a chebyshev method for the solution of boundary value problems abstract an expansion procedure using the chebyshev polynomials as base functions is proposed the method yields more accurate results than either of the galerkin or tau methods as indicated from solving the orr sommerfeld equation for both the plane poiseuille flow and the blasius velocity profile the chebyshev approximation is also applied to resolve the radial dependence of the flow field for a circular cylinder or a sphere in a uniform flow

convergence of the shifted qr algorithm on 3 3 normal matrices this paper concerns the convergence properties of the shifted qr algorithm on 3 3 normal hessenberg matrices the algorithm is viewed as an iteration on one dimensional subspaces a class of matrices is characterized for which hqr2 is unable to approximate a solution

high lewis number combustion wavefronts a perturbative melnikov analysis the wavefronts associated with a one dimensional combustion model with arrhenius kinetics and no heat loss are analyzed within the high lewis number perturbative limit this situation in which fuel diffusivity is small in comparison to that of heat is appropriate for highly dense fluids a formula for the wavespeed is established by a nonstandard application of melnikov s method and slow manifold theory from dynamical systems and compared to numerical results a simple characterization of the wavespeed correction is obtained it is proportional to the ratio between the exothermicity parameter and the lewis number the perturbation method developed herein is also applicable to more general coupled reaction diffusion equations with strongly differing diffusivities the stability of the wavefronts is also tested using a numerical evans function method

a bundle view of boundary value problems generalizing the gardner jones bundle abstract holomorphic families of linear ordinary differential equations on a finite interval with prescribed parameter dependent boundary conditions are considered from a geometrical viewpoint the gardner jones bundle which was introduced for linearized reaction diffusion equations is generalized and applied to this abstract class of dependent boundary value problems where is a complex eigenvalue parameter the fundamental analytical object of such boundary value problems bvps is the characteristic determinant and it is proved that any characteristic determinant on a jordan curve can be characterized geometrically as the determinant of a transition function associated with the gardner jones bundle the topology of the bundle represented by the chern number then yields precise information about the number of eigenvalues in a prescribed subset of the complex plane this result shows that the gardner jones bundle is an intrinsic geometric property of such dependent bvps the bundle framework is applied to examples from hydrodynamic stability theory and the linearized complex ginzburg landau equation

the gap lemma and geometric criteria for instability of viscous shock profiles an obstacle in the use of evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis since the evans function has in general been constructed only away from the essential spectrum a notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles here we prove a general theorem the gap lemma concerning the analytic continuation of the evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave this allows geometric stability theory to be applied in many cases where it could not be applied previously we demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves a necessary geometric condition for stability is determined in terms of the sign of a certain melnikov integral of the associated viscous profile this sign can easily be evaluated numerically we also compute it analytically for solutions of several important classes of systems in particular we show for a wide class of systems that homoclinic solitary waves are linearly unstable confirming these as the first known examples of unstable viscous shock waves we also show that strong heteroclinic undercompressive waves are sometimes unstable similar stability conditions are also derived for lax and overcompressive shocks and for nn

asymptotic behavior near planar transition fronts for the cahn hilliard equation abstract we consider the asymptotic behavior of perturbations of planar wave solutions arising in the cahn hilliard equation in space dimensions d 2 such equations are well known to arise in the study of spinodal decomposition a phenomenon in which rapid cooling of a homogeneously mixed binary alloy causes separation to occur resolving the mixture into regions in which one component or the other is dominant with these regions separated by steep transition layers a critical feature of the cahn hilliard equation in one space dimension is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis a feature that is known to complicate the step from spectral to nonlinear stability the analysis of planar waves in multiple space dimensions is further complicated by the fact that the leading eigenvalue for this linearized operator leading in the case of stability moves into the negative real half plane with cubic scaling 3 where r d 1 denotes a fourier variable associated with spatial components transverse to the planar wave under the assumption of spectral stability described in terms of an appropriate evans function we develop detailed asymptotics for perturbations from planar wave solutions establishing asymptotic stability for initial perturbations decaying at an appropriate algebraic rate in an l 1 norm of the transverse variables

a chebyshev collocation method for solving two phase flow stability problems this paper describes a chebyshev collocation method for solving the eigenvalue problem that governs the stability of parallel two phase flow the method is based on the expansion of the eigenfunctions in terms of chebyshev polynomials point collocation and the subsequent solution of the resulting generalized eigenvalue problem with the qz algorithm we concentrate on the question how to handle difficulties that arise when these standard techniques are applied to the stability problem of a thin film of liquid that is sheared by a gas after discussing this specific problem in detail it is argued that the method of solution can readily be applied to other two phase flow configurations as well

graphical krein signature theory and evans krein functions two concepts evidently very different in nature have proved to be useful in analytical and numerical studies of spectral stability in nonlinear wave theory i the krein signature of an eigenvalue a quantity usually defined in terms of the relative orientation of certain subspaces that is capable of detecting the structural instability of imaginary eigenvalues and hence their potential for moving into the right half plane leading to dynamical instability under perturbation of the system and ii the evans function an analytic function detecting the location of eigenvalues one might expect these two concepts to be related but unfortunately examples demonstrate that there is no way in general to deduce the krein signature of an eigenvalue from the evans function for example by studying derivatives of the latter the purpose of this paper is to recall and popularize a simple graphical interpretation of the krein signature well known in the spectral theory of polynomial operator pencils once esta

rayleigh quotient iteration for nonsymmetric matrices rayleigh quotient iteration is an iterative algorithm for the calculation of approximate eigenvectors of a matrix given a matrix the algorithm supplies a function whose iteration of an initial vector v0 produces a sequence of vectors vn if the matrix is symmetric then for almost any choice of v0 the sequence will converge to an eigenvector at an eventually cubic rate in this paper we show that there exist open sets of real matrices each of which possesses an open set of initial vectors for which the algorithm will not converge to an eigenspace the proof employs techniques from dynamical systems and bifurcation theory

asymptotic behavior near transition fronts for equations of generalized cahn hilliard form we consider the asymptotic behavior of perturbations of standing wave solutions arising in evolutionary pde of generalized cahn hilliard form in one space dimension such equations are well known to arise in the study of spinodal decomposition a phenomenon in which the rapid cooling of a homogeneously mixed binary alloy causes separation to occur resolving the mixture into its two components with their concentrations separated by sharp transition layers motivated by work of bricmont kupiainen and taskinen 5 we regard the study of standing waves as an interesting step toward understanding the dynamics of these transitions a critical feature of the cahn hilliard equation is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis a feature that is known to complicate the step from spectral to nonlinear stability under the assumption of spectral stability described in terms of an appropriate evans function we develop detailed asymptotics for perturbations from standing wave solutions establishing phase asymptotic orbital stability for initial perturbations decaying with appropriate algebraic rate

ruga formation instabilities of a graded stiffness boundary layer in a neo hookean solid we present an analysis of ruga formation instabilities arising in a graded stiffness boundary layer of a neo hookean half space caused by lateral plane strain compression in this study we repres

an automatic orthonormalization method for solving stiff boundary value problems abstract a new initial value method is described based on a remark by drury for solving stiff linear differential two point cigenvalue and boundary value problems the method is extremely reliable it is especially suitable for high order differential systems and it is capable of accommodating realms of stiffness which other methods cannot reach the key idea behind the method is to decompose the stiff differential operator into two non stiff operators one of which is nonlinear the nonlinear one is specially chosen so that it advances an orthonormal frame indeed the method is essentially a kind of automatic orthonormalization the second is auxiliary but it is needed to determine the required function the usefulness of the method is demonstrated by calculating some eigenfunctions for an orr sommerfeld problem when the reynolds number is as large as 10

recent developments in the algebraic multiblock method for euler equations abstract in this paper the algebraic multiblock method is presented and applied to the solution of the euler equations used to model inviscid compressible flow problems the method employs two extreme grids aimed at combining multigrid and domain decomposition ideas the fine grid is used by the discretization method the coarse grid results from the fine grid and defines the domain partitioning of the solution procedure the multigrid concept is practically simulated by applying the idea of the substructuring technique in which the inner finite volumes of a block are expressed by their boundary volumes through a matrix inversion process the conjugate gradient squared method is used for the approximate solution of numerous large block pentadiagonal systems emerging at each time step additionally efficient preconditioning and smoothing are employed to accelerate the iterative solution the proposed method is applied for the solution of the time marching euler equations using an implicit upwind finite volume method euflex numerical results and comparisons show the efficiency of the proposed method even for sequential processing

the compound matrix method for ordinary differential systems abstract a generalization of the compound matrix method is presented to deal with eigenvalue and boundary value problems involving unstable systems of ordinary differential equations details are given for fourth and sixth order problems next is shown that a simple equivalence relation exists between the compounds of the solution matrices of the eigenvalue problems and their adjoints and how this relation can be exploited to simplify the calculation of the adjoint eigenfunctions is discussed using the orr sommerfeld problem as an example it is also shown how the techniques used in the derivation of certain auxiliary systems which play a crucial role in the generalization of the compound matrix method can also provide an alternate method for the direct computation of certain quantities defined in terms of the eigenfunctions finally there is a brief discussion of the relationship between the compound matrix method anf the riccati method

stability and instability of solitary waves of the fifth order kdv equation a numerical framework abstract the spectral problem associated with the linearization about solitary waves of the generalized fifth order kdv equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to eigenvalues a numerical framework based on a fast robust shooting algorithm on exterior algebra spaces is introduced the complete algorithm has several new features including a rigorous numerical algorithm for choosing starting values a new method for numerical analytic continuation of starting vectors the role of the grassmannian g 2 c 5 in choosing the numerical integrator and the role of the hodge star operator for relating 2 c 5 and 3 c 5 and deducing a range of numerically computable forms for the evans function the algorithm is illustrated by computing the stability and instability of solitary waves of the fifth order kdv equation with polynomial nonlinearity

cohomology of twisted tensor products abstract it is well known that the cohomology of a tensor product is essentially the tensor product of the cohomologies we look at twisted tensor products and investigate to which extent this is still true we give an explicit description of the ext algebra of the tensor product of two modules and under certain additional conditions describe an essential part of the hochschild cohomology ring of a twisted tensor product as an application we characterize precisely when the cohomology groups over a quantum complete intersection are finitely generated over the hochschild cohomology ring moreover both for quantum complete intersections and in related cases we obtain a lower bound for the representation dimension of the algebra

evaluation of eigenfunctions from compound matrix variables in non linear elasticity ii sixth order systems we show how the compound matrix method can be extended to give eigenfunctions as well as eigenvalues to bifurcation problems in non linear elasticity the non trivial boundary conditions create some difficulties and we find that sixth order systems for elasticity problems will require a shooting method for two functions of two unknown parameters over and above the calculations required for comparable problems in fluids

an introduction to finite element methods for transient advection problems abstract an introduction is presented to time accurate finite element methods recently developed for solving unsteady problems governed by linear and nonlinear advection equations this includes methods making an explicit use of the characteristic curves methods based on the taylor series in time and least squares methods

the close relationships between methods for solving two point boundary value problems the well conditioning of a boundary value problem is shown to be related to bounding two quantities one involving the boundary conditions and the other involving the green s function in the case of a solution dichotomy they are related to known stability results these results easily explain why difficulties arise using superposition and reduced superposition it is shown how multiple shooting overcomes these difficulties by relating its matrix conditioning to the underlying boundary value problem some factorization methods for the multiple shooting method are considered these show the close relationships between multiple shooting the stabilized march and invariant imbedding and one factorization leads to an efficient new way to implement each of these three methods many of the results apply for collocation and finite difference methods too finally it is shown both theroetically and computationally when matrix compactification can lead to difficulty

pointwise green function bounds for shock profiles of systems with real viscosity following the pointwise semigroup approach of zh mz 1 we establish sharp pointwise green function bounds and consequent linearized stability for viscous shock profiles of general hyperbolic parabolic systems of conservation laws of dissipative type under the necessary assumptions z 1 z 3 z 4 of spectral stability i e stable point spectrum of the linearized operator about the wave transversality of the connecting profile and hyperbolic stability of the corresponding ideal shock of the associated inviscid system with no additional assumptions on the structure or strength of the shock these bounds are used in a companion paper mz 2 to establish nonlinear stability of small amplitude lax shocks of symmetrizable hyperbolic parabolic systems

numerical testing of the stability of viscous shock waves a new theoretical evans function condition is used as the basis of a numerical test of viscous shock wave stability accuracy of the method is demonstrated through comparison against exact solutions a convergence study and evaluation of approximate error equations robustness is demonstrated by applying the method to waves for which no current analytic results apply highly nonlinear waves from the cubic model and strong shocks from gas dynamics an interesting aspect of the analysis is the need to incorporate features from the analytic evans function theory for purposes of numerical stability for example we find it necessary for numerical accuracy to solve odes on the space of wedge products

constructing quantum groups and hopf algebras from coverings abstract given a group graded free associative algebra we show that in many cases the path algebra associated to the covering coming from the grading has a hopf algebra structure our structure on the path algebra is that of a quantum group far most of these constructions adding more restrictions we create some finite dimensional quotients which inherit the structure of a hoof algebra usually a quantum group from the path algebra

classification of pointed rank one hopf algebras abstract we classify pointed rank one hopf algebras over fields of prime characteristic which are generated as algebras by the first term of the coradical filtration we obtain three types of hopf algebras presented by generators and relations for hopf algebras with semi simple coradical only the first and second type appears we determine the indecomposable projective modules for certain classes of pointed rank one hopf algebras

instability of the hocking stewartson pulse and its implications for three dimensional poiseuille flow the linear stability problem for the hocking stewartson pulse obtained by linearizing the complex ginzburg landau cgl equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to stability exponents a numerical algorithm based on the compound matrix method is developed for computing the evans function using values in the cgl equation associated with spanwise modulation of plane poiseuille flow we show that the hocking stewartson pulse associated with points along the neutral curve is always linearly unstable due to a real positive eigenvalue implications for the spanwise structure of nonlinear poiseuille problem between parallel plates are also discussed

convergence of the francis shifted qr algorithm on normal matrices we examine global convergence properties of the francis shifted qr algorithm on real normal hessenberg matrices it is shown that the algorithm will almost always produce a decoupling eigenvalue conditions are identified which assure decoupling in particular a sufficient condition is that a normal matrix has more than four real eigenvalues

classification of connected hopf algebras of dimension p 3 i abstract let p be a prime and k be an algebraically closed field of characteristic p in this paper we provide the classification of connected hopf algebras of dimension p 3 except for the case when the primitive space of the hopf algebra is a two dimensional abelian restricted lie algebra each isomorphism class is presented by generators x y z with relations and hopf algebra structures let be the multiplicative group of p 2 p 1 th roots of unity when the primitive space is one dimensional and p is odd there is an infinite family of isomorphism classes which is naturally parameterized by a k 1

spectral stability of noncharacteristic isentropic navier stokes boundary layers building on the work of barker humpherys lafitte rudd and zumbrun in the shock wave case we study stability of compressive or shock like boundary layers of the isentropic compressible navier stokes equations with law pressure by a combination of asymptotic ode estimates and numerical evans function computations our analytical results include convergence of the evans function in the shock and large amplitude limits and stability in the large amplitude limit the first rigorous stability result for other than the nearly constant case for all gamma geqq 1 together with these analytical results our numerical investigations indicate stability for 1 3 for all compressive boundary layers independent of amplitude save for inflow layers in the characteristic limit not treated expansive inflow boundary layers have been shown to be stable for all amplitudes by matsumura and nishihara using energy estimates besides the parameter of amplitude appearing in the shock case the boundary layer case features an additional parameter measuring displacement of the background profile which greatly complicates the resulting case structure inflow boundary layers turn out to have quite delicate stability in both large displacement shock and large amplitude limits necessitating the additional use of a mod two stability index studied earlier by serre and zumbrun in order to decide stability

spectral stability of ideal gas shock layers extending recent results in the isentropic case we use a combination of asymptotic ode estimates and numerical evans function computations to examine the spectral stability of shock wave solutions of the compressible navier stokes equations with ideal gas equation of state our main results are that in appropriately rescaled coordinates the evans function associated with the linearized operator about the wave i converges in the large amplitude limit to the evans function for a limiting shock profile of the same equations for which internal energy vanishes at one end state and ii has no unstable positive real part zeros outside a uniform ball thus the rescaled eigenvalue ode for the set of all shock waves augmented with the nonphysical limiting case form a compact family of boundary value problems that can be conveniently investigated numerically an extensive numerical evans function study yields one dimensional spectral stability independent of amplitude for gas constant in 1 2 3 and ratio of heat conduction to viscosity coefficient within 0 2 5 1 4 1 47 for air other values may be treated similarly but were not considered the method of analysis extends also to the multi dimensional case a direction that we shall pursue in a future work

graphs with relations coverings and group graded algebras the paper studies the interrelationship between coverings of finite directed graphs and gradings of the path algebras associated to the directed graphs to include gradings of all basic finite dimensional algebras over an algebraically closed field a theory of coverings of graphs with relations is introduced the object of this paper is to relate group gradings on algebras to coverings of a graph which is associated to the algebra the linking of the theories allows one to relate purely algebraic questions to questions in algebraic topology group theory or combinatorics in the representation theory of artin algebras the association to each algebra of a finite directed graph called the quiver of the algebra has been a useful tool the reason that the quiver of such an algebra is of interest is that there is a natural definition of representations of the quiver so that the category of representations of the quiver satisfying certain relations is equivalent to the category of finitely generated modules over the algebra 1 gives a slight extension of these concepts to finitely generated algebras the main emphasis of the paper is to show that the theory of coverings of graphs with relations introduced by c riedtmann 9 and expanded by p gabriel 2 and the theory of group graded algebras are essentially the same although the original connection between coverings and gradings was inspired by the similarity of results for z graded artin algebras 3 4 and p gabriel s announced results 2 the context of this paper is more general and deals with all finitely generated algebras over a field we associate to each such algebra a finite directed graph which we still call a quiver of the algebra we show that for each regular covering t of a quiver t0 of an algebra a with certain prescribed restrictions we get a g grading of the algebra a where g is the automorphism group of the covering y over ro conversely given a certain type of g grading of a where g is a group we construct a regular covering y of the quiver t0 of the algebra such that g is isomorphic to the automorphism group of t over ro furthermore if y is a regular covering of t0 with automorphism group g we show that the category of representations of y satisfying a certain set of relations is equivalent to the category of finite dimensional graded g modules we

a shooting method for nonlinear heat transfer using automatic differentiation the steady state temperature distribution in a cylinder of unit radius is modeled by a nonlinear two point boundary value problem with an endpoint singularity we present a simple shooting method for the solution of this problem it is well known that shooting methods solve initial value problems repeatedly until the boundary conditions are satisfied for the solution of the initial value problems the method of this paper uses a technique known as automatic differentiation and obtains a taylor series expansion for the solution automatic differentiation is the process of obtaining the exact values of derivatives needed in the taylor series expansion using recursive formulas derived from the governing differential equation itself thus the method does not face the need to deal with step size issues or the need to carry out lengthy algebraic manipulations for obtaining higher order derivatives the method successfully reproduces the solutions obtained by previous researchers

computing connecting orbits via an improved algorithm for continuing invariant subspaces a successive continuation method for locating connecting orbits in parametrized systems of autonomous odes was considered in numer algorithms 14 1997 pp 103 124 in this paper we present an improved algorithm for locating and continuing connecting orbits which includes a new algorithm for the continuation of invariant subspaces the latter algorithm is of independent interest and can be used in contexts different than the present one

the cauchy problem for 1 d linear nonconservative hyperbolic systems with possibly expansive discontinuity of the coefficient a viscous approach abstract in this paper we consider linear nonconservative cauchy systems with discontinuous coefficients across the noncharacteristic hypersurface x 0 for the sake of simplicity we restrict ourselves to piecewise constant hyperbolic operators of the form t a x x with a x a 1 x 0 a 1 x 0 where a m n r under assumptions incorporating a sharp spectral stability assumption we prove that a unique solution is successfully singled out by a vanishing viscosity approach due to our framework which includes systems with expansive discontinuities of the coefficient the selected small viscosity solution satisfies an unusual hyperbolic problem which is well posed even though it does not satisfy in general a uniform lopatinski condition in addition based on a detailed analysis of our stability assumption explicit examples of 2 2 systems checking our assumptions are given our result is new and contains both a stability result and a description of the boundary layers forming at any order two kinds of boundary layers form each polarized on specific linear subspaces in direct sum

hyman s method revisited the qr algorithm is considered one of the most reliable methods for computing matrix eigenpairs however it is unable to detect multiple eigenvalues and jordan blocks matlab s eigensolver returns heavily perturbed eigenvalues and eigenvectors in such cases and there is no hint for possible principal vectors this paper calls attention to hyman s method as it is applicable for computing principal vectors and higher derivatives of the characteristic polynomial that may help to estimate multiplicity an important information for more reliable computation we suggest a test matrix collection for jordan blocks the first numerical tests with these matrices reveal that the computational problems are deeper than expected at the beginning of this work

stability of viscous detonations for majda s model abstract using analytical and numerical evans function techniques we examine the spectral stability of strong detonation wave solutions of a version of majda s scalar model for a reacting gas mixture with an arrhenius type ignition function we introduce an energy estimate to limit possible unstable eigenvalues to a compact region in the unstable complex half plane and we use a numerical approximation of the evans function to search for possible unstable eigenvalues in this region our results show for the parameter values tested that these waves are spectrally stable combining these numerical results with the pointwise green function analysis of lyng raoofi texier zumbrun g lyng m raoofi b texier k zumbrun pointwise green function bounds and stability of combustion waves j differential equations 233 2 2007 654 698 we conclude that these waves are nonlinearly stable this represents the first demonstration of nonlinear stability for detonation wave solutions of a majda type model without a smallness assumption notably our results indicate that for this simplified scalar model there does not occur either in a normal parameter range or in the limit of high activation energy hopf bifurcation to galloping or pulsating solutions as is observed in the full reactive navier stokes equations this answers in the negative for this model a question posed by majda as to whether such scalar detonation models capture this aspect of detonation behavior

high order numerical method for two point boundary value problems in this paper the two point boundary value problem is transformed into general first order ordinary differential equation system through introduction of conditions of an integral character to supplement the simultaneous set of first order equations a new discrete approximation of a high order compact difference scheme is presented for the first order system it is a block bidiagonal profile and removes the limits of other high order discrete schemes at the interval ends the numerical tests of a seventh order compact difference scheme show that the proposed scheme is very convenient and efficient for linear and nonlinear two point boundary value problems

numerical computation of solitary waves in infinite cylindrical domains the numerical computation of solitary waves to semilinear elliptic equations in infinite cylindrical domains is investigated rather than solving on the infinite cylinder the equation is approximated by a boundary value problem on a finite cylinder convergence and stability results for this approach are given it is also shown that galerkin approximations can be used to compute solitary waves of the elliptic problem on the finite cylinder in addition it is demonstrated that the aforementioned procedures simplify in cases where the elliptic equation admits an additional reversibility structure finally the theoretical predictions are compared with numerical computations in particular post buckling of an infinitely long cylindrical shell under axial compression is considered it is shown numerically that for a fixed spatial truncation the error in the truncation scales with the length of the cylinder as predicted theoretically

on swapping diagonal blocks in real schur form we discuss a new version of an existing algorithm for reordering the eigenvalues on the diagonal of a matrix in real schur form by performing an orthogonal similarity transformation a detailed error analysis and software description are presented numerical examples show the superiority of our algorithm over previous algorithms

computing lyapunov exponents on a stiefel manifold the problem of numerical computation of a few lyapunov exponents les of finite dimensional dynamical systems is considered from the viewpoint of the differential geometry of stiefel manifolds whether one computes one many or all les of a continuous dynamical system by time integration discrete or continuous orthonormalization is essential for stable numerical integration a differential geometric view of continuous orthogonalization suggests that one restricts the linearized vectorfield to a stiefel manifold however the stiefel manifold is not in general an attracting submanifold of the ambient euclidean space it is a constraint manifold with a weak numerical invariant new numerical algorithms for this problem are then designed which use the fiber bundle characterization of these manifolds this framework leads to a new class of systems for continuous orthogonalization which have strong numerical invariance properties and the strong skew symmetry property numerical integration of these new systems with geometric integrators leads to a new class of numerical methods for computing a few les which preserve orthonormality to machine accuracy this idea is also taken a step further by making the stiefel manifold an attracting invariant manifold in which case standard explicit runge kutta algorithms can be used this leads to an algorithm which requires only marginally more computation than a standard explicit integration without orthogonalization these class of methods should be particularly effective for computing a few les for large dimension dynamical systems the new schemes are straightforward to implement a test case is presented for illustration and an example from dynamical systems is presented where a few les are computed for an array of coupled oscillators

numerical solution of two point boundary value problems abstract this paper presents an original formulation of two point boundary value and eigenvalue problems expressed as a system of first order equations the fundamental difference between the new method and other methods based on a first order approach is the introduction of conditions of an integral character to supplement the simultaneous set of first order equations which are hence never regarded as an initial value problem the consideration of integral conditions leads to establish a class of linear multipoint schemes for the numerical solution of boundary value problems for ordinary differential equations furthermore the global character of the integral conditions nonlocality combined with the block structure of the system of algebraic equations allow dealing with stiff problems by means of the classical procedure of iterative refinement introduced by wilkinson the properties of the numerical schemes are illustrated by the solution of linear and nonlinear problems and by the accurate and efficient determination of some eigensolutions of a difficult problem of hydrodynamic stability the proposed method is conceptually simpler and numerically more convenient than existing initial value methods while still retaining all the advantages of a formulation based on a first order system

stability of bright solitary wave solutions to perturbed nonlinear schrodinger equations abstract the propagation of pulses in ideal nonlinear optical fibers without loss is governed by the nonlinear schrodinger equation nls when considering realistic fibers one must examine perturbed nls equations with the particular perturbation depending on the physical situation that is being modeled a common example is the complex ginzburg landau equation cgl which is a dissipative perturbation it is known that some of the stable bright solitons of the nls survive a dissipative perturbation such as the cgl given that a wave persists it is then important to determine its stability with respect to the perturbed nls a major difficulty in analyzing the stability of solitary waves upon adding dissipative terms is that eigenvalues may bifurcate out of the essential spectrum since the essential spectrum of the nls is located on the imaginary axis such eigenvalues may lead to an unstable wave in fact eigenvalues can pop out of the essential spectrum even if the unperturbed problem has no eigenvalue embedded in the essential spectrum here we present a technique which can be used to track these bifurcating eigenvalues as a consequence we are able to locate the spectrum for bright solitary wave solutions to various perturbed nonlinear schrodinger equations and determine precise conditions on parameters for which the waves are stable in addition we show that a particular perturbation the parametrically forced nls equation supports stable multi bump solitary waves the technique for tracking eigenvalues which bifurcate from the essential spectrum is very general and should therefore be applicable to a larger class of problems than those presented here

spectral algorithms for vector elliptic equations in a spherical gap abstract this paper is devoted to the numerical solution of three dimensional elliptic equations of a vector field in the region contained between two concentric spheres the method of truncated series expansion in orthogonal functions is considered using chebyshev polynomials and vector spherical harmonics spectral algorithms are described for solving two point boundary value problems of the vector modes of the poisson helmholtz and biharmonic equations supplemented by boundary conditions typical of fluid dynamical applications the accuracy of the algorithms is illustrated by computing some analytical single mode examples the proposed algorithms when combined with efficient transform techniques can be used to solve multi mode nonlinear problems

unitary integrators and applications to continuous orthonormalization techniques in this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed such systems have skew hermitian coefficient matrices in the linear case and a

stability of isentropic navier stokes shocks in the high mach number limit by a combination of asymptotic ode estimates and numerical evans function calculations we establish stability of viscous shock solutions of the isentropic compressible navier stokes equations with law pressure i in the limit as mach number m goes to infinity for any 1 proved analytically and ii for m 2 500 gamma in 1 2 5 or m 13 000 gamma in 2 5 3 demonstrated numerically this builds on and completes earlier studies by matsumura nishihara and barker humpherys rudd zumbrun establishing stability for low and intermediate mach numbers respectively indicating unconditional stability independent of shock amplitude of viscous shock waves for law gas dynamics in the range gamma in 1 3 other values may be treated similarly but have not been checked numerically the main idea is to establish convergence of the evans function in the high mach number limit to that of a pressureless or infinitely compressible gas with additional upstream boundary condition determined by a boundary layer analysis recall that low mach number behavior is formally incompressible

on the compuation of lyapunov exponents for continuous dynamical systems in this paper we consider discrete and continuous qr algorithms for computing all of the lyapunov exponents of a regular dynamical system we begin by reviewing theoretical results for regular systems and present general perturbation results for lyapunov exponents we then present the algorithms give an error analysis of them and describe their implementation finally we give several numerical examples and some conclusions

the instability of the hocking stewartson pulse and its geometric phase in the hopf bundle this work demonstrates an innovative numerical method for counting and locating eigenvalues with the evans function utilizing the geometric phase in the hopf bundle the technique calculates the winding of the evans function about a contour in the spectral plane describing the eigenvalues enclosed by the contour for the hocking stewartson pulse of the complex ginzburg landau equation locating eigenvalues with the geometric phase in the hopf bundle was proposed by way 2009 and proven by grudzien et al 2016 way demonstrated his proposed method for the hocking stewartson pulse and this manuscript redevelops this example as in the proof of the method in grudzien et al 2016 modifying his numerical shooting argument and introduces new numerical results concerning the phase transition

continuation of invariant subspaces in large bifurcation problems we summarize an algorithm for computing a smooth orthonormal basis for an invariant subspace of a parameter dependent matrix and describe how to extend it for numerical bifurcation analysis we adapt the continued subspace to track behavior relevant to bifurcations and use projection methods to deal with large problems to test our ideas we have integrated our code into matcont a program for numerical continuation and bifurcation analysis

numerical solution of eigenvalue problems using spectral techniques abstract two algorithms based on spectral chebyshev and pseudospectral chebyshev methods are presented for solving difficult eigenvalue problems that are valid over connected domains coupled through interfacial conditions to demonstrate the applicability of these methods we have examined the eigenvalue problems that describe the linear stability of two superposed newtonian and inelastic power law fluids in plane poiseuille flow for a selected range of parameters both algorithms provide accurate results and the pseudospectral code appears to be more efficient in handling linear stability problems

the orr sommerfeld equation on a manifold the most effective and widely used methods for integrating the orr sommerfeld equation by shooting are the continuous orthogonalization method and the compound matrix method in this paper we consider this problem from a differential geometric point of view a new definition of orthogonalization is presented restriction of the orr sommerfeld to a complex stiefel manifold and this definition leads to a new formulation of continuous orthogonalization which differs in a precise and interesting geometric way from existing orthogonalization routines present orthogonalization methods based on davey9s algorithm are shown to have a different differential geometric interpretation restriction of the orr sommerfeld equation to a complex grassmanian manifold this leads us to introduce the concept of a grassmanian integrator which preserves linear independence and not necessarily orthogonality using properties of grassmanian manifolds and their tangent spaces a new grassmanian integrator is introduced that generalizes davey9s algorithm furthermore it is shown that the compound matrix method is a dual grassmanian integrator it uses plucker coordinates for integrating on a grassmanian manifold and this characterization suggests a new algorithm for constructing the compound matrices extension of the differential geometric framework to general systems of linear ordinary differential equations is also discussed

dualising complexes and twisted hochschild co homology for noetherian hopf algebras we show that many noetherian hopf algebras a have a rigid dualising complex r with r a1 d here d is the injective dimension of the algebra and is a certain k algebra automorphism of a unique up to an inner automorphism in honour of the finite dimensional theory which is hereby generalised we call the nakayama automorphism of a we prove that s2 where s is the antipode of a and is the left winding automorphism of a determined by the left integral of a the hochschild homology and cohomology groups with coefficients in a suitably twisted free bimodule are shown to be non zero in the top dimension d when a is an artin schelter regular noetherian hopf algebra of global dimension d twisted poincare duality holds in this setting as is deduced from a theorem of van den bergh calculating for a using also the opposite coalgebra structure we determine a formula for s4 generalising a 1976 formula of radford for a finite dimensional applications of the results to the cases where a is pi an enveloping algebra a quantum group a quantised function algebra and a group algebra are outlined

an efficient shooting algorithm for evans function calculations in large systems abstract in evans function computations of the spectra of asymptotically constant coefficient linear operators a basic issue is the efficient and numerically stable computation of subspaces evolving according to the associated eigenvalue ode for small systems a fast shooting algorithm may be obtained by representing subspaces as single exterior products j c alexander r sachs linear instability of solitary waves of a boussinesq type equation a computer assisted computation nonlinear world 2 4 1995 471 507 l q brin numerical testing of the stability of viscous shock waves ph d thesis indiana university bloomington 1998 l q brin numerical testing of the stability of viscous shock waves math comp 70 235 2001 1071 1088 l q brin k zumbrun analytically varying eigenvectors and the stability of viscous shock waves in seventh workshop on partial differential equations part i 2001 rio de janeiro mat contemp 22 2002 19 32 t j bridges g derks g gottwald stability and instability of solitary waves of the fifth order kdv equation a numerical framework physica d 172 1 4 2002 190 216 for large systems however the dimension of the exterior product space quickly becomes prohibitive growing as n k where n is the dimension of the system written as a first order ode and k typically n 2 is the dimension of the subspace we resolve this difficulty by the introduction of a simple polar coordinate algorithm representing pure monomial products as scalar multiples of orthonormal bases for which the angular equation is a numerically optimized version of the continuous orthogonalization method of drury davey a davey an automatic orthonormalization method for solving stiff boundary value problems j comput phys 51 2 1983 343 356 l o drury numerical solution of orr sommerfeld type equations j comput phys 37 1 1980 133 139 and the radial equation is evaluable by quadrature notably the polar coordinate method preserves the important property of analyticity with respect to parameters

continuous orthonormalization for boundary value problems abstract continuous orthonormalization describes an initial value method for linear 2 point boundary value problems which provides an orthogonal basis for the solution space at all points of the interval in this paper the equations of continuous orthonormalization are derived with elementary projection arguments to provide geometric insight and motivate some modifications of an earlier algorithm the method is then applied to some oscillatory and stiff boundary value problems to demonstrate that it is simple to use problem independent and as adaptive as the initial value code which is used to integrate the equations of continuous orthonormalization

efficient computation of analytic bases in evans function analysis of large systems in evans function computations of the spectra of asymptotically constant coefficient linearized operators of large systems a problem that becomes important is the efficient computation of global analytically varying bases for invariant subspaces of the limiting coefficient matrices in the case that the invariant subspace is spectrally separated from its complementary invariant subspace we propose an efficient numerical implementation of a standard projection based algorithm of kato for which the key step is the solution of an associated sylvester problem this may be recognized as the analytic cousin of a c k algorithm developed by dieci and collaborators based on orthogonal projection rather than eigenprojection as in our case for a one dimensional subspace it reduces essentially to an algorithm of bridges derks and gottwald based on path finding and continuation methods

linear convergence in the shifted qr algorithm global and asymptotic convergence properties for the qr algorithm with francis double shift are established for certain orthogonal similarity classes of 4 x 4 real matrices it is shown that in each of the classes every unreduced hessenberg matrix will decouple and that the rate of decoupling is almost always linear the effect of the eispack exceptional shift strategy is shown to be negligible

on one dimensional stretching functions for finite difference calculations abstract the class of one dimensional stretching functions used in finite difference calculations is studied for solutions containing a highly localized region of rapid variation simple criteria for a stretching function are derived using a truncation error analysis these criteria are used to investigate two types of stretching functions one is an interior stretching function for which the location and slope of an interior clustering region are specified the simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine it was first employed by thomas et al aiaa j 10 1972 887 the other type of function is a two sided stretching function for which the arbitrary slopes at the two ends of the one dimensional interval are specified the simplest such general function is found to be one based on the inverse tangent the special case where the slopes were both equal and greater than one was first employed by roberts the general two sided function has many applications in the construction of finite difference grids examples of such applications are found in the listed references

numerical solution of eigenvalue problem using the compound matrix method abstract an algorithm based on the compound matrix method is presented for solving difficult eigen value problems details are given for systems of ordinary differential equations of fourth order that are valid over connected domains coupled through interfacial conditions as an example we examine the linear stability of two superposed fluids in plane poiseuille flow and consider both interfacial and shear modes the orr sommerfeld system that describes linear stability is solved for a selected range of parameters

exponential dichotomy on the real line svd and qr methods abstract in this work we show when and how techniques based on the singular value decomposition svd and the qr decomposition of a fundamental matrix solution can be used to infer if a system enjoys or not exponential dichotomy on the whole real line

geometric phase in the hopf bundle and the stability of non linear waves we develop a stability index for the travelling waves of non linear reaction diffusion equations using the geometric phase induced on the hopf bundle s2n 1 cns2n 1 cn this can be viewed as an alternative formulation of the winding number calculation of the evans function whose zeroes correspond to the eigenvalues of the linearization of reaction diffusion operators about the wave the stability of a travelling wave can be determined by the existence of eigenvalues of positive real part for the linear operator our method of geometric phase for locating and counting eigenvalues is inspired by the numerical results in way s dynamics in the hopf bundle the geometric phase and implications for dynamical systems way 2009 we provide a detailed proof of the relationship between the phase and eigenvalues for dynamical systems defined on c2c2 and sketch the proof of the method of geometric phase for cncn and its generalization to boundary value problems implementing the numerical method modified from way 2009 we conclude with open questions inspired from the results

numerical implementation of complex orthogonalization parallel transport on stiefel bundles and analyticity abstract numerical integration of complex linear systems of odes depending analytically on an eigenvalue parameter are considered complex orthogonalization which is required to stabilize the numerical integration results in non analytic systems it is shown that properties of eigenvalues are still efficiently recoverable by extracting information from a non analytic characteristic function the orthonormal systems are constructed using the geometry of stiefel bundles different forms of continuous orthogonalization in the literature are shown to correspond to different choices of connection one form on the stiefel bundle for the numerical integration gauss legendre runge kutta algorithms are the principal choice for preserving orthogonality and performance results are shown for a range of glrk methods the theory and methods are tested by application to example boundary value problems including the orr sommerfeld equation in hydrodynamic stability

decoupling and stability of algorithms for boundary value problems the ordinary differential equations occurring in linear boundary value problems characteristically have both stable and unstable solution modes therefore a stable numerical algorithm should avoid both forward and backward integration of solutions on large intervals it is shown that most methods like multiple shooting collocation invariant imbedding and difference methods derive their stability from the fact that they all decouple the continuous or the discrete problem sooner or later for instance when solving a linear system this decoupling is related to the dichotomy of the ordinary differential equations in fact it turns out that the inherent initial value instability is an important prerequisite for a stable utilization of the decoupled representations from which the solutions are computed how this stability is related to the use of the boundary conditions is also investigated

combustion waves in hydraulically resistant porous media in a special parameter regime abstract in this paper we study the stability of fronts in a reduction of a well known pde system that is used to model the combustion in hydraulically resistant porous media more precisely we consider the original pde system under the assumption that one of the parameters of the model the lewis number is chosen in a specific way and with initial conditions of a specific form for a class of initial conditions then the number of unknown functions is reduced from three to two for the reduced system the existence of combustion fronts follows from the existence results for the original system the stability of these fronts is studied here by a combination of energy estimates and numerical evans function computations and nonlinear analysis when applicable we then lift the restriction on the initial conditions and show that the stability results obtained for the reduced system extend to the fronts in the full system considered for that specific value of the lewis number the fronts that we investigate are proved to be either absolutely unstable or convectively unstable on the nonlinear level

on the structure and cohomology ring of connected hopf algebras abstract let p be a prime and k be a field of characteristic p we investigate the algebra structure and the structure of the cohomology ring for the connected hopf algebras of dimension p 3 which appear in the classification obtained in 20 the list consists of 23 algebras together with two infinite families we identify the morita type of the algebra and in almost all cases this is sufficient to clarify the structure of the cohomology ring

numerical exterior algebra and the compound matrix method the compound matrix method which was first proposed for numerically integrating systems of differential equations in hydrodynamic stability on k 2 3 dimensional subspaces of mathbb c n by using compound matrices as coordinates is reformulated in a coordinate free way using exterior algebra spaces bigwedge k mathbb c n

convergence of algorithms of decomposition type for the eigenvalue problem we develop the theory of convergence of a generic gr algorithm for the matrix eigenvalue problem that includes the qr lr sr and other algorithms as special cases our formulation allows for shifts of origin and multiple gr steps the convergence theory is based on the idea that the gr algorithm performs nested subspace iteration with a change of coordinate system at each step thus the convergence of the gr algorithm depends on the convergence of certain sequences of subspaces it also depends on the quality of the coordinate transformation matrices as measured by their condition numbers we show that with a certain obvious shifting strategy the gr algorithm typically has a quadratic asymptotic convergence rate for matrices possessing certain special types of structure cubic convergence can be achieved