complex scaling circle criteria for lure systems abstractthis study reports alternative frequency domain interpretation and implementation of the circle criteria for absolute stability in lure systems by means of complex scaling and the argument principle by lure system a feedback configuration with a nominal lti model subject to sector nonlinearities is meant as usual firstly the complex scaling stability criterion is proved for asymptotic stability in lti feedback connections which dispenses open loop poles and contour locus pre orientation and possesses bounded loci without prior frequency sweeping secondly a novel frequency domain interpretation for positive realness of transfer functions is developed and employed for claiming the complex scaling circle criteria which accommodate various sector nonlinearities with unified conditions the new circle criteria are conformable in both multivariable and scalar cases implementable graphically and tractable numerically besides being a frequency complex domain analytical technique thirdly the re

weak and strong detonation profiles for a qualitative model abstract in this article the unidirectional travelling wave fronts for a qualitative model in combustion are studied the model is considered for a simple r p kind of chemical reaction by proving a general theorem of ode s the weak and strong detonations for exothermic reactions are shown to exist the uniqueness of these waves are also considered

nonlinear stability of an undercompressive shock for complex burgers equation though there is strong numerical evidence for the stability of undercompressive shocks their stability has not been verified analytically in particular the energy methods used to analyze stability of standard shocks do not apply here we present the first proof of stability for a particular undercompresive shock the real burgers shock considered as a solution of the complex burgers equation our analysis is by direct calculation of the green s function for the linearized equations combined with pointwise estimates of nonlinear effects a benefit of this method is to obtain fairly detailed information about the solution includingl1 behavior and rates of decay in different regions of space

generalized difference methods for a nonlinear dirichlet problem the paper is devoted to the construction of generalized difference methods on irregular meshes for second order nonlinear elliptic equations based on the generalized galerkin method the error estimate is proved

stability of viscous scalar shock fronts in several dimensions we prove nonlinear stability of planar shock front solutions for viscous scalar conservation laws in two or more space dimensions the proof uses the integrated equation and an effective equation for the motion of the front itself we derive energy estimates that balance terms from the integrated equation with terms from the front motion equation in this paper we prove that viscous shock profiles for scalar conservation laws are stable in two or more space dimensions these multidimensional stability questions are separate from their one dimensional analogues because of the possibility of transverse instabilities such as those that occur in combustion fronts lu and in shock waves with phase changes the proof here is a rigorous version of arguments that are used to derive effective equations such as the kuramoto sivashinsky equation to describe the behavior of fronts the one dimensional stability for scalar conservation laws was proven by ii in and oleinik io using the integrated equation see below and a max imum principle another proof based on weighted norms and spectral theory for the linearized problem was given by sattinger s the multidimensional stability proof below has more in common with the stability proofs for systems of conservation laws in one space dimension begun by kawashima and mat sumura km and goodman go and completed by liu li these proofs use l2 energy estimates for the integrated equation we consider equations of the form ut f u x g u y uxx uyy where f u is a strictly convex function of u

a fourth order compact scheme for the helmholtz equation alpha interpolation of fem and fdm stencils we propose a fourth order compact scheme on structured meshes for the helmholtz equation given by r f x 2 0 the scheme consists of taking the alpha interpolation of the galerkin finite element method and the classical central finite difference method in 1d this scheme is identical to the alpha interpolation method j comput appl math 1982 8 1 15 19 and in 2d making the choice 0 5 we recover the generalized fourth order compact pade approximation j comput phys 1995 119 252 270 comput meth appl mech engrg 1998 163 343 358 therein using the parameter 2 we follow siam rev 2000 42 3 451 484 comput meth appl mech engrg 1995 128 325 359 for the analysis of this scheme and its performance on square meshes is compared with that of the quasi stabilized fem comput meth appl mech engrg 1995 128 325 359 in particular we show that the relative phase error of the numerical solution and the local truncation error of this scheme for plane wave solutions diminish at the rate o l 4 where l represent the wavenumber and the mesh size respectively an expression for the parameter is given that minimizes the maximum relative phase error in a sense that will be explained in section 4 5 convergence studies of the error in the l2 norm the h1 semi norm and the l euclidean norm are done and the pollution effect is found to be small copyright 2010 john wiley sons ltd

eigenvalues of the schrodinger equation by the alpha interpolation method the alpha interpolation method was derived to solve numerically the two dimensional helmholtz equation more efficiently than the finite element method we develop the method further to calculate the eigenvalues of the schrodinger equation in an axisymmetrical field and derive a new numerical method to efficiently compute the eigenvalues of the eigenproblem for the alpha interpolation method using the eigenvalues and eigenvectors of the corresponding standard eigenproblem as numerical examples we compute the eigen energy levels of three problems in quantum mechanics by the new method and compare the results with the exact values or previous calculations

a galerkin least squares finite element method for the two dimensional helmholtz equation in this paper a galerkin least squares gls finite element method in which residuals in least squares form are added to the standard galerkin variational equation is developed to solve the helmholtz equation in two dimensions an important feature of gls methods is the introduction of a local mesh parameter that may be designed to provide accurate solutions with relatively coarse meshes previous work has accomplished this for the one dimensional helmholtz equation using dispersion analysis in this paper the selection of the gls mesh parameter for two dimensions is considered and leads to elements that exhibit improved phase accuracy for any given direction of wave propagation an optimal gls mesh parameter is determined using two dimensional fourier analysis in general problems the direction of wave propagation will not be known a priori in this case an optimal gls parameter is found which reduces phase error for all possible wave vector orientations over elements the optimal gls parameters are derived for both consistent and lumped mass approximations several numerical examples are given and the results compared with those obtained from the galerkin method the extension of gls to higher order quadratic interpolations is also presented

dynamics and stability of a weak detonation wave one dimensional weak detonation waves of a basic reactive shock wave model are proved to be nonlinearly stable i e initially perturbed waves tend asymptotically to translated weak detonation waves this model system was derived as the low mach number limit of the one component reactive navier stokes equations by majda and roytburd siam j sci stat comput 43 1086 1118 1983 and its weak detonation waves have been numerically observed as stable the analysis shows in particular the key role of the new nonlinear dynamics of the position of the shock wave the shock translation solves a nonlinear integral equation obtained by green s function techniques and its solution is estimated by observing that the kernel can be split into a dominating convolution operator and a remainder the inverse operator of the convolution and detailed properties of the traveling wave reduce by monotonicity the remainder to a small l1 perturbation

arbitrarily shaped hollow waveguide analysis by the alpha interpolation method an efficient numerical method named the alpha interpolation method aim has been developed for solving hollow waveguide problems the aim is a simpler and more systematic procedure than the finite element method and when the optimum interpolation parameter is chosen for any number of linear triangular elements the cutoff wavenumbers or the exact eigenvalues may be estimated by the aim numerical results and comparisons are given

a generalized finite element method for solving the helmholtz equation in two dimensions with minimal pollution when using the galerkin fem for solving the helmholtz equation in two dimensions the error of the corresponding solution differs substantially from the error of the best approximation and this effect increases with higher wave number k in this paper we will design a generalized finite element method gfem for the helmholtz equation such that the pollution effect is minimal

low gain integral control of well posed linear infinite dimensional systems with input and output nonlinearities abstract time varying low gain integral control strategies are presented for asymptotic tracking of constant reference signals in the context of exponentially stable well posed linear infinite dimensional single input single output systems subject to globally lipschitz nondecreasing input and output nonlinearities it is shown that applying error feedback using an integral controller ensures that the tracking error is small in a certain sense provided that a the steady state gain of the linear part of the system is positive b the reference value r is feasible in an entirely natural sense and c the positive gain function t k t is ultimately sufficiently small and not of class l 1 under a weak restriction on the initial data it is shown that a b and c ensure asymptotic tracking if additionally the impulse response of the linear part of the system is a finite signed borel measure the global lipschitz assumption on the output nonlinearity may be considerably relaxed

a high resolution petrov galerkin method for the convection diffusion reaction problem part ii a multidimensional extension abstract a multidimensional extension of the hrpg method doi 10 1016 j cma 2009 10 009 using the lowest order block finite elements is presented first we design a nondimensional element number that quantifies the characteristic layers which are found only in higher dimensions this is done by matching the width of the characteristic layers to the width of the parabolic layers found for a fictitious 1d reaction diffusion problem the nondimensional element number is then defined using this fictitious reaction coefficient the diffusion coefficient and an appropriate element size next we introduce anisotropic element length vectors li and the stabilization parameters i i are calculated along these li except for the modification to include the new dimensionless number that quantifies the characteristic layers the definitions of i i are a direct extension of their counterparts in 1d using i i and li objective characteristic tensors associated with the hrpg method are defined the numerical artifacts across the characteristic layers are manifested as the gibbs phenomenon hence we treat them just like the artifacts formed across the parabolic layers in the reaction dominant case several 2d examples are presented that support the design objective stabilization with high resolution

unified analysis of finite volume methods for second order elliptic problems we establish a general framework for analyzing the class of finite volume methods which employ continuous or totally discontinuous trial functions and piecewise constant test functions under the framework optimal order convergence in the h 1 and l 2 norms can be obtained in a natural and systematic way for classical finite volume methods and new finite volume methods such as discontinuous finite volume methods applied to second order elliptic problems

high order finite difference methods for the helmholtz equation high order finite difference methods for solving the helmholtz equation are developed and analyzed in one and two dimensions on uniform grids the standard pointwise representation has a second order accurate local truncation error we also study two schemes which have a fourth order accurate local truncation error one of the high order schemes is based on generalizations of the pade approximation the second scheme is based on high order approximation to the derivative calculated from the helmholtz equation itself a symmetric high order representation is developed for a neumann boundary condition numerical results are presented on model problems approximated with the developed schemes

existence of chapman jouguet detonation for a viscous combustion model in siam j appl math 41 1981 70 93 majda proposed a model for the interaction between chemical reactions and compressible fluid dynamics this model is a low mach number limit of the one component reactive navier stokes equations siam j appl math 43 1983 1086 1118 and was extended to the case where the diffusion coefficient is positive by larrouturou nonlinear anal 77 2001 405 418 in this paper the existence of a one dimensional chapman jouguet detonation wave or equivalently a heteroclinic orbit for the extended model is proven the proof is based on an application of topological arguments to a system of ordinary differential equations which is obtained from the partial differential equations describing the interaction

accurate finite difference methods for time harmonic wave propagation finite difference methods for solving problems of time harmonic acoustics are developed and analyzed multi dimensional inhomogeneous problems with variable possibly discontinuous coefficients are considered accounting for the effects of employing non uniform grids a weighted average representation is less sensitive to transition in wave resolution due to variable wave numbers or non uniform grids than the standard pointwise representation further enhancement in method performance is obtained by basing the stencils on generalizations of pade approximation or generalized definitions of the derivative reducing spurious dispersion anisotropy and reflection and by improving the representation of source terms the resulting schemes have fourth order accurate local truncation error on uniform grids and third order in the non uniform case guidelines for discretization pertaining to grid orientation and resolution are presented

remarks on a model for combustion waves discussion d un modele unidimensionnel simplifie propose par majda pour l etude qualitative des ondes de combustion

stability of rarefaction waves in viscous media we study the time asymptotic behavior of weak rarefaction waves of systems of conservation laws describing one dimensional viscous media with strictly hyperbolic flux functions our main result is to show that solutions of perturbed rarefaction data converge to an approximate burgers rarefaction wave for initial perturbations w 0 with small mass and localized as w 0 x mathcal o x 1 the proof proceeds by iteration of a pointwise ansatz for the error using integral representations of its various components based on green s functions we estimate the green s functions by careful use of the hopf cole transformation combined with a refined parametrix method as a consequence of our method we also obtain rates of decay and detailed pointwise estimates for the error this pointwise method has been used successfully in studying stability of shock and constant state solutions new features in the rarefaction case are time varying coefficients in the linearized equations and error waves of unbounded mass mathcal o log t these diffusion waves have amplitude mathcal o t 1 2logt in linear degenerate transversal fields and mathcal o t 1 2logt in genuinely nonlinear transversal fields a distinction which is critical in the stability proof

a petrov galerkin formulation for the alpha interpolation of fem and fdm stencils applications to the helmholtz equation summary a new petrov galerkin pg method involving two parameters namely 1 and 2 is presented which yields the following schemes on rectangular meshes i a compact stencil obtained by the linear interpolation of the galerkin fem and the classical central finite difference method fdm should the parameters be equal that is 1 2 and ii the nonstandard compact stencil presented in int j numer meth engng 2011 86 18 46 for the helmholtz equation if the parameters are distinct that is 1 2 the nonstandard compact stencil is obtained by taking the linear interpolation of the diffusive terms specified by 1 and the mass terms specified by 2 that appear in the stencils obtained by the standard galerkin fem and the classical central fdm respectively on square meshes these two schemes were shown to provide solutions to the helmholtz equation that have a dispersion accuracy of fourth and sixth order respectively int j numer meth engng 2011 86 18 46 the objective of this paper is to study the performance of this pg method for the helmholtz equation using nonuniform meshes and the treatment of natural boundary conditions copyright 2011 john wiley sons ltd

absolute stability and integral control for infinite dimensional discrete time systems abstract we derive absolute stability results of popov and circle criterion types for infinite dimensional discrete time systems in an input output setting our results apply to feedback systems in which the linear part is the series interconnection of an l 2 stable linear system and an integrator and the nonlinearity satisfies a sector condition which allows for saturation and deadzone effects the absolute stability theory is then used to prove tracking and disturbance rejection results for integral control schemes in the presence of input and output nonlinearities applications of the input output theory to state space systems are also provided

the alpha interpolation method for the solution of an eigenvalue problem abstract we present a weighted residual finite element method for the solution of an eigenvalue problem as a test function we take a linear combination of two functions which belong to different spaces we call this method the alpha interpolation method aim for the eigenvalue problem we compare the aim with the standard galerkin finite element method sgfem

a mortar finite element method using dual spaces for the lagrange multiplier the mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries in the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which serves as a space of lagrange multipliers in this paper this lagrange multiplier space is replaced by a dual space without losing the optimality of the method the advantage of this new approach is that the matching condition is much easier to realize in particular all the basis functions of the new method are supported in a few elements the mortar map can be represented by a diagonal matrix in the standard mortar method a linear system of equations must be solved the problem is considered in a positive definite nonconforming variational as well as an equivalent saddle point formulation

discrete time and sampled data low gain control of infinite dimensional linear systems in the presence of input hysteresis we introduce a general class of causal dynamic discrete time nonlinearities which have certain monotonicity and lipschitz continuity properties in particular the discretizations of a large class of continuous time hysteresis operators obtained by applying the standard sampling and hold operations belong to this class it is shown that closing the loop around a power stable linear infinite dimensional discrete time single input single output system subject to an input nonlinearity from the class under consideration and compensated by a discrete time integral controller guarantees asymptotic tracking of constant reference signals provided that a the positive integrator gain is sufficiently small and b the reference value is feasible in a very natural sense we apply this result in the development of sampled data low gain integral control for exponentially stable regular linear infinite dimensional continuous time systems subject to input hysteresis

a quasi optimal petrov galerkin method for helmholtz problem a petrov galerkin finite element formulation is introduced for helmholtz problem in two dimensions using polynomial weighting functions at each node of the mesh a global basis function for the weighting space is obtained adding to the bilinear c 0 lagrangian weighting function linear combinations of polynomial bubbles defined on a macroelement containing this node quasi optimal weighting functions with the same support of the corresponding global test functions are obtained after computing the coefficients of these linear combinations attending to optimality criteria this is done numerically through a simple preprocessing technique naturally applied to non uniform and unstructured meshes in particular for uniform mesh a quasi optimal interior stencil of the same order of the quasi stabilized finite element method stencil derived by babuska et al comput methods appl mech engrg 1995 128 325 359 is obtained numerical results are presented illustrating the great stability and accuracy of this formulation with uniform and non uniform meshes

pointwise convergence to shock waves for viscous conservation laws we are interested in the pointwise behavior of the perturbations of shock waves for viscous conservation laws it is shown that besides a translation of the shock waves and of linear and nonlinear diffusion waves of heat and burgers equations a perturbation also gives rise to algebraically decaying terms which measure the coupling of waves of different characteristic families our technique is a combination of time asymptotic expansion construction of approximate green functions and analysis of nonlinear wave interactions the pointwise estimates yield optimal lp convergence of the perturbation to the shock and diffusion waves 1 p the new approach of obtaining pointwise estimates based on the green functions for the linearized system and the analysis of nonlinear wave interactions is also useful for studying the stability of waves of distinct types and nonclassical shocks these are being explored elsewhere 1997 john wiley sons inc

quasi optimal finite difference method for helmholtz problem on unstructured grids a quasi optimal finite difference method qofd is proposed for the helmholtz problem the stencils coefficients are obtained numerically by minimizing a least squares functional of the local truncation error for plane wave solutions in any direction in one dimension this approach leads to a nodally exact scheme with no truncation error for uniform or non uniform meshes in two dimensions when applied to a uniform cartesian grid a 9 point sixth order scheme is derived with the same truncation error of the quasi stabilized finite element method qsfem introduced by babuska et al comp meth appl mech eng 1995 128 325 359 similarly a 27 point sixth order stencil is derived in three dimensions the qofd formulation proposed here is naturally applied on uniform non uniform and unstructured meshes in any dimension numerical results are presented showing optimal rates of convergence and reduced pollution effects for large values of the wave number copyright 2009 john wiley sons ltd

discrete time low gain control of uncertain infinite dimensional systems using a frequency domain analysis it is shown that the application of a feedback controller of the form k z 1 or kz z 1 where k spl isin r to a power stable infinite dimensional discrete time system with square transfer function matrix g z will result in a power stable closed loop system which achieves asymptotic tracking of arbitrary constant reference signals provided that i all the eigenvalues of g 1 have positive real parts and ii the gain parameter k is positive and sufficiently small moreover if g 1 is positive definite we show how the gain parameter gain x can be tuned adaptively the resulting adaptive tracking controllers are universal in the sense that they apply to any power stable system with g 1 0 in particular they are not based on system identification or plant parameter estimation algorithms nor is the injection of probing signals required finally we apply these discrete time results to obtain adaptive sampled data low gain controllers for the class of regular systems a rather general class of infinite dimensional continuous time systems for which convenient representations are known to exist both in state space and in frequency domain we emphasize that our results guarantee not only asymptotic tracking at the sampling instants but also in the sampling interval

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