompressible navier stokes oseen and stokes equations and show how to avoid singularities associated with null spaces by choosing specific boundary conditions the theoretical results are derived for a general form of energy stable boundary conditions and applied to a few commonly used ones the analysis is done on a system that simultaneously covers the nonlinear incompressible navier stokes the oseen and the stokes equations when the spectrum of the spatial operator is investigated we restrict the analysis to the oseen and stokes equations the continuous analysis carries over to the discrete setting by using the summation by parts framework

stability of explicit navier stokes procedures using k e and k e algebraic reynolds stress turbulence models abstract a three dimensional explicit navier stokes procedure has been developed for application to compressible turbulent flows including rotation effects in the present work a numerical stability analysis of the discrete coupled system of seven governing equations is presented order of magnitude arguments are presented for flow and geometric properties typical of internal flows including turbomachinery applications to ascertain the relative importance of grid stretching rotation and turbulence source terms and effective diffusivity on the stability of the scheme it is demonstrated through both analysis and corroborative numerical experiments that 1 it is quite feasible to incorporate efficiently a two equation k turbulence model in an explicit time marching scheme provided certain numerical stability constraints are enforced 2 the role of source terms due to system rotation on the stability of the numerical scheme is not significant when appropriate grids are used and realistic rotor angular velocities are specified 3 the direct role of source terms in the turbulence transport equations on the stability of the numerical scheme is not significant when appropriate grids are used and realistic freestream turbulence quantities are specified except in the earliest stages of iteration a result which is contrary to that generally perceived 4 there is no advantage to numerically coupling the two equation model system to the mean flow equation system in regard to convergence or accuracy 5 for some flow configurations including turbomachinery blade rows it is useful to incorporate the influence of artificial dissipation in the prescription of a local timestep 6 explicit implementation of an algebraic reynolds stress model arsm is intrinsically stable provided that the discrete two equation transport model which provides the necessary values of k and is itself stable

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 stable penalty method for the compressible navier stokes equations iii multidimensional domain decomposition schemes this paper concluding the trilogy develops schemes for the stable solution of wave dominated unsteady problems in general three dimensional domains the schemes utilize a spectral approximation in each subdomain and asymptotic stability of the semidiscrete schemes is established the complex computational domains are constructed by using nonoverlapping quadrilaterals in the two dimensional case and hexahedrals in the three dimensional space to illustrate the ideas underlying the multidomain method a stable scheme for the solution of the three dimensional linear advection diffusion equation in general curvilinear coordinates is developed the analysis suggests a novel yet simple stable treatment of geometric singularities like edges and vertices the theoretical results are supported by a two dimensional implementation of the scheme the main part of the paper is devoted to the development of a spectral multidomain scheme for the compressible navier stokes equations on conservation form and a unified approach for dealing with the open boundaries and subdomain boundaries is presented well posedness and asymptotic stability of the semidiscrete scheme is established in a general curvilinear volume with special attention given to a hexahedral domain the treatment includes a stable procedure for dealing with boundary conditions at a solid wall the efficacy of the scheme for the compressible navier stokes equations is illustrated by obtaining solutions to subsonic and supersonic boundary layer flows with various types of boundary conditions the results are found to agree with the solution of the compressible boundary layer equations

strict stability of high order compact implicit finite difference schemes the role of boundary conditions for hyperbolic pdes ii abstract this paper deals with the problem of systems of hyperbolic pdes in one and two space dimensions using the theory of part i 7