spectral stability of small amplitude viscous shock waves in several space dimensions a planar viscous shock profile of a hyperbolic parabolic system of conservation laws is a steady solution in a moving coordinate frame the asymptotic stability of viscous profiles and the related vanishing viscosity limit are delicate questions already in the well understood case of one space dimension and even more so in the case of several space dimensions it is a natural idea to study the stability of viscous profiles by analyzing the spectrum of the linearization about the profile the evans function method provides a geometric dynamical systems framework to study the eigenvalue problem in this approach eigenvalues correspond to zeros of an essentially analytic function mathcal e rho lambda rho omega which detects nontrivial intersections of the so called stable and unstable spaces that is spaces of solutions that decay on one or the other side of the shock wave respectively in a series of pioneering papers kevin zumbrun and collaborators have established in various contexts that spectral stability that is the non vanishing of mathcal e rho lambda rho omega and the non vanishing of the lopatinski kreiss majda function imply nonlinear stability of viscous shock profiles in several space dimensions in this paper we show that these conditions hold true for small amplitude extreme shocks under natural assumptions this is done by exploiting the slow fast nature of the small amplitude limit which was used in a previous paper by the authors to prove spectral stability of small amplitude shock waves in one space dimension geometric singular perturbation methods are applied to decompose the stable and unstable spaces into subbundles with good control over their limiting behavior three qualitatively different regimes are distinguished that relate the small strength epsilon of the shock wave to appropriate ranges of values of the spectral parameters various rescalings are used to overcome apparent degeneracies in the problem caused by loss of hyperbolicity or lack of transversality

taylor approximation of second degree for complex mode in asymmetrical damped system algorithms for the first and second order sensitivities of the complex mode for an asymmetrical damped linear discrete dynamic system with respect to design parameters are presented the first an

eigenderivative analysis of asymmetric non conservative systems in general the damping matrix of a dynamic system or structure is such that it can not be simultaneously diagonalized with the mass and stiffness matrices by any linear transformation for this reason the eigenvalues and eigenvectors and consequently their derivatives become complex expressions for the first and second order derivatives of the eigenvalues and eigenvectors of these linear non conservative systems are given traditional restrictions of symmetry and positive definiteness have not been imposed on the mass damping and stiffness matrices the results are derived in terms of the eigenvalues and left and right eigenvectors of the second order system so that the undesirable use of the first order representation of the equations of motion can be avoided the usefulness of the derived expressions is demonstrated by considering a non proportionally damped two degree of freedom symmetric system and a damped rigid rotor on flexible supports copyright 2001 john wiley sons ltd

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

calculating derivatives of repeated and nonrepeated eigenvalues without explicit use of eigenvectors the analysis of inverse problems in parametric model updating often require the sensitivities of eigenvalues the calculation of these sensitivities is mathematically related to the derivatives of the eigenvalues with respect to the model parameters a common method to calculate these derivatives is the nelson method which requires the eigenvectors the method introduced in this paper is derived from the characteristic equation of the underlying general eigenvalue problem and allows the derivatives of eigenvalues with respect to the model parameters to be calculated without explicit use of the eigenvectors the method is extended for the case of repeated eigenvalues which leads to restrictions on the parameterization for repeated eigenvalues of multiplicity two these restrictions are formulated expicitly applications and limitations of the method are demonstrated by examples

multidimensional viscous shocks ii the small viscosity limit in this paper we prove the existence of curved multidimensional viscous shocks and also justify the small viscosity limit starting with a curved multidimensional inviscid shock solution to a system of hyperbolic conservation laws we show that the shock can be obtained as a small viscosity limit of solutions to an associated parabolic problem viscous shocks the two main hypotheses are a natural evans function assumption on the viscous profile together with a restriction on how much the shock can deviate from flatness the main tools are a conjugation lemma that removes xn dependence from the linearization of the parabolic problem about the viscous profile new degenerate kreiss type symmetrizers used to prove an l2 estimate for the linearized problem and a finite regularity calculus of semiclassical and mixed type classical semiclassical pseudodifferential operators 2003 wiley periodicals inc

well posedness in smooth function spaces for moving boundary 1 d compressible euler equations in physical vacuum we prove well posedness for the 3 d compressible euler equations with moving physical vacuum boundary with an equation of state given by the so called gamma gas law for gamma 1 the physical vacuum singularity requires the sound speed c to go to zero as the square root of the distance to the moving boundary and thus creates a degenerate and characteristic hyperbolic free boundary system wherein the density vanishes on the free boundary the uniform kreiss lopatinskii condition is violated and manifest derivative loss ensues nevertheless we are able to establish the existence of unique solutions to this system on a short time interval which are smooth in sobolev spaces all the way to the moving boundary and our estimates have no derivative loss with respect to initial data our proof is founded on an approximation of the euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term chosen to preserve as much of the geometric structure of the euler equations as possible we first construct solutions to this degenerate parabolic regularization using a new higher order hardy type inequality we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter solutions to the compressible euler equations are found in the limit as the artificial viscosity tends to zero our regular solutions can be viewed as degenerate viscosity solutions out methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws

stability and asymptotic behavior of periodic traveling wave solutions of viscous conservation laws in several dimensions under natural spectral stability assumptions motivated by previous investigations of the associated spectral stability problem we determine sharp lp estimates on the linearized solution operator about a multidimensional planar periodic wave of a system of conservation laws with viscosity yielding linearized l1 lp lp stability for all p geqq 2 and dimensions d geqq 1 and nonlinear l1 hs lp hs stability and l2 asymptotic behavior for p geqq 2 and d geqq 3 the behavior can in general be rather complicated involving both convective that is wave like and diffusive effects

a general non iterative riemann solver for godunov s method abstract godunov s method is characterized by the use of a riemann problem solution to resolve discontinuities at the interface between cells the major drawback of this method is the difficulty and the high cost of solving the nonlinear riemann problem exactly especially for materials with complex equations of state this paper describes a simplified and noniterative approximate riemann solver which is characterized by only two material dependent parameters for a given material these parameters are the local speed of sound and a parameter which is directly related to the shock density ratio in the limit of strong shocks these parameters are conveniently obtained from a linear fit to the experimental data for the shock hugoniot in various materials the approximate riemann solver retains the essential quadratic nonlinearity which enables it to deal with the whole range of cases from weak sound waves to strong shocks

maximal positive boundary value problems as limits of singular perturbation problems we study three types of singular perturbations of a symmetric positive system of partial differential equations on a domain 2 c r in all cases the limiting behavior is given by the solution of a maximal positive boundary value problem in the sense of friedrichs the perturbation is either a second order elliptic term or a term large on the complement of q the first corresponds to a sort of viscosity and the second to physical systems with vastly different properties in i2 and outside u the results show that in the limit of zero viscosity or infinitely large difference the behavior is described by a maximal positive boundary value problem in 2 the boundary condition is determined in a simple way from the system and the singular terms 1 introduction in this paper we study boundary value roblems for singularly perturbed systems of partial differential equations the unperturbed system is of first order and positive symmetric in the sense of friedrichs we study three sorts of singular perturbations 1 the addition of e times a positive second order elliptic system in a domain fi 2 3 the addition of xp or p d dt with a 1 and p strictly positive in the exterior of n and zero in s2 in the first case we study the solutions he to the dirichlet problem in s2 bardos brezis and brezis 1 have shown that in this case ue u weakly in l2 fi where u is uniquely determined as the solution of a maximal positive boundary value problem for the unperturbed opera tor here one has the phenomenon of loss of boundary conditions in the limit and the existence of boundary layers for uc near 3i2 we prove several more refined estimates on ue in particular a uniform bound in l 2 fi for any rj 0 these estimates yield a proof of the strong convergence of uc to u in all hs u for s this singular perturbation problem is well known and much studied in contrast to the problems of the second sort mentioned above in the latter problems we study the cauchy problems

existence and stability of compressible current vortex sheets in three dimensional magnetohydrodynamics compressible vortex sheets are fundamental waves along with shocks and rarefaction waves in entropy solutions to multidimensional hyperbolic systems of conservation laws understanding the behavior of compressible vortex sheets is an important step towards our full understanding of fluid motions and the behavior of entropy solutions for the euler equations in two dimensional gas dynamics the classical linearized stability analysis on compressible vortex sheets predicts stability when the mach number m sqrt 2 and instability when m sqrt 2 for the euler equations in three dimensions every compressible vortex sheet is violently unstable and this instability is the analogue of the kelvin helmholtz instability for incompressible fluids the purpose of this paper is to understand whether compressible vortex sheets in three dimensions which are unstable in the regime of pure gas dynamics become stable under the magnetic effect in three dimensional magnetohydrodynamics mhd one of the main features is that the stability problem is equivalent to a free boundary problem whose free boundary is a characteristic surface which is more delicate than noncharacteristic free boundary problems another feature is that the linearized problem for current vortex sheets in mhd does not meet the uniform kreiss lopatinskii condition these features cause additional analytical difficulties and especially prevent a direct use of the standard picard iteration to the nonlinear problem in this paper we develop a nonlinear approach to deal with these difficulties in three dimensional mhd we first carefully formulate the linearized problem for the current vortex sheets to show rigorously that the magnetic effect makes the problem weakly stable and establish energy estimates especially high order energy estimates in terms of the nonhomogeneous terms and variable coefficients then we exploit these results to develop a suitable iteration scheme of the nash moser hormander type to deal with the loss of the order of derivative in the nonlinear level and establish its convergence which leads to the existence and stability of compressible current vortex sheets locally in time in three dimensional mhd

sensitivity analysis of non conservative eigensystems an expression for the derivatives of eigenvalues and eigenvectors of non conservative systems is presented contrary to previous methods that use state space form 2n space to consider damping proposed method solves the eigenpair derivatives of damped system explicitly the computation size of n order is maintained and the eigenpair derivatives are obtained simultaneously from one equation so that it is efficient in cpu time and storage capacity moreover this method can be extended to asymmetric non conservative damped systems although additional problems are generated contrary to the eigenpair sensitivity methods of symmetric systems in asymmetric case an algebraic method for the eigenpair derivatives can be obtained through similar procedure the proposed expression is derived by combining the differentiations of the eigenvalue problem and normalization condition into one linear algebraic equation the numerical stability is proved by showing non singularity of the proposed equation and the efficiency of the derived expression is illustrated by considering a cantilever beam with lumped dampers and a whirling beam

a unified coordinate system for solving the three dimensional euler equations it is well known that the use of eulerian coordinates for shock capturing methods results in badly smeared slip lines and that lagrangian coordinates while capable of producing sharp slip line resolution may result in severe grid deformation causing inaccuracy and even breakdown of computation a unified coordinate system is introduced in which the flow variables are considered to be functions of time and of some permanent identification of pseudo particles which move with velocity hq q being the velocity of fluid particles it includes the eulerian coordinates as a special case when h 0 and the lagrangian when h 1 for two dimensional inviscid flow the free function h is chosen so as to preserve the grid angles this results in a coordinate system which avoids excessive numerical diffusion across slip lines in the eulerian coordinates and avoids severe grid deformation in the lagrangian coordinates yet it retains sharp resolution of slip lines especially for steady flow furthermore the two dimensional unsteady euler equations of gasdynamics in the unified coordinates are found to be hyperbolic for all values of h except when h 1 i e lagrangian in the latter case the euler equations are only weakly hyperbolic lacking one eigenvector although all eigenvalues are real the consequences of this deficiency of the lagrangian coordinates are pointed out in connection with numerical computation

derivatives of complex eigenvectors using nelson s method 6shyy w an adaptive grid method for navier stokes flow computation applied mathematics and computation vol 21 no 3 1987 pp 201 219 7jeng y n andliou s c modi edmultipleone dimensional adaptive grid method numerical heat transfer part b vol 15 no 2 1989 pp 241 247 8jeng y n and lee z s revisit to the modi ed muliple onedimensional adaptive grid method numerical heat transfer part b vol 29 no 3 1996 pp 305 323 9bockelie m j and smith r e an adaptive grid method for computing the high speed 3d viscous flow about a re entry vehicle aiaa paper 92 2685 june 1992 10yee h c and harten a implicit tvd schemes for hyperbolic conservation laws in curvilinear coordinates aiaa journal vol 25 no 2 1987 pp 266 274 11jeng y n and liou y c adaptive grid generation by elliptic equations with grid control at all of the boundaries numerical heat transfer part b vol 23 no 2 1993 pp 135 151

convex entropy hopf bifurcation and viscous and inviscid shock stability we consider by a combination of analytical and numerical techniques some basic questions regarding the relations between inviscid and viscous stability and the existence of a convex entropy specifically for a system possessing a convex entropy in particular for the equations of gas dynamics with a convex equation of state we ask 1 can inviscid instability occur 2 can viscous instability not detected by inviscid theory occur 3 can there occur the necessarily viscous effect of hopf bifurcation or galloping instability and perhaps most important from a practical point of view 4 as shock amplitude is increased from the stable weak amplitude limit can there occur a first transition from viscous stability to instability that is not detected by inviscid theory we show that 1 does occur for strictly hyperbolic genuinely nonlinear gas dynamics with certain convex equations of state while 2 and 3 do occur for an artifically constructed system with convex viscosity compatible entropy we do not know of an example for which 4 occurs leaving this as a key open question in viscous shock theory related to the principal eigenvalue property of sturm liouville and related operators in analogy with and partly proceeding close to the analysis of smith on non uniqueness of the riemann problem we obtain convenient criteria for shock in stability in the form of necessary and sufficient conditions on the equation of state

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

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

natural frequency and mode shape sensitivities of damped systems part ii multiple natural frequencies abstract an efficient algorithm with proven numerical stability is derived for computation of eigenvalue and eigenvector derivatives of damped vibratory systems with multiple eigenvalues in the proposed method adjacent eigenvectors and orthonormal condition are used to compose an algebraic equation whose order is n m n m where n is the number of co ordinates and m the number of multiplicity of a multiple natural frequency the mode shape derivatives of the damped systems can be obtained by solving the algebraic equation the method can be consistently applied to both structural systems with structural design parameters and mechanical systems with lumped design parameters as an example of a structural system to demonstrate the theory of the proposed methods and its possibilities in the case of multiple eigenvalues the finite element model of the cantilever beam is considered and also a 5 dof mechanical system in the case of a non proportionally damped system the design parameter of the cantilever beam is its height and that of the 5 dof mechanical system is a spring

a note on the unified coordinate system for computing shock waves in 1999 j comput phys 153 596 hui and his co workers proposed a unified coordinate system for computing compressible flows with discontinuous solutions in their coordinate system there is a free parameter h such that the traditional eulerian approach and lagrangian approach correspond to the particular cases h 0 and h 1 respectively hence this approach unifies the two classical methods for describing fluid flows in this note we consider a one dimensional problem and we show that there is a parameter range within 0

a stochastic scaled boundary finite element method abstract by extending the existing scaled boundary finite element method sbfem into the random field problem a stochastic scaled boundary finite element method ssbfem is developed in this paper to predict the structural responses with randomly distributed material properties the analyzed domain is discretized into a number of polygons modeled as sub domains of the sbfem so that the non homogeneous material properties are possible to be considered in establishment of the constitutive equation the random field represented by the karhunen loeve kl expansion is then directly incorporated into both the equilibrium equation of the sbfem and its sensitivity analysis formulation the perturbation method is employed to calculate the statistical moments of the structural responses and a semi analytical sensitivity analysis method is used to efficiently calculate the gradients involved particularly the proposed method is successfully applied to the stochastic fracture analysis with spatially varying random material properties four numerical examples are investigated to demonstrate the validity of the proposed method

eigensensitivity analysis for asymmetric nonviscous systems t he eigensensitivities of mechanical systems with respect to structural design parameters have become an integral part of many engineering design methodologies including optimization structural health monitoring structural reliability model updating dynamic modification reanalysis techniques and many other applications fox and kapoor 1 computed the derivative of each eigenvector as a linear combination of all of the undamped eigenvectors later adhikari and friswell 2 and adhikari 3 extended the modal method to the more general asymmetric systems with viscous and nonviscous damping respectively nelson 4 presented a method which requires only the eigenvector of interest by expressing the derivative of each undamped eigenvector as a particular solution and a homogeneous solution later friswell and adhikari 5 extended nelson s method to symmetric and asymmetric systems with viscous damping recently adhikari and friswell 6 extendednelson s method to symmetric and asymmetric nonviscously damped systems fox and kapoor 1 also suggested a direct algebraic method to calculate the eigensensitivity for symmetric undamped systems by solving a nonsingular linear system of algebraic equations lee et al 7 derived an efficient algebraic method which has a compact linear system with a symmetric coefficient matrix for symmetric systems with viscous damping later guedria et al 8 extended the algebraic method to general asymmetric viscous damped systems chouchane et al 9 reviewed the algebraic method and extended their method to the second order and high order derivatives of eigensolutions li et al 10 extended the algebraic method to symmetric and asymmetric nonviscously damped systems xu andwu 11 proposed a new normalization and presented a method for the computation of eigensolution derivatives of asymmetric systemswith viscously damping recently mirzaeifar et al 12 proposed a new method based on a combination of algebraic and modal methods for generally asymmetric viscously damped systems more recently li et al 13 proposed a method of design sensitivity analysis of asymmetric viscously damped systems with distinct and repeated eigenvalues which can compute the left and right eigenvector derivatives separately and independently all of the methods mentioned previously compute the eigensensitivities of asymmetric damped systems by using the left eigenvector however these methods have disadvantages in computational cost and storage capacity for the left eigenvector should be calculated to avoid using the left eigenvector an algebraic method is presented 14 which does not require the left eigenvector for asymmetric damped systems but this method is restricted to the case of viscous damping it should be noted that the coefficientmatrices of the algebraicmethodmay be ill conditioned due to the components of the additional constraints and system matrices in the coefficient matrices are not all of the same order of magnitude in addition 15 the normalization adapted in 14 and 12 will fail in some cases because it can equal zero or a very small number this note will present a method which is well conditioned and can calculate the eigensensitivity of asymmetric nonviscous damped systems without using the left eigenvector considering an n degree of freedom linear system with nonviscous viscoelastic damping 3 6 10

zero dissipation limit to strong contact discontinuity for the 1 d compressible navier stokes equations abstract in this paper we study the zero dissipation limit problem for the one dimensional compressible navier stokes equations we prove that if the solution of the inviscid euler equations is piecewise constants with a contact discontinuity then there exist smooth solutions to the navier stokes equations which converge to the inviscid solution away from the contact discontinuity at a rate of 1 4 as the heat conductivity coefficient tends to zero provided that the viscosity is of higher order than the heat conductivity without loss of generality we set 0 here we have no need to restrict the strength of the contact discontinuity to be small

a direct algebraic method for eigensolution sensitivity computation of damped asymmetric systems in general the derivative of an eigenvector of a vibrating symmetric system is the solution of a singular problem further complications are encountered in dealing with asymmetric damped systems for which the left and right eigenvectors and their derivatives become distinct and complex several approaches have been proposed to overcome this singularity such as nelson s method and the modal method in the present work a new approach is presented for calculating simultaneously the derivatives of the eigenvalues and their associated derivatives of the left and right eigenvectors for asymmetric damped systems with the proposed method the exact eigenderivatives can be obtained by solving a first order linear algebraic system of equations the method is applied on a 104 dof ventilator rotor system which is used as an example of an asymmetric damped system with distinct eigenvalues the diameter of the shaft has been chosen as the design parameter the comparison of the computational time shows that the proposed method is more efficient than both nelson s approach and the modal method copyright 2006 john wiley sons ltd

on the stability of traveling waves in weighted l spaces abstract i present new techniques and results concerning the stability of travelling waves to semilinear parabolic equations i am concerned with those problems in which the essential spectrum of the linearization about the wave contains 0 in these circumstances the stability results of d henry in lecture notes in mathematics vol 840 springer verlag new york 1981 and p bates and c jones dynamics rep 2 1989 1 38 are not applicable as the spectral information does not yield an exponential time decay estimate on the semigroup i show that under certain relatively easily computable conditions the wave is stable in polynomially weighted l spaces furthermore the rate at which the perturbation decays to the wave as t depends on the growth rate of the polynomial near infinity the method of proof uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula estimates for the semigroup are derived through a new technique for estimating the resolvent

well posedness in smooth function spaces for the moving boundary three dimensional compressible euler equations in physical vacuum we prove well posedness for the three dimensional compressible euler equations with moving physical vacuum boundary with an equation of state given by p c for 1 the physical vacuum singularity requires the sound speed c to go to zero as the square root of the distance to the moving boundary and thus creates a degenerate and characteristic hyperbolic free boundary system wherein the density vanishes on the free boundary the uniform kreiss lopatinskii condition is violated and manifest derivative loss ensues nevertheless we are able to establish the existence of unique solutions to this system on a short time interval which are smooth in sobolev spaces all the way to the moving boundary and our estimates have no derivative loss with respect to initial data our proof is founded on an approximation of the euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term chosen to preserve as much of the geometric structure of the euler equations as possible we first construct solutions to this degenerate parabolic regularization using a higher order version of hardy s inequality we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter solutions to the compressible euler equations are found in the limit as the artificial viscosity tends to zero our regular solutions can be viewed as degenerate viscosity solutions our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws

conditional stability of unstable viscous shocks abstract continuing a line of investigation initiated by texier and zumbrun on dynamics of viscous shock and detonation waves we show that a linearly unstable lax type viscous shock solution of a semilinear strictly parabolic system of conservation laws possesses a translation invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small l 1 h 2 perturbations converging time asymptotically to a translate of the unperturbed wave that is for a shock with p unstable eigenvalues we establish conditional stability on a codimension p manifold of initial data with sharp rates of decay in all l p for p 0 we recover the result of unconditional stability obtained by howard mascia and zumbrun

hyperbolic boundary value problems for symmetric systems with variable multiplicities abstract we extend the kreiss majda theory of stability of hyperbolic initial boundary value and shock problems to a class of systems notably including the equations of magnetohydrodynamics mhd for which majda s block structure condition does not hold namely simultaneously symmetrizable systems with characteristics of variable multiplicity satisfying at points of variable multiplicity either a totally nonglancing or a nonglancing and linearly splitting condition at the same time we give a simple characterization of the block structure condition as geometric regularity of characteristics defined as analyticity of associated eigenprojections the totally nonglancing or nonglancing and linearly splitting conditions are generically satisfied in the simplest case of crossings of two characteristics and likewise for our main physical examples of mhd or maxwell equations for a crystal together with previous analyses of spectral stability carried out by gardner kruskal and blokhin trakhinin this yields immediately a number of new results of nonlinear inviscid stability of shock waves in mhd in the cases of parallel or transverse magnetic field and recovers the sole previous nonlinear result obtained by blokhin trakhinin by direct dissipative integral methods of stability in the zero magnetic field limit we also discuss extensions to the viscous case

nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case abstract extending previous results of oh zumbrun and johnson zumbrun we show that spectral stability implies linearized and nonlinear stability of spatially periodic traveling wave solutions of viscous systems of conservation laws for systems of generic type removing a restrictive assumption that wave speed be constant to first order along the manifold of nearby periodic solutions key to our analysis is a nonlinear cancellation estimate observed by johnson and zumbrun along with a detailed understanding of the whitham averaged system the latter motivates a careful analysis of the bloch perturbation expansion near zero frequency and suggests factoring out an appropriate translational modulation of the underlying wave allowing us to derive the sharpened low frequency estimates needed to close the nonlinear iteration arguments

existence and stability of multidimensional shock fronts in the vanishing viscosity limit in this paper we present a new approach to the study of linear and nonlinear stability of inviscid multidimensional shock waves under small viscosity perturbation yielding optimal estimates and eventually an extension to the viscous case of the celebrated theorem of majda on existence and stability of multidimensional shock waves more precisely given a curved lax shock solution u0 to a hyperbolic system of conservation laws we construct nearby viscous shock solutions ue to a parabolic viscous perturbation of the hyperbolic system which converge to u0 as viscosity e 0 and satisfy an appropriate conormal version of majda s stability estimate

existence and stability of noncharacteristic boundary layers for the compressible navier stokes and viscous mhd equations for a general class of hyperbolic parabolic systems including the compressible navier stokes and compressible mhd equations we prove existence and stability of noncharacteristic viscous boundary layers for a variety of boundary conditions including classical navier stokes boundary conditions our first main result using the abstract framework established by the authors in the companion work gues et al in j differ equ 244 309 387 2008 is to show that existence and stability of arbitrary amplitude exact boundary layer solutions follow from a uniform spectral stability condition on layer profiles that is expressible in terms of an evans function uniform evans stability whenever this condition holds we give a rigorous description of the small viscosity limit as the solution of a hyperbolic problem with residual boundary conditions our second is to show that uniform evans stability for small amplitude layers is equivalent to evans stability of the limiting constant layer which in turn can be checked by a linear algebraic computation finally for a class of symmetric dissipative systems including the physical examples mentioned above we carry out energy estimates showing that constant and thus small amplitude layers always satisfy uniform evans stability this yields existence of small amplitude multi dimensional boundary layers for the compressible navier stokes and mhd equations for both equations these appear to be the first such results in the compressible case

boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems abstract in this paper we study viscous perturbations of quasilinear hyperbolic systems in several dimensions as the viscosity goes to zero the boundary is noncharacteristic for the hyperbolic system we in particular describe the boundary layer which arises near the boundary and give a sufficient condition for the convergence of the solution to the solution of some mixed hyperbolic problem with some nonlinear maximal dissipative boundary conditions a counterexample is given when this condition is not satisfied and the solution blows up as the viscosity goes to 0

calculation of eigensolution derivatives for nonviscously damped systems using nelson s method a method to calculate the derivatives of the eigenvalues and eigenvectors of multiple degree of freedom damped linear dynamic systems with respect to arbitrary design parameters is presented in contrast to the traditional viscous damping model a more general nonviscous damping model is considered the nonviscous damping model is such that the damping forces depend on the past history of velocities via convolution integrals over given kernel functions because of the general nature of the damping eigensolutions are generally complex valued and eigenvectors do not satisfy the classical orthogonality relationship the proposed method to calculate the eigenvector derivative depends only on the eigenvector concerned numerical examples are provided to illustrate the derived results nomenclature c j constant associated with the derivative of u j d s dynamic stiffness matrix d j constant associated with the derivative of v j g s laplace transform of g t g t nonviscous damping matrix k stiffness matrix l laplace transform of m mass matrix m total number of eigenvectors n number of degrees of freedom of the system n number of relaxation parameters p design parameter rank rank of a matrix s laplace domain parameter u j jth eigenvector t displacement vector v j jth adjoint left eigenvector x j vector associated with the derivative of u j yj vector associated with the derivative of v j j normalization constant for the jth eigenvector j jth eigenvalue k relaxation parameters subscripts t matrix transpose derivative with respect to s

improved approximate methods for calculating frequency response function matrix and response of mdof systems with viscoelastic hereditary terms as we know it is difficult and unnecessary to obtain all the eigenpairs of a large scaled viscoelastic nonviscous or hysteretic damping systems which means that the mode truncation scheme is generally used and the mode truncated error is therefore introduced this study is aimed at eliminating the influence of the unavailable modes on the dynamic response of mdof systems with viscoelastic hereditary terms the energy dissipation terms of the system depend on the past history of motion via convolution integrals over some kernel functions therefore the system is a nonviscously damped system which has been considered as the most generalized damping model within the scope of a linear mechanical analysis to approximate frequency response function frf matrix and response without using the unavailable modes we suggest two methods which attempt to approximate the influence of the unavailable modes in terms of the lower modes and system matrices by using the first one or two terms of neumann expansion of the contribution of the unavailable modes in contrast with the frf matrix approximated in terms of the first two terms of neumann expansion these procedures cannot be extended to further high order terms since all of them will be affected by the frequency dependent variation of damping matrix from previous terms finally an example is shown that the two presented methods can make the mode truncated error reduce and may be used to approximate the influence of nonviscous modes contributed to frf matrix due to the fact that the nonviscous modes are difficult to be obtained accurately even if a small scaled model is used for some eigensolution methods

viscous limits for piecewise smooth solutions to systems of conservation laws in this paper we study the zero dissipation problem for a general system of conservation laws with positive viscosity it is shown that if the solution of the problem with zero viscosity is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition then there exist solutions to the corresponding system with viscosity that converge to the solutions of the system without viscosity away from shock discontinuities at a rate of order e as the viscosity coefficient e goes to zero the proof uses a matched asymptotic analysis and an energy estimate related to the stability theory for viscous shock profiles

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

eliminating the modal truncation problem encountered in frequency responses of viscoelastic systems increasing the number of degrees of freedom used in finite element analysis for mechanical and structural systems with viscoelastic damping the need to consider the modal truncation problem of viscoelastic systems is more than ever before the higher modes may be unnecessary to obtain in dynamic analysis for engineering applications for viscoelastic systems the modal truncation problem may be more frequently encountered since the nonviscous modes are difficult or even impossible to be found accurately even if a small scaled problem is considered for some eigensolution methods this study aims at eliminating the influence of the higher modes on the frequency responses of viscoelastically damped systems a method is presented by making the equilibrium equations of motion into a subspace equation spanned in terms of the columns of a projection basis obtained by considering the use of the contribution of the lower modes and the first two terms of the neumann expansion of the contribution of the unavailable modes finally three example studies are provided to illustrate the effectiveness of the derived results it is shown that the proposed method can reduce the modal truncation error significantly

viscous boundary value problems for symmetric systems with variable multiplicities abstract extending investigations of metivier and zumbrun in the hyperbolic case we treat stability of viscous shock and boundary layers for viscous perturbations of multidimensional hyperbolic systems with characteristics of variable multiplicity specifically the construction of symmetrizers in the low frequency regime where variable multiplicity plays a role at the same time we extend the boundary layer theory to real or partially parabolic viscosities neumann or mixed type parabolic boundary conditions and systems with nonconservative form in addition proving a more fundamental version of the zumbrun serre rousset theorem valid for variable multiplicities characterizing the limiting hyperbolic system and boundary conditions as a nonsingular limit of a reduced viscous system the new effects of viscosity are seen to be surprisingly subtle in particular viscous coupling of crossing hyperbolic modes may induce a destabilizing effect we illustrate the theory with applications to magnetohydrodynamics

a comparison of first and second order rezoned and lagrangian godunov solutions abstract in this paper we present solutions from a godunov type algorithm to a selection of hydrodynamics test problems included in this selection are implosions in planar cylindrical and spherical coordinates piston problems with a variety of mesh configurations including the difficult saltzman problem interacting blast waves and a spherical expansion on a rectangular mesh these solutions have been obtained using caveat a three dimensional arbitrary lagrangian eulerian ale hydrodynamics code that employs a godunov method of solution with an approximate riemann solver first and second order solutions are presented for most of the problems and rezoned and lagrangian solutions to the implosion noh problem are compared

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

computation of eigensolution derivatives for nonviscously damped systems using the algebraic method t he computation of the eigensolution derivatives plays a significant role in dynamic model updating structural design optimization structural dynamic modification damage detection andmany other applications themethods to calculate eigensolution derivatives are well established for undamped and viscous damped systems these common methods can be divided into the modal method nelson s method and the algebraic method fox and kapoor 1 first proposed the modal method for symmetric undamped systems by approximating the derivative of each eigenvector as a linear combination of all undamped eigenvectors later adhikari and friswell 2 and adhikari 3 extended the modal method to the more general asymmetric systems with viscous and nonviscous damping respectively to simplify the computation of eigensolution derivatives nelson 4 proposed a method which requires only the eigenvector of interest by expressing the derivative of each eigenvector as a particular solution and a homogeneous solution for symmetric undamped systems later friswell and adhikari 5 extended nelson s method to symmetric and asymmetric systemswith viscous damping recently adhikari and friswell 6 extended nelson s method to symmetric and asymmetric nonviscously damped systems however nelson s method is lengthy and clumsy for programming lee et al 7 derived an efficient algebraic method which has a compact form to compute the eigensolution derivatives by solving a nonsingular linear system of algebraic equations for symmetric systems with viscous damping later guedria et al 8 extended the algebraic method to general asymmetric systems with viscous damping recently chouchane et al 9 wrote an excellent review of the algebraic method for symmetric and asymmetric systems with viscous damping and extended their method to the second order and high order derivatives of eigensolutions in this note the algebraic method will be extended to symmetric and asymmetric systems with nonviscous damping the equations of motion describing free vibration of ann degreeof freedom dof linear system with nonviscous viscoelastic damping can be expressed by 3 6

spectral stability of small shock waves this paper is the first in a series of articles that study the eigenvalue problem for small amplitude shock waves in systems of conservation laws with viscosity or relaxation the papers show in various contexts that in the zero amplitude limit appropriately scaled versions of the associated evans bundles converge to suspensions of evans bundles for fixed shock waves in related reduced systems with lower dimensional state space in this article the new geometric framework establishing this connection is introduced in the simplest context that of shock waves associated with a simple genuinely nonlinear mode in systems with identity viscosity in one space dimension

multidimensional viscous shocks i degenerate symmetrizers and long time stability received by the editors september 18 2002 2000 mathematics subject classification primary 35l60 secondary 35b35 35b45 35k55 35l65 76l05

analytical solutions in the time domain for vibration problems of discrete viscoelastic systems in this study isotropic homogeneous hereditary materials are considered the material is supposed to be in an isothermal state

viscous limit to contact discontinuity for the 1 d compressible navier stokes equations abstract in this paper we study the zero dissipation limit problem for the one dimensional compressible navier stokes equations we prove that if the solution of the inviscid euler equations is piecewise constants with a contact discontinuity then there exist smooth solutions to the navier stokes equations which converge to the inviscid solution away from the contact discontinuity at a rate of 3 4 as the heat conductivity coefficient tends to zero provided that the viscosity is higher order than the heat conductivity or the same order as here we have no need to restrict the strength of the contact discontinuity to be small

symmetric euler and navier stokes shocks in stationary barotropic flow on a bounded domain abstract we construct stationary solutions to the barotropic compressible euler and navier stokes equations in several space dimensions with spherical or cylindrical symmetry for given dirichlet data on a sphere or a cylinder we first construct smooth and radially symmetric solutions to the euler equations in the exterior domain on the other hand stationary smooth solutions in the interior domain necessarily become sonic and cannot be continued beyond a critical inner radius we then use these solutions to construct entropy satisfying shocks for the euler equations in the region between two concentric spheres or cylinders next we construct smooth navier stokes solutions converging to the previously constructed euler shocks in the small viscosity limit in the process we introduce a new technique for constructing smooth solutions which exhibit a fast transition in the interior to a class of two point boundary problems

evans functions jost functions and fredholm determinants the principal results of this paper consist of an intrinsic definition of the evans function in terms of newly introduced generalized matrix valued jost solutions for general first order matrix valued differential equations on the real line and a proof of the fact that the evans function a finite dimensional determinant by construction coincides with a modified fredholm determinant associated with a birman schwinger type integral operator up to an explicitly computable nonvanishing factor

derivative of eigensolutions of nonviscously damped linear systems derivatives of eigenvalues and eigenvectors of multiple degree of freedom damped linear dynamic systems with respect to arbitrary design parameters are presented in contrast to the traditional viscous damping model a more general nonviscous damping model is considered the nonviscous damping model is such that the damping forces depend on the past history of velocities via convolution integrals over some kernel functions because of the general nature of the damping eigensolutions are generally complex valued and eigenvectors do not satisfy any orthogonality relationship it is shown that under such general conditions the derivative of eigensolutions can be expressed in a way similar to that of undamped or viscously damped systems numerical examples are provided to illustrate the derived results

calculation of eigenpair derivatives for asymmetric damped systems with distinct and repeated eigenvalues summary an algorithm is derived for the computation of eigenpair derivatives of asymmetric quadratic eigenvalue problem with distinct and repeated eigenvalues in the proposed method the eigenvector derivatives of the damped systems are divided into a particular solution and a homogeneous solution by introducing an additional normalization condition we construct two extended systems of linear equations with nonsingular coefficient matrices to calculate the particular solution the method is numerically stable and the homogeneous solutions are computed by the second order derivatives of the eigenequations two numerical examples are used to illustrate the validity of the proposed method copyright 2015 john wiley sons ltd

stability of one dimensional boundary layers by using green s functions the aim of this paper is to investigate the stability of one dimensional boundary layers of parabolic systems as the viscosity goes to 0 in the noncharacteristic case and more precisely to prove that spectral stability implies linear and nonlinear stability of approximate solutions in particular we replace the smallness condition obtained by the energy method 10 13 by a weaker spectral condition 2001 john wiley sons inc

stability of the travelling wave solution of the fitzhugh nagumo system on demontre la stabilite relative au systeme complet d equations aux derivees partielles de ces ondes de propagation

nonclassical multidimensional viscous and inviscid shocks extending our earlier work on lax type shocks of systems of conservation laws we establish existence and stability of curved multidimensional shock fronts in the vanishing viscosity limit for general lax or undercompressive type shock waves of nonconservative hyperbolic systems with parabolic regularization the hyperbolic equations may be of variable multiplicity and the parabolic regularization may be of real or partially parabolic type we prove an existence result for inviscid nonconservative shocks that extends to multidimensional shocks a one dimensional result of x lin proved by quite different methods in addition we construct families of smooth viscous shocks converging to a given inviscid shock as viscosity goes to zero thereby justifying the small viscosity limit for multidimensional nonconservative shocks in our previous work on shocks we made use of conservative form especially in parts of the low frequency analysis thus most of the new analysis of this paper is concentrated in this area by adopting the more general nonconservative viewpoint we are able to shed new light on both the viscous and inviscid theory for example we can now provide a clearer geometric motivation for the low frequency analysis in the viscous case also we show that one may in the treatment of inviscid stability of nonclassical and or nonconservative shocks remove an apparently restrictive technical assumption made by mokrane and coulombel in their work on respectively shocktype nonconservative boundary problems and conservative undercompressive shocks another advantage of the nonconservative perspective is that lax and undercompressive shocks can be treated by exactly the same analysis

derivatives of complex eigenvectors with distinct and repeated eigenvalues in this paper we investigate the computation of the first order derivatives of complex eigenvectors for general non defective eigensystems a new normalization condition is proposed with which we can compute unique first order derivatives of arbitrary differentiable eigenvectors of systems with distinct and repeated eigenvalues we also present an efficient algorithm to compute the particular solutions to the governing equations of the first order derivatives of eigenvectors finally numerical examples are included to demonstrate the validity of the proposed method copyright 2007 john wiley sons ltd

symmetric positive systems with boundary characteristic of constant multiplicity the theory of maximal positive boundary value problems for symmetric positive systems is developed assuming that the boundary is characteristic of constant multiplicity no such hypothesis is needed on a neighborhood of the boundary both regularity theorems and mixed initial boundary value problems are discussed many classical ideas are sharpened in the process

galloping instability of viscous shock waves abstract motivated by physical and numerical observations of time oscillatory galloping spinning and cellular instabilities of detonation waves we study poincare hopf bifurcation of traveling wave solutions of viscous conservation laws the main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum preventing standard reduction to a finite dimensional center manifold we overcome this by direct lyapunov schmidt reduction using detailed pointwise bounds on the linearized solution operator to carry out a nonstandard implicit function construction in the absence of a spectral gap the key computation is a space time stability estimate on the transverse linearized solution operator reminiscent of duhamel estimates carried out on the full solution operator in the study of nonlinear stability of spectrally stable traveling waves

a unified coordinates approach to computational fluid dynamics a unified coordinate system is introduced for computational fluid dynamics in which the grid moves with velocity hq q being fluid velocity it includes the eulerian coordinates as a special case when h 0 and the lagrangian when h 1 by suitably choosing the free function h h 1 for 1d flow h chosen to preserve grid angles for 2d flow and to preserve grid skewness for 3d flow the unified coordinate system is shown in a large number of examples involving the euler equations to be superior to both eulerian and lagrangian ones in resolving flow discontinuities shocks and especially slip lines this approach is also successfully extended to shallow water waves and viscous flow

derivatives of eigenvalues and eigenvectors of a general complex matrix a survey of methods for sensitivity analysis of the algebraic eigenvalue problem for non hermitian matrices is presented in addition a modification of one method based on a better normalizing condition is proposed methods are classified as direct or adjoint and are evaluated for efficiency operation counts are presented in terms of matrix size number of design variables and number of eigenvalues and eigenvectors of interest the effect of the sparsity of the matrix and its derivatives is also considered and typical solution times are given general guidelines are established for the selection of the most efficient method

zero dissipation limit of solutions with shocks for systems of hyperbolic conservation laws we consider the convergence of solutions of conservation laws with viscosity to solutions having shocks of hyperbolic conservation laws without viscosity as the viscosity tends to zero our analysis reveals a rich structure of nonlinear wave interactions due to the presence of shocks and initial layers these interactions generate four different wave patterns initial layers shock layers diffusion waves and coupling waves we study the propagation and interactions of the four wave patterns by a detailed pointwise analysis